Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
|---|---|---|---|---|---|
/-
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, Matthew Robert Ballard
-/
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
i... | Mathlib/NumberTheory/Padics/PadicVal.lean | 777 | 778 | theorem padicValInt_dvd_iff (n : ℕ) (a : ℤ) : (p : ℤ) ^ n ∣ a ↔ a = 0 ∨ n ≤ padicValInt p a := by |
rw [padicValInt, ← Int.natAbs_eq_zero, ← padicValNat_dvd_iff, ← Int.natCast_dvd, Int.natCast_pow]
|
/-
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.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
#align_import algebra.group_with_zero.basic from "leanprov... | Mathlib/Algebra/GroupWithZero/Basic.lean | 110 | 111 | theorem eq_zero_of_zero_eq_one (h : (0 : M₀) = 1) (a : M₀) : a = 0 := by |
rw [← mul_one a, ← h, mul_zero]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import... | Mathlib/RingTheory/Nilpotent/Lemmas.lean | 87 | 90 | theorem isNilpotent_mulLeft_iff (a : A) : IsNilpotent (mulLeft R a) ↔ IsNilpotent a := by |
constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
simp only [mulLeft_eq_zero_iff, pow_mulLeft] at hn ⊢ <;>
exact hn
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Johan Commelin
-/
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
import Mathlib.CategoryTheory.Monoidal.Functorial
import Mathlib.CategoryTheory.Monoidal.Type... | Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | 132 | 149 | theorem right_unitality (X : Type u) :
(ρ_ ((free R).obj X)).hom =
(𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom := by |
-- Porting note (#11041): broken ext
apply TensorProduct.ext
apply Finsupp.lhom_ext'
intro x
apply LinearMap.ext_ring
apply LinearMap.ext_ring
apply Finsupp.ext
intro x'
-- Porting note (#10934): used to be dsimp [ε, μ]
let q : X →₀ R := ((ρ_ (of R (X →₀ R))).hom) (Finsupp.single x 1 ⊗ₜ[R] 1)
cha... |
/-
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.Instances
import Mathlib.RingTheory.Ideal.Colon
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import... | Mathlib/RingTheory/PrincipalIdealDomain.lean | 460 | 461 | theorem Irreducible.coprime_iff_not_dvd {p n : R} (hp : Irreducible p) :
IsCoprime p n ↔ ¬p ∣ n := by | rw [← isRelPrime_iff_isCoprime, hp.isRelPrime_iff_not_dvd]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.A... | Mathlib/RingTheory/AdjoinRoot.lean | 237 | 240 | theorem adjoinRoot_eq_top : Algebra.adjoin R ({root f} : Set (AdjoinRoot f)) = ⊤ := by |
refine Algebra.eq_top_iff.2 fun x => ?_
induction x using AdjoinRoot.induction_on with
| ih p => exact (Algebra.adjoin_singleton_eq_range_aeval R (root f)).symm ▸ ⟨p, aeval_eq p⟩
|
/-
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.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.bas... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 204 | 212 | theorem not_facet_iff_subface (hs : s ∈ K.faces) : s ∉ K.facets ↔ ∃ t, t ∈ K.faces ∧ s ⊂ t := by |
refine ⟨fun hs' : ¬(_ ∧ _) => ?_, ?_⟩
· push_neg at hs'
obtain ⟨t, ht⟩ := hs' hs
exact ⟨t, ht.1, ⟨ht.2.1, fun hts => ht.2.2 (Subset.antisymm ht.2.1 hts)⟩⟩
· rintro ⟨t, ht⟩ ⟨hs, hs'⟩
have := hs' ht.1 ht.2.1
rw [this] at ht
exact ht.2.2 (Subset.refl t)
|
/-
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.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import ... | Mathlib/Data/Set/Basic.lean | 1,972 | 1,973 | theorem inter_insert_of_mem (h : a ∈ s) : s ∩ insert a t = insert a (s ∩ t) := by |
rw [insert_inter_distrib, insert_eq_of_mem h]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.... | Mathlib/Logic/Basic.lean | 309 | 309 | theorem xor_comm (a b : Prop) : Xor' a b = Xor' b a := by | simp [Xor', and_comm, or_comm]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.InsertNth
#align_import group_theory.f... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 559 | 559 | theorem invRev_length : (invRev L₁).length = L₁.length := by | simp [invRev]
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | Mathlib/Probability/Martingale/Upcrossing.lean | 742 | 749 | theorem Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part [IsFiniteMeasure μ]
(a b : ℝ) (hf : Submartingale f ℱ μ) (N : ℕ) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[fun ω => (f N ω - a)⁺] := by |
by_cases hab : a < b
· exact mul_integral_upcrossingsBefore_le_integral_pos_part_aux hf hab
· rw [not_lt, ← sub_nonpos] at hab
exact le_trans (mul_nonpos_of_nonpos_of_nonneg hab (by positivity))
(integral_nonneg fun ω => posPart_nonneg _)
|
/-
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.Data.Finset.NAry
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Pointwise.Finite
import Mathlib.Data.Set.Pointwise.SMul
... | Mathlib/Data/Finset/Pointwise.lean | 1,358 | 1,359 | theorem preimage_mul_left_one' : preimage 1 (a⁻¹ * ·) (mul_right_injective _).injOn = {a} := by |
rw [preimage_mul_left_one, inv_inv]
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.Equiv
#align_import topology... | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 565 | 576 | theorem uniformSpace_eq_iInf_precomp_of_cover {δ : ι → Type*} (φ : Π i, δ i → α)
(h_cover : ∃ I : Set ι, I.Finite ∧ ⋃ i ∈ I, range (φ i) = univ) :
𝒰(α, β, _) = ⨅ i, .comap (ofFun ∘ (· ∘ φ i) ∘ toFun) 𝒰(δ i, β, _) := by |
ext : 1
simp_rw [iInf_uniformity, uniformity_comap]
refine le_antisymm (le_iInf fun i ↦ tendsto_iff_comap.mp UniformFun.precomp_uniformContinuous) ?_
rcases h_cover with ⟨I, I_finite, I_cover⟩
refine Filter.hasBasis_iInf (fun i : ι ↦ UniformFun.hasBasis_uniformity (δ i) β |>.comap _)
|>.le_basis_iff (U... |
/-
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, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENa... | Mathlib/RingTheory/Multiplicity.lean | 273 | 276 | theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by |
rw [← pow_one a]
apply pow_dvd_of_le_multiplicity
simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h
|
/-
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.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a5... | Mathlib/Algebra/MvPolynomial/Variables.lean | 98 | 99 | theorem mem_vars (i : σ) : i ∈ p.vars ↔ ∃ d ∈ p.support, i ∈ d.support := by |
classical simp only [vars_def, Multiset.mem_toFinset, mem_degrees, mem_support_iff, exists_prop]
|
/-
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.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
#align_import logic.equiv.local_equ... | Mathlib/Logic/Equiv/PartialEquiv.lean | 332 | 334 | theorem symm_symm : e.symm.symm = e := by |
cases e
rfl
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johan Commelin
-/
import Mathlib.RingTheory.IntegralClosure
#align_import field_theory.minpoly.basic from "leanprover-community/mathlib"@"df0098f0db291900600f32070f6abb3e1... | Mathlib/FieldTheory/Minpoly/Basic.lean | 237 | 246 | theorem eq_X_sub_C_of_algebraMap_inj (a : A) (hf : Function.Injective (algebraMap A B)) :
minpoly A (algebraMap A B a) = X - C a := by |
nontriviality A
refine (unique' A _ (monic_X_sub_C a) ?_ ?_).symm
· rw [map_sub, aeval_C, aeval_X, sub_self]
simp_rw [or_iff_not_imp_left]
intro q hl h0
rw [← natDegree_lt_natDegree_iff h0, natDegree_X_sub_C, Nat.lt_one_iff] at hl
rw [eq_C_of_natDegree_eq_zero hl] at h0 ⊢
rwa [aeval_C, map_ne_zero_iff ... |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
impor... | Mathlib/LinearAlgebra/Vandermonde.lean | 72 | 74 | theorem vandermonde_transpose_mul_vandermonde {n : ℕ} (v : Fin n → R) (i j) :
((vandermonde v)ᵀ * vandermonde v) i j = ∑ k : Fin n, v k ^ (i + j : ℕ) := by |
simp only [vandermonde_apply, Matrix.mul_apply, Matrix.transpose_apply, pow_add]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 238 | 241 | theorem coeff_add_monomial_mul (a : R) :
coeff R (m + n) (monomial R m a * φ) = a * coeff R n φ := by |
rw [coeff_monomial_mul, if_pos, add_tsub_cancel_left]
exact le_add_right le_rfl
|
/-
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
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib... | Mathlib/Deprecated/Submonoid.lean | 232 | 237 | theorem list_prod_mem (hs : IsSubmonoid s) : ∀ {l : List M}, (∀ x ∈ l, x ∈ s) → l.prod ∈ s
| [], _ => hs.one_mem
| a :: l, h =>
suffices a * l.prod ∈ s by simpa
have : a ∈ s ∧ ∀ x ∈ l, x ∈ s := by | simpa using h
hs.mul_mem this.1 (list_prod_mem hs this.2)
|
/-
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.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Int.ModEq
import Mat... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 99 | 103 | theorem cycleOf_apply_apply_zpow_self (f : Perm α) (x : α) (k : ℤ) :
cycleOf f x ((f ^ k) x) = (f ^ (k + 1) : Perm α) x := by |
rw [SameCycle.cycleOf_apply]
· rw [add_comm, zpow_add, zpow_one, mul_apply]
· exact ⟨k, rfl⟩
|
/-
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, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction... | Mathlib/Algebra/Polynomial/Derivative.lean | 115 | 116 | theorem derivative_X_sq : derivative (X ^ 2 : R[X]) = C 2 * X := by |
rw [derivative_X_pow, Nat.cast_two, pow_one]
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 342 | 343 | theorem Fork.app_one_eq_ι_comp_right (s : Fork f g) : s.π.app one = s.ι ≫ g := by |
rw [← s.app_zero_eq_ι, ← s.w right, parallelPair_map_right]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Basic
import Mathlib.Tactic.Coe
import Mathlib.Tactic.Lift
imp... | Mathlib/Data/PNat/Defs.lean | 266 | 275 | theorem div_coe (m k : ℕ+) :
(div m k : ℕ) = ite ((m : ℕ) % (k : ℕ) = 0) ((m : ℕ) / (k : ℕ)).pred ((m : ℕ) / (k : ℕ)) := by |
dsimp [div, modDiv]
cases (m : ℕ) % (k : ℕ) with
| zero =>
rw [if_pos rfl]
rfl
| succ n =>
rw [if_neg n.succ_ne_zero]
rfl
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlg... | Mathlib/Algebra/Lie/Nilpotent.lean | 595 | 603 | theorem Equiv.lieModule_isNilpotent_iff (f : L ≃ₗ⁅R⁆ L₂) (g : M ≃ₗ[R] M₂)
(hfg : ∀ x m, ⁅f x, g m⁆ = g ⁅x, m⁆) : IsNilpotent R L M ↔ IsNilpotent R L₂ M₂ := by |
constructor <;> intro h
· have hg : Surjective (g : M →ₗ[R] M₂) := g.surjective
exact f.surjective.lieModuleIsNilpotent hg hfg
· have hg : Surjective (g.symm : M₂ →ₗ[R] M) := g.symm.surjective
refine f.symm.surjective.lieModuleIsNilpotent hg fun x m => ?_
rw [LinearEquiv.coe_coe, LieEquiv.coe_to_lieH... |
/-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
import Mathlib.Logic.Basic
#align_import logic.function.basic from "leanprover-community/mathli... | Mathlib/Logic/Function/Basic.lean | 1,055 | 1,059 | theorem eq_mp_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mp h) := by |
-- TODO: mathlib3 uses `eq_rec_on_bijective`, difference in elaboration here
-- due to `@[macro_inline]` possibly?
cases h
exact ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
|
/-
Copyright (c) 2024 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
/-!
# Final functors wit... | Mathlib/CategoryTheory/Filtered/Final.lean | 130 | 136 | theorem IsFiltered.of_exists_of_isFiltered_of_fullyFaithful [IsFiltered D] [F.Full] [F.Faithful]
(h : ∀ d, ∃ c, Nonempty (d ⟶ F.obj c)) : IsFiltered C :=
{ IsFilteredOrEmpty.of_exists_of_isFiltered_of_fullyFaithful F h with
nonempty := by |
have : Nonempty D := IsFiltered.nonempty
obtain ⟨c, -⟩ := h (Classical.arbitrary D)
exact ⟨c⟩ }
|
/-
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.Data.Multiset.Nodup
#align_import data.multiset.sum from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
/-!
# Disjoint sum of ... | Mathlib/Data/Multiset/Sum.lean | 55 | 60 | theorem inl_mem_disjSum : inl a ∈ s.disjSum t ↔ a ∈ s := by |
rw [mem_disjSum, or_iff_left]
-- Porting note: Previous code for L62 was: simp only [exists_eq_right]
· simp only [inl.injEq, exists_eq_right]
rintro ⟨b, _, hb⟩
exact inr_ne_inl hb
|
/-
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.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 623 | 625 | theorem prod_sigma {σ : α → Type*} (s : Finset α) (t : ∀ a, Finset (σ a)) (f : Sigma σ → β) :
∏ x ∈ s.sigma t, f x = ∏ a ∈ s, ∏ s ∈ t a, f ⟨a, s⟩ := by |
simp_rw [← disjiUnion_map_sigma_mk, prod_disjiUnion, prod_map, Function.Embedding.sigmaMk_apply]
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 853 | 854 | theorem ciSup_const [hι : Nonempty ι] {a : α} : ⨆ _ : ι, a = a := by |
rw [iSup, range_const, csSup_singleton]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.... | Mathlib/Topology/Compactness/Compact.lean | 648 | 662 | theorem Tendsto.isCompact_insert_range_of_cocompact {f : X → Y} {y}
(hf : Tendsto f (cocompact X) (𝓝 y)) (hfc : Continuous f) : IsCompact (insert y (range f)) := by |
intro l hne hle
by_cases hy : ClusterPt y l
· exact ⟨y, Or.inl rfl, hy⟩
simp only [clusterPt_iff, not_forall, ← not_disjoint_iff_nonempty_inter, not_not] at hy
rcases hy with ⟨s, hsy, t, htl, hd⟩
rcases mem_cocompact.1 (hf hsy) with ⟨K, hKc, hKs⟩
have : f '' K ∈ l := by
filter_upwards [htl, le_princi... |
/-
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, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDo... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 374 | 382 | theorem degree_add_div (hq0 : q ≠ 0) (hpq : degree q ≤ degree p) :
degree q + degree (p / q) = degree p := by |
have : degree (p % q) < degree (q * (p / q)) :=
calc
degree (p % q) < degree q := EuclideanDomain.mod_lt _ hq0
_ ≤ _ := degree_le_mul_left _ (mt (div_eq_zero_iff hq0).1 (not_lt_of_ge hpq))
conv_rhs =>
rw [← EuclideanDomain.div_add_mod p q, degree_add_eq_left_of_degree_lt this, degree_mul]
|
/-
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.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import ... | Mathlib/Data/Set/Basic.lean | 2,055 | 2,060 | theorem subset_insert_iff {s t : Set α} {x : α} :
s ⊆ insert x t ↔ s ⊆ t ∨ (x ∈ s ∧ s \ {x} ⊆ t) := by |
rw [← diff_singleton_subset_iff]
by_cases hx : x ∈ s
· rw [and_iff_right hx, or_iff_right_of_imp diff_subset.trans]
rw [diff_singleton_eq_self hx, or_iff_left_of_imp And.right]
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Basic
#align_import probability.probability_mass_function.monad from "leanprover-community/mathlib"@"4ac69... | Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 255 | 259 | theorem bindOnSupport_eq_zero_iff (b : β) :
p.bindOnSupport f b = 0 ↔ ∀ (a) (ha : p a ≠ 0), f a ha b = 0 := by |
simp only [bindOnSupport_apply, ENNReal.tsum_eq_zero, mul_eq_zero, or_iff_not_imp_left]
exact ⟨fun h a ha => Trans.trans (dif_neg ha).symm (h a ha),
fun h a ha => Trans.trans (dif_neg ha) (h a ha)⟩
|
/-
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.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 457 | 457 | theorem inner_re_symm (x y : E) : re ⟪x, y⟫ = re ⟪y, x⟫ := by | rw [← inner_conj_symm, conj_re]
|
/-
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.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedS... | Mathlib/Analysis/RCLike/Basic.lean | 374 | 377 | theorem add_conj (z : K) : z + conj z = 2 * re z :=
calc
z + conj z = re z + im z * I + (re z - im z * I) := by | rw [re_add_im, conj_eq_re_sub_im]
_ = 2 * re z := by rw [add_add_sub_cancel, two_mul]
|
/-
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.InitTail
#align_import ring_theory.witt_vector.truncated from "leanprover-community/mathlib"@"acbe099ced8be9c97... | Mathlib/RingTheory/WittVector/Truncated.lean | 166 | 167 | theorem truncateFun_out (x : TruncatedWittVector p n R) : x.out.truncateFun n = x := by |
simp only [WittVector.truncateFun, coeff_out, mk_coeff]
|
/-
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.Algebra.Order.Ring.Rat
import Mathlib.SetTheory.Cardinal.Basic
#align_import data.rat.denumerable from "leanprover-community/mathlib"@"dde670c9a3f503647fd... | Mathlib/Data/Rat/Denumerable.lean | 45 | 45 | theorem Cardinal.mkRat : #ℚ = ℵ₀ := by | simp only [mk_eq_aleph0]
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Computability.Encoding
import Mathlib.Logic.Small.List
import Mathlib.ModelTheory.Syntax
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import model... | Mathlib/ModelTheory/Encoding.lean | 122 | 151 | theorem card_sigma : #(Σn, L.Term (Sum α (Fin n))) = max ℵ₀ #(Sum α (Σi, L.Functions i)) := by |
refine le_antisymm ?_ ?_
· rw [mk_sigma]
refine (sum_le_iSup_lift _).trans ?_
rw [mk_nat, lift_aleph0, mul_eq_max_of_aleph0_le_left le_rfl, max_le_iff,
ciSup_le_iff' (bddAbove_range _)]
· refine ⟨le_max_left _ _, fun i => card_le.trans ?_⟩
refine max_le (le_max_left _ _) ?_
rw [← add_... |
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Patrick Massot, Scott Morrison
-/
import Mathlib.CategoryTheory.Adjunction.Reflective
import Mathlib.CategoryTheory.ConcreteCategory.UnbundledHom
import Mathlib.CategoryTh... | Mathlib/Topology/Category/UniformSpace.lean | 210 | 215 | theorem extension_comp_coe {X : UniformSpaceCat} {Y : CpltSepUniformSpace}
(f : toUniformSpace (CpltSepUniformSpace.of (Completion X)) ⟶ toUniformSpace Y) :
extensionHom (completionHom X ≫ f) = f := by |
apply Subtype.eq
funext x
exact congr_fun (Completion.extension_comp_coe f.property) x
|
/-
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.Basic
/-!
# Subsingleton
Defines the predicate `Subsingleton s : Prop`, saying that `s` has at most one element.
Also defi... | Mathlib/Data/Set/Subsingleton.lean | 99 | 104 | theorem exists_eq_singleton_iff_nonempty_subsingleton :
(∃ a : α, s = {a}) ↔ s.Nonempty ∧ s.Subsingleton := by |
refine ⟨?_, fun h => ?_⟩
· rintro ⟨a, rfl⟩
exact ⟨singleton_nonempty a, subsingleton_singleton⟩
· exact h.2.eq_empty_or_singleton.resolve_left h.1.ne_empty
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanpro... | Mathlib/RingTheory/Ideal/Operations.lean | 689 | 691 | theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by |
rw [sup_comm] at h
rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm]
|
/-
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... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 96 | 96 | theorem volume_singleton {a : ℝ} : volume ({a} : Set ℝ) = 0 := by | simp [volume_val]
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean | 276 | 279 | theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by |
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Yaël Dillies
-/
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.Group.Units
import Ma... | Mathlib/Algebra/Order/Ring/Defs.lean | 1,175 | 1,182 | theorem mul_self_pos {a : α} : 0 < a * a ↔ a ≠ 0 := by |
constructor
· rintro h rfl
rw [mul_zero] at h
exact h.false
· intro h
cases' h.lt_or_lt with h h
exacts [mul_pos_of_neg_of_neg h h, mul_pos h h]
|
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
/-!
# Some results on free modules over rings satisfying strong rank condition
T... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 284 | 295 | theorem rank_eq_one_iff [Nontrivial E] [Module.Free F S] : Module.rank F S = 1 ↔ S = ⊥ := by |
refine ⟨fun h ↦ Subalgebra.eq_bot_of_rank_le_one h.le, ?_⟩
rintro rfl
obtain ⟨κ, b⟩ := Module.Free.exists_basis (R := F) (M := (⊥ : Subalgebra F E))
refine le_antisymm ?_ ?_
· have := lift_rank_range_le (Algebra.linearMap F E)
rwa [← one_eq_range, rank_self, lift_one, lift_le_one_iff] at this
· by_cont... |
/-
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
-/
import Mathlib.Computability.NFA
#align_import computability.epsilon_NFA from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
/-!
... | Mathlib/Computability/EpsilonNFA.lean | 212 | 214 | theorem toεNFA_correct (M : NFA α σ) : M.toεNFA.accepts = M.accepts := by |
rw [εNFA.accepts, εNFA.eval, toεNFA_evalFrom_match]
rfl
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "lean... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,297 | 1,299 | theorem HasFTaylorSeriesUpTo.ofLe (h : HasFTaylorSeriesUpTo n f p) (hmn : m ≤ n) :
HasFTaylorSeriesUpTo m f p := by |
rw [← hasFTaylorSeriesUpToOn_univ_iff] at h ⊢; exact h.of_le hmn
|
/-
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.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@... | Mathlib/Topology/Basic.lean | 773 | 776 | theorem compl_frontier_eq_union_interior :
(frontier s)ᶜ = interior s ∪ interior sᶜ := by |
rw [frontier_eq_inter_compl_interior]
simp only [compl_inter, compl_compl]
|
/-
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
#align_import geometry.euclidean.angle.oriented.rig... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 428 | 431 | theorem tan_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle (x - y) x) * ‖x‖ = ‖y‖ := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
exact (-o).tan_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two h
|
/-
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.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedS... | Mathlib/Analysis/RCLike/Basic.lean | 583 | 584 | theorem div_re_ofReal {z : K} {r : ℝ} : re (z / r) = re z / r := by |
rw [div_eq_inv_mul, div_eq_inv_mul, ← ofReal_inv, re_ofReal_mul]
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison
-/
import Mathlib.Algebra.Field.Subfield
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.T... | Mathlib/Topology/Algebra/Field.lean | 112 | 114 | theorem IsLocalMin.inv {f : α → β} {a : α} (h1 : IsLocalMin f a) (h2 : ∀ᶠ z in 𝓝 a, 0 < f z) :
IsLocalMax f⁻¹ a := by |
filter_upwards [h1, h2] with z h3 h4 using(inv_le_inv h4 h2.self_of_nhds).mpr h3
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-communit... | Mathlib/MeasureTheory/Group/Integral.lean | 165 | 168 | theorem integral_smul_eq_self {μ : Measure α} [SMulInvariantMeasure G α μ] (f : α → E) {g : G} :
(∫ x, f (g • x) ∂μ) = ∫ x, f x ∂μ := by |
have h : MeasurableEmbedding fun x : α => g • x := (MeasurableEquiv.smul g).measurableEmbedding
rw [← h.integral_map, map_smul]
|
/-
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, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Multiplicit... | Mathlib/Algebra/Polynomial/Div.lean | 599 | 612 | theorem modByMonic_X_sub_C_eq_C_eval (p : R[X]) (a : R) : p %ₘ (X - C a) = C (p.eval a) := by |
nontriviality R
have h : (p %ₘ (X - C a)).eval a = p.eval a := by
rw [modByMonic_eq_sub_mul_div _ (monic_X_sub_C a), eval_sub, eval_mul, eval_sub, eval_X,
eval_C, sub_self, zero_mul, sub_zero]
have : degree (p %ₘ (X - C a)) < 1 :=
degree_X_sub_C a ▸ degree_modByMonic_lt p (monic_X_sub_C a)
have :... |
/-
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.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Data.Fintype.Card
#align_import data.finset.noncomm_prod f... | Mathlib/Data/Finset/NoncommProd.lean | 64 | 67 | theorem noncommFoldr_cons (s : Multiset α) (a : α) (h h') (b : β) :
noncommFoldr f (a ::ₘ s) h b = f a (noncommFoldr f s h' b) := by |
induction s using Quotient.inductionOn
simp
|
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheor... | Mathlib/CategoryTheory/Generator.lean | 93 | 98 | theorem isSeparating_op_iff (𝒢 : Set C) : IsSeparating 𝒢.op ↔ IsCoseparating 𝒢 := by |
refine ⟨fun h𝒢 X Y f g hfg => ?_, fun h𝒢 X Y f g hfg => ?_⟩
· refine Quiver.Hom.op_inj (h𝒢 _ _ fun G hG h => Quiver.Hom.unop_inj ?_)
simpa only [unop_comp, Quiver.Hom.unop_op] using hfg _ (Set.mem_op.1 hG) _
· refine Quiver.Hom.unop_inj (h𝒢 _ _ fun G hG h => Quiver.Hom.op_inj ?_)
simpa only [op_comp,... |
/-
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.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.... | Mathlib/Order/Interval/Set/OrderIso.lean | 98 | 99 | theorem image_Ico (e : α ≃o β) (a b : α) : e '' Ico a b = Ico (e a) (e b) := by |
rw [e.image_eq_preimage, e.symm.preimage_Ico, e.symm_symm]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.M... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 308 | 319 | theorem comp_mul_left (hf : IntervalIntegrable f volume a b) (c : ℝ) :
IntervalIntegrable (fun x => f (c * x)) volume (a / c) (b / c) := by |
rcases eq_or_ne c 0 with (hc | hc); · rw [hc]; simp
rw [intervalIntegrable_iff'] at hf ⊢
have A : MeasurableEmbedding fun x => x * c⁻¹ :=
(Homeomorph.mulRight₀ _ (inv_ne_zero hc)).closedEmbedding.measurableEmbedding
rw [← Real.smul_map_volume_mul_right (inv_ne_zero hc), IntegrableOn, Measure.restrict_smul,... |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.... | Mathlib/Order/UpperLower/Basic.lean | 1,312 | 1,314 | theorem map_Iic (f : α ≃o β) (a : α) : map f (Iic a) = Iic (f a) := by |
ext
simp
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriente... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 68 | 68 | theorem oangle_zero_left (x : V) : o.oangle 0 x = 0 := by | simp [oangle]
|
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
/-!
# The length function, reduced words, and descents
Throughout this file, `B` is a type and `... | Mathlib/GroupTheory/Coxeter/Length.lean | 281 | 289 | theorem exists_leftDescent_of_ne_one {w : W} (hw : w ≠ 1) : ∃ i : B, cs.IsLeftDescent w i := by |
rcases cs.exists_reduced_word w with ⟨ω, h, rfl⟩
have h₁ : ω ≠ [] := by rintro rfl; simp at hw
rcases List.exists_cons_of_ne_nil h₁ with ⟨i, ω', rfl⟩
use i
rw [IsLeftDescent, ← h, wordProd_cons, simple_mul_simple_cancel_left]
calc
ℓ (π ω') ≤ ω'.length := cs.length_wordProd_le ω'
_ ... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `μ` on a type `α` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 899 | 905 | theorem comap_apply (μ : Measure β) (s : Set α) : comap f μ s = μ (f '' s) :=
calc
comap f μ s = comap f μ (f ⁻¹' (f '' s)) := by | rw [hf.injective.preimage_image]
_ = (comap f μ).map f (f '' s) := (hf.map_apply _ _).symm
_ = μ (f '' s) := by
rw [hf.map_comap, restrict_apply' hf.measurableSet_range,
inter_eq_self_of_subset_left (image_subset_range _ _)]
|
/-
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, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 128 | 129 | theorem natDegree_of_subsingleton [Subsingleton R] : natDegree p = 0 := by |
rw [Subsingleton.elim p 0, natDegree_zero]
|
/-
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.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mat... | Mathlib/SetTheory/Ordinal/Exponential.lean | 379 | 382 | theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o := by |
rcases eq_or_ne b 0 with (rfl | hb)
· simpa using Ordinal.pos_iff_ne_zero.2 ho
· exact (mod_lt _ <| opow_ne_zero _ hb).trans_le (opow_log_le_self _ ho)
|
/-
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.Algebra.GCDMonoid.Basic
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Polynomial.Eisenstein.Basic
#align_import algebra.gcd_monoid.in... | Mathlib/Algebra/GCDMonoid/IntegrallyClosed.lean | 23 | 30 | theorem IsLocalization.surj_of_gcd_domain [GCDMonoid R] (M : Submonoid R) [IsLocalization M A]
(z : A) : ∃ a b : R, IsUnit (gcd a b) ∧ z * algebraMap R A b = algebraMap R A a := by |
obtain ⟨x, ⟨y, hy⟩, rfl⟩ := IsLocalization.mk'_surjective M z
obtain ⟨x', y', hx', hy', hu⟩ := extract_gcd x y
use x', y', hu
rw [mul_comm, IsLocalization.mul_mk'_eq_mk'_of_mul]
convert IsLocalization.mk'_mul_cancel_left (M := M) (S := A) _ _ using 2
rw [Subtype.coe_mk, hy', ← mul_comm y', mul_assoc]; conv... |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
/-!
# Some results on free modules over rings satisfying strong rank condition
T... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 262 | 274 | theorem eq_bot_of_rank_le_one (h : Module.rank F S ≤ 1) [Module.Free F S] : S = ⊥ := by |
nontriviality E
obtain ⟨κ, b⟩ := Module.Free.exists_basis (R := F) (M := S)
by_cases h1 : Module.rank F S = 1
· refine bot_unique fun x hx ↦ Algebra.mem_bot.2 ?_
rw [← b.mk_eq_rank'', eq_one_iff_unique, ← unique_iff_subsingleton_and_nonempty] at h1
obtain ⟨h1⟩ := h1
obtain ⟨y, hy⟩ := (bijective_alg... |
/-
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.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import ... | Mathlib/Data/DFinsupp/Basic.lean | 635 | 636 | theorem single_eq_same {i b} : (single i b : Π₀ i, β i) i = b := by |
simp only [single_apply, dite_eq_ite, ite_true]
|
/-
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.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 1,102 | 1,111 | theorem snd_map_prod_mk₀ {X : α → β} {Y : α → γ} {μ : Measure α} (hX : AEMeasurable X μ) :
(μ.map fun a => (X a, Y a)).snd = μ.map Y := by |
by_cases hY : AEMeasurable Y μ
· ext1 s hs
rw [Measure.snd_apply hs, Measure.map_apply_of_aemeasurable (hX.prod_mk hY) (measurable_snd hs),
Measure.map_apply_of_aemeasurable hY hs, ← univ_prod, mk_preimage_prod, preimage_univ,
univ_inter]
· have : ¬AEMeasurable (fun x ↦ (X x, Y x)) μ := by
... |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.... | Mathlib/Data/Matrix/Rank.lean | 140 | 142 | theorem rank_submatrix [Fintype m] (A : Matrix m m R) (e₁ e₂ : n ≃ m) :
rank (A.submatrix e₁ e₂) = rank A := by |
simpa only [reindex_apply] using rank_reindex e₁.symm e₂.symm A
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 479 | 480 | theorem symmDiff_assoc : a ∆ b ∆ c = a ∆ (b ∆ c) := by |
rw [symmDiff_symmDiff_left, symmDiff_symmDiff_right]
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 2,261 | 2,263 | theorem inr_pushoutLeftPushoutInrIso_inv :
pushout.inr ≫ (pushoutLeftPushoutInrIso f g g').inv = pushout.inr := by |
rw [Iso.comp_inv_eq, inr_pushoutLeftPushoutInrIso_hom]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 395 | 397 | theorem get?_set (a : α) {m n} (l : List α) :
(set l m a).get? n = if m = n then (fun _ => a) <$> l.get? n else l.get? n := by |
by_cases m = n <;> simp [*, get?_set_eq, get?_set_ne]
|
/-
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.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#a... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 526 | 534 | theorem IsCycle.of_pow {n : ℕ} (h1 : IsCycle (f ^ n)) (h2 : f.support ⊆ (f ^ n).support) :
IsCycle f := by |
have key : ∀ x : α, (f ^ n) x ≠ x ↔ f x ≠ x := by
simp_rw [← mem_support, ← Finset.ext_iff]
exact (support_pow_le _ n).antisymm h2
obtain ⟨x, hx1, hx2⟩ := h1
refine ⟨x, (key x).mp hx1, fun y hy => ?_⟩
cases' hx2 ((key y).mpr hy) with i _
exact ⟨n * i, by rwa [zpow_mul]⟩
|
/-
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.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Ring.Multiset
import Mathlib.Algebra.Field.Defs
import Mathlib.Data.Finty... | Mathlib/Algebra/BigOperators/Ring.lean | 180 | 201 | theorem prod_add (f g : ι → α) (s : Finset ι) :
∏ i ∈ s, (f i + g i) = ∑ t ∈ s.powerset, (∏ i ∈ t, f i) * ∏ i ∈ s \ t, g i := by |
classical
calc
∏ i ∈ s, (f i + g i) =
∏ i ∈ s, ∑ p ∈ ({True, False} : Finset Prop), if p then f i else g i := by simp
_ = ∑ p ∈ (s.pi fun _ => {True, False} : Finset (∀ a ∈ s, Prop)),
∏ a ∈ s.attach, if p a.1 a.2 then f a.1 else g a.1 := prod_sum _ _ _
_ = ∑ t ∈ s.powerset, (∏ a ∈ t, ... |
/-
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, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,096 | 1,100 | theorem natDegree_mul_le {p q : R[X]} : natDegree (p * q) ≤ natDegree p + natDegree q := by |
apply natDegree_le_of_degree_le
apply le_trans (degree_mul_le p q)
rw [Nat.cast_add]
apply add_le_add <;> apply degree_le_natDegree
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Sébastien Gouëzel, Alex J. Best
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.Group.Nat
import ... | Mathlib/Algebra/BigOperators/Group/List.lean | 477 | 478 | theorem prod_reverse_noncomm : ∀ L : List G, L.reverse.prod = (L.map fun x => x⁻¹).prod⁻¹ := by |
simp [prod_inv_reverse]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriente... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 1,048 | 1,060 | theorem oangle_sign_smul_add_smul_smul_add_smul (x y : V) (r₁ r₂ r₃ r₄ : ℝ) :
(o.oangle (r₁ • x + r₂ • y) (r₃ • x + r₄ • y)).sign =
SignType.sign (r₁ * r₄ - r₂ * r₃) * (o.oangle x y).sign := by |
by_cases hr₁ : r₁ = 0
· rw [hr₁, zero_smul, zero_mul, zero_add, zero_sub, Left.sign_neg,
oangle_sign_smul_left, add_comm, oangle_sign_smul_add_smul_right, oangle_rev,
Real.Angle.sign_neg, sign_mul, mul_neg, mul_neg, neg_mul, mul_assoc]
· rw [← o.oangle_sign_smul_add_right (r₁ • x + r₂ • y) (r₃ • x + ... |
/-
Copyright (c) 2023 Bulhwi Cha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bulhwi Cha, Mario Carneiro
-/
import Batteries.Data.Char
import Batteries.Data.List.Lemmas
import Batteries.Data.String.Basic
import Batteries.Tactic.Lint.Misc
import Batteries.Tactic.SeqF... | .lake/packages/batteries/Batteries/Data/String/Lemmas.lean | 678 | 679 | theorem map_eq (f : Char → Char) (s) : map f s = ⟨s.1.map f⟩ := by |
simpa using mapAux_of_valid f [] s.1
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Oleksandr Manzyuk
-/
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Eq... | Mathlib/CategoryTheory/Monoidal/Bimod.lean | 317 | 325 | theorem actRight_one' : (_ ◁ T.one) ≫ actRight P Q = (ρ_ _).hom := by |
refine (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 ?_
dsimp [X]
-- Porting note: had to replace `rw` by `erw`
slice_lhs 1 2 =>erw [← whisker_exchange]
slice_lhs 2 3 => rw [π_tensor_id_actRight]
slice_lhs 1 2 => rw [associator_naturality_right]
slice_lhs 2 3 => rw [← MonoidalCategory.whiskerL... |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic... | Mathlib/Algebra/Ring/Hom/Defs.lean | 467 | 468 | theorem toMonoidWithZeroHom_eq_coe (f : α →+* β) : (f.toMonoidWithZeroHom : α → β) = f := by |
rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.ZornAtoms
#align_import order.filter.ultrafilter from "leanprover-community... | Mathlib/Order/Filter/Ultrafilter.lean | 205 | 207 | theorem finite_biUnion_mem_iff {is : Set β} {s : β → Set α} (his : is.Finite) :
(⋃ i ∈ is, s i) ∈ f ↔ ∃ i ∈ is, s i ∈ f := by |
simp only [← sUnion_image, finite_sUnion_mem_iff (his.image s), exists_mem_image]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yourong Zang
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mat... | Mathlib/Analysis/Complex/RealDeriv.lean | 171 | 185 | theorem conformalAt_iff_differentiableAt_or_differentiableAt_comp_conj {f : ℂ → ℂ} {z : ℂ} :
ConformalAt f z ↔
(DifferentiableAt ℂ f z ∨ DifferentiableAt ℂ (f ∘ conj) (conj z)) ∧ fderiv ℝ f z ≠ 0 := by |
rw [conformalAt_iff_isConformalMap_fderiv]
rw [isConformalMap_iff_is_complex_or_conj_linear]
apply and_congr_left
intro h
have h_diff := h.imp_symm fderiv_zero_of_not_differentiableAt
apply or_congr
· rw [differentiableAt_iff_restrictScalars ℝ h_diff]
rw [← conj_conj z] at h_diff
rw [differentiableAt... |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris Van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_imp... | Mathlib/MeasureTheory/Measure/Regular.lean | 361 | 366 | theorem _root_.Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ] {ε : ℝ≥0∞}
(hε : ε ≠ 0) : ∃ U, U ⊇ A ∧ IsOpen U ∧ μ U ≤ μ A + ε := by |
rcases eq_or_ne (μ A) ∞ with (H | H)
· exact ⟨univ, subset_univ _, isOpen_univ, by simp only [H, _root_.top_add, le_top]⟩
· rcases A.exists_isOpen_lt_add H hε with ⟨U, AU, U_open, hU⟩
exact ⟨U, AU, U_open, hU.le⟩
|
/-
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.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathli... | Mathlib/Probability/StrongLaw.lean | 106 | 111 | theorem truncation_eq_self {f : α → ℝ} {A : ℝ} {x : α} (h : |f x| < A) :
truncation f A x = f x := by |
simp only [truncation, indicator, Set.mem_Icc, id, Function.comp_apply, ite_eq_left_iff]
intro H
apply H.elim
simp [(abs_lt.1 h).1, (abs_lt.1 h).2.le]
|
/-
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.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 446 | 449 | theorem support_swap_iff (x y : α) : support (swap x y) = {x, y} ↔ x ≠ y := by |
refine ⟨fun h => ?_, fun h => support_swap h⟩
rintro rfl
simp [Finset.ext_iff] at h
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mitchell Rowett, Scott Morrison, Johan Commelin, Mario Carneiro,
Michael Howes
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Deprecated.Submonoid
#al... | Mathlib/Deprecated/Subgroup.lean | 475 | 478 | theorem trivial_ker_iff_eq_one {f : G → H} (hf : IsGroupHom f) :
ker f = trivial G ↔ ∀ x, f x = 1 → x = 1 := by |
rw [Set.ext_iff]; simp [ker];
exact ⟨fun h x hx => (h x).1 hx, fun h x => ⟨h x, fun hx => by rw [hx, hf.map_one]⟩⟩
|
/-
Copyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Abhimanyu Pallavi Sudhir, Yury Kudryashov
-/
import Mathlib.Order.Filter.Ultrafilter
import Mathlib.Order.Filter.Germ
#align_import order.filter.filter_product from "leanpro... | Mathlib/Order/Filter/FilterProduct.lean | 166 | 167 | theorem const_min [LinearOrder β] (x y : β) : (↑(min x y : β) : β*) = min ↑x ↑y := by |
rw [min_def, map₂_const]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Alex Kontorovich
-/
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be4... | Mathlib/Order/Filter/Pi.lean | 229 | 230 | theorem mem_coprodᵢ_iff {s : Set (∀ i, α i)} :
s ∈ Filter.coprodᵢ f ↔ ∀ i : ι, ∃ t₁ ∈ f i, eval i ⁻¹' t₁ ⊆ s := by | simp [Filter.coprodᵢ]
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Fin.Basic
namespace Fin
attribute [norm_cast] val_last
protected theorem le_antisymm_iff {x y : Fin n} : x = y ↔ x ≤ y ∧ y ≤ x :=
Fin.ext_i... | .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean | 87 | 90 | theorem foldl_eq_foldl_list (f : α → Fin n → α) (x) : foldl n f x = (list n).foldl f x := by |
induction n generalizing x with
| zero => rw [foldl_zero, list_zero, List.foldl_nil]
| succ n ih => rw [foldl_succ, ih, list_succ, List.foldl_cons, List.foldl_map]
|
/-
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.Analysis.Complex.Polynomial
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingT... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 168 | 169 | theorem place_conjugate (φ : K →+* ℂ) : place (conjugate φ) = place φ := by |
ext; simp only [place_apply, norm_eq_abs, abs_conj, conjugate_coe_eq]
|
/-
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.Option.Basic
import Mathlib.Data.Prod.PProd
import Mathlib.Logic.Equiv.Basic
#align_import logic.embedding.basic from "leanprover... | Mathlib/Logic/Embedding/Basic.lean | 163 | 166 | theorem equiv_toEmbedding_trans_symm_toEmbedding {α β : Sort*} (e : α ≃ β) :
e.toEmbedding.trans e.symm.toEmbedding = Embedding.refl _ := by |
ext
simp
|
/-
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.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
#a... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 140 | 145 | theorem inner_condexpL2_eq_inner_fun (hm : m ≤ m0) (f g : α →₂[μ] E)
(hg : AEStronglyMeasurable' m g μ) :
⟪(condexpL2 E 𝕜 hm f : α →₂[μ] E), g⟫₂ = ⟪f, g⟫₂ := by |
symm
rw [← sub_eq_zero, ← inner_sub_left, condexpL2]
simp only [mem_lpMeas_iff_aeStronglyMeasurable'.mpr hg, orthogonalProjection_inner_eq_zero f g]
|
/-
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.CategoryTheory.Limits.KanExtension
import Mathlib.CategoryTheory.Sites.CoverPreserving
import Mathlib.CategoryTheory.Sites.Sheafification
#align_import cate... | Mathlib/CategoryTheory/Sites/CoverLifting.lean | 287 | 294 | theorem ran_isSheaf_of_isCocontinuous (ℱ : Sheaf J A) :
Presheaf.IsSheaf K ((ran G.op).obj ℱ.val) := by |
intro X U S hS x hx
constructor; swap
· apply RanIsSheafOfIsCocontinuous.gluedSection ℱ hS hx
constructor
· apply RanIsSheafOfIsCocontinuous.gluedSection_isAmalgamation
· apply RanIsSheafOfIsCocontinuous.gluedSection_is_unique
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
#a... | Mathlib/Geometry/Euclidean/Triangle.lean | 279 | 285 | theorem angle_eq_angle_of_dist_eq {p1 p2 p3 : P} (h : dist p1 p2 = dist p1 p3) :
∠ p1 p2 p3 = ∠ p1 p3 p2 := by |
rw [dist_eq_norm_vsub V p1 p2, dist_eq_norm_vsub V p1 p3] at h
unfold angle
convert angle_sub_eq_angle_sub_rev_of_norm_eq h
· exact (vsub_sub_vsub_cancel_left p3 p2 p1).symm
· exact (vsub_sub_vsub_cancel_left p2 p3 p1).symm
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.... | Mathlib/Order/Filter/AtTopBot.lean | 564 | 573 | theorem Eventually.atTop_of_arithmetic {p : ℕ → Prop} {n : ℕ} (hn : n ≠ 0)
(hp : ∀ k < n, ∀ᶠ a in atTop, p (n * a + k)) : ∀ᶠ a in atTop, p a := by |
simp only [eventually_atTop] at hp ⊢
choose! N hN using hp
refine ⟨(Finset.range n).sup (n * N ·), fun b hb => ?_⟩
rw [← Nat.div_add_mod b n]
have hlt := Nat.mod_lt b hn.bot_lt
refine hN _ hlt _ ?_
rw [ge_iff_le, Nat.le_div_iff_mul_le hn.bot_lt, mul_comm]
exact (Finset.le_sup (f := (n * N ·)) (Finset.m... |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Heather Macbeth, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.Alternating.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.Topology.Algebra.Module.M... | Mathlib/Topology/Algebra/Module/Alternating/Basic.lean | 627 | 630 | theorem alternatization_apply_toAlternatingMap :
(alternatization f).toAlternatingMap = MultilinearMap.alternatization f.1 := by |
ext v
simp [alternatization_apply_apply, MultilinearMap.alternatization_apply, (· ∘ ·)]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 121 | 124 | theorem lineMap_slope_slope_sub_div_sub (f : k → PE) (a b c : k) (h : a ≠ c) :
lineMap (slope f a b) (slope f b c) ((c - b) / (c - a)) = slope f a c := by |
field_simp [sub_ne_zero.2 h.symm, ← sub_div_sub_smul_slope_add_sub_div_sub_smul_slope f a b c,
lineMap_apply_module]
|
/-
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.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 608 | 612 | theorem biSup_add_biSup_le' {ι ι'} {p : ι → Prop} {q : ι' → Prop} (hp : ∃ i, p i) (hq : ∃ j, q j)
{f : ι → ℝ≥0∞} {g : ι' → ℝ≥0∞} {a : ℝ≥0∞} (h : ∀ i, p i → ∀ j, q j → f i + g j ≤ a) :
((⨆ (i) (_ : p i), f i) + ⨆ (j) (_ : q j), g j) ≤ a := by |
simp_rw [biSup_add' hp, add_biSup' hq]
exact iSup₂_le fun i hi => iSup₂_le (h i hi)
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 501 | 503 | theorem mul_le_mul_of_mul_div_le (h : a * (b / c) ≤ d) (hc : 0 < c) : b * a ≤ d * c := by |
rw [← mul_div_assoc] at h
rwa [mul_comm b, ← div_le_iff hc]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
/-!
# Map... | Mathlib/RingTheory/Ideal/Maps.lean | 791 | 803 | theorem map_radical_of_surjective {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R}
(h : RingHom.ker f ≤ I) : map f I.radical = (map f I).radical := by |
rw [radical_eq_sInf, radical_eq_sInf]
have : ∀ J ∈ {J : Ideal R | I ≤ J ∧ J.IsPrime}, RingHom.ker f ≤ J := fun J hJ => h.trans hJ.left
convert map_sInf hf this
refine funext fun j => propext ⟨?_, ?_⟩
· rintro ⟨hj, hj'⟩
haveI : j.IsPrime := hj'
exact
⟨comap f j, ⟨⟨map_le_iff_le_comap.1 hj, comap... |
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