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) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.RingTheor... | variable [GCDMonoid R]
theorem IsBezout.span_gcd_eq_span_gcd (x y : R) :
span {GCDMonoid.gcd x y} = span {IsBezout.gcd x y} := by
rw [Ideal.span_singleton_eq_span_singleton]
exact associated_of_dvd_dvd
| Mathlib/RingTheory/PrincipalIdealDomain.lean | 426 | 431 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Yongle Hu
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.RingTheory.Ideal.Pointwise
import Mathlib.RingTheory.Ideal.Quot... | Mathlib/RingTheory/Ideal/Over.lean | 361 | 382 | |
/-
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.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | rw [ennrealRatEmbed, Encodable.encodek]; rfl
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 758 | 758 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | theorem assert_defined {p : Prop} {f : p → Part α} : ∀ h : p, (f h).Dom → (assert p f).Dom :=
Exists.intro
| Mathlib/Data/Part.lean | 558 | 559 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... |
theorem volume_real_Ioo_of_le {a b : ℝ} (hab : a ≤ b) : volume.real (Ioo a b) = b - a := by
simp [hab]
@[simp]
| Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 100 | 104 |
/-
Copyright (c) 2021 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.Convex.Extreme
import Mathlib.Analysis.Convex.Function
import Mathlib.Topology.Algebra.Module.LinearMap
import Mathlib... | Iff.rfl
theorem exposedPoints_subset : A.exposedPoints 𝕜 ⊆ A := fun _ hx => hx.1
@[simp]
theorem exposedPoints_empty : (∅ : Set E).exposedPoints 𝕜 = ∅ :=
| Mathlib/Analysis/Convex/Exposed.lean | 181 | 186 |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
import Mathlib.Algebra.Star.Rat
/-!
# Self-adjoint, sk... |
@[simp] protected theorem one : IsSelfAdjoint (1 : R) :=
| Mathlib/Algebra/Star/SelfAdjoint.lean | 172 | 173 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Gaussian intege... | rw [abs_natCast_norm]
exact Int.add_one_le_of_lt (norm_pos.2 hy)))
instance instNontrivial : Nontrivial ℤ[i] :=
⟨⟨0, 1, by decide⟩⟩
instance : EuclideanDomain ℤ[i] :=
| Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 221 | 227 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... |
section InsertErase
@[simp] theorem ncard_insert_of_not_mem {a : α} (h : a ∉ s) (hs : s.Finite := by toFinite_tac) :
(insert a s).ncard = s.ncard + 1 := by
| Mathlib/Data/Set/Card.lean | 594 | 598 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Degree.Monomial
/-!
# Erase the... | eraseLead_add_of_degree_lt_right (degree_lt_degree pq)
| Mathlib/Algebra/Polynomial/EraseLead.lean | 179 | 180 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Filter.Prod
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Filter.Finite
import Mathlib.Order.Filter.Bases.Basic
/... | · exact lift_iInf hg
end lift
| Mathlib/Order/Filter/Lift.lean | 192 | 194 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.CharP.Pi
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.CharP.Subring
import Mathlib.Analysis.Specia... | ⟨fun hfg => by
simp_rw [← map_eq_zero hv] at hfg ⊢; contrapose! hfg; rw [Valuation.map_mul]
exact mul_ne_zero hfg.1 hfg.2⟩
exact NoZeroDivisors.to_isDomain _
end PreTilt
/-- The tilt of a field, as defined in Perfectoid Spaces by Peter Scholze, as in
[scholze2011perfectoid]. Given a field `K` with v... | Mathlib/RingTheory/Perfection.lean | 581 | 600 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... |
theorem isCycleOn_swap [DecidableEq α] (hab : a ≠ b) : (swap a b).IsCycleOn {a, b} :=
⟨bijOn_swap (by simp) (by simp), fun x hx y hy => by
rw [Set.mem_insert_iff, Set.mem_singleton_iff] at hx hy
obtain rfl | rfl := hx <;> obtain rfl | rfl := hy
· exact ⟨0, by rw [zpow_zero, coe_one, id]⟩
· exact ⟨1, ... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 704 | 717 |
/-
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 | 2,217 | 2,219 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | rw [← Set.image2_add, Set.image2_image_left, Set.image2_image_right]
exact Set.image_prod fun m m₂ => f m + g m₂
| Mathlib/LinearAlgebra/Prod.lean | 239 | 241 |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Junyan Xu
-/
import Mathlib.Data.DFinsupp.Defs
/-!
# Locus of unequal values of finitely supported dependent functions
Let `N : α → Type*` be a type family, assume that `N ... | @[simp]
theorem neLocus_self_sub_right : neLocus f (f - g) = g.support := by
| Mathlib/Data/DFinsupp/NeLocus.lean | 145 | 146 |
/-
Copyright (c) 2019 Minchao Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Minchao Wu, Mario Carneiro
-/
import Mathlib.Computability.Halting
/-!
# Strong reducibility and degrees.
This file defines the notions of computable many-one reduction and one-one
reduc... | theorem ManyOneReducible.trans {α β γ} [Primcodable α] [Primcodable β] [Primcodable γ]
{p : α → Prop} {q : β → Prop} {r : γ → Prop} : p ≤₀ q → q ≤₀ r → p ≤₀ r
| Mathlib/Computability/Reduce.lean | 52 | 53 |
/-
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... |
theorem sum_div_factorial_le {α : Type*} [Field α] [LinearOrder α] [IsStrictOrderedRing α]
| Mathlib/Data/Complex/Exponential.lean | 341 | 342 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.PreservesHomology
import Mathlib.Algebra.Homology.ShortComplex.Abelian
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import... | hS.mono_g' _
lemma LeftHomologyData.exact_iff_epi_f' [S.HasHomology] (h : LeftHomologyData S) :
S.Exact ↔ Epi h.f' := by
constructor
| Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 316 | 320 |
/-
Copyright (c) 2023 Yaël Dillies, Vladimir Ivanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Ivanov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.... |
@[simp] lemma truncatedSup_empty (a : α) : truncatedSup ∅ a = ⊤ := truncatedSup_of_not_mem (by simp)
@[simp] lemma truncatedSup_singleton (b a : α) : truncatedSup {b} a = if a ≤ b then b else ⊤ := by
simp [truncatedSup]; split_ifs <;> simp [Finset.filter_true_of_mem, *]
| Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | 125 | 130 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... | left_inv _ := Subsingleton.elim _ _
right_inv _ := Subsingleton.elim _ _
/-- When lowered to a prop, `Matrix.invertibleOfSubmatrixEquivInvertible` forms an `iff`. -/
@[simp]
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 689 | 693 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Data.DFinsupp.Submonoid
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp.SumProd
import Mathlib.LinearAlgebra.LinearIn... | @[simp]
theorem mapRange.linearMap_id :
(mapRange.linearMap fun i => (LinearMap.id : β₂ i →ₗ[R] _)) = LinearMap.id := by
ext
simp [linearMap]
| Mathlib/LinearAlgebra/DFinsupp.lean | 212 | 216 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | closure E = ⋂ (δ : ℝ) (_ : 0 < δ), cthickening δ E := by
rw [← cthickening_zero]
exact cthickening_eq_iInter_cthickening E
/-- The closure of a set equals the intersection of its open thickenings of positive radii
accumulating at zero. -/
theorem closure_eq_iInter_thickening' (E : Set α) (s : Set ℝ) (hs₀ : s ⊆... | Mathlib/Topology/MetricSpace/Thickening.lean | 494 | 507 |
/-
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.Shift.CommShift
/-!
# Functors from a category to a category with a shift
Given a category `C`, and a category `D` equipped with a shift by a mo... | lemma shiftIso_add_hom_app (n m a a' a'' : A) (ha' : n + a = a') (ha'' : m + a' = a'') (X : C) :
(F.shiftIso (m + n) a a'' (by rw [add_assoc, ha', ha''])).hom.app X =
(shiftFunctorAdd D m n).hom.app ((F.functor a'').obj X) ≫
((F.shiftIso m a' a'' ha'').hom.app X)⟦n⟧' ≫
(F.shiftIso n a a' ha').... | Mathlib/CategoryTheory/Shift/SingleFunctors.lean | 56 | 61 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.OrderOfElement
/-!
# Multiplicative... | /-!
### Equivalence of multiplicative characters with homomorphisms on units
We show that restriction / extension by zero gives an equivalence
between `MulChar R R'` and `Rˣ →* R'ˣ`.
-/
| Mathlib/NumberTheory/MulChar/Basic.lean | 138 | 143 |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | card_roots (X_pow_sub_C_ne_zero hn a)
_ = n := degree_X_pow_sub_C hn a
| Mathlib/Algebra/Polynomial/Roots.lean | 268 | 269 |
/-
Copyright (c) 2023 Paul Reichert. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Reichert, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Affine.AddTorsorBases
/-!
# Intrinsic frontier and interior
This file defines the intrinsic frontier, interior and closur... | Mathlib/Analysis/Convex/Intrinsic.lean | 112 | 112 | |
/-
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.RingTheory.WittVector.StructurePolynomial
/-!
# Witt vectors
In this file we define the type of `p`-typical Witt vectors and ring op... | rw [Finset.mem_range] at hin
rw [IH _ hin (Nat.pos_of_ne_zero hi0), zero_pow (pow_ne_zero _ hp.1.ne_zero), mul_zero]
· rw [Finset.mem_range]; intro; contradiction
| Mathlib/RingTheory/WittVector/Defs.lean | 225 | 228 |
/-
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.... | /-- Assume `k = l`. If two functions defined on `Fin k` and `Fin l` are equal on each element,
then they coincide (in the heq sense). -/
| Mathlib/Data/Fin/Basic.lean | 121 | 122 |
/-
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.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | exact H i hi
| @tail j k _ hjk ih =>
obtain ⟨p, hpt, hip, hjp, hp⟩ := ih hjk.2
refine ⟨insert k p, insert_subset_iff.mpr ⟨hj, hpt⟩, mem_insert_of_mem k hip,
mem_insert k p, ?_⟩
rw [biUnion_insert]
refine (H k hj).union' (hjk.1.mono ?_) hp
rw [inter_comm]
| Mathlib/Topology/Connected/Basic.lean | 157 | 164 |
/-
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.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | Mathlib/Data/Set/Image.lean | 1,643 | 1,644 | |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | Nat.cast_ne_zero.mpr mult_ne_zero
theorem one_le_mult {w : InfinitePlace K} : (1 : ℝ) ≤ mult w := by
rw [← Nat.cast_one, Nat.cast_le]
exact mult_pos
open scoped Classical in
theorem card_filter_mk_eq [NumberField K] (w : InfinitePlace K) : #{φ | mk φ = w} = mult w := by
conv_lhs =>
congr; congr; ext
r... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 469 | 481 |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 591 | 593 | |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Adjunction.Unique
import Mathlib.CategoryTheory.Adjunction.Reflective
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.Categ... | (sheafificationAdjunction J D).unit.app P = toSheafify J P := rfl
| Mathlib/CategoryTheory/Sites/Sheafification.lean | 96 | 97 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | rw [← smul_assoc, smul_eq_mul, ← div_eq_mul_inv, div_self (norm_pos_iff.mp hk), one_smul]
theorem ball_zero_absorbs_ball_zero (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) := by
rcases exists_pos_lt_mul hr₁ r₂ with ⟨r, hr₀, hr⟩
| Mathlib/Analysis/Seminorm.lean | 924 | 928 |
/-
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.Algebra.Ring.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
import Mathlib.Al... |
protected theorem add_lt_add_left (a b : ℤ√d) (h : a < b) (c) : c + a < c + b := fun h' =>
h (Zsqrtd.le_of_add_le_add_left _ _ _ h')
theorem nonneg_smul {a : ℤ√d} {n : ℕ} (ha : Nonneg a) : Nonneg ((n : ℤ√d) * a) := by
rw [← Int.cast_natCast n]
exact
match a, nonneg_cases ha, ha with
| _, ⟨x, y, Or.inl r... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 657 | 676 |
/-
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... |
lemma shiftFunctorCompIsoId_zero_zero_inv_app (X : C) :
(shiftFunctorCompIsoId C 0 0 (add_zero 0)).inv.app X =
(shiftFunctorZero C A).inv.app X ≫ ((shiftFunctorZero C A).inv.app X)⟦0⟧' := by
simp [shiftFunctorCompIsoId, shiftFunctorAdd'_zero_add_hom_app]
end
section
| Mathlib/CategoryTheory/Shift/Basic.lean | 493 | 501 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Homology.ShortComplex.Retract
import Mathlib.CategoryTheory.MorphismProperty.Composition
/-!
#... | Mathlib/Algebra/Homology/QuasiIso.lean | 352 | 360 | |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Prod
/-!
# N-ary images of sets
This file defines `Set.image2`, the binary image of sets.
This is mostly useful to define pointwise oper... | theorem image_image2_right_comm {f : α → β' → γ} {g : β → β'} {f' : α → β → δ} {g' : δ → γ}
(h_right_comm : ∀ a b, f a (g b) = g' (f' a b)) :
image2 f s (t.image g) = (image2 f' s t).image g' :=
(image_image2_distrib_right fun a b => (h_right_comm a b).symm).symm
| Mathlib/Data/Set/NAry.lean | 248 | 252 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | refine ⟨fun n => s n.unpair.1 ×ˢ t n.unpair.2, fun n => mem_image2_of_mem (h1s _) (h1t _), ?_⟩
rw [iUnion_unpair_prod, h2s, h2t, univ_prod_univ]
| Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 345 | 353 |
/-
Copyright (c) 2024 Theodore Hwa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
-/
import Mathlib.Logic.Hydra
import Mathlib.SetTheory.Surreal.Basic
/-!
### Surreal multiplication
In... | intros a ih ha
replace ih : ∀ a', ArgsRel a' a → P124 a' := fun a' hr ↦ ih a' hr (hr.numeric_closed ha)
cases a with
/- P1 -/
| P1 x y =>
rw [Args.numeric_P1] at ha
exact P1_of_ih ih ha.1 ha.2
| P24 x₁ x₂ y =>
have h₁₂ := ih₁₂ ih
have h₂₁ := ih₂₁ ih
have h4 := ih4 ih
obtain ⟨h₁₂x, h₁... | Mathlib/SetTheory/Surreal/Multiplication.lean | 422 | 448 |
/-
Copyright (c) 2019 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.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
import Mathlib.CategoryTheory.Limits.Shape... | Mathlib/CategoryTheory/Limits/Shapes/Images.lean | 1,021 | 1,024 | |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Data.Fintype.Parity
import Mathlib.LinearAl... | ring
theorem mul_smul' (x y : GL(2, ℝ)⁺) (z : ℍ) : smulAux (x * y) z = smulAux x (smulAux y z) := by
ext1
change _ / _ = (_ * (_ / _) + _) / _
rw [denom_cocycle]
field_simp [denom_ne_zero]
| Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean | 246 | 252 |
/-
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.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | BiheytingAlgebra (α × β) where
__ := instHeytingAlgebra
| Mathlib/Order/Heyting/Basic.lean | 974 | 975 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Algebra.Group.Idempotent
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.PUnit
import Mathlib.Order.Hom.BoundedLattic... | instance : BooleanRing PUnit :=
| Mathlib/Algebra/Ring/BooleanRing.lean | 106 | 106 |
/-
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.Logic.Relator
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
import Mathlib.Logic.Basic
import Mathl... | | tail _ hcd hac => exact TransGen.tail hac (h _ _ hcd)
theorem TransGen.lift' {p : β → β → Prop} {a b : α} (f : α → β)
| Mathlib/Logic/Relation.lean | 480 | 482 |
/-
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.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... |
/-- Given a function `g : β → γ` and a simple function `f : α →ₛ β`, `f.map g` return the simple
function `g ∘ f : α →ₛ γ` -/
def map (g : β → γ) (f : α →ₛ β) : α →ₛ γ :=
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 247 | 250 |
/-
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... | Mathlib/Data/Finsupp/Defs.lean | 1,349 | 1,355 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... | simp only [comp_v _ _ h p _ q rfl (by omega), zero_v, comp_zero]
@[simp]
protected lemma comp_add {n₁ n₂ n₁₂ : ℤ} (z₁ : Cochain F G n₁) (z₂ z₂' : Cochain G K n₂)
(h : n₁ + n₂ = n₁₂) : z₁.comp (z₂ + z₂') h = z₁.comp z₂ h + z₁.comp z₂' h := by
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 344 | 348 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... |
@[simp]
theorem canonicalEquiv_self : canonicalEquiv S P P = RingEquiv.refl _ := by
rw [← canonicalEquiv_trans_canonicalEquiv S P P]
convert (canonicalEquiv S P P).symm_trans_self
exact (canonicalEquiv_symm S P P).symm
| Mathlib/RingTheory/FractionalIdeal/Operations.lean | 249 | 254 |
/-
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.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.AlgebraicGeometry.Limits
import Mathlib.CategoryTheory.MorphismPrope... | (∀ i, P (f ∣_ U i)) → P f) :
IsLocalAtTarget P := by
refine ⟨inferInstance, fun {X Y} f 𝒰 ↦ ⟨?_, fun H ↦ of_sSup_eq_top f _ 𝒰.iSup_opensRange ?_⟩⟩
· exact fun H i ↦ (P.arrow_mk_iso_iff (morphismRestrictOpensRange f _)).mp (restrict _ _ H)
· exact fun i ↦ (P.arrow_mk_iso_iff (morphismRestrictOpensRan... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 131 | 141 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... | have hderiv : derivative f = C b := by
simp [hn, f, map_add derivative, derivative_C, derivative_X_pow]
rw [hderiv, right_distrib, ← add_assoc, neg_mul, mul_comm, neg_add_cancel, zero_add,
← map_mul, hb, map_one]
/-- If `R` is of characteristic `p`, `p ∣ n` and `b` is a unit,
then `a * X ^ n + b * X + c` i... | Mathlib/FieldTheory/Separable.lean | 268 | 284 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | theorem Filter.EventuallyEq.congr_continuousWithinAt_of_insert (h : f =ᶠ[𝓝[insert x s] x] g) :
ContinuousWithinAt f s x ↔ ContinuousWithinAt g s x :=
⟨fun h' ↦ h'.congr_of_eventuallyEq_insert h.symm,
fun h' ↦ h'.congr_of_eventuallyEq_insert h⟩
| Mathlib/Topology/ContinuousOn.lean | 871 | 874 |
/-
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.Algebra.Polynomial.Identities
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.Topology.A... | have : (F.derivative.eval soln + q * h) * h = 0 :=
Eq.symm
(calc
0 = F.eval (soln + h) := by simp [h, hev]
_ = F.derivative.eval soln * h + q * h ^ 2 := by rw [hq, eval_soln, zero_add]
_ = (F.derivative.eval soln + q * h) * h := by rw [sq, right_distrib, mul_assoc]
)
have :... | Mathlib/NumberTheory/Padics/Hensel.lean | 406 | 438 |
/-
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.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.Antichain
import Mathlib.Order.SetNotation
/-!
# Order-connected sets
We say that a set `s :... | theorem OrdConnected.dual {s : Set α} (hs : OrdConnected s) :
OrdConnected (OrderDual.ofDual ⁻¹' s) :=
⟨fun _ hx _ hy _ hz => hs.out hy hx ⟨hz.2, hz.1⟩⟩
| Mathlib/Order/Interval/Set/OrdConnected.lean | 126 | 129 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Data.List.Iterate
import Mathlib.GroupTheory.Perm.Cycle.Basic
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Tactic.Group
/-!
# ... | -/
def cycleFactorsFinset : Finset (Perm α) :=
(truncCycleFactors f).lift
(fun l : { l : List (Perm α) // l.prod = f ∧ (∀ g ∈ l, IsCycle g) ∧ l.Pairwise Disjoint } =>
l.val.toFinset)
fun ⟨_, hl⟩ ⟨_, hl'⟩ =>
List.toFinset_eq_of_perm _ _
| Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 485 | 491 |
/-
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 ε₁ ⊆ closedBall y ε₂ := fun z hz =>
calc
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 515 | 516 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Ira Fesefeldt
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.Order.CompleteLattice.Basic
import Mathlib.Order.Iterate
import M... |
@[simps! ωSup_coe]
instance omegaCompletePartialOrder : OmegaCompletePartialOrder (α →o β) :=
OmegaCompletePartialOrder.lift OrderHom.coeFnHom OrderHom.ωSup (fun _ _ h => h) fun _ => rfl
end OrderHom
| Mathlib/Order/OmegaCompletePartialOrder.lean | 518 | 524 |
/-
Copyright (c) 2023 Paul Reichert. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Reichert, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Affine.AddTorsorBases
/-!
# Intrinsic frontier and interior
This file defines the intrinsic frontier, interior and closur... | rw [intrinsicFrontier, preimage_coe_affineSpan_singleton, frontier_univ, image_empty]
| Mathlib/Analysis/Convex/Intrinsic.lean | 120 | 120 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | · -- ** direction 2: empty orthogonal complement implies maximal
simp only [Subset.antisymm_iff]
rintro h u (huv : v ⊆ u) hu
refine ⟨?_, huv⟩
intro x hxu
refine ((mt (h x)) (hu.ne_zero ⟨x, hxu⟩)).imp_symm ?_
intro hxv y hy
have hxv' : (⟨x, hxu⟩ : u) ∉ ((↑) ⁻¹' v : Set u) := by simp [huv, h... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 1,366 | 1,425 |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
/-!
# Exterior Algebras
We construct the e... | ιInv.comp (map f).toLinearMap = f.comp ιInv := by
letI : Module Rᵐᵒᵖ M := Module.compHom _ ((RingHom.id R).fromOpposite mul_comm)
haveI : IsCentralScalar R M := ⟨fun r m => rfl⟩
letI : Module Rᵐᵒᵖ N := Module.compHom _ ((RingHom.id R).fromOpposite mul_comm)
| Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 404 | 407 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Riccardo Brasca, Eric Rodriguez
-/
import Mathlib.Data.PNat.Prime
import Mathlib.NumberTheory.Cyclotomic.Basic
import Mathlib.RingTheory.Adjoin.PowerBasis
import Mathlib... | convert Submodule.finrank_le (Subalgebra.toSubmodule (adjoin K {z}))
rw [show Nat.lcm p q = (k : ℕ) from rfl] at hirr
simpa using (IsCyclotomicExtension.finrank (Algebra.adjoin K {z}) hirr).symm
end IsCyclotomicExtension
end NoOrder
section Norm
namespace IsPrimitiveRoot
section Field
variable {K} [Field K]... | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | 195 | 209 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 431 | 436 | |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | simpa [NNRat.cast_smul_eq_nnqsmul] using nnnorm_smul (q : K) x
| Mathlib/Analysis/RCLike/Basic.lean | 636 | 636 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Powerset
import Mathlib.Algebra.NoZeroSMulDivisors.Pi
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Finty... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,652 | 1,655 | |
/-
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.Algebra.Pi
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
impo... |
@[simp]
theorem toFinsupp_algebraMap (r : R) : (algebraMap R A[X] r).toFinsupp = algebraMap R _ r :=
show toFinsupp (C (algebraMap _ _ r)) = _ by
| Mathlib/Algebra/Polynomial/AlgebraMap.lean | 58 | 61 |
/-
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.Logic.Relator
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
import Mathlib.Logic.Basic
import Mathl... | · exact head hac hcb
theorem total_of_right_unique (U : Relator.RightUnique r) (ab : ReflTransGen r a b)
(ac : ReflTransGen r a c) : ReflTransGen r b c ∨ ReflTransGen r c b := by
induction ab with
| Mathlib/Logic/Relation.lean | 353 | 357 |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... | refine logb_lt_logb_of_base_lt_one b_pos b_lt_one (abs_pos.2 hy.ne) ?_
rwa [abs_of_neg hy, abs_of_neg hx, neg_lt_neg_iff]
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 328 | 329 |
/-
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... | Mathlib/Computability/TuringMachine.lean | 982 | 985 | |
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.PFunctor.Univariate.Basic
/-!
# M-types
M types are potentially infinite tree-like structures. They are defined
as the greatest fixpoint of a polynomi... | let Q : M P × M P → Prop := fun p => R p.fst p.snd
bisim' Q Prod.fst Prod.snd
(fun p Qp =>
let ⟨a, f, f', hx, hy, h'⟩ := h p.fst p.snd Qp
⟨a, f, f', hx, hy, fun i => ⟨⟨f i, f' i⟩, h' i, rfl, rfl⟩⟩)
⟨x, y⟩ Rxy
theorem corec_unique (g : α → P α) (f : α → M P) (hyp : ∀ x, M.dest (f x) = P.map f (g... | Mathlib/Data/PFunctor/Univariate/M.lean | 626 | 641 |
/-
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.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Ma... | rw [← cancel_mono (adj.counit.app <| F.obj j)]
dsimp [α']
simp only [Category.comp_id, Adjunction.counit_naturality_assoc, Category.id_comp,
Adjunction.counit_naturality, Category.assoc, Functor.map_comp]
let β := isoWhiskerLeft F' (asIso adj.counit) ≪≫ F'.rightUnitor
let hl := (IsColimit.precompo... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 411 | 465 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Gaussian intege... | x / y = ⟨round ((x * star y).re / norm y : ℚ), round ((x * star y).im / norm y : ℚ)⟩ :=
show Zsqrtd.mk _ _ = _ by simp [div_eq_mul_inv]
| Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 162 | 163 |
/-
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.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | theorem oangle_left_eq_arccos_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₃ p₁ p₂ = Real.arccos (dist p₁ p₂ / dist p₁ p₃) := by
have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
| Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 535 | 537 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finsets in product types
This file defines finset constru... | for `a, b ∈ s`. -/
def offDiag :=
| Mathlib/Data/Finset/Prod.lean | 271 | 272 |
/-
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.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.Over
/-!
# Restriction of Schemes and Morphisms
## Main definition
- `AlgebraicGeometry.Schem... | lemma stalkIso_inv {X : Scheme.{u}} (U : X.Opens) (x : U) :
(U.stalkIso x).inv = U.ι.stalkMap x := by
rw [← Category.comp_id (U.stalkIso x).inv, Iso.inv_comp_eq]
apply TopCat.Presheaf.stalk_hom_ext
intro W hxW
simp only [Category.comp_id, U.germ_stalkIso_hom_assoc]
convert (Scheme.stalkMap_germ U.ι (U.ι '... | Mathlib/AlgebraicGeometry/Restrict.lean | 152 | 167 |
/-
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.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Nat.Notation
import Mathlib.Logic.Function.Basic
/-!
# Inductive type variant o... | let ⟨elems, compl⟩ := instFintype n
{ elems := elems.map ⟨Fin2.fs, @fs.inj _⟩ |>.cons .fz (by simp)
| Mathlib/Data/Fin/Fin2.lean | 151 | 152 |
/-
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... | (Exists.intro s₂ rfl)
theorem interleave_even_odd (s₁ : Stream' α) : even s₁ ⋈ odd s₁ = s₁ :=
eq_of_bisim (fun s' s => s' = even s ⋈ odd s)
| Mathlib/Data/Stream/Init.lean | 416 | 419 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# L'Hôpital's rule for 0/0 indeterminate forms
In this file, we ... | (hfa : Tendsto f (𝓝[>] a) (𝓝 0)) (hga : Tendsto g (𝓝[>] a) (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) :
Tendsto (fun x => f x / g x) (𝓝[>] a) l := by
have sub : ∀ x ∈ Ioo a b, Ioo a x ⊆ Ioo a b := fun x hx =>
Ioo_subset_Ioo (le_refl a) (le_of_lt hx.2)
have hg : ∀ x ∈ Ioo a b, g ... | Mathlib/Analysis/Calculus/LHopital.lean | 51 | 92 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mat... | variable [Semiring R] [PartialOrder R] [IsOrderedRing R] {f : ι → R} {s : Finset ι}
lemma sum_sq_le_sq_sum_of_nonneg (hf : ∀ i ∈ s, 0 ≤ f i) :
∑ i ∈ s, f i ^ 2 ≤ (∑ i ∈ s, f i) ^ 2 := by
simp only [sq, sum_mul_sum]
refine sum_le_sum fun i hi ↦ ?_
| Mathlib/Algebra/Order/BigOperators/Ring/Finset.lean | 88 | 93 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... | ∃ a ∈ s, x ∈ (f a).support := by
contrapose! hx
| Mathlib/GroupTheory/Perm/Support.lean | 383 | 384 |
/-
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.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | · exact mul_lt_top ha.lt_top (finMeasSupp_iff.1 hm _ ha0)
theorem of_lintegral_ne_top {f : α →ₛ ℝ≥0∞} (h : f.lintegral μ ≠ ∞) : f.FinMeasSupp μ := by
refine finMeasSupp_iff.2 fun b hb => ?_
rw [f.lintegral_eq_of_subset' (Finset.subset_insert b _)] at h
refine ENNReal.lt_top_of_mul_ne_top_right ?_ hb
exact ... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 1,118 | 1,124 |
/-
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.MeasureTheory.Function.LocallyIntegrable
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Me... | lemma isHaarMeasure_eq_of_isProbabilityMeasure [LocallyCompactSpace G] (μ' μ : Measure G)
[IsProbabilityMeasure μ] [IsProbabilityMeasure μ'] [IsHaarMeasure μ] [IsHaarMeasure μ'] :
μ' = μ := by
have : CompactSpace G := by
by_contra H
rw [not_compactSpace_iff] at H
simpa using measure_univ_of_isMulL... | Mathlib/MeasureTheory/Measure/Haar/Unique.lean | 663 | 678 |
/-
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.Localization.CalculusOfFractions.Fractions
import Mathlib.CategoryTheory.Localization.HasLocalization
import Mathlib.CategoryTheory.Preadditive.Ad... | lemma neg'_add'_self (f : L.obj X ⟶ L.obj Y) :
add' W (neg' W f) f = L.map 0 := by
obtain ⟨α, rfl⟩ := exists_leftFraction L W f
have := inverts L W _ α.hs
rw [add'_eq W _ _ (LeftFraction₂.mk (-α.f) α.f α.s α.hs) (neg'_eq W _ _ rfl) rfl]
simp only [← cancel_mono (L.map α.s), LeftFraction.map_comp_map_s, ← L.... | Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean | 161 | 167 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | lemma exists_norm_mul_eq_self (x : K) : ∃ c, ‖c‖ = 1 ∧ c * ‖x‖ = x := by
obtain rfl | hx := eq_or_ne x 0
| Mathlib/Analysis/RCLike/Basic.lean | 485 | 486 |
/-
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... | simp only [Multiset.count_add, add_le_add_iff_left]
convert Multiset.count_le_of_le j <| degrees_X' j
rw [Multiset.count_singleton_self]
@[deprecated (since := "2024-12-01")] alias degreeOf_mul_X_eq := degreeOf_mul_X_self
theorem degreeOf_mul_X_eq_degreeOf_add_one_iff (j : σ) (f : MvPolynomial σ R) :
degree... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 303 | 312 |
/-
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... | rw [show x = F (0 : 𝕜) by simp [F]] at hf
have A : HasDerivWithinAt F (0 + (1 : 𝕜) • v) (F ⁻¹' s) 0 :=
((hasDerivAt_const (0 : 𝕜) x).add ((hasDerivAt_id' (0 : 𝕜)).smul_const v)).hasDerivWithinAt
simp only [one_smul, zero_add] at A
| Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 262 | 265 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Power
import M... | have A : Ico 0 R ⊆ Ioo (-R) R :=
fun x hx ↦ ⟨(neg_lt_zero.2 (hx.1.trans_lt hx.2)).trans_le hx.1, hx.2⟩
have B : Ioo 0 R ⊆ Ioo (-R) R := Subset.trans Ioo_subset_Ico_self A
-- First we prove that 1-4 are equivalent using 2 → 3 → 4, 1 → 3, and 2 → 1
tfae_have 1 → 3 := fun ⟨a, ha, H⟩ ↦ ⟨a, ha, H.isBigO⟩
tfae_... | Mathlib/Analysis/SpecificLimits/Normed.lean | 72 | 77 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement ope... | ⟨fun H => by
apply le_antisymm
· conv_lhs => rw [← sup_inf_sdiff y x]
apply sup_le_sup_right
rwa [inf_eq_right.2 hx]
· apply le_trans
· apply sup_le_sup_right hz
· rw [sup_sdiff_left],
| Mathlib/Order/BooleanAlgebra.lean | 245 | 252 |
/-
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... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,411 | 2,413 | |
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Triangle.Basic
/-!
# Construct a tripartite graph from its triangles
This file contains the constru... | lemma exists_mem_toTriangle {x y : α ⊕ β ⊕ γ} (hxy : (graph t).Adj x y) :
∃ z ∈ t, x ∈ toTriangle z ∧ y ∈ toTriangle z := by cases hxy <;> exact ⟨_, ‹_›, by simp⟩
| Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean | 164 | 165 |
/-
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... | rwa [restrict_apply₀, inter_comm]
exact AEStronglyMeasurable.nullMeasurableSet_le hf.1 aestronglyMeasurable_const
haveI := Fact.mk hμ₁.lt_top
refine (integral_sub_average (μ.restrict s) f).not_gt ?_
refine (setIntegral_pos_iff_support_of_nonneg_ae ?_ ?_).2 ?_
· refine measure_mono_null (fun x hx ↦ ?_) H... | Mathlib/MeasureTheory/Integral/Average.lean | 487 | 493 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Subsemigroup.Defs
import M... | f = g :=
eq_of_eqOn_top <| hs ▸ eqOn_closure h
| Mathlib/Algebra/Group/Subsemigroup/Basic.lean | 254 | 255 |
/-
Copyright (c) 2022 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Analysis.Convex.Cone.Closure
import Mathlib.Analysis.InnerProductSpace.Adjoint
/-!
# Proper cones
We define a *proper cone* as a closed, pointed cone. ... | variable [Semiring 𝕜] [PartialOrder 𝕜] [IsOrderedRing 𝕜]
[AddCommGroup E] [PartialOrder E] [IsOrderedAddMonoid E] [Module 𝕜 E] [OrderedSMul 𝕜 E]
| Mathlib/Analysis/Convex/Cone/Proper.lean | 87 | 88 |
/-
Copyright (c) 2021 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Yaël Dillies, Anthony DeRossi
-/
import Mathlib.Computability.NFA
import Mathlib.Data.List.ReduceOption
/-!
# Epsilon Nondeterministic Finite Automata
This file contains th... | intro ⟨n, h⟩
induction n generalizing s₂
· rw [List.replicate_zero] at h
apply IsPath.eq_of_nil at h
solve_by_elim
· simp_rw [List.replicate_add, isPath_append, List.replicate_one, isPath_singleton] at h
obtain ⟨t, _, _⟩ := h
solve_by_elim [εClosure.step]
theorem mem_evalFrom_if... | Mathlib/Computability/EpsilonNFA.lean | 198 | 208 |
/-
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.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... | /-- Replacing the first point by one on the same line does not change twice the oriented angle. -/
theorem _root_.Collinear.two_zsmul_oangle_eq_left {p₁ p₁' p₂ p₃ : P}
(h : Collinear ℝ ({p₁, p₂, p₁'} : Set P)) (hp₁p₂ : p₁ ≠ p₂) (hp₁'p₂ : p₁' ≠ p₂) :
(2 : ℤ) • ∡ p₁ p₂ p₃ = (2 : ℤ) • ∡ p₁' p₂ p₃ := by
by_cases ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 545 | 550 |
/-
Copyright (c) 2020 Jannis Limperg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jannis Limperg
-/
import Mathlib.Data.List.Induction
/-!
# Lemmas about List.*Idx functions.
Some specification lemmas for `List.mapIdx`, `List.mapIdxM`, `List.foldlIdx` and `List.fo... |
set_option linter.deprecated false in
@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
theorem foldlIdx_eq_foldl_enum (f : ℕ → α → β → α) (a : α) (bs : List β) :
foldlIdx f a bs = foldl (fun a p ↦ f p.fst a p.snd) a (enum bs) := by
simp only [foldlIdx, foldlIdxSpec, foldlIdx_eq_foldlIdxSpe... | Mathlib/Data/List/Indexes.lean | 151 | 172 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
/-!
# The category of functors and natural transf... | aesop_cat
end NatTrans
| Mathlib/CategoryTheory/Functor/Category.lean | 132 | 134 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Eric Wieser
-/
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Tactic.AdaptationNote
import Mathlib.LinearAlgebra.Multilinear.Curry
/-!
# Tenso... | toFun := lift (constOfIsEmpty R _ 1)
invFun r := r • tprod R (@isEmptyElim _ _ _)
left_inv x := by
refine x.induction_on ?_ ?_
· intro x y
-- Note: https://github.com/leanprover-community/mathlib4/pull/8386 had to change `map_smulₛₗ` into `map_smulₛₗ _`
simp only [map_smulₛₗ _, RingHom.id_appl... | Mathlib/LinearAlgebra/PiTensorProduct.lean | 796 | 803 |
/-
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 | 2,748 | 2,751 | |
/-
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.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
/-!
# Properties of cyclic permutations constructed... | -- c[0, 2] * c[1, 3]
#eval (c[1, 2, 3] * c[0, 1, 2] : Perm (Fin 4))
-- c[0, 2] * c[1, 3]
#eval (c[1, 2, 3] * c[0, 1, 2] * c[3, 1] * c[0, 2] : Perm (Fin 4))
| Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 485 | 490 |
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