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 |
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/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Init.Order.Defs
set_option autoImplicit true
structure UFModel (n) where
parent : Fin n → Fin n
rank : Nat ... | Mathlib/Data/UnionFind.lean | 196 | 198 | theorem rank_eq (self : UnionFind α) {n} {m : UFModel n} (H : m.Models self.arr)
{i} (h : i < self.size) : self.rank i = m.rank i := by |
simp [rank, h, H.rank_eq]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathli... | Mathlib/Data/ZMod/Basic.lean | 611 | 614 | theorem coe_intCast {n : ℕ} (a : ℤ) : cast (a : ZMod n) = a % n := by |
cases n
· rw [Int.ofNat_zero, Int.emod_zero, Int.cast_id]; rfl
· rw [← val_intCast, val]; rfl
|
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import control.traversable.lemmas from "leanprover-community/mathlib"@"3342d1b2178381196... | Mathlib/Control/Traversable/Lemmas.lean | 70 | 73 | theorem map_traverse (x : t α) : map f <$> traverse g x = traverse (map f ∘ g) x := by |
rw [map_eq_traverse_id f]
refine (comp_traverse (pure ∘ f) g x).symm.trans ?_
congr; apply Comp.applicative_comp_id
|
/-
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, Mario Carneiro
-/
import Mathlib.Data.List.Count
import Mathlib.Data.List.Dedup
import Mathlib.Data.List.InsertNth
import Mathlib.Data.List.Lat... | Mathlib/Data/List/Perm.lean | 335 | 358 | theorem cons_subperm_of_mem {a : α} {l₁ l₂ : List α} (d₁ : Nodup l₁) (h₁ : a ∉ l₁) (h₂ : a ∈ l₂)
(s : l₁ <+~ l₂) : a :: l₁ <+~ l₂ := by |
rcases s with ⟨l, p, s⟩
induction s generalizing l₁ with
| slnil => cases h₂
| @cons r₁ r₂ b s' ih =>
simp? at h₂ says simp only [mem_cons] at h₂
cases' h₂ with e m
· subst b
exact ⟨a :: r₁, p.cons a, s'.cons₂ _⟩
· rcases ih d₁ h₁ m p with ⟨t, p', s'⟩
exact ⟨t, p', s'.cons _⟩
| @c... |
/-
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 Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a7... | Mathlib/Data/List/Sigma.lean | 767 | 771 | theorem dlookup_kunion_right {a} {l₁ l₂ : List (Sigma β)} (h : a ∉ l₁.keys) :
dlookup a (kunion l₁ l₂) = dlookup a l₂ := by |
induction l₁ generalizing l₂ with
| nil => simp
| cons _ _ ih => simp_all [not_or]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Control.EquivFunctor
import Mathlib.Data.Option.Basic
import Mathlib.Data.Subtype
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Cases
#align_import ... | Mathlib/Logic/Equiv/Option.lean | 230 | 235 | theorem optionSubtype_apply_symm_apply
[DecidableEq β] (x : β)
(e : { e : Option α ≃ β // e none = x })
(b : { y : β // y ≠ x }) : ↑((optionSubtype x e).symm b) = (e : Option α ≃ β).symm b := by |
dsimp only [optionSubtype]
simp
|
/-
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.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | Mathlib/Topology/Order/DenselyOrdered.lean | 52 | 61 | theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by |
apply Subset.antisymm
· exact closure_minimal Ioo_subset_Icc_self isClosed_Icc
· cases' hab.lt_or_lt with hab hab
· rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]
have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab
simp only [insert_subset_iff, singleton_subset_iff]
exact ⟨(isGLB... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebr... | Mathlib/LinearAlgebra/Lagrange.lean | 547 | 549 | theorem eval_nodal_at_node {i : ι} (hi : i ∈ s) : eval (v i) (nodal s v) = 0 := by |
rw [eval_nodal]
exact s.prod_eq_zero hi (sub_self (v i))
|
/-
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.SetTheory.Cardinal.Basic
import Mathlib.Topology.MetricSpace.Closeds
import Mathlib.Topology.MetricSpace.Completion
import Mathlib.Topology.Metri... | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | 142 | 145 | theorem GHSpace.toGHSpace_rep (p : GHSpace) : toGHSpace p.Rep = p := by |
change toGHSpace (Quot.out p : NonemptyCompacts ℓ_infty_ℝ) = p
rw [← eq_toGHSpace]
exact Quot.out_eq p
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Range
#align_import data.list.nat_antidiagonal from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601... | Mathlib/Data/List/NatAntidiagonal.lean | 38 | 47 | theorem mem_antidiagonal {n : ℕ} {x : ℕ × ℕ} : x ∈ antidiagonal n ↔ x.1 + x.2 = n := by |
rw [antidiagonal, mem_map]; constructor
· rintro ⟨i, hi, rfl⟩
rw [mem_range, Nat.lt_succ_iff] at hi
exact Nat.add_sub_cancel' hi
· rintro rfl
refine ⟨x.fst, ?_, ?_⟩
· rw [mem_range]
omega
· exact Prod.ext rfl (by simp only [Nat.add_sub_cancel_left])
|
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subri... | Mathlib/LinearAlgebra/Ray.lean | 476 | 478 | theorem neg_units_smul (u : Rˣ) (v : Module.Ray R M) : -u • v = -(u • v) := by |
induction v using Module.Ray.ind
simp only [smul_rayOfNeZero, Units.smul_def, Units.val_neg, neg_smul, neg_rayOfNeZero]
|
/-
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.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mat... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 200 | 204 | theorem gc_set :
@GaloisConnection (Set R) (Set (PrimeSpectrum R))ᵒᵈ _ _ (fun s => zeroLocus s) fun t =>
vanishingIdeal t := by |
have ideal_gc : GaloisConnection Ideal.span _ := (Submodule.gi R R).gc
simpa [zeroLocus_span, Function.comp] using ideal_gc.compose (gc R)
|
/-
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 | 1,946 | 1,955 | theorem insert_diff_of_not_mem (s) (h : a ∉ t) : insert a s \ t = insert a (s \ t) := by |
classical
ext x
by_cases h' : x ∈ t
· have : x ≠ a := by
intro H
rw [H] at h'
exact h h'
simp [h, h', this]
· simp [h, h']
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Sub... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 735 | 744 | theorem mul_mem_add_closure (ha : a ∈ AddSubmonoid.closure (S : Set R))
(hb : b ∈ AddSubmonoid.closure (S : Set R)) : a * b ∈ AddSubmonoid.closure (S : Set R) := by |
revert a
apply @AddSubmonoid.closure_induction _ _ _
(fun z => ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * z ∈ AddSubmonoid.closure ↑S)
_ hb <;> clear hb b
· exact fun r hr b hb => MulMemClass.mul_right_mem_add_closure hb hr
· exact fun _ => by simp only [mul_zero, (AddSubmonoid.closure (S : Set R))... |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most one bas... | Mathlib/Data/Matroid/Constructions.lean | 211 | 215 | theorem uniqueBaseOn_basis_iff (hI : I ⊆ E) (hX : X ⊆ E) :
(uniqueBaseOn I E).Basis J X ↔ J = X ∩ I := by |
rw [basis_iff_mem_maximals]
simp_rw [uniqueBaseOn_indep_iff', ← subset_inter_iff, ← le_eq_subset, Iic_def, maximals_Iic,
mem_singleton_iff, inter_eq_self_of_subset_left hI, inter_comm I]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Basic
#align_import data.multiset.locally_finite from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
/-!... | Mathlib/Order/Interval/Multiset.lean | 254 | 257 | theorem Ico_filter_lt_of_le_right [DecidablePred (· < c)] (hcb : c ≤ b) :
((Ico a b).filter fun x => x < c) = Ico a c := by |
rw [Ico, ← Finset.filter_val, Finset.Ico_filter_lt_of_le_right hcb]
rfl
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.BigOperators.Fin
im... | Mathlib/Algebra/BigOperators/Finsupp.lean | 472 | 477 | theorem univ_sum_single_apply' [AddCommMonoid M] [Fintype α] (i : α) (m : M) :
∑ j : α, single j m i = m := by |
-- Porting note: rewrite due to leaky classical in lean3
simp_rw [single, coe_mk]
classical rw [Finset.sum_pi_single]
simp
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.Tactic.Ring
#align_import data.fintype.perm from "leanprover-community/mathlib"... | Mathlib/Data/Fintype/Perm.lean | 145 | 146 | theorem card_perms_of_finset : ∀ s : Finset α, (permsOfFinset s).card = s.card ! := by |
rintro ⟨⟨l⟩, hs⟩; exact length_permsOfList l
|
/-
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.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7... | Mathlib/Data/Seq/WSeq.lean | 1,204 | 1,205 | theorem productive_congr {s t : WSeq α} (h : s ~ʷ t) : Productive s ↔ Productive t := by |
simp only [productive_iff]; exact forall_congr' fun n => terminates_congr <| get?_congr h _
|
/-
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 | 103 | 104 | theorem derivative_C_mul_X_sq (a : R) : derivative (C a * X ^ 2) = C (a * 2) * X := by |
rw [derivative_C_mul_X_pow, Nat.cast_two, pow_one]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Monic
#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594b... | Mathlib/Algebra/Polynomial/BigOperators.lean | 239 | 240 | theorem coeff_zero_prod : (∏ i ∈ s, f i).coeff 0 = ∏ i ∈ s, (f i).coeff 0 := by |
simpa using coeff_zero_multiset_prod (s.1.map f)
|
/-
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.Fintype.Card
import Mathlib.Order.UpperLower.Basic
#align_import combinatorics.set_family.intersecting from "leanprover-community/mathlib"@"d90e4e186... | Mathlib/Combinatorics/SetFamily/Intersecting.lean | 81 | 92 | theorem intersecting_iff_pairwise_not_disjoint :
s.Intersecting ↔ (s.Pairwise fun a b => ¬Disjoint a b) ∧ s ≠ {⊥} := by |
refine ⟨fun h => ⟨fun a ha b hb _ => h ha hb, ?_⟩, fun h a ha b hb hab => ?_⟩
· rintro rfl
exact intersecting_singleton.1 h rfl
have := h.1.eq ha hb (Classical.not_not.2 hab)
rw [this, disjoint_self] at hab
rw [hab] at hb
exact
h.2
(eq_singleton_iff_unique_mem.2
⟨hb, fun c hc => not_n... |
/-
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
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.In... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 123 | 124 | theorem image_const_add_Icc : (fun x => a + x) '' Icc b c = Icc (a + b) (a + c) := by |
simp only [add_comm a, image_add_const_Icc]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#alig... | Mathlib/RingTheory/Ideal/Basic.lean | 167 | 168 | theorem isCompactElement_top : CompleteLattice.IsCompactElement (⊤ : Ideal α) := by |
simpa only [← span_singleton_one] using Submodule.singleton_span_isCompactElement 1
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_im... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 101 | 105 | theorem dist_eq_iff_eq_sq_sinh (hr : 0 ≤ r) :
dist z w = r ↔ dist (z : ℂ) w ^ 2 / (4 * z.im * w.im) = sinh (r / 2) ^ 2 := by |
rw [dist_eq_iff_eq_sinh, ← sq_eq_sq, div_pow, mul_pow, sq_sqrt, mul_assoc]
· norm_num
all_goals positivity
|
/-
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.Analysis.Calculus.FDeriv.Measurable
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calc... | Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 1,216 | 1,234 | theorem integral_eq_sub_of_hasDerivAt_of_tendsto (hab : a < b) {fa fb}
(hderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hint : IntervalIntegrable f' volume a b)
(ha : Tendsto f (𝓝[>] a) (𝓝 fa)) (hb : Tendsto f (𝓝[<] b) (𝓝 fb)) :
∫ y in a..b, f' y = fb - fa := by |
set F : ℝ → E := update (update f a fa) b fb
have Fderiv : ∀ x ∈ Ioo a b, HasDerivAt F (f' x) x := by
refine fun x hx => (hderiv x hx).congr_of_eventuallyEq ?_
filter_upwards [Ioo_mem_nhds hx.1 hx.2] with _ hy
unfold_let F
rw [update_noteq hy.2.ne, update_noteq hy.1.ne']
have hcont : ContinuousOn... |
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 1,051 | 1,053 | theorem mk_d_2_0 : (mk X₀ X₁ X₂ d₀ d₁ s succ).d 1 2 = d₁ := by |
change ite (2 = 1 + 1) (d₁ ≫ 𝟙 X₂) 0 = d₁
rw [if_pos rfl, Category.comp_id]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Option.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sigma.Basic
import Mathlib... | Mathlib/Logic/Equiv/Basic.lean | 337 | 339 | theorem sumCongr_refl : Equiv.sumCongr (Equiv.refl α) (Equiv.refl β) = Equiv.refl (Sum α β) := by |
ext i
cases i <;> rfl
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import probability.process.stopping from "leanp... | Mathlib/Probability/Process/Stopping.lean | 959 | 963 | theorem integrable_stoppedValue_of_mem_finset (hτ : IsStoppingTime ℱ τ)
(hu : ∀ n, Integrable (u n) μ) {s : Finset ι} (hbdd : ∀ ω, τ ω ∈ s) :
Integrable (stoppedValue u τ) μ := by |
simp_rw [← memℒp_one_iff_integrable] at hu ⊢
exact memℒp_stoppedValue_of_mem_finset hτ hu hbdd
|
/-
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.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vanderm... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 164 | 189 | theorem hasseDeriv_comp (k l : ℕ) :
(@hasseDeriv R _ k).comp (hasseDeriv l) = (k + l).choose k • hasseDeriv (k + l) := by |
ext i : 2
simp only [LinearMap.smul_apply, comp_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply,
mul_one, monomial_eq_zero_iff, sum_monomial_index, mul_zero, ←
tsub_add_eq_tsub_tsub, add_comm l k]
rw_mod_cast [nsmul_eq_mul]
rw [← Nat.cast_mul]
congr 2
by_cases hikl : i < k + l
· rw [ch... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 605 | 643 | theorem isLocalStructomorphWithinAt_localInvariantProp [ClosedUnderRestriction G] :
LocalInvariantProp G G (IsLocalStructomorphWithinAt G) :=
{ is_local := by |
intro s x u f hu hux
constructor
· rintro h hx
rcases h hx.1 with ⟨e, heG, hef, hex⟩
have : s ∩ u ∩ e.source ⊆ s ∩ e.source := by mfld_set_tac
exact ⟨e, heG, hef.mono this, hex⟩
· rintro h hx
rcases h ⟨hx, hux⟩ with ⟨e, heG, hef, hex⟩
refine ⟨e.restr (int... |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | Mathlib/SetTheory/Game/Nim.lean | 227 | 230 | theorem nim_fuzzy_zero_of_ne_zero {o : Ordinal} (ho : o ≠ 0) : nim o ‖ 0 := by |
rw [Impartial.fuzzy_zero_iff_lf, nim_def, lf_zero_le]
rw [← Ordinal.pos_iff_ne_zero] at ho
exact ⟨(Ordinal.principalSegOut ho).top, by simp⟩
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 981 | 982 | theorem mem_iInf {ι : Sort*} {S : ι → Subgroup G} {x : G} : (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by |
simp only [iInf, mem_sInf, Set.forall_mem_range]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 773 | 776 | theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ := by |
simp only [iUnion_and, @iUnion_comm _ ι']
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
imp... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 351 | 360 | theorem exp_neg_two_mul_le {x : ℝ} (hx : 0 < x) : exp (-2 * x) < exp (2 - ⌈x⌉₊) / ⌈x⌉₊ := by |
have h₁ := ceil_lt_add_one hx.le
have h₂ : 1 - x ≤ 2 - ⌈x⌉₊ := by linarith
calc
_ ≤ exp (1 - x) / (x + 1) := ?_
_ ≤ exp (2 - ⌈x⌉₊) / (x + 1) := by gcongr
_ < _ := by gcongr
rw [le_div_iff (add_pos hx zero_lt_one), ← le_div_iff' (exp_pos _), ← exp_sub, neg_mul,
sub_neg_eq_add, two_mul, sub_add_a... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury G. Kudryashov
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"... | Mathlib/Data/Sum/Basic.lean | 66 | 67 | theorem getRight_eq_getRight? (h₁ : x.isRight) (h₂ : x.getRight?.isSome) :
x.getRight h₁ = x.getRight?.get h₂ := by | simp [← getRight?_eq_some_iff]
|
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.pow fr... | Mathlib/Algebra/Regular/Pow.lean | 54 | 58 | theorem IsRightRegular.pow_iff {n : ℕ} (n0 : 0 < n) :
IsRightRegular (a ^ n) ↔ IsRightRegular a := by |
refine ⟨?_, IsRightRegular.pow n⟩
rw [← Nat.succ_pred_eq_of_pos n0, pow_succ']
exact IsRightRegular.of_mul
|
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
/-!
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 102 | 105 | theorem smul_fixedBy (g h: G) :
h • fixedBy α g = fixedBy α (h * g * h⁻¹) := by |
ext a
simp_rw [Set.mem_smul_set_iff_inv_smul_mem, mem_fixedBy, mul_smul, smul_eq_iff_eq_inv_smul h]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | Mathlib/Analysis/Calculus/MeanValue.lean | 767 | 772 | theorem exists_hasDerivAt_eq_slope : ∃ c ∈ Ioo a b, f' c = (f b - f a) / (b - a) := by |
obtain ⟨c, cmem, hc⟩ : ∃ c ∈ Ioo a b, (b - a) * f' c = (f b - f a) * 1 :=
exists_ratio_hasDerivAt_eq_ratio_slope f f' hab hfc hff' id 1 continuousOn_id
fun x _ => hasDerivAt_id x
use c, cmem
rwa [mul_one, mul_comm, ← eq_div_iff (sub_ne_zero.2 hab.ne')] at hc
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Order.RelClasses
import Mathlib.Order.Interval.Set.Basic
#align_import order.bounded from "leanprover-community/mathlib"@"aba5... | Mathlib/Order/Bounded.lean | 309 | 311 | theorem unbounded_inter_not (H : ∀ a b, ∃ m, ∀ c, r c a ∨ r c b → r c m) (a : α) :
Unbounded r (s ∩ { b | ¬r b a }) ↔ Unbounded r s := by |
simp_rw [← not_bounded_iff, bounded_inter_not H]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Da... | Mathlib/GroupTheory/OrderOfElement.lean | 1,022 | 1,043 | theorem orderOf_dvd_card : orderOf x ∣ Fintype.card G := by |
classical
have ft_prod : Fintype ((G ⧸ zpowers x) × zpowers x) :=
Fintype.ofEquiv G groupEquivQuotientProdSubgroup
have ft_s : Fintype (zpowers x) := @Fintype.prodRight _ _ _ ft_prod _
have ft_cosets : Fintype (G ⧸ zpowers x) :=
@Fintype.prodLeft _ _ _ ft_prod ⟨⟨1, (zpowers x).one_mem⟩⟩
h... |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Chris Hughes, Michael Howes
-/
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Group.Semiconj.Units
#align_import algebra.group.conj from "leanprover-community... | Mathlib/Algebra/Group/Conj.lean | 199 | 204 | theorem map_surjective {f : α →* β} (hf : Function.Surjective f) :
Function.Surjective (ConjClasses.map f) := by |
intro b
obtain ⟨b, rfl⟩ := ConjClasses.mk_surjective b
obtain ⟨a, rfl⟩ := hf b
exact ⟨ConjClasses.mk a, rfl⟩
|
/-
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 | 124 | 130 | theorem restrict_eq_self (h : s ⊆ t) : μ.restrict t s = μ s :=
(le_iff'.1 restrict_le_self s).antisymm <|
calc
μ s ≤ μ (toMeasurable (μ.restrict t) s ∩ t) :=
measure_mono (subset_inter (subset_toMeasurable _ _) h)
_ = μ.restrict t s := by |
rw [← restrict_apply (measurableSet_toMeasurable _ _), measure_toMeasurable]
|
/-
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.Tactic.SeqFocus
/-! ## Ordering -/
namespace Ordering
@[simp] theorem swap_swap {o : Ordering} : o.swap.swap = o := by cases o <;> rfl
@[simp] th... | .lake/packages/batteries/Batteries/Classes/Order.lean | 137 | 138 | theorem LECmp.cmp_iff_ge [LE α] [LECmp (α := α) cmp] : cmp x y ≠ .lt ↔ y ≤ x := by |
rw [← OrientedCmp.cmp_ne_gt, LECmp.cmp_iff_le]
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.ordinal.principal from "leanprover-community/mathlib"@"31b269b6093548394... | Mathlib/SetTheory/Ordinal/Principal.lean | 233 | 241 | theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :
Principal (· + ·) (a^b) := by |
rcases principal_add_iff_zero_or_omega_opow.1 ha with (rfl | ⟨c, rfl⟩)
· rcases eq_or_ne b 0 with (rfl | hb)
· rw [opow_zero]
exact principal_add_one
· rwa [zero_opow hb]
· rw [← opow_mul]
exact principal_add_omega_opow _
|
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68a... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 220 | 228 | theorem le_of_mem_compression_of_not_mem (h : a ∈ 𝓒 u v s) (ha : a ∉ s) : u ≤ a := by |
rw [mem_compression] at h
obtain h | ⟨-, b, hb, hba⟩ := h
· cases ha h.1
unfold compress at hba
split_ifs at hba with h
· rw [← hba, le_sdiff]
exact ⟨le_sup_right, h.1.mono_right h.2⟩
· cases ne_of_mem_of_not_mem hb ha hba
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, E. W. Ayers
-/
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Yoneda
import Mathlib.Data.Set.L... | Mathlib/CategoryTheory/Sites/Sieves.lean | 288 | 290 | theorem arrows_ext : ∀ {R S : Sieve X}, R.arrows = S.arrows → R = S := by |
rintro ⟨_, _⟩ ⟨_, _⟩ rfl
rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
/-!
... | Mathlib/Data/ENNReal/Real.lean | 397 | 399 | theorem smul_toNNReal (a : ℝ≥0) (b : ℝ≥0∞) : (a • b).toNNReal = a * b.toNNReal := by |
change ((a : ℝ≥0∞) * b).toNNReal = a * b.toNNReal
simp only [ENNReal.toNNReal_mul, ENNReal.toNNReal_coe]
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 614 | 623 | theorem stepNormal.is_ret (c k v) : ∃ k' v', stepNormal c k v = Cfg.ret k' v' := by |
induction c generalizing k v with
| cons _f fs IHf _IHfs => apply IHf
| comp f _g _IHf IHg => apply IHg
| case f g IHf IHg =>
rw [stepNormal]
simp only []
cases v.headI <;> [apply IHf; apply IHg]
| fix f IHf => apply IHf
| _ => exact ⟨_, _, rfl⟩
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.GroupTheory.Submonoid.Inverses
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Loc... | Mathlib/RingTheory/Localization/InvSubmonoid.lean | 108 | 112 | theorem span_invSubmonoid : Submodule.span R (invSubmonoid M S : Set S) = ⊤ := by |
rw [eq_top_iff]
rintro x -
rcases IsLocalization.surj'' M x with ⟨r, m, rfl⟩
exact Submodule.smul_mem _ _ (Submodule.subset_span (toInvSubmonoid M S m).prop)
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_im... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 136 | 139 | theorem cosh_dist' (z w : ℍ) :
Real.cosh (dist z w) = ((z.re - w.re) ^ 2 + z.im ^ 2 + w.im ^ 2) / (2 * z.im * w.im) := by |
field_simp [cosh_dist, Complex.dist_eq, Complex.sq_abs, normSq_apply]
ring
|
/-
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.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis
impo... | Mathlib/RingTheory/Adjoin/Basic.lean | 79 | 80 | theorem adjoin_attach_biUnion [DecidableEq A] {α : Type*} {s : Finset α} (f : s → Finset A) :
adjoin R (s.attach.biUnion f : Set A) = ⨆ x, adjoin R (f x) := by | simp [adjoin_iUnion]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Antichain
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.RelIso.Set
#align_import order.minimal ... | Mathlib/Order/Minimal.lean | 459 | 463 | theorem RelEmbedding.maximals_preimage_eq (f : r ↪r s) (y : Set β) :
maximals r (f ⁻¹' y) = f ⁻¹' maximals s (y ∩ range f) := by |
convert (f.inter_preimage_maximals_eq univ y).symm
· simp [univ_inter]
· simp [inter_comm]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Si... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 548 | 565 | theorem commutative_of_cyclic_center_quotient [IsCyclic H] (f : G →* H) (hf : f.ker ≤ center G)
(a b : G) : a * b = b * a :=
let ⟨⟨x, y, (hxy : f y = x)⟩, (hx : ∀ a : f.range, a ∈ zpowers _)⟩ :=
IsCyclic.exists_generator (α := f.range)
let ⟨m, hm⟩ := hx ⟨f a, a, rfl⟩
let ⟨n, hn⟩ := hx ⟨f b, b, rfl⟩
have... | simpa [Subtype.ext_iff] using hm
have hn : x ^ n = f b := by simpa [Subtype.ext_iff] using hn
have ha : y ^ (-m) * a ∈ center G :=
hf (by rw [f.mem_ker, f.map_mul, f.map_zpow, hxy, zpow_neg x m, hm, inv_mul_self])
have hb : y ^ (-n) * b ∈ center G :=
hf (by rw [f.mem_ker, f.map_mul, f.map_zpow, hxy, zpow... |
/-
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.Order.AbsoluteValue
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Ring.Pi
import Ma... | Mathlib/Algebra/Order/CauSeq/Basic.lean | 479 | 480 | theorem add_equiv_add {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :
f1 + g1 ≈ f2 + g2 := by | simpa only [← add_sub_add_comm] using add_limZero hf hg
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.LinearAlgebra.Span
import Mathlib.LinearAlgebra.BilinearMap
#align_import algebra.module.submodule.bilinear from "leanprover-community/mathlib"@"60... | Mathlib/Algebra/Module/Submodule/Bilinear.lean | 146 | 150 | theorem map₂_iSup_left (f : M →ₗ[R] N →ₗ[R] P) (s : ι → Submodule R M) (t : Submodule R N) :
map₂ f (⨆ i, s i) t = ⨆ i, map₂ f (s i) t := by |
suffices map₂ f (⨆ i, span R (s i : Set M)) (span R t) = ⨆ i, map₂ f (span R (s i)) (span R t) by
simpa only [span_eq] using this
simp_rw [map₂_span_span, ← span_iUnion, map₂_span_span, Set.image2_iUnion_left]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 839 | 847 | theorem balance_sz_dual {l r}
(H : (∃ l', Raised (@size α l) l' ∧ BalancedSz l' (@size α r)) ∨
∃ r', Raised r' (size r) ∧ BalancedSz (size l) r') :
(∃ l', Raised l' (size (dual r)) ∧ BalancedSz l' (size (dual l))) ∨
∃ r', Raised (size (dual l)) r' ∧ BalancedSz (size (dual r)) r' := by |
rw [size_dual, size_dual]
exact
H.symm.imp (Exists.imp fun _ => And.imp_right BalancedSz.symm)
(Exists.imp fun _ => And.imp_right BalancedSz.symm)
|
/-
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.Topology.MetricSpace.Antilipschitz
#align_import topology.metric_space.isometry from "leanprover-community/mathlib"@"b1859b6d4636fdbb78c5d5cefd2... | Mathlib/Topology/MetricSpace/Isometry.lean | 562 | 564 | theorem completeSpace_iff (e : α ≃ᵢ β) : CompleteSpace α ↔ CompleteSpace β := by |
simp only [completeSpace_iff_isComplete_univ, ← e.range_eq_univ, ← image_univ,
isComplete_image_iff e.isometry.uniformInducing]
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.List.OfFn
import Mathlib.Data.Set.Pointwise.Basic
#align_import data.set.pointwise.list_of_fn from "lean... | Mathlib/Data/Set/Pointwise/ListOfFn.lean | 52 | 54 | theorem mem_pow {a : α} {n : ℕ} :
a ∈ s ^ n ↔ ∃ f : Fin n → s, (List.ofFn fun i ↦ (f i : α)).prod = a := by |
rw [← mem_prod_list_ofFn, List.ofFn_const, List.prod_replicate]
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Decomposition
import Mathlib.Tactic.FinCases
#align_import algebraic_topology.dold_kan.degeneracies from "leanprover-community/mathlib... | Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean | 57 | 115 | theorem σ_comp_P_eq_zero (X : SimplicialObject C) {n q : ℕ} (i : Fin (n + 1)) (hi : n + 1 ≤ i + q) :
X.σ i ≫ (P q).f (n + 1) = 0 := by |
revert i hi
induction' q with q hq
· intro i (hi : n + 1 ≤ i)
exfalso
linarith [Fin.is_lt i]
· intro i (hi : n + 1 ≤ i + q + 1)
by_cases h : n + 1 ≤ (i : ℕ) + q
· rw [P_succ, HomologicalComplex.comp_f, ← assoc, hq i h, zero_comp]
· replace hi : n = i + q := by
obtain ⟨j, hj⟩ := le_i... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathli... | Mathlib/FieldTheory/Separable.lean | 467 | 470 | theorem card_rootSet_eq_natDegree [Algebra F K] {p : F[X]} (hsep : p.Separable)
(hsplit : Splits (algebraMap F K) p) : Fintype.card (p.rootSet K) = p.natDegree := by |
simp_rw [rootSet_def, Finset.coe_sort_coe, Fintype.card_coe]
rw [Multiset.toFinset_card_of_nodup (nodup_roots hsep.map), ← natDegree_eq_card_roots hsplit]
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 726 | 782 | theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
(h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
(ht : MeasurableSet t) (h''t : μ t ≠ ∞) :
Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) := by |
refine tendsto_order.2 ⟨fun a' ha' => (ENNReal.not_lt_zero ha').elim, fun ε (εpos : 0 < ε) => ?_⟩
rcases eq_or_ne (μ t) 0 with (h't | h't)
· filter_upwards with r
suffices H : μ (s ∩ ({x} + r • t)) = 0 by
rw [H]; simpa only [ENNReal.zero_div] using εpos
apply le_antisymm _ (zero_le _)
calc
... |
/-
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, Winston Yin
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Topology.MetricSpace.Contracting
#align_import analysis.ODE.picard_lindelof fr... | Mathlib/Analysis/ODE/PicardLindelof.lean | 279 | 291 | theorem hasDerivWithinAt_next (t : Icc v.tMin v.tMax) :
HasDerivWithinAt (f.next ∘ v.proj) (v t (f t)) (Icc v.tMin v.tMax) t := by |
haveI : Fact ((t : ℝ) ∈ Icc v.tMin v.tMax) := ⟨t.2⟩
simp only [(· ∘ ·), next_apply]
refine HasDerivWithinAt.const_add _ ?_
have : HasDerivWithinAt (∫ τ in v.t₀..·, f.vComp τ) (f.vComp t) (Icc v.tMin v.tMax) t :=
integral_hasDerivWithinAt_right (f.intervalIntegrable_vComp _ _)
(f.continuous_vComp.stro... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,449 | 1,458 | theorem iteratedFDerivWithin_sum_apply {ι : Type*} {f : ι → E → F} {u : Finset ι} {i : ℕ} {x : E}
(hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (h : ∀ j ∈ u, ContDiffOn 𝕜 i (f j) s) :
iteratedFDerivWithin 𝕜 i (∑ j ∈ u, f j ·) s x =
∑ j ∈ u, iteratedFDerivWithin 𝕜 i (f j) s x := by |
induction u using Finset.cons_induction with
| empty => ext; simp [hs, hx]
| cons a u ha IH =>
simp only [Finset.mem_cons, forall_eq_or_imp] at h
simp only [Finset.sum_cons]
rw [iteratedFDerivWithin_add_apply' h.1 (ContDiffOn.sum h.2) hs hx, IH h.2]
|
/-
Copyright (c) 2023 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Prod
import Mathlib.Tactic.Common
/-!
# Lemmas about the divisibility relation in product (sem... | Mathlib/Algebra/Divisibility/Prod.lean | 35 | 36 | theorem pi_dvd_iff {x y : ∀ i, G i} : x ∣ y ↔ ∀ i, x i ∣ y i := by |
simp_rw [dvd_def, Function.funext_iff, Classical.skolem]; rfl
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 163 | 166 | theorem fix_eval (f) : (fix f).eval =
PFun.fix fun v => (f.eval v).map fun v =>
if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tail := by |
simp [eval]
|
/-
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.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 1,311 | 1,313 | theorem take_one_drop_eq_of_lt_length {l : List α} {n : ℕ} (h : n < l.length) :
(l.drop n).take 1 = [l.get ⟨n, h⟩] := by |
rw [drop_eq_get_cons h, take, take]
|
/-
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 | 71 | 73 | theorem vars_def [DecidableEq σ] (p : MvPolynomial σ R) : p.vars = p.degrees.toFinset := by |
rw [vars]
convert 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.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 651 | 658 | theorem exists_mem_frontier_infDist_compl_eq_dist {E : Type*} [NormedAddCommGroup E]
[NormedSpace ℝ E] [FiniteDimensional ℝ E] {x : E} {s : Set E} (hx : x ∈ s) (hs : s ≠ univ) :
∃ y ∈ frontier s, Metric.infDist x sᶜ = dist x y := by |
rcases Metric.exists_mem_closure_infDist_eq_dist (nonempty_compl.2 hs) x with ⟨y, hys, hyd⟩
rw [closure_compl] at hys
refine ⟨y, ⟨Metric.closedBall_infDist_compl_subset_closure hx <|
Metric.mem_closedBall.2 <| ge_of_eq ?_, hys⟩, hyd⟩
rwa [dist_comm]
|
/-
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.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
... | Mathlib/Combinatorics/SimpleGraph/Clique.lean | 207 | 213 | theorem is3Clique_iff_exists_cycle_length_three :
(∃ s : Finset α, G.IsNClique 3 s) ↔ ∃ (u : α) (w : G.Walk u u), w.IsCycle ∧ w.length = 3 := by |
classical
simp_rw [is3Clique_iff, isCycle_def]
exact
⟨(fun ⟨_, a, _, _, hab, hac, hbc, _⟩ => ⟨a, cons hab (cons hbc (cons hac.symm nil)), by aesop⟩),
(fun ⟨_, .cons hab (.cons hbc (.cons hca nil)), _, _⟩ => ⟨_, _, _, _, hab, hca.symm, hbc, rfl⟩)⟩
|
/-
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.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.... | Mathlib/RingTheory/Trace.lean | 169 | 176 | theorem trace_prod_apply [Module.Free R S] [Module.Free R T] [Module.Finite R S] [Module.Finite R T]
(x : S × T) : trace R (S × T) x = trace R S x.fst + trace R T x.snd := by |
nontriviality R
let f := (lmul R S).toLinearMap.prodMap (lmul R T).toLinearMap
have : (lmul R (S × T)).toLinearMap = (prodMapLinear R S T S T R).comp f :=
LinearMap.ext₂ Prod.mul_def
simp_rw [trace, this]
exact trace_prodMap' _ _
|
/-
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 | 419 | 420 | theorem mul_le_of_one_le_right (ha : a ≤ 0) (h : 1 ≤ b) : a * b ≤ a := by |
simpa only [mul_one] using mul_le_mul_of_nonpos_left h ha
|
/-
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 | 162 | 163 | theorem sameCycle_pow_right {n : ℕ} : SameCycle f x ((f ^ n) y) ↔ SameCycle f x y := by |
rw [← zpow_natCast, sameCycle_zpow_right]
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathli... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 158 | 169 | theorem deleteFar_iff :
G.DeleteFar p r ↔ ∀ ⦃H : SimpleGraph _⦄ [DecidableRel H.Adj],
H ≤ G → p H → r ≤ G.edgeFinset.card - H.edgeFinset.card := by |
classical
refine ⟨fun h H _ hHG hH ↦ ?_, fun h s hs hG ↦ ?_⟩
· have := h (sdiff_subset (t := H.edgeFinset))
simp only [deleteEdges_sdiff_eq_of_le hHG, edgeFinset_mono hHG, card_sdiff,
card_le_card, coe_sdiff, coe_edgeFinset, Nat.cast_sub] at this
exact this hH
· classical
simpa [card_sdiff hs... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 440 | 444 | theorem ListBlank.map_modifyNth {Γ Γ'} [Inhabited Γ] [Inhabited Γ'] (F : PointedMap Γ Γ')
(f : Γ → Γ) (f' : Γ' → Γ') (H : ∀ x, F (f x) = f' (F x)) (n) (L : ListBlank Γ) :
(L.modifyNth f n).map F = (L.map F).modifyNth f' n := by |
induction' n with n IH generalizing L <;>
simp only [*, ListBlank.head_map, ListBlank.modifyNth, ListBlank.map_cons, ListBlank.tail_map]
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topol... | Mathlib/Topology/EMetricSpace/Basic.lean | 1,160 | 1,163 | theorem countable_closure_of_compact {s : Set γ} (hs : IsCompact s) :
∃ t, t ⊆ s ∧ t.Countable ∧ s = closure t := by |
rcases subset_countable_closure_of_compact hs with ⟨t, hts, htc, hsub⟩
exact ⟨t, hts, htc, hsub.antisymm (closure_minimal hts hs.isClosed)⟩
|
/-
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 | 1,021 | 1,022 | theorem oangle_sign_sub_left_swap (x y : V) : (o.oangle (x - y) x).sign = (o.oangle x y).sign := by |
rw [oangle_sign_sub_left_eq_neg, o.oangle_rev y x, Real.Angle.sign_neg]
|
/-
Copyright (c) 2023 Koundinya Vajjha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Koundinya Vajjha, Thomas Browning
-/
import Mathlib.NumberTheory.Harmonic.Defs
import Mathlib.NumberTheory.Padics.PadicNumbers
/-!
The nth Harmonic number is not an integer. We for... | Mathlib/NumberTheory/Harmonic/Int.lean | 42 | 46 | theorem harmonic_not_int {n : ℕ} (h : 2 ≤ n) : ¬ (harmonic n).isInt := by |
apply padicNorm.not_int_of_not_padic_int 2
rw [padicNorm.eq_zpow_of_nonzero (harmonic_pos (ne_zero_of_lt h)).ne',
padicValRat_two_harmonic, neg_neg, zpow_natCast]
exact one_lt_pow one_lt_two (Nat.log_pos one_lt_two h).ne'
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
/-!
# Moments and m... | Mathlib/Probability/Moments.lean | 210 | 210 | theorem mgf_neg : mgf (-X) μ t = mgf X μ (-t) := by | simp_rw [mgf, Pi.neg_apply, mul_neg, neg_mul]
|
/-
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
-/
import Mathlib.Geometry.Manifold.Algebra.Structures
import Mathlib.Geometry.Manifold.BumpFunction
import Mathlib.Topology.MetricSpace.PartitionOfUnity
import ... | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | 410 | 430 | theorem exists_isSubordinate [T2Space M] [SigmaCompactSpace M] (hs : IsClosed s)
(hU : ∀ x ∈ s, U x ∈ 𝓝 x) :
∃ (ι : Type uM) (f : SmoothBumpCovering ι I M s), f.IsSubordinate U := by |
-- First we deduce some missing instances
haveI : LocallyCompactSpace H := I.locallyCompactSpace
haveI : LocallyCompactSpace M := ChartedSpace.locallyCompactSpace H M
-- Next we choose a covering by supports of smooth bump functions
have hB := fun x hx => SmoothBumpFunction.nhds_basis_support I (hU x hx)
r... |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Order.Group.WithTop
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.V... | Mathlib/RingTheory/HahnSeries/Summable.lean | 96 | 106 | theorem isPWO_iUnion_support_powers [LinearOrderedCancelAddCommMonoid Γ] [Ring R] [IsDomain R]
{x : HahnSeries Γ R} (hx : 0 < addVal Γ R x) : (⋃ n : ℕ, (x ^ n).support).IsPWO := by |
apply (x.isWF_support.isPWO.addSubmonoid_closure _).mono _
· exact fun g hg => WithTop.coe_le_coe.1 (le_trans (le_of_lt hx) (addVal_le_of_coeff_ne_zero hg))
refine Set.iUnion_subset fun n => ?_
induction' n with n ih <;> intro g hn
· simp only [Nat.zero_eq, pow_zero, support_one, Set.mem_singleton_iff] at hn... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 1,748 | 1,749 | theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by |
rw [sInter_eq_biInter, preimage_iInter₂]
|
/-
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.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import measure_theory.function.egorov from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f79... | Mathlib/MeasureTheory/Function/Egorov.lean | 98 | 107 | theorem exists_notConvergentSeq_lt (hε : 0 < ε) (hf : ∀ n, StronglyMeasurable (f n))
(hg : StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ∞)
(hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :
∃ j : ι, μ (s ∩ notConvergentSeq f g n j) ≤ ENNReal.ofReal (ε * 2⁻¹ ^ n) := by |
have ⟨N, hN⟩ := (ENNReal.tendsto_atTop ENNReal.zero_ne_top).1
(measure_notConvergentSeq_tendsto_zero hf hg hsm hs hfg n) (ENNReal.ofReal (ε * 2⁻¹ ^ n)) (by
rw [gt_iff_lt, ENNReal.ofReal_pos]
exact mul_pos hε (pow_pos (by norm_num) n))
rw [zero_add] at hN
exact ⟨N, (hN N le_rfl).2⟩
|
/-
Copyright (c) 2021 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Monoidal.Free.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.DiscreteCategory
#align_import category_theory.m... | Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean | 91 | 95 | theorem inclusion_map {X Y : N C} (f : X ⟶ Y) :
inclusion.map f = eqToHom (congr_arg _ (Discrete.ext _ _ (Discrete.eq_of_hom f))) := by |
rcases f with ⟨⟨⟩⟩
cases Discrete.ext _ _ (by assumption)
apply inclusion.map_id
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
#align_import data.matri... | Mathlib/Data/Matrix/Notation.lean | 501 | 504 | theorem vec3_eq {a₀ a₁ a₂ b₀ b₁ b₂ : α} (h₀ : a₀ = b₀) (h₁ : a₁ = b₁) (h₂ : a₂ = b₂) :
![a₀, a₁, a₂] = ![b₀, b₁, b₂] := by |
subst_vars
rfl
|
/-
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.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.... | Mathlib/Data/Rat/Lemmas.lean | 255 | 257 | theorem inv_natCast_den_of_pos {a : ℕ} (ha0 : 0 < a) : (a : ℚ)⁻¹.den = a := by |
rw [← Int.ofNat_inj, ← Int.cast_natCast a, inv_intCast_den_of_pos]
rwa [Int.natCast_pos]
|
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara
-/
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 79 | 86 | theorem norm_cderiv_lt (hr : 0 < r) (hfM : ∀ w ∈ sphere z r, ‖f w‖ < M)
(hf : ContinuousOn f (sphere z r)) : ‖cderiv r f z‖ < M / r := by |
obtain ⟨L, hL1, hL2⟩ : ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L := by
have e1 : (sphere z r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le
have e2 : ContinuousOn (fun w => ‖f w‖) (sphere z r) := continuous_norm.comp_continuousOn hf
obtain ⟨x, hx, hx'⟩ := (isCompact_sphere z r).exists_isMaxOn e1 e2
ex... |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 209 | 211 | theorem toIocMod_apply_left (a : α) : toIocMod hp a a = a + p := by |
rw [toIocMod_eq_iff hp, Set.right_mem_Ioc]
exact ⟨lt_add_of_pos_right _ hp, -1, by simp⟩
|
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
/-!
# Separable closure
This file contains basics about the (relative) separable closure of a fie... | Mathlib/FieldTheory/SeparableClosure.lean | 384 | 385 | theorem IsSeparable.sepDegree_eq [IsSeparable F E] : sepDegree F E = Module.rank F E := by |
rw [sepDegree, (separableClosure.eq_top_iff F E).2 ‹_›, IntermediateField.rank_top']
|
/-
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.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 1,109 | 1,110 | theorem erase_add_left_neg {a : α} (s) {t : Multiset α} (h : a ∉ t) :
(s + t).erase a = s.erase a + t := by | rw [add_comm, erase_add_right_neg s h, add_comm]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Adjunction.Over
#align_import categ... | Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 330 | 334 | theorem diagonalObjPullbackFstIso_inv_fst_snd {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :
(diagonalObjPullbackFstIso f g).inv ≫ pullback.fst ≫ pullback.snd =
pullback.fst ≫ pullback.fst := by |
delta diagonalObjPullbackFstIso
simp
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align... | Mathlib/Computability/Ackermann.lean | 251 | 256 | theorem ack_succ_right_le_ack_succ_left (m n : ℕ) : ack m (n + 1) ≤ ack (m + 1) n := by |
cases' n with n n
· simp
· rw [ack_succ_succ]
apply ack_mono_right m (le_trans _ <| add_add_one_le_ack _ n)
omega
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.Bin... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 170 | 173 | theorem pullbackZeroZeroIso_inv_fst (X Y : C) [HasBinaryProduct X Y] :
(pullbackZeroZeroIso X Y).inv ≫ pullback.fst = prod.fst := by |
dsimp [pullbackZeroZeroIso]
simp
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Grou... | Mathlib/RingTheory/Coprime/Basic.lean | 326 | 328 | theorem add_mul_right_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (x + z * y) y := by |
rw [mul_comm]
exact h.add_mul_left_left z
|
/-
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
#al... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 436 | 436 | theorem arccos_le_pi_div_two {x} : arccos x ≤ π / 2 ↔ 0 ≤ x := by | simp [arccos]
|
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.... | Mathlib/Data/Sign.lean | 358 | 362 | theorem sign_eq_one_iff : sign a = 1 ↔ 0 < a := by |
refine ⟨fun h => ?_, fun h => sign_pos h⟩
by_contra hn
rw [sign_apply, if_neg hn] at h
split_ifs at h
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.MeasureTheory.Covering.OneDim
import Mathlib.Order.Monotone.Extension
#align_import analysis.calcul... | Mathlib/Analysis/Calculus/Monotone.lean | 67 | 131 | theorem StieltjesFunction.ae_hasDerivAt (f : StieltjesFunction) :
∀ᵐ x, HasDerivAt f (rnDeriv f.measure volume x).toReal x := by |
/- Denote by `μ` the Stieltjes measure associated to `f`.
The general theorem `VitaliFamily.ae_tendsto_rnDeriv` ensures that `μ [x, y] / (y - x)` tends
to the Radon-Nikodym derivative as `y` tends to `x` from the right. As `μ [x,y] = f y - f (x^-)`
and `f (x^-) = f x` almost everywhere, this gives differ... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Data.Multiset.Nodup
import Mathlib.Data.List.NatAntidiagonal
#align_import data.multiset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28... | Mathlib/Data/Multiset/NatAntidiagonal.lean | 42 | 43 | theorem card_antidiagonal (n : ℕ) : card (antidiagonal n) = n + 1 := by |
rw [antidiagonal, coe_card, List.Nat.length_antidiagonal]
|
/-
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.Init.Algebra.Classes
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.BoundedOrder
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Lift... | Mathlib/Order/WithBot.lean | 1,143 | 1,144 | theorem coe_lt_coe : (a : WithTop α) < b ↔ a < b := by |
simp only [← toDual_lt_toDual_iff, toDual_apply_coe, WithBot.coe_lt_coe, toDual_lt_toDual]
|
/-
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.Init.Data.Ordering.Lemmas
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.NormNum
#ali... | Mathlib/SetTheory/Ordinal/Notation.lean | 224 | 225 | theorem NFBelow.fst {e n a b} (h : NFBelow (ONote.oadd e n a) b) : NF e := by |
cases' h with _ _ _ _ eb _ h₁ h₂ h₃; exact ⟨⟨_, h₁⟩⟩
|
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