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/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 696 | 698 | |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
/-!
# Variance of random ... | congr
exact h.integral_mul_of_integrable (hX.integrable one_le_two) (hY.integrable one_le_two)
_ = variance X μ + variance Y μ := by simp only [variance_def', hX, hY, Pi.pow_apply]; ring
-- Porting note: supplied `MeasurableSpace Ω` argument of `hs`, `h` by unification
/-- The variance of a finite sum ... | Mathlib/Probability/Variance.lean | 288 | 306 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.Basis.Submodule
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix... | ext i j
rw [Matrix.add_apply, e.toMatrix_apply, Pi.add_apply, LinearEquiv.map_add]
rfl
| Mathlib/LinearAlgebra/Matrix/Basis.lean | 132 | 134 |
/-
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 ... |
lemma Sigma.map'_eq {f : α → C} {g : β → C} [HasCoproduct f] [HasCoproduct g]
{p p' : α → β} {q : ∀ (a : α), f a ⟶ g (p a)} {q' : ∀ (a : α), f a ⟶ g (p' a)}
(hp : p = p') (hq : ∀ (a : α), q a ≫ eqToHom (hp ▸ rfl) = q' a) :
Sigma.map' p q = Sigma.map' p' q' := by
| Mathlib/CategoryTheory/Limits/Shapes/Products.lean | 482 | 486 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Determinant
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Ri... |
/-- This lemma follows from the finite correctness proof, the determinant equality, and
by simplifying the difference. -/
theorem sub_convs_eq {ifp : IntFractPair K}
(stream_nth_eq : IntFractPair.stream v n = some ifp) :
let g := of v
let B := (g.contsAux (n + 1)).b
let pB := (g.contsAux n).b
v - ... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 322 | 357 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
import Mathlib.MeasureTheory.Measure.Real
/-! # Conditional expectation in L1
This file contains ... | simp only [Pi.smul_apply, hy]
@[deprecated (since := "2025-01-21")] alias condexpIndL1Fin_smul' := condExpIndL1Fin_smul'
theorem norm_condExpIndL1Fin_le (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
‖condExpIndL1Fin hm hs hμs x‖ ≤ μ.real s * ‖x‖ := by
rw [L1.norm_eq_integral_norm, ← ENNReal.toReal_ofReal (... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 128 | 143 |
/-
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 ... | Mathlib/Data/DFinsupp/NeLocus.lean | 160 | 161 | |
/-
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.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tact... | ∏ i, ∏ j ∈ Ioi i, f j i * f i j = ∏ i, ∏ j ∈ {i}ᶜ, f j i := by
simp_rw [← Ioi_disjUnion_Iio, prod_disjUnion, prod_mul_distrib]
congr 1
rw [prod_sigma', prod_sigma']
refine prod_nbij' (fun i ↦ ⟨i.2, i.1⟩) (fun i ↦ ⟨i.2, i.1⟩) ?_ ?_ ?_ ?_ ?_ <;> simp
end LinearOrder
| Mathlib/Algebra/BigOperators/Intervals.lean | 79 | 85 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | /-- The inner product of a nonzero vector with a negative multiple of
itself, divided by the product of their norms, has value -1. -/
theorem real_inner_div_norm_mul_norm_eq_neg_one_of_ne_zero_of_neg_mul {x : F} {r : ℝ} (hx : x ≠ 0)
(hr : r < 0) : ⟪x, r • x⟫_ℝ / (‖x‖ * ‖r • x‖) = -1 := by
rw [real_inner_smul_self... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 656 | 660 |
/-
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.Transfer
/-!
# The Schur-Zassenhaus Theorem
In this file we prove the Schur-Zassenhaus theorem.
## Main results
- `Subgroup.exists_ri... |
include h2 in
/-- Do not use this lemma: It is made obsolete by `exists_right_complement'_of_coprime` -/
| Mathlib/GroupTheory/SchurZassenhaus.lean | 233 | 235 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
/-!... | · rw [not_le] at hx₁
rw [arccos_of_le_neg_one hx₁.le, sin_pi, sqrt_eq_zero_of_nonpos]
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 365 | 366 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma, Oliver Nash
-/
import Mathlib.Algebra.Group.Action.Opposite
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.GroupWithZero.Associated
import M... | ⟨hr _, by simp +contextual⟩
| Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | 129 | 130 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Fintype.Sigma
/-!
# Darts in graphs
A `Dart` or half-edge or bond in a graph is an ordered pair of adja... | theorem Dart.ext_iff (d₁ d₂ : G.Dart) : d₁ = d₂ ↔ d₁.toProd = d₂.toProd := by
cases d₁; cases d₂; simp
| Mathlib/Combinatorics/SimpleGraph/Dart.lean | 33 | 34 |
/-
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... |
instance : DecidableEq (Word M) :=
Function.Injective.decidableEq fun _ _ => Word.ext
instance : DecidableEq (CoprodI M) :=
Equiv.decidableEq Word.equiv
| Mathlib/GroupTheory/CoprodI.lean | 609 | 615 |
/-
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.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
/-!
# Slope of a function
In this file we define the slope of a function `f : k... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 124 | 127 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.O... |
@[simp, mfld_simps]
theorem coe_coe_fst (hb : b ∈ e'.baseSet) : (e' y).1 = b :=
| Mathlib/Topology/FiberBundle/Trivialization.lean | 203 | 205 |
/-
Copyright (c) 2023 Luke Mantle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Mantle, Jake Levinson
-/
import Mathlib.RingTheory.Polynomial.Hermite.Basic
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathli... | field_simp [Real.exp_ne_zero]
rw [← @smul_eq_mul ℝ _ ((-1) ^ n), ← inv_smul_eq_iff₀, mul_assoc, smul_eq_mul, ← inv_pow, ←
neg_inv, inv_one]
exact pow_ne_zero _ (by norm_num)
theorem hermite_eq_deriv_gaussian' (n : ℕ) (x : ℝ) : aeval x (hermite n) =
(-1 : ℝ) ^ n * deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2... | Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean | 58 | 64 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
/-!
# Triangle inequality for `Lp`-seminorm
In this file w... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 185 | 193 | |
/-
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... | rw [keys, kerase, erase_eq_eraseP, eraseP_map, Function.comp_def]
congr
theorem kerase_kerase {a a'} {l : List (Sigma β)} :
(kerase a' l).kerase a = (kerase a l).kerase a' := by
by_cases h : a = a'
· subst a'; rfl
induction' l with x xs
· rfl
· by_cases a' = x.1
· subst a'
simp [kerase_cons... | Mathlib/Data/List/Sigma.lean | 429 | 443 |
/-
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.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Order.Nonneg.Basic
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
imp... | Mathlib/Data/NNRat/Defs.lean | 423 | 424 | |
/-
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.Partrec
import Mathlib.Data.Option.Basic
/-!
# Gödel Numbering for Partial Recursive Functions.
This file defines `Nat.Partrec.Code`, a... | refine ⟨fun h => ?_, ?_⟩
· induction h with
| zero => exact ⟨zero, rfl⟩
| Mathlib/Computability/PartrecCode.lean | 525 | 527 |
/-
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.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Anal... | theorem y_pos_of_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (D_lt_one : D < 1)
(hx : x ∈ ball (0 : E) (1 + D)) : 0 < y D x := by
simp only [mem_ball_zero_iff] at hx
refine (integral_pos_iff_support_of_nonneg (w_mul_φ_nonneg D x) ?_).2 ?_
· have F_comp : HasCompactSupport (w D) := w_compact_support E Dpos
hav... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 384 | 399 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | Mathlib/Data/List/Rotate.lean | 640 | 648 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Ordering.Lemmas
import Mathlib.Data.PNat.Basic
import Mathlib.SetTheory.Ordinal.Principal
import Ma... | | oadd' {e n a eb b} : NFBelow e eb → NFBelow a (repr e) → repr e < b → NFBelow (oadd e n a) b
/-- A normal form ordinal notation has the form
`ω ^ a₁ * n₁ + ω ^ a₂ * n₂ + ⋯ + ω ^ aₖ * nₖ`
where `a₁ > a₂ > ⋯ > aₖ` and all the `aᵢ` are also in normal form.
We will essentially only be interested in normal form ordi... | Mathlib/SetTheory/Ordinal/Notation.lean | 176 | 188 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.O... | theorem apply_symm_apply {x : B × F} (hx : x ∈ e.target) : e (e.toPartialEquiv.symm x) = x :=
e.toPartialEquiv.right_inv hx
| Mathlib/Topology/FiberBundle/Trivialization.lean | 145 | 147 |
/-
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... | lemma _root_.CovBy.card_multiset (h : s ⋖ t) : card s ⋖ card t := by
obtain ⟨a, rfl⟩ := h.exists_multiset_cons; rw [card_cons]; exact covBy_succ _
| Mathlib/Data/Finset/Grade.lean | 40 | 41 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Determinant
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Ri... | | zero => exact Or.inl <| le_refl 1
| succ n => exact Or.inr (Or.resolve_left hyp n.succ_ne_zero)
exact fib_le_of_contsAux_b this
/-! As a simple consequence, we can now derive that all denominators are nonnegative. -/
theorem zero_le_of_contsAux_b : 0 ≤ ((of v).contsAux n).b := by
let g := of v
induct... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 252 | 267 |
/-
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... | d ∣ ordCompl[p] n := by
| Mathlib/Data/Nat/Factorization/Basic.lean | 253 | 253 |
/-
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... | @[to_additive]
theorem mulAntidiagonal_mono_right (h : u ⊆ t) :
| Mathlib/Data/Finset/MulAntidiagonal.lean | 72 | 73 |
/-
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
/... |
theorem ball_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).ball x r = f ⁻¹' p.ball (f x) r := by
| Mathlib/Analysis/Seminorm.lean | 736 | 738 |
/-
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.Analysis.Seminorm
import Mathlib.GroupTheory.GroupAction.Pointwise
/-!
# The Minkowski functional, normed field version
In this file we define `(eg... |
lemma le_egauge_prod (s : Set E) (t : Set F) (a : E) (b : F) :
max (egauge 𝕜 s a) (egauge 𝕜 t b) ≤ egauge 𝕜 (s ×ˢ t) (a, b) :=
max_le (mapsTo_fst_prod.egauge_le 𝕜 (MulActionHom.fst 𝕜 E F) (a, b))
(MapsTo.egauge_le 𝕜 (MulActionHom.snd 𝕜 E F) mapsTo_snd_prod (a, b))
end SMul
| Mathlib/Analysis/Convex/EGauge.lean | 83 | 89 |
/-
Copyright (c) 2019 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.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Rat
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.PNat.Defs... |
theorem num_den_mk {q : ℚ} {n d : ℤ} (hd : d ≠ 0) (qdf : q = n /. d) :
∃ c : ℤ, n = c * q.num ∧ d = c * q.den := by
obtain rfl | hn := eq_or_ne n 0
· simp [qdf]
have : q.num * d = n * ↑q.den := by
| Mathlib/Data/Rat/Lemmas.lean | 33 | 38 |
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux, Kim Morrison, Oliver Nash
-/
import Mathlib.Algebra.Group.Action.Defs
import Mathlib.Tactic.Abel
/-! # The `noncomm_ring` tactic
Solve goals in not necessarily commutativ... | lemma mul_nat_lit_eq_nsmul [n.AtLeastTwo] : r * ofNat(n) = OfNat.ofNat n • r := by
simp only [nsmul_eq_mul', Nat.cast_ofNat]
| Mathlib/Tactic/NoncommRing.lean | 28 | 29 |
/-
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... |
/-- Composing a `List.map` with another `List.map` is equal to
a single `List.map` of composed functions.
-/
@[simp]
theorem map_comp_map (g : β → γ) (f : α → β) : map g ∘ map f = map (g ∘ f) := by
| Mathlib/Data/List/Basic.lean | 728 | 733 |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
-/
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Logic.F... | fastFibAux (bit false n) =
let p := fastFibAux n
(p.1 * (2 * p.2 - p.1), p.2 ^ 2 + p.1 ^ 2) := by
rw [fastFibAux, binaryRec_eq]
· rfl
· simp
theorem fast_fib_aux_bit_tt (n : ℕ) :
| Mathlib/Data/Nat/Fib/Basic.lean | 186 | 193 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Defs
import Mathlib.Tactic.Linter.Deprecate... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 1,085 | 1,086 | |
/-
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 | 1,599 | 1,601 | |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.CDF
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
/-! # Gamma distributions over ℝ
Define the gamm... | /-- A Lebesgue Integral from -∞ to y can be expressed as the sum of one from -∞ to 0 and 0 to x -/
lemma lintegral_Iic_eq_lintegral_Iio_add_Icc {y z : ℝ} (f : ℝ → ℝ≥0∞) (hzy : z ≤ y) :
∫⁻ x in Iic y, f x = (∫⁻ x in Iio z, f x) + ∫⁻ x in Icc z y, f x := by
rw [← Iio_union_Icc_eq_Iic hzy, lintegral_union measurable... | Mathlib/Probability/Distributions/Gamma.lean | 29 | 35 |
/-
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.SpecificLimits.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Results on discretized exponentials
We state several auxi... | _ ≤ c ^ i - 1 := by
gcongr
simpa only [← div_eq_mul_inv, one_le_div cpos, pow_one] using le_self_pow₀ hc.le hi
_ ≤ ⌊c ^ i⌋₊ := (Nat.sub_one_lt_floor _).le
/-- The sum of `1/⌊c^i⌋₊^2` above a threshold `j` is comparable to `1/j^2`, up to a multiplicative
constant. -/
theorem sum_div_nat_floor_pow_sq... | Mathlib/Analysis/SpecificLimits/FloorPow.lean | 271 | 280 |
/-
Copyright (c) 2024 Etienne Marion. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Etienne Marion
-/
import Mathlib.Data.Finite.Prod
import Mathlib.MeasureTheory.SetSemiring
/-!
# Algebra of sets
In this file we define the notion of algebra of sets and give its bas... | /-- If a family of sets `𝒜` is contained in an algebra of sets `ℬ`, then so is the algebra of sets
generated by `𝒜`. -/
theorem generateSetAlgebra_subset {ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ)
(hℬ : IsSetAlgebra ℬ) : generateSetAlgebra 𝒜 ⊆ ℬ := by
intro s hs
induction hs with
| base t t_mem => exact h t_mem
| em... | Mathlib/MeasureTheory/SetAlgebra.lean | 151 | 158 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Closed.Cartesian
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Adjunction.FullyFaithful... | theorem expComparison_whiskerLeft {A A' : C} (f : A' ⟶ A) :
(expComparison F A).whiskerBottom (pre (F.map f)) =
(expComparison F A').whiskerTop (pre f) := by
unfold expComparison pre
| Mathlib/CategoryTheory/Closed/Functor.lean | 100 | 103 |
/-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Sites.Coherent.Comparison
import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves
import Mathlib.CategoryTheory.Sites.Coherent... | rfl
lemma coverPreserving : haveI := F.reflects_preregular
CoverPreserving (regularTopology _) (regularTopology _) F :=
| Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean | 188 | 191 |
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | /-!
## Redundant state optimization
The following section are collection of rewrites to simplify, or even get rid, redundant
| Mathlib/Data/Vector/MapLemmas.lean | 224 | 227 |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Divisibility.Hom
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Group.Nat.Hom
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Alg... | lemma _root_.nsmul_eq_mul' (a : α) (n : ℕ) : n • a = a * n := by
induction n with
| Mathlib/Data/Nat/Cast/Basic.lean | 62 | 63 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... |
/-! ### Two-side inverses -/
namespace InvOn
lemma _root_.Set.invOn_id (s : Set α) : InvOn id id s s := ⟨s.leftInvOn_id, s.rightInvOn_id⟩
lemma comp (hf : InvOn f' f s t) (hg : InvOn g' g t p) (fst : MapsTo f s t)
| Mathlib/Data/Set/Function.lean | 857 | 863 |
/-
Copyright (c) 2022 Yaël Dillies, Ella Yu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Ella Yu
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Fintype.Prod
import Mathlib.Algebra.Group.Poin... |
variable [CommMonoid α]
@[to_additive] lemma mulEnergy_comm (s t : Finset α) : Eₘ[s, t] = Eₘ[t, s] := by
rw [mulEnergy, ← Finset.card_map (Equiv.prodComm _ _).toEmbedding, map_filter]
simp [-Finset.card_map, eq_comm, mulEnergy, mul_comm, map_eq_image, Function.comp_def]
end CommMonoid
section CommGroup
variabl... | Mathlib/Combinatorics/Additive/Energy.lean | 159 | 170 |
/-
Copyright (c) 2024 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Lezeau, Calle Sönne
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.CommSq
/-!
# HomLift
Given a functor `p : 𝒳 ⥤ 𝒮`, this file provides API for... | subst_hom_lift p f φ; simp
| Mathlib/CategoryTheory/FiberedCategory/HomLift.lean | 85 | 86 |
/-
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 | 507 | 508 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | theorem measure_diff_null (ht : μ t = 0) : μ (s \ t) = μ s :=
(measure_mono diff_subset).antisymm <| calc
| Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 93 | 94 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... | def equivOpposite : Submodule R Aᵐᵒᵖ ≃+* (Submodule R A)ᵐᵒᵖ where
toFun p := op <| p.comap (↑(opLinearEquiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ)
invFun p := p.unop.comap (↑(opLinearEquiv R : A ≃ₗ[R] Aᵐᵒᵖ).symm : Aᵐᵒᵖ →ₗ[R] A)
| Mathlib/Algebra/Algebra/Operations.lean | 652 | 654 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... | theorem one_lt_rpow_of_pos_of_lt_one_of_neg {x : ℝ≥0∞} {z : ℝ} (hx1 : 0 < x) (hx2 : x < 1)
(hz : z < 0) : 1 < x ^ z := by
lift x to ℝ≥0 using ne_of_lt (lt_of_lt_of_le hx2 le_top)
simp only [coe_lt_one_iff, coe_pos] at hx1 hx2 ⊢
simp [← coe_rpow_of_ne_zero (ne_of_gt hx1), NNReal.one_lt_rpow_of_pos_of_lt_one_of... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 857 | 861 |
/-
Copyright (c) 2023 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# D... | theorem doubleFactorial_eq_prod_odd :
∀ n : ℕ, (2 * n + 1)‼ = ∏ i ∈ Finset.range n, (2 * (i + 1) + 1)
| 0 => rfl
| n + 1 => by
rw [Finset.prod_range_succ, ← doubleFactorial_eq_prod_odd _, mul_comm (2 * n + 1)‼,
(by ring : 2 * (n + 1) + 1 = 2 * n + 1 + 2)]
| Mathlib/Data/Nat/Factorial/DoubleFactorial.lean | 71 | 76 |
/-
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
/... | lift_lift_same_le_lift
theorem lift_lift'_same_eq_lift' {g : Set α → Set α → Set β} (hg₁ : ∀ s, Monotone fun t => g s t)
(hg₂ : ∀ t, Monotone fun s => g s t) :
| Mathlib/Order/Filter/Lift.lean | 294 | 297 |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Multiset coercion to type
This module defines a `CoeSort` instance for multi... | split
· rename_i h
obtain ⟨rfl, h2⟩ := h
simp [← h2]
· simp
right_inv := by
| Mathlib/Data/Multiset/Fintype.lean | 263 | 268 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 1,364 | 1,367 | |
/-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Category.ModuleCat.Projective
import Mathlib.AlgebraicTopology.ExtraDegeneracy
import Mathlib.CategoryTheory.Abelian.Ext
import Mathlib.G... |
/-- The universal cover of the classifying space of `G` as a simplicial set, augmented by the map
from `Fin 1 → G` to the terminal object in `Type u`. -/
def compForgetAugmented : SimplicialObject.Augmented (Type u) :=
SimplicialObject.augment (classifyingSpaceUniversalCover G ⋙ forget _) (terminal _)
(terminal.... | Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean | 401 | 410 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.StronglyMeasura... | variable (F p μ)
/-- `lpMeasSubgroup` and `Lp F p (μ.trim hm)` are isometric. -/
noncomputable def lpMeasSubgroupToLpTrimIso [Fact (1 ≤ p)] (hm : m ≤ m0) :
lpMeasSubgroup F m p μ ≃ᵢ Lp F p (μ.trim hm) where
toFun := lpMeasSubgroupToLpTrim F p μ hm
invFun := lpTrimToLpMeasSubgroup F p μ hm
| Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 387 | 393 |
/-
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.Data.Finset.Lattice.Prod
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Setoid.Basic
import Mathlib.Order.Atoms
impor... | ext i
| Mathlib/Order/Partition/Finpartition.lean | 501 | 501 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.OrderDual
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!... | @[to_additive exists_lt_of_sum_lt]
lemma exists_lt_of_prod_lt' [LinearOrder M] [MulRightMono M]
[MulLeftMono M] {l : List ι} (f g : ι → M)
(h : (l.map f).prod < (l.map g).prod) : ∃ i ∈ l, f i < g i := by
contrapose! h
| Mathlib/Algebra/Order/BigOperators/Group/List.lean | 97 | 101 |
/-
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.MvPolynomial.Expand
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.FiniteDimensi... |
section
variable (K σ)
/-- `MvPolynomial.eval` as a `K`-linear map. -/
@[simps]
| Mathlib/FieldTheory/Finite/Polynomial.lean | 110 | 116 |
/-
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... | {v : E} (hv : v ∈ Kᗮ) : K.orthogonalProjection v = 0 := by
ext
convert eq_orthogonalProjection_of_mem_orthogonal (K := K) _ _ <;> simp [hv]
/-- The projection into `U` from an orthogonal submodule `V` is the zero map. -/
theorem IsOrtho.orthogonalProjection_comp_subtypeL {U V : Submodule 𝕜 E}
[U.HasOrthog... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 813 | 824 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.Normed.Group.Int
import Mathlib.Analysis.Normed.Group.Subgroup
import Mathlib.Analysis.Normed.Group.Uniform
/-!
# Normed groups homomorphisms... | Mathlib/Analysis/Normed/Group/Hom.lean | 954 | 956 | |
/-
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.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... | let m := f.natDegree - 1
have hm : m + 1 = f.natDegree := tsub_add_cancel_of_le f_nat_degree_pos
have h2 := coeff_derivative f m
rw [Polynomial.ext_iff] at h
rw [h m, coeff_zero, ← Nat.cast_add_one, ← nsmul_eq_mul', eq_comm, smul_eq_zero] at h2
replace h2 := h2.resolve_left m.succ_ne_zero
rw [hm, ← leadin... | Mathlib/Algebra/Polynomial/Derivative.lean | 235 | 244 |
/-
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 | 2,373 | 2,378 | |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Eq... | next => skip
rw [← neg_def, neg_sub]
| Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | 358 | 359 |
/-
Copyright (c) 2020 Kevin Buzzard, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathl... | refine Equiv.trans (isLimitMapConeForkEquiv _ _) ?_
refine (IsLimit.postcomposeHomEquiv e _).symm.trans
(IsLimit.equivIsoLimit (Fork.ext (Iso.refl _) ?_))
dsimp [Equalizer.forkMap, forkMap, e, Fork.ι]
simp only [id_comp, map_lift_piComparison]
-- Remark : this lemma uses `A'` not `A`; `A'` is `A` but with ... | Mathlib/CategoryTheory/Sites/Sheaf.lean | 657 | 663 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Kim Morrison, Adam Topaz
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.Cat... | variable [ConcreteCategory.{max v w} C FC]
open WidePullback
open WidePullbackShape
theorem widePullback_ext {B : C} {ι : Type w} {X : ι → C} (f : ∀ j : ι, X j ⟶ B)
[HasWidePullback B X f] [PreservesLimit (wideCospan B X f) (forget C)]
| Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean | 229 | 236 |
/-
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
/-!
# ... | mem_list_cycles_iff hl'.left hl'.right.left
| Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 556 | 557 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | refine rel_iSup_tsum μ measure_empty (· ≤ ·) (fun t ↦ ?_) _
calc
μ (⋃ i, t i) = μ (⋃ i, disjointed t i) := by rw [iUnion_disjointed]
_ ≤ ∑' i, μ (disjointed t i) :=
OuterMeasureClass.measure_iUnion_nat_le _ _ (disjoint_disjointed _)
_ ≤ ∑' i, μ (t i) := by gcongr; exact disjointed_subset ..
| Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 63 | 69 |
/-
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... |
lemma ih1 (ih : ∀ a, ArgsRel a (Args.P1 x y) → P124 a) : IH1 x y := by
rintro x₁ x₂ y' h₁ h₂ (rfl|hy) <;> apply ih (Args.P24 _ _ _)
on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)
| Mathlib/SetTheory/Surreal/Multiplication.lean | 220 | 223 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | simp
theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by
constructor <;> simp +contextual [subset_rfl]
theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') :
G'.deleteEdges s' ≤ G'.deleteEdges s := by
constructor <;> simp +contextual only [deleteEdges_verts, deleteEdges_adj,
true_and, and_im... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,084 | 1,098 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... |
protected theorem union {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') :
| Mathlib/Logic/Equiv/PartialEquiv.lean | 372 | 373 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to... | theorem inverse_domain : (inverse f).domain = LinearMap.range f.toFun := by
rw [inverse, Submodule.toLinearPMap_domain, ← graph_map_snd_eq_range,
← LinearEquiv.fst_comp_prodComm, Submodule.map_comp]
rfl
variable (hf : LinearMap.ker f.toFun = ⊥)
include hf
/-- The graph of the inverse generates a `LinearPMap`.... | Mathlib/LinearAlgebra/LinearPMap.lean | 967 | 977 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 1,912 | 1,916 | |
/-
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.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
import Mathlib.MeasureTheory.Measure.Prod
impo... | def mass (μ : FiniteMeasure Ω) : ℝ≥0 := μ univ
@[simp] theorem apply_le_mass (μ : FiniteMeasure Ω) (s : Set Ω) : μ s ≤ μ.mass := by
simpa using apply_mono μ (subset_univ s)
| Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 168 | 171 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Heather Macbeth
-/
import Mathlib.Topology.FiberBundle.Trivialization
import Mathlib.Topology.Order.LeftRightNhds
/-!
# Fiber bundles
Mathemat... | theorem totalSpaceMk_isClosedEmbedding [T1Space B] (x : B) :
IsClosedEmbedding (@TotalSpace.mk B F E x) :=
⟨totalSpaceMk_isEmbedding F E x, by
rw [TotalSpace.range_mk]
exact isClosed_singleton.preimage <| continuous_proj F E⟩
| Mathlib/Topology/FiberBundle/Basic.lean | 272 | 276 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Countable
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.FunProp.Attr
import Mathlib.Tactic.Mea... | scoped notation "Measurable[" mα ", " mβ "]" => @Measurable _ _ mα mβ
end MeasureTheory
section MeasurableFunctions
@[measurability]
theorem measurable_id {_ : MeasurableSpace α} : Measurable (@id α) := fun _ => id
@[fun_prop, measurability]
theorem measurable_id' {_ : MeasurableSpace α} : Measurable fun a : α => a... | Mathlib/MeasureTheory/MeasurableSpace/Defs.lean | 481 | 491 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 1,431 | 1,434 | |
/-
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.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Complex.Convex
import Mathlib.D... | theorem integrable_cexp_neg_mul_sq {b : ℂ} (hb : 0 < b.re) :
Integrable fun x : ℝ => cexp (-b * (x : ℂ) ^ 2) := by
refine ⟨(Complex.continuous_exp.comp
| Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 152 | 154 |
/-
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.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Additive operations on derivative... | simp only [hasFDerivAtFilter_iff_isLittleO] at *
convert IsLittleO.sum h
| Mathlib/Analysis/Calculus/FDeriv/Add.lean | 332 | 333 |
/-
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... | unfold clPrehaar; rw [IsClosed.closure_subset_iff]
· rintro _ ⟨U, _, rfl⟩; apply prehaar_empty
· apply continuous_iff_isClosed.mp this; exact isClosed_singleton
| Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 370 | 372 |
/-
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.Data.Fin.Tuple.Sort
import Mathlib.Order.WellFounded
import Mathlib.Order.PiLex
import Mathlib.Data.Finite.Prod
/-!
# "Bubble sort" induction
We implem... | theorem bubble_sort_induction' {n : ℕ} {α : Type*} [LinearOrder α] {f : Fin n → α}
{P : (Fin n → α) → Prop} (hf : P f)
(h : ∀ (σ : Equiv.Perm (Fin n)) (i j : Fin n),
i < j → (f ∘ σ) j < (f ∘ σ) i → P (f ∘ σ) → P (f ∘ σ ∘ Equiv.swap i j)) :
P (f ∘ sort f) := by
letI := @Preorder.lift _ (Lex (Fin n → ... | Mathlib/Data/Fin/Tuple/BubbleSortInduction.lean | 34 | 44 |
/-
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.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | theorem IsBase.diff_infinite_comm (hB₁ : M.IsBase B₁) (hB₂ : M.IsBase B₂) :
(B₁ \ B₂).Infinite ↔ (B₂ \ B₁).Infinite :=
infinite_iff_infinite_of_encard_eq_encard (hB₁.encard_diff_comm hB₂)
| Mathlib/Data/Matroid/Basic.lean | 488 | 490 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# zip & unzip
This file provides results about `List.zipWith`, `List.zip` and `List.unzip` (definitions are in
core ... | Mathlib/Data/List/Zip.lean | 262 | 268 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.Separation.Regular
import Mathlib.Topology.UniformSpace.Basic
/-!
# Hausdorff... | Mathlib/Topology/UniformSpace/Separation.lean | 329 | 334 | |
/-
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
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... |
@[simp]
theorem homTensorHomMap_apply (f : M →ₗ[R] P) (g : N →ₗ[R] Q) :
homTensorHomMap R M N P Q (f ⊗ₜ g) = map f g :=
rfl
@[simp]
theorem map₂_apply_tmul (f : M →ₗ[R] P →ₗ[R] Q) (g : N →ₗ[R] S →ₗ[R] T) (m : M) (n : N) :
map₂ f g (m ⊗ₜ n) = map (f m) (g n) := rfl
@[simp]
theorem map_zero_left (g : N →ₗ[R]... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 848 | 859 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Support
/-! # Interactions between `R[X]` and `Rᵐᵒᵖ[X]`
This file contains the basic API for "pushing through" the isomorph... | opRingEquiv_symm_monomial 1 1
theorem opRingEquiv_symm_C_mul_X_pow (r : Rᵐᵒᵖ) (n : ℕ) :
| Mathlib/RingTheory/Polynomial/Opposites.lean | 68 | 70 |
/-
Copyright (c) 2017 Mario Carneiro. 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.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 1,244 | 1,248 | |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
/-!
# The layer cake formula / Cavalieri's principle / tail probability formula
In this file we prove the f... | rw [measure_zero_iff_ae_nmem]
filter_upwards [f_bdd] with a ha using not_lt.mpr ha
rw [measureReal_def, ENNReal.toReal_eq_zero_iff]
exact Or.inl <| measure_mono_null (fun a ha ↦ lt_of_lt_of_le htM ha) obs
end LayercakeIntegral
end MeasureTheory
| Mathlib/MeasureTheory/Integral/Layercake.lean | 553 | 565 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... |
@[simp, mfld_simps]
theorem transPartialEquiv_trans (e : α ≃ β) (f' : PartialEquiv β γ) (f'' : PartialEquiv γ δ) :
(e.transPartialEquiv f').trans f'' = e.transPartialEquiv (f'.trans f'') := by
| Mathlib/Logic/Equiv/PartialEquiv.lean | 936 | 939 |
/-
Copyright (c) 2024 Miyahara Kō. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Miyahara Kō
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Defs
import Mathlib.Data.Set.Function
/-!
# iterate
Proves various lemmas about `List.iterate`.
-/
variab... | @[simp]
theorem mem_iterate {f : α → α} {a : α} {n : ℕ} {b : α} :
b ∈ iterate f a n ↔ ∃ m < n, b = f^[m] a := by
| Mathlib/Data/List/Iterate.lean | 38 | 40 |
/-
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... | theorem product_image_fst [DecidableEq α] (ht : t.Nonempty) : (s ×ˢ t).image Prod.fst = s := by
ext i
| Mathlib/Data/Finset/Prod.lean | 72 | 73 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Order.Group.Multiset
import Mathlib.Data.Setoid.Basic
import Mathlib.Data.Vector.Basic
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Tactic.Ap... |
instance decidableMem [DecidableEq α] (a : α) (s : Sym α n) : Decidable (a ∈ s) :=
s.1.decidableMem _
| Mathlib/Data/Sym/Basic.lean | 154 | 156 |
/-
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... | theorem lineDeriv_smul {c : 𝕜} : lineDeriv 𝕜 f x (c • v) = c • lineDeriv 𝕜 f x v := by
rcases eq_or_ne c 0 with rfl|hc
| Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 542 | 543 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Topology.Instances.AddCircle
/-!
# The additive circle as a norm... | theorem coe_real_preimage_closedBall_eq_iUnion (x ε : ℝ) :
(↑) ⁻¹' closedBall (x : AddCircle p) ε = ⋃ z : ℤ, closedBall (x + z • p) ε := by
rcases eq_or_ne p 0 with (rfl | hp)
· simp [iUnion_const]
ext y
simp only [dist_eq_norm, mem_preimage, mem_closedBall, zsmul_eq_mul, mem_iUnion, Real.norm_eq_abs,
←... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 142 | 164 |
/-
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 | 2,390 | 2,392 | |
/-
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... | section NumberField
variable [NumberField K]
open scoped Classical in
/-- The number of infinite real places of the number field `K`. -/
noncomputable abbrev nrRealPlaces := card { w : InfinitePlace K // IsReal w }
@[deprecated (since := "2024-10-24")] alias NrRealPlaces := nrRealPlaces
open scoped Classical in
/--... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 604 | 619 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,872 | 1,873 | |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
/-!
# Jensen's inequality... | theorem ConcaveOn.le_map_sum (hf : ConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
(h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) :
(∑ i ∈ t, w i • f (p i)) ≤ f (∑ i ∈ t, w i • p i) :=
ConvexOn.map_sum_le (β := βᵒᵈ) hf h₀ h₁ hmem
| Mathlib/Analysis/Convex/Jensen.lean | 69 | 72 |
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov, Winston Yin
-/
import Mathlib.Algebra.Group.End
import Mathlib.Topology.EMetricSpace.Diam
/-!
# Lipschitz co... | @[simp] lemma locallyLipschitzOn_univ : LocallyLipschitzOn univ f ↔ LocallyLipschitz f := by
simp [LocallyLipschitzOn, LocallyLipschitz]
| Mathlib/Topology/EMetricSpace/Lipschitz.lean | 86 | 88 |
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