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) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
imp... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 125 | 128 | theorem braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
(f ⊗ g) ≫ (braiding Y Y').hom = (braiding X X').hom ≫ (g ⊗ f) := by |
rw [tensorHom_def' f g, tensorHom_def g f]
simp_rw [Category.assoc, braiding_naturality_left, braiding_naturality_right_assoc]
|
/-
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.Data.Countable.Basic
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.Interval.Finset.Defs
#align_import orde... | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 250 | 259 | theorem toZ_iterate_succ_of_not_isMax (n : ℕ) (hn : ¬IsMax (succ^[n] i0)) :
toZ i0 (succ^[n] i0) = n := by |
let m := (toZ i0 (succ^[n] i0)).toNat
have h_eq : succ^[m] i0 = succ^[n] i0 := iterate_succ_toZ _ (le_succ_iterate _ _)
by_cases hmn : m = n
· nth_rw 2 [← hmn]
rw [Int.toNat_eq_max, toZ_of_ge (le_succ_iterate _ _), max_eq_left]
exact Int.natCast_nonneg _
suffices IsMax (succ^[n] i0) from absurd this ... |
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 499 | 500 | theorem concat_ne_nil {u v : V} (p : G.Walk u v) (h : G.Adj v u) : p.concat h ≠ nil := by |
cases p <;> simp [concat]
|
/-
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.Algebra.Pi
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
#align_im... | Mathlib/Algebra/Polynomial/AlgebraMap.lean | 131 | 136 | theorem eval₂_algebraMap_X {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A] (p : R[X])
(f : R[X] →ₐ[R] A) : eval₂ (algebraMap R A) (f X) p = f p := by |
conv_rhs => rw [← Polynomial.sum_C_mul_X_pow_eq p]
simp only [eval₂_eq_sum, sum_def]
simp only [f.map_sum, f.map_mul, f.map_pow, eq_intCast, map_intCast]
simp [Polynomial.C_eq_algebraMap]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Sh... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 184 | 187 | theorem cocycle_snd_fst_snd (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.snd ≫ pullback.fst ≫ pullback.snd =
pullback.snd ≫ pullback.fst ≫ pullback.snd := by |
simp only [pullback.condition_assoc, t'_snd_fst_snd]
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 853 | 853 | theorem model_singleton_iff {φ : L.Sentence} : M ⊨ ({φ} : L.Theory) ↔ M ⊨ φ := by | simp
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Fin.Tuple.Sort
import Mathlib.Order.WellFounded
#align_import data.fin.tuple.bubble_sort_induction from "leanprover-community/mathlib"@"bf2428c9486... | Mathlib/Data/Fin/Tuple/BubbleSortInduction.lean | 34 | 44 | theorem bubble_sort_induction' {n : ℕ} {α : Type*} [LinearOrder α] {f : Fin n → α}
{P : (Fin n → α) → Prop} (hf : P f)
(h : ∀ (σ : Equiv.Perm (Fin n)) (i j : Fin n),
i < j → (f ∘ σ) j < (f ∘ σ) i → P (f ∘ σ) → P (f ∘ σ ∘ Equiv.swap i j)) :
P (f ∘ sort f) := by |
letI := @Preorder.lift _ (Lex (Fin n → α)) _ fun σ : Equiv.Perm (Fin n) => toLex (f ∘ σ)
refine
@WellFounded.induction_bot' _ _ _ (IsWellFounded.wf : WellFounded (· < ·))
(Equiv.refl _) (sort f) P (fun σ => f ∘ σ) (fun σ hσ hfσ => ?_) hf
obtain ⟨i, j, hij₁, hij₂⟩ := antitone_pair_of_not_sorted' hσ
ex... |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
#align_import ring_theory.fractional_ideal from "lean... | Mathlib/RingTheory/FractionalIdeal/Basic.lean | 404 | 405 | theorem coe_one : (↑(1 : FractionalIdeal S P) : Submodule R P) = 1 := by |
rw [coe_one_eq_coeSubmodule_top, coeSubmodule_top]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathli... | Mathlib/Data/Nat/Digits.lean | 290 | 293 | theorem ofDigits_one (L : List ℕ) : ofDigits 1 L = L.sum := by |
induction' L with _ _ ih
· rfl
· simp [ofDigits, List.sum_cons, ih]
|
/-
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.Finset.Sort
import Mathlib.Data.Vector.Basic
import Mathlib.Logic.Denumerable
#align_import logic.equiv.list from "leanprover-community/mathlib"@... | Mathlib/Logic/Equiv/List.lean | 269 | 269 | theorem list_ofNat_zero : ofNat (List α) 0 = [] := by | rw [← @encode_list_nil α, ofNat_encode]
|
/-
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 | 553 | 575 | theorem strong_law_aux5 :
∀ᵐ ω, (fun n : ℕ => ∑ i ∈ range n, truncation (X i) i ω - ∑ i ∈ range n, X i ω) =o[atTop]
fun n : ℕ => (n : ℝ) := by |
have A : (∑' j : ℕ, ℙ {ω | X j ω ∈ Set.Ioi (j : ℝ)}) < ∞ := by
convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2
ext1 j
exact (hident j).measure_mem_eq measurableSet_Ioi
have B : ∀ᵐ ω, Tendsto (fun n : ℕ => truncation (X n) n ω - X n ω) atTop (𝓝 0) := by
filter_upwards [ae_eventually_not_m... |
/-
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 | 216 | 221 | theorem subset_union {H K L : S} : (H : Set G) ⊆ K ∪ L ↔ H ≤ K ∨ H ≤ L := by |
refine ⟨fun h ↦ ?_, fun h x xH ↦ h.imp (· xH) (· xH)⟩
rw [or_iff_not_imp_left, SetLike.not_le_iff_exists]
exact fun ⟨x, xH, xK⟩ y yH ↦ (h <| mul_mem xH yH).elim
((h yH).resolve_left fun yK ↦ xK <| (mul_mem_cancel_right yK).mp ·)
(mul_mem_cancel_left <| (h xH).resolve_left xK).mp
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import field_theory.finite.pol... | Mathlib/FieldTheory/Finite/Polynomial.lean | 137 | 150 | theorem map_restrict_dom_evalₗ : (restrictDegree σ K (Fintype.card K - 1)).map (evalₗ K σ) = ⊤ := by |
cases nonempty_fintype σ
refine top_unique (SetLike.le_def.2 fun e _ => mem_map.2 ?_)
refine ⟨∑ n : σ → K, e n • indicator n, ?_, ?_⟩
· exact sum_mem fun c _ => smul_mem _ _ (indicator_mem_restrictDegree _)
· ext n
simp only [_root_.map_sum, @Finset.sum_apply (σ → K) (fun _ => K) _ _ _ _ _, Pi.smul_apply... |
/-
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.Topology.Sets.Closeds
#align_import topology.noetherian_space from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Noetheri... | Mathlib/Topology/NoetherianSpace.lean | 139 | 142 | theorem noetherianSpace_set_iff (s : Set α) :
NoetherianSpace s ↔ ∀ t, t ⊆ s → IsCompact t := by |
simp only [noetherianSpace_iff_isCompact, embedding_subtype_val.isCompact_iff,
Subtype.forall_set_subtype]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
#align_import order.heyting.basic from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# Heyting algebr... | Mathlib/Order/Heyting/Basic.lean | 332 | 333 | theorem himp_inf_himp_inf_le : (b ⇨ c) ⊓ (a ⇨ b) ⊓ a ≤ c := by |
simpa using @himp_le_himp_himp_himp
|
/-
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 | 1,550 | 1,552 | theorem insert.valid [IsTotal α (· ≤ ·)] [@DecidableRel α (· ≤ ·)] (x : α) {t} (h : Valid t) :
Valid (Ordnode.insert x t) := by |
rw [insert_eq_insertWith]; exact insertWith.valid _ _ (fun _ _ => ⟨le_rfl, le_rfl⟩) h
|
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.NormalC... | Mathlib/FieldTheory/SeparableDegree.lean | 333 | 337 | theorem natSepDegree_eq_of_splits (h : f.Splits (algebraMap F E)) :
f.natSepDegree = (f.aroots E).toFinset.card := by |
rw [aroots, ← (SplittingField.lift f h).comp_algebraMap, ← map_map,
roots_map _ ((splits_id_iff_splits _).mpr <| SplittingField.splits f),
Multiset.toFinset_map, Finset.card_image_of_injective _ (RingHom.injective _), natSepDegree]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.linear_pmap from "leanprover-community/mathlib"@"8b981918a93bc45a8600de608cde7944a80d92... | Mathlib/LinearAlgebra/LinearPMap.lean | 771 | 774 | theorem mem_graph_iff (f : E →ₗ.[R] F) {x : E × F} :
x ∈ f.graph ↔ ∃ y : f.domain, (↑y : E) = x.1 ∧ f y = x.2 := by |
cases x
simp_rw [mem_graph_iff', Prod.mk.inj_iff]
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | Mathlib/Analysis/MeanInequalities.lean | 560 | 569 | theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by |
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne'
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable
have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']
have hB : B = (∑' i : ι, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']
refine ⟨(∑' ... |
/-
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.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dea... | Mathlib/Data/Multiset/Bind.lean | 352 | 355 | theorem coe_sigma (l₁ : List α) (l₂ : ∀ a, List (σ a)) :
(@Multiset.sigma α σ l₁ fun a => l₂ a) = l₁.sigma l₂ := by |
rw [Multiset.sigma, List.sigma, ← coe_bind]
simp
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Ma... | Mathlib/Analysis/Analytic/Basic.lean | 1,251 | 1,256 | theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by |
refine NNReal.summable_of_le
(fun n => ?_) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
|
/-
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 | 75 | 76 | theorem volume_val (s) : volume s = StieltjesFunction.id.measure s := by |
simp [volume_eq_stieltjes_id]
|
/-
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 | 213 | 216 | theorem intCast_div (a b : ℤ) (h : b ∣ a) : ((a / b : ℤ) : ℚ) = a / b := by |
rcases h with ⟨c, rfl⟩
rw [mul_comm b, Int.mul_ediv_assoc c (dvd_refl b), Int.cast_mul,
intCast_div_self, Int.cast_mul, mul_div_assoc]
|
/-
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.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 665 | 666 | theorem iUnion_inter_closedBall_nat (s : Set α) (x : α) : ⋃ n : ℕ, s ∩ closedBall x n = s := by |
rw [← inter_iUnion, iUnion_closedBall_nat, inter_univ]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Init.Order.LinearOrder
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Subtype
import Mathlib.Tactic.Spread
import Mathlib.Tactic.Conv... | Mathlib/Order/Basic.lean | 1,325 | 1,332 | theorem lt_iff : x < y ↔ x.1 < y.1 ∧ x.2 ≤ y.2 ∨ x.1 ≤ y.1 ∧ x.2 < y.2 := by |
refine ⟨fun h ↦ ?_, ?_⟩
· by_cases h₁ : y.1 ≤ x.1
· exact Or.inr ⟨h.1.1, LE.le.lt_of_not_le h.1.2 fun h₂ ↦ h.2 ⟨h₁, h₂⟩⟩
· exact Or.inl ⟨LE.le.lt_of_not_le h.1.1 h₁, h.1.2⟩
· rintro (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩)
· exact ⟨⟨h₁.le, h₂⟩, fun h ↦ h₁.not_le h.1⟩
· exact ⟨⟨h₁, h₂.le⟩, fun h ↦ h₂.not_le h.2⟩
|
/-
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.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 192 | 193 | theorem dist_triangle_left (x y z : α) : dist x y ≤ dist z x + dist z y := by |
rw [dist_comm z]; apply dist_triangle
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 969 | 982 | theorem isLimit_add_iff {a b} : IsLimit (a + b) ↔ IsLimit b ∨ b = 0 ∧ IsLimit a := by |
constructor <;> intro h
· by_cases h' : b = 0
· rw [h', add_zero] at h
right
exact ⟨h', h⟩
left
rw [← add_sub_cancel a b]
apply sub_isLimit h
suffices a + 0 < a + b by simpa only [add_zero] using this
rwa [add_lt_add_iff_left, Ordinal.pos_iff_ne_zero]
rcases h with (h | ⟨rfl, ... |
/-
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.CategoryTheory.MorphismProperty.Composition
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
import Mathlib.CategoryTheory.Category.Quiv
#align_impor... | Mathlib/CategoryTheory/Localization/Construction.lean | 306 | 307 | theorem natTransExtension_hcomp {F G : W.Localization ⥤ D} (τ : W.Q ⋙ F ⟶ W.Q ⋙ G) :
𝟙 W.Q ◫ natTransExtension τ = τ := by | aesop_cat
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
#align_import order.heyting.basic from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# Heyting algebr... | Mathlib/Order/Heyting/Basic.lean | 300 | 300 | theorem himp_inf_self (a b : α) : (a ⇨ b) ⊓ a = b ⊓ a := by | rw [inf_comm, inf_himp, inf_comm]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Johan Commelin
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.... | Mathlib/RingTheory/TensorProduct/Basic.lean | 90 | 92 | theorem baseChange_zero : baseChange A (0 : M →ₗ[R] N) = 0 := by |
ext
simp [baseChange_eq_ltensor]
|
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Init.Core
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#ali... | Mathlib/NumberTheory/Cyclotomic/Basic.lean | 282 | 287 | theorem equiv {C : Type*} [CommRing C] [Algebra A C] [h : IsCyclotomicExtension S A B]
(f : B ≃ₐ[A] C) : IsCyclotomicExtension S A C := by |
letI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra
haveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective
haveI : IsScalarTower A B C := IsScalarTower.of_algHom f.toAlgHom
exact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)
|
/-
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.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
/-!
# Infinite sums and products over `ℕ` and `ℤ`
This file contains lemma... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 124 | 132 | theorem tprod_iSup_decode₂ [CompleteLattice α] (m : α → M) (m0 : m ⊥ = 1) (s : β → α) :
∏' i : ℕ, m (⨆ b ∈ decode₂ β i, s b) = ∏' b : β, m (s b) := by |
rw [← tprod_extend_one (@encode_injective β _)]
refine tprod_congr fun n ↦ ?_
rcases em (n ∈ Set.range (encode : β → ℕ)) with ⟨a, rfl⟩ | hn
· simp [encode_injective.extend_apply]
· rw [extend_apply' _ _ _ hn]
rw [← decode₂_ne_none_iff, ne_eq, not_not] at hn
simp [hn, m0]
|
/-
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 | 482 | 486 | theorem exists_associated_pow_of_mul_eq_pow' {a b c : R} (hab : IsCoprime a b) {k : ℕ}
(h : a * b = c ^ k) : ∃ d : R, Associated (d ^ k) a := by |
classical
letI := IsBezout.toGCDDomain R
exact exists_associated_pow_of_mul_eq_pow ((gcd_isUnit_iff _ _).mpr hab) h
|
/-
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.LeftRightNhds
/-!
# Properties of LUB and GLB in an order topology
-/
open Set Filter TopologicalSpa... | Mathlib/Topology/Order/IsLUB.lean | 186 | 192 | theorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)]
(htx : IsLUB t x) (ht : t.Nonempty) :
∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by |
by_cases h : x ∈ t
· exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩
· rcases htx.exists_seq_strictMono_tendsto_of_not_mem h ht with ⟨u, hu⟩
exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩
|
/-
Copyright (c) 2023 Ali Ramsey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ali Ramsey, Eric Wieser
-/
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.LinearAlgebra.TensorProduct.Basic
/-!
# Coalgebras
In this file we define... | Mathlib/RingTheory/Coalgebra/Basic.lean | 145 | 151 | theorem comul_comp_snd :
comul ∘ₗ .snd R A B = TensorProduct.map (.snd R A B) (.snd R A B) ∘ₗ comul := by |
ext : 1
· rw [comp_assoc, snd_comp_inl, comp_zero, comp_assoc, comul_comp_inl, ← comp_assoc,
← TensorProduct.map_comp, snd_comp_inl, TensorProduct.map_zero_left, zero_comp]
· rw [comp_assoc, snd_comp_inr, comp_id, comp_assoc, comul_comp_inr, ← comp_assoc,
← TensorProduct.map_comp, snd_comp_inr, Tenso... |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Eric Wieser, Jeremy Avigad, Johan Commelin
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Linear... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 406 | 413 | theorem det_fromBlocks₂₂ (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α)
(D : Matrix n n α) [Invertible D] :
(Matrix.fromBlocks A B C D).det = det D * det (A - B * ⅟ D * C) := by |
have : fromBlocks A B C D =
(fromBlocks D C B A).submatrix (Equiv.sumComm _ _) (Equiv.sumComm _ _) := by
ext (i j)
cases i <;> cases j <;> rfl
rw [this, det_submatrix_equiv_self, det_fromBlocks₁₁]
|
/-
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.FunLike.Equiv
import Mathlib.Data.Quot
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Logic.Unique
import Mathlib.T... | Mathlib/Logic/Equiv/Defs.lean | 338 | 339 | theorem cast_eq_iff_heq {α β} (h : α = β) {a : α} {b : β} : Equiv.cast h a = b ↔ HEq a b := by |
subst h; simp [coe_refl]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 947 | 952 | theorem L1.integral_of_fun_eq_integral {f : α → G} (hf : Integrable f μ) :
∫ a, (hf.toL1 f) a ∂μ = ∫ a, f a ∂μ := by |
by_cases hG : CompleteSpace G
· simp only [MeasureTheory.integral, hG, L1.integral]
exact setToFun_toL1 (dominatedFinMeasAdditive_weightedSMul μ) hf
· simp [MeasureTheory.integral, hG]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.ca... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 1,029 | 1,031 | theorem Filter.EventuallyEq.fderivWithin_eq (hs : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
fderivWithin 𝕜 f₁ s x = fderivWithin 𝕜 f s x := by |
simp only [fderivWithin, hs.hasFDerivWithinAt_iff hx]
|
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Algebra.SMulWithZero
imp... | Mathlib/Algebra/Module/Defs.lean | 499 | 499 | theorem smul_ne_zero_iff : c • x ≠ 0 ↔ c ≠ 0 ∧ x ≠ 0 := by | rw [Ne, smul_eq_zero, not_or]
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.Normed.Group.Basic
#align_import analysis.normed.group.hom from "leanprover-community/mathlib"@"3c4225288b55380a90df078ebae0991080b12393"
/-... | Mathlib/Analysis/Normed/Group/Hom.lean | 685 | 687 | theorem comp_zero (f : NormedAddGroupHom V₂ V₃) : f.comp (0 : NormedAddGroupHom V₁ V₂) = 0 := by |
ext
exact map_zero f
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTh... | Mathlib/RingTheory/Localization/Basic.lean | 676 | 679 | theorem map_map {A : Type*} [CommSemiring A] {U : Submonoid A} {W} [CommSemiring W] [Algebra A W]
[IsLocalization U W] {l : P →+* A} (hl : T ≤ U.comap l) (x : S) :
map W l hl (map Q g hy x) = map W (l.comp g) (fun x hx => hl (hy hx)) x := by |
rw [← map_comp_map (Q := Q) hy hl]; rfl
|
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed... | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 175 | 193 | theorem minpoly_eq_pow_coprime {m : ℕ} (hcop : Nat.Coprime m n) :
minpoly ℤ μ = minpoly ℤ (μ ^ m) := by |
revert n hcop
refine UniqueFactorizationMonoid.induction_on_prime m ?_ ?_ ?_
· intro h hn
congr
simpa [(Nat.coprime_zero_left _).mp hn] using h
· intro u hunit _ _
congr
simp [Nat.isUnit_iff.mp hunit]
· intro a p _ hprime
intro hind h hcop
rw [hind h (Nat.Coprime.coprime_mul_left hcop... |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mat... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 511 | 519 | theorem count_mono {I J} (hI : I ≠ 0) (h : I ≤ J) : count K v J ≤ count K v I := by |
by_cases hJ : J = 0
· exact (hI (FractionalIdeal.le_zero_iff.mp (h.trans hJ.le))).elim
have := FractionalIdeal.mul_le_mul_left h J⁻¹
rw [inv_mul_cancel hJ, FractionalIdeal.le_one_iff_exists_coeIdeal] at this
obtain ⟨J', hJ'⟩ := this
rw [← mul_inv_cancel_left₀ hJ I, ← hJ', count_mul K v hJ, le_add_iff_nonne... |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 209 | 211 | theorem gcd_zero_left (a : R) : gcd 0 a = a := by |
rw [gcd]
exact if_pos rfl
|
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-c... | Mathlib/RingTheory/Discriminant.lean | 136 | 139 | theorem discr_not_zero_of_basis [IsSeparable K L] (b : Basis ι K L) :
discr K b ≠ 0 := by |
rw [discr_def, traceMatrix_of_basis, ← LinearMap.BilinForm.nondegenerate_iff_det_ne_zero]
exact traceForm_nondegenerate _ _
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Scott Morrison, Alex Keizer
-/
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.fin_range from "leanprover-community/mat... | Mathlib/Data/List/FinRange.lean | 54 | 55 | theorem ofFn_eq_map {n} {f : Fin n → α} : ofFn f = (finRange n).map f := by |
rw [← ofFn_id, map_ofFn, Function.comp_id]
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
imp... | Mathlib/FieldTheory/Adjoin.lean | 732 | 736 | theorem toSubalgebra_iSup_of_directed (dir : Directed (· ≤ ·) t) :
(iSup t).toSubalgebra = ⨆ i, (t i).toSubalgebra := by |
cases isEmpty_or_nonempty ι
· simp_rw [iSup_of_empty, bot_toSubalgebra]
· exact SetLike.ext' ((coe_iSup_of_directed dir).trans (Subalgebra.coe_iSup_of_directed dir).symm)
|
/-
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.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
... | Mathlib/Algebra/Polynomial/Coeff.lean | 243 | 245 | theorem card_support_binomial {k m : ℕ} (h : k ≠ m) {x y : R} (hx : x ≠ 0) (hy : y ≠ 0) :
card (support (C x * X ^ k + C y * X ^ m)) = 2 := by |
rw [support_binomial h hx hy, card_insert_of_not_mem (mt mem_singleton.mp h), card_singleton]
|
/-
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.Group.Support
import Mathlib.Order.WellFoundedSet
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e126... | Mathlib/RingTheory/HahnSeries/Basic.lean | 252 | 256 | theorem orderTop_eq_top_iff {x : HahnSeries Γ R} : orderTop x = ⊤ ↔ x = 0 := by |
constructor
· contrapose!
exact ne_zero_iff_orderTop.mp
· simp_all only [orderTop_zero, implies_true]
|
/-
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 | 116 | 119 | theorem primeSpectrumProd_symm_inl_asIdeal (x : PrimeSpectrum R) :
((primeSpectrumProd R S).symm <| Sum.inl x).asIdeal = Ideal.prod x.asIdeal ⊤ := by |
cases x
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, Yury Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducib... | Mathlib/Topology/Connected/Basic.lean | 142 | 145 | theorem IsPreconnected.union' {s t : Set α} (H : (s ∩ t).Nonempty) (hs : IsPreconnected s)
(ht : IsPreconnected t) : IsPreconnected (s ∪ t) := by |
rcases H with ⟨x, hxs, hxt⟩
exact hs.union x hxs hxt ht
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTh... | Mathlib/MeasureTheory/Function/LpSpace.lean | 440 | 444 | theorem norm_le_of_ae_bound [IsFiniteMeasure μ] {f : Lp E p μ} {C : ℝ} (hC : 0 ≤ C)
(hfC : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) : ‖f‖ ≤ measureUnivNNReal μ ^ p.toReal⁻¹ * C := by |
lift C to ℝ≥0 using hC
have := nnnorm_le_of_ae_bound hfC
rwa [← NNReal.coe_le_coe, NNReal.coe_mul, NNReal.coe_rpow] at this
|
/-
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 | 802 | 805 | theorem Raised.add_left (k) {n m} (H : Raised n m) : Raised (k + n) (k + m) := by |
rcases H with (rfl | rfl)
· exact Or.inl rfl
· exact Or.inr rfl
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.App... | Mathlib/FieldTheory/Finite/Basic.lean | 357 | 370 | theorem roots_X_pow_card_sub_X : roots (X ^ q - X : K[X]) = Finset.univ.val := by |
classical
have aux : (X ^ q - X : K[X]) ≠ 0 := X_pow_card_sub_X_ne_zero K Fintype.one_lt_card
have : (roots (X ^ q - X : K[X])).toFinset = Finset.univ := by
rw [eq_univ_iff_forall]
intro x
rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
sub_eq_zero,... |
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "lea... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 309 | 315 | theorem leftDistributor_ext₂_left {J : Type} [Fintype J]
{X Y Z : C} {f : J → C} {g h : X ⊗ (Y ⊗ ⨁ f) ⟶ Z}
(w : ∀ j, (X ◁ (Y ◁ biproduct.ι f j)) ≫ g = (X ◁ (Y ◁ biproduct.ι f j)) ≫ h) :
g = h := by |
apply (cancel_epi (α_ _ _ _).hom).mp
ext
simp [w]
|
/-
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.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Nontrivial.Defs
import Mathlib.Tactic.SplitIfs
#align_import algebra.group... | Mathlib/Algebra/GroupWithZero/Defs.lean | 290 | 290 | theorem zero_eq_mul : 0 = a * b ↔ a = 0 ∨ b = 0 := by | rw [eq_comm, mul_eq_zero]
|
/-
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 | 238 | 242 | theorem norm_div_sin_oangle_add_left_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.sin (o.oangle (x + y) y) = ‖x + y‖ := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two h
|
/-
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.Algebra.Group.Pi.Lemmas
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Homeomorph
#align_import topology.algebra.group_with_ze... | Mathlib/Topology/Algebra/GroupWithZero.lean | 342 | 348 | theorem continuousAt_zpow₀ (x : G₀) (m : ℤ) (h : x ≠ 0 ∨ 0 ≤ m) :
ContinuousAt (fun x => x ^ m) x := by |
cases' m with m m
· simpa only [Int.ofNat_eq_coe, zpow_natCast] using continuousAt_pow x m
· simp only [zpow_negSucc]
have hx : x ≠ 0 := h.resolve_right (Int.negSucc_lt_zero m).not_le
exact (continuousAt_pow x (m + 1)).inv₀ (pow_ne_zero _ hx)
|
/-
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 | 198 | 204 | theorem natDegree_derivative_lt {p : R[X]} (hp : p.natDegree ≠ 0) :
p.derivative.natDegree < p.natDegree := by |
rcases eq_or_ne (derivative p) 0 with hp' | hp'
· rw [hp', Polynomial.natDegree_zero]
exact hp.bot_lt
· rw [natDegree_lt_natDegree_iff hp']
exact degree_derivative_lt fun h => hp (h.symm ▸ natDegree_zero)
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
#align_import algebra.star.self_adjoint from "leanpro... | Mathlib/Algebra/Star/SelfAdjoint.lean | 165 | 166 | theorem conjugate {x : R} (hx : IsSelfAdjoint x) (z : R) : IsSelfAdjoint (z * x * star z) := by |
simp only [isSelfAdjoint_iff, star_mul, star_star, mul_assoc, hx.star_eq]
|
/-
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.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathl... | Mathlib/Algebra/MvPolynomial/Basic.lean | 795 | 807 | theorem coeff_mul_monomial' (m) (s : σ →₀ ℕ) (r : R) (p : MvPolynomial σ R) :
coeff m (p * monomial s r) = if s ≤ m then coeff (m - s) p * r else 0 := by |
classical
split_ifs with h
· conv_rhs => rw [← coeff_mul_monomial _ s]
congr with t
rw [tsub_add_cancel_of_le h]
· contrapose! h
rw [← mem_support_iff] at h
obtain ⟨j, -, rfl⟩ : ∃ j ∈ support p, j + s = m := by
simpa [Finset.add_singleton]
using Finset.add_subset_add_left support_... |
/-
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 | 294 | 297 | theorem rank_eq_finrank_span_row [LinearOrderedField R] [Finite m] (A : Matrix m n R) :
A.rank = finrank R (Submodule.span R (Set.range A)) := by |
cases nonempty_fintype m
rw [← rank_transpose, rank_eq_finrank_span_cols, transpose_transpose]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Scott Morrison, Alex Keizer
-/
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.fin_range from "leanprover-community/mat... | Mathlib/Data/List/FinRange.lean | 30 | 34 | theorem finRange_succ_eq_map (n : ℕ) : finRange n.succ = 0 :: (finRange n).map Fin.succ := by |
apply map_injective_iff.mpr Fin.val_injective
rw [map_cons, map_coe_finRange, range_succ_eq_map, Fin.val_zero, ← map_coe_finRange, map_map,
map_map]
simp only [Function.comp, Fin.val_succ]
|
/-
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 | 1,515 | 1,519 | theorem bijective_iff_bijective_of_iUnion_eq_univ :
Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) := by |
rw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU,
surjective_iff_surjective_of_iUnion_eq_univ hU]
simp [Bijective, forall_and]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate f... | Mathlib/Data/List/Rotate.lean | 53 | 53 | theorem rotate'_zero (l : List α) : l.rotate' 0 = l := by | cases l <;> rfl
|
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Simon Hudon
-/
import Batteries.Data.List.Lemmas
import Batteries.Tactic.Classical
import Mathlib.Tactic.TypeStar
import Mathlib.Mathport.Rename
#align_import data.lis... | Mathlib/Data/List/TFAE.lean | 63 | 71 | theorem tfae_of_cycle {a b} {l : List Prop} (h_chain : List.Chain (· → ·) a (b :: l))
(h_last : getLastD l b → a) : TFAE (a :: b :: l) := by |
induction l generalizing a b with
| nil => simp_all [tfae_cons_cons, iff_def]
| cons c l IH =>
simp only [tfae_cons_cons, getLastD_cons, tfae_singleton, and_true, chain_cons, Chain.nil] at *
rcases h_chain with ⟨ab, ⟨bc, ch⟩⟩
have := IH ⟨bc, ch⟩ (ab ∘ h_last)
exact ⟨⟨ab, h_last ∘ (this.2 c (.head... |
/-
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.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Localization.NumDen
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomi... | Mathlib/RingTheory/Polynomial/RationalRoot.lean | 49 | 54 | theorem num_isRoot_scaleRoots_of_aeval_eq_zero [UniqueFactorizationMonoid A] {p : A[X]} {x : K}
(hr : aeval x p = 0) : IsRoot (scaleRoots p (den A x)) (num A x) := by |
apply isRoot_of_eval₂_map_eq_zero (IsFractionRing.injective A K)
refine scaleRoots_aeval_eq_zero_of_aeval_mk'_eq_zero ?_
rw [mk'_num_den]
exact hr
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.RingTheory.Noetherian
#align_import algebra.lie.subalgebra from "leanprover-community/mathlib"@"6d584f1709bedbed9175bd9350d... | Mathlib/Algebra/Lie/Subalgebra.lean | 689 | 694 | theorem lieSpan_le {K} : lieSpan R L s ≤ K ↔ s ⊆ K := by |
constructor
· exact Set.Subset.trans subset_lieSpan
· intro hs m hm
rw [mem_lieSpan] at hm
exact hm _ hs
|
/-
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 | 165 | 173 | theorem compression_self (u : α) (s : Finset α) : 𝓒 u u s = s := by |
unfold compression
convert union_empty s
· ext a
rw [mem_filter, compress_self, and_self_iff]
· refine eq_empty_of_forall_not_mem fun a ha ↦ ?_
simp_rw [mem_filter, mem_image, compress_self] at ha
obtain ⟨⟨b, hb, rfl⟩, hb'⟩ := ha
exact hb' hb
|
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Co... | Mathlib/Data/Fin/Basic.lean | 995 | 999 | theorem succ_ne_last_of_lt {p i : Fin n} (h : i < p) : succ i ≠ last n := by |
cases n
· exact i.elim0
· rw [succ_ne_last_iff, Ne, ext_iff]
exact ((le_last _).trans_lt' h).ne
|
/-
Copyright (c) 2021 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic... | Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean | 565 | 570 | theorem Ideal.IsHomogeneous.toIdeal_homogeneousHull_eq_self (h : I.IsHomogeneous 𝒜) :
(Ideal.homogeneousHull 𝒜 I).toIdeal = I := by |
apply le_antisymm _ (Ideal.le_toIdeal_homogeneousHull _ _)
apply Ideal.span_le.2
rintro _ ⟨i, x, rfl⟩
exact h _ x.prop
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | Mathlib/Topology/ContinuousFunction/Compact.lean | 146 | 147 | theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by |
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
|
/-
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.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "lea... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 604 | 606 | theorem infDist_zero_of_mem_closure (hx : x ∈ closure s) : infDist x s = 0 := by |
rw [← infDist_closure]
exact infDist_zero_of_mem hx
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 794 | 813 | theorem surj_on_of_inj_on_of_ncard_le {t : Set β} (f : ∀ a ∈ s, β) (hf : ∀ a ha, f a ha ∈ t)
(hinj : ∀ a₁ a₂ ha₁ ha₂, f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂) (hst : t.ncard ≤ s.ncard)
(ht : t.Finite := by | toFinite_tac) :
∀ b ∈ t, ∃ a ha, b = f a ha := by
intro b hb
set f' : s → t := fun x ↦ ⟨f x.1 x.2, hf _ _⟩
have finj : f'.Injective := by
rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy
simp only [f', Subtype.mk.injEq] at hxy ⊢
apply hinj _ _ hx hy hxy
have hft := ht.fintype
have hft' := Fintype.ofInjective f' fi... |
/-
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.MeasureTheory.Measure.AEDisjoint
import Mathlib.MeasureTheory.Constructions.EventuallyMeasurable
#align_import measur... | Mathlib/MeasureTheory/Measure/NullMeasurable.lean | 266 | 273 | theorem measure_iUnion {m0 : MeasurableSpace α} {μ : Measure α} [Countable ι] {f : ι → Set α}
(hn : Pairwise (Disjoint on f)) (h : ∀ i, MeasurableSet (f i)) :
μ (⋃ i, f i) = ∑' i, μ (f i) := by |
rw [measure_eq_extend (MeasurableSet.iUnion h),
extend_iUnion MeasurableSet.empty _ MeasurableSet.iUnion _ hn h]
· simp [measure_eq_extend, h]
· exact μ.empty
· exact μ.m_iUnion
|
/-
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 | 1,835 | 1,838 | theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by |
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Data.Finsupp.Fintype
import Mathlib.Data.Int.Absolut... | Mathlib/RingTheory/Ideal/Norm.lean | 365 | 376 | theorem absNorm_span_singleton (r : S) :
absNorm (span ({r} : Set S)) = (Algebra.norm ℤ r).natAbs := by |
rw [Algebra.norm_apply]
by_cases hr : r = 0
· simp only [hr, Ideal.span_zero, Algebra.coe_lmul_eq_mul, eq_self_iff_true, Ideal.absNorm_bot,
LinearMap.det_zero'', Set.singleton_zero, _root_.map_zero, Int.natAbs_zero]
letI := Ideal.fintypeQuotientOfFreeOfNeBot (span {r}) (mt span_singleton_eq_bot.mp hr)
... |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 267 | 270 | theorem comp_right_fderiv {f : F → G} {x : E} :
fderiv 𝕜 (f ∘ iso) x = (fderiv 𝕜 f (iso x)).comp (iso : E →L[𝕜] F) := by |
rw [← fderivWithin_univ, ← fderivWithin_univ, ← iso.comp_right_fderivWithin, preimage_univ]
exact uniqueDiffWithinAt_univ
|
/-
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.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | Mathlib/Analysis/Convex/Gauge.lean | 175 | 181 | theorem gauge_lt_eq (absorbs : Absorbent ℝ s) (a : ℝ) :
{ x | gauge s x < a } = ⋃ r ∈ Set.Ioo 0 (a : ℝ), r • s := by |
ext
simp_rw [mem_setOf, mem_iUnion, exists_prop, mem_Ioo, and_assoc]
exact
⟨exists_lt_of_gauge_lt absorbs, fun ⟨r, hr₀, hr₁, hx⟩ =>
(gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 1,423 | 1,425 | theorem preimage_Iio : ((↑) : ℤ → α) ⁻¹' Set.Iio a = Set.Iio ⌈a⌉ := by |
ext
simp [lt_ceil]
|
/-
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 | 1,113 | 1,115 | theorem iUnion_Icc_add_intCast : ⋃ n : ℤ, Icc (a + n) (a + n + 1) = Set.univ := by |
simpa only [zsmul_one, Int.cast_add, Int.cast_one, ← add_assoc] using
iUnion_Icc_add_zsmul zero_lt_one a
|
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.L... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 521 | 523 | theorem kahler_comp_rightAngleRotation' (x y : E) :
-(Complex.I * (Complex.I * o.kahler x y)) = o.kahler x y := by |
linear_combination -o.kahler x y * Complex.I_sq
|
/-
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.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import M... | Mathlib/Order/WellFoundedSet.lean | 834 | 865 | theorem subsetProdLex [PartialOrder α] [Preorder β] {s : Set (α ×ₗ β)}
(hα : ((fun (x : α ×ₗ β) => (ofLex x).1)'' s).IsPWO)
(hβ : ∀ a, {y | toLex (a, y) ∈ s}.IsPWO) : s.IsPWO := by |
intro f hf
rw [isPWO_iff_exists_monotone_subseq] at hα
obtain ⟨g, hg⟩ : ∃ (g : (ℕ ↪o ℕ)), Monotone fun n => (ofLex f (g n)).1 :=
hα (fun n => (ofLex f n).1) (fun k => mem_image_of_mem (fun x => (ofLex x).1) (hf k))
have hhg : ∀ n, (ofLex f (g 0)).1 ≤ (ofLex f (g n)).1 := fun n => hg n.zero_le
by_cases hc... |
/-
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.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 761 | 763 | theorem preimage_const_mul_Ico_of_neg (a b : α) {c : α} (h : c < 0) :
(c * ·) ⁻¹' Ico a b = Ioc (b / c) (a / c) := by |
simpa only [mul_comm] using preimage_mul_const_Ico_of_neg a b h
|
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.LinearAlgebra.Matrix.Spec... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 56 | 66 | theorem Gamma_mem (N : ℕ) (γ : SL(2, ℤ)) : γ ∈ Gamma N ↔ ((↑ₘγ 0 0 : ℤ) : ZMod N) = 1 ∧
((↑ₘγ 0 1 : ℤ) : ZMod N) = 0 ∧ ((↑ₘγ 1 0 : ℤ) : ZMod N) = 0 ∧ ((↑ₘγ 1 1 : ℤ) : ZMod N) = 1 := by |
rw [Gamma_mem']
constructor
· intro h
simp [← SL_reduction_mod_hom_val N γ, h]
· intro h
ext i j
rw [SL_reduction_mod_hom_val N γ]
fin_cases i <;> fin_cases j <;> simp only [h]
exacts [h.1, h.2.1, h.2.2.1, h.2.2.2]
|
/-
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 | 243 | 243 | theorem oangle_neg_neg (x y : V) : o.oangle (-x) (-y) = o.oangle x y := by | simp [oangle]
|
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Scott Morrison, Eric Rodriguez
-/
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count fr... | Mathlib/Data/Nat/Count.lean | 91 | 91 | theorem count_one : count p 1 = if p 0 then 1 else 0 := by | simp [count_succ]
|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheo... | Mathlib/CategoryTheory/Monoidal/Functor.lean | 171 | 174 | theorem LaxMonoidalFunctor.right_unitality_inv (F : LaxMonoidalFunctor C D) (X : C) :
(ρ_ (F.obj X)).inv ≫ F.obj X ◁ F.ε ≫ F.μ X (𝟙_ C) = F.map (ρ_ X).inv := by |
rw [Iso.inv_comp_eq, F.right_unitality, Category.assoc, Category.assoc, ← F.toFunctor.map_comp,
Iso.hom_inv_id, F.toFunctor.map_id, comp_id]
|
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Andrew Yang,
Johannes Hölzl, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.Module.Submodule.Lattice
import Ma... | Mathlib/Algebra/Module/Submodule/RestrictScalars.lean | 116 | 117 | theorem restrictScalars_eq_top_iff {p : Submodule R M} : restrictScalars S p = ⊤ ↔ p = ⊤ := by |
simp [SetLike.ext_iff]
|
/-
Copyright (c) 2024 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.NumberTheory.ModularForms.EisensteinSeries.UniformConvergence
import Mathlib.Analysis.Complex.UpperHalfPlane.Manifold
import Mathlib.Analysis.Complex.... | Mathlib/NumberTheory/ModularForms/EisensteinSeries/MDifferentiable.lean | 54 | 65 | theorem eisensteinSeries_SIF_MDifferentiable {k : ℤ} {N : ℕ} (hk : 3 ≤ k) (a : Fin 2 → ZMod N) :
MDifferentiable 𝓘(ℂ) 𝓘(ℂ) (eisensteinSeries_SIF a k) := by |
intro τ
suffices DifferentiableAt ℂ (↑ₕeisensteinSeries_SIF a k) τ.1 by
convert MDifferentiableAt.comp τ (DifferentiableAt.mdifferentiableAt this) τ.mdifferentiable_coe
exact funext fun z ↦ (comp_ofComplex (eisensteinSeries_SIF a k) z).symm
refine DifferentiableOn.differentiableAt ?_
((isOpen_lt cont... |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 565 | 569 | theorem orthogonalProjection_eq_zero_iff {v : E} : orthogonalProjection K v = 0 ↔ v ∈ Kᗮ := by |
refine ⟨fun h ↦ ?_, fun h ↦ Subtype.eq <| eq_orthogonalProjection_of_mem_orthogonal
(zero_mem _) ?_⟩
· simpa [h] using sub_orthogonalProjection_mem_orthogonal (K := K) v
· simpa
|
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanpro... | Mathlib/GroupTheory/Schreier.lean | 85 | 89 | theorem closure_mul_image_eq_top (hR : R ∈ rightTransversals (H : Set G)) (hR1 : (1 : G) ∈ R)
(hS : closure S = ⊤) : closure ((R * S).image fun g =>
⟨g * (toFun hR g : G)⁻¹, mul_inv_toFun_mem hR g⟩ : Set H) = ⊤ := by |
rw [eq_top_iff, ← map_subtype_le_map_subtype, MonoidHom.map_closure, Set.image_image]
exact (map_subtype_le ⊤).trans (ge_of_eq (closure_mul_image_eq hR hR1 hS))
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measu... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 174 | 246 | theorem exists_goodδ :
∃ δ : ℝ, 0 < δ ∧ δ < 1 ∧ ∀ s : Finset E, (∀ c ∈ s, ‖c‖ ≤ 2) →
(∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → s.card ≤ multiplicity E := by |
classical
/- This follows from a compactness argument: otherwise, one could extract a converging
subsequence, to obtain a `1`-separated set in the ball of radius `2` with cardinality
`N = multiplicity E + 1`. To formalize this, we work with functions `Fin N → E`.
-/
by_contra! h
set N := multiplic... |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
import Mathlib.Ord... | Mathlib/Order/Filter/Bases.lean | 985 | 990 | theorem map_sigma_mk_comap {π : α → Type*} {π' : β → Type*} {f : α → β}
(hf : Function.Injective f) (g : ∀ a, π a → π' (f a)) (a : α) (l : Filter (π' (f a))) :
map (Sigma.mk a) (comap (g a) l) = comap (Sigma.map f g) (map (Sigma.mk (f a)) l) := by |
refine (((basis_sets _).comap _).map _).eq_of_same_basis ?_
convert ((basis_sets l).map (Sigma.mk (f a))).comap (Sigma.map f g)
apply image_sigmaMk_preimage_sigmaMap hf
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 857 | 870 | theorem iSup_approx_apply [TopologicalSpace β] [CompleteLattice β] [OrderClosedTopology β] [Zero β]
[MeasurableSpace β] [OpensMeasurableSpace β] (i : ℕ → β) (f : α → β) (a : α) (hf : Measurable f)
(h_zero : (0 : β) = ⊥) : ⨆ n, (approx i f n : α →ₛ β) a = ⨆ (k) (_ : i k ≤ f a), i k := by |
refine le_antisymm (iSup_le fun n => ?_) (iSup_le fun k => iSup_le fun hk => ?_)
· rw [approx_apply a hf, h_zero]
refine Finset.sup_le fun k _ => ?_
split_ifs with h
· exact le_iSup_of_le k (le_iSup (fun _ : i k ≤ f a => i k) h)
· exact bot_le
· refine le_iSup_of_le (k + 1) ?_
rw [approx_appl... |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Order.SupIndep
import Mathlib.Order.Atoms
#align_import order.partition.finpa... | Mathlib/Order/Partition/Finpartition.lean | 457 | 462 | theorem mem_avoid : c ∈ (P.avoid b).parts ↔ ∃ d ∈ P.parts, ¬d ≤ b ∧ d \ b = c := by |
simp only [avoid, ofErase, mem_erase, Ne, mem_image, exists_prop, ← exists_and_left,
@and_left_comm (c ≠ ⊥)]
refine exists_congr fun d ↦ and_congr_right' <| and_congr_left ?_
rintro rfl
rw [sdiff_eq_bot_iff]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 1,693 | 1,694 | theorem inter_insert_of_mem {s₁ s₂ : Finset α} {a : α} (h : a ∈ s₁) :
s₁ ∩ insert a s₂ = insert a (s₁ ∩ s₂) := by | rw [inter_comm, insert_inter_of_mem h, inter_comm]
|
/-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_imp... | Mathlib/Analysis/Convex/Slope.lean | 145 | 173 | theorem strictConvexOn_of_slope_strict_mono_adjacent (hs : Convex 𝕜 s)
(hf :
∀ {x y z : 𝕜},
x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) < (f z - f y) / (z - y)) :
StrictConvexOn 𝕜 s f :=
LinearOrder.strictConvexOn_of_lt hs fun x hx z hz hxz a b ha hb hab => by
let y := a * x + b... |
rw [← one_mul x, ← hab, add_mul]
exact add_lt_add_left ((mul_lt_mul_left hb).2 hxz) _
have hyz : y < z := by
rw [← one_mul z, ← hab, add_mul]
exact add_lt_add_right ((mul_lt_mul_left ha).2 hxz) _
have : (f y - f x) * (z - y) < (f z - f y) * (y - x) :=
(div_lt_div_iff (sub_pos.2 hx... |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Trim
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
#align_import measure_theory.measure.ae_measurable fr... | Mathlib/MeasureTheory/Measure/AEMeasurable.lean | 374 | 388 | theorem MeasureTheory.Measure.restrict_map_of_aemeasurable {f : α → δ} (hf : AEMeasurable f μ)
{s : Set δ} (hs : MeasurableSet s) : (μ.map f).restrict s = (μ.restrict <| f ⁻¹' s).map f :=
calc
(μ.map f).restrict s = (μ.map (hf.mk f)).restrict s := by |
congr 1
apply Measure.map_congr hf.ae_eq_mk
_ = (μ.restrict <| hf.mk f ⁻¹' s).map (hf.mk f) := Measure.restrict_map hf.measurable_mk hs
_ = (μ.restrict <| hf.mk f ⁻¹' s).map f :=
(Measure.map_congr (ae_restrict_of_ae hf.ae_eq_mk.symm))
_ = (μ.restrict <| f ⁻¹' s).map f := by
apply c... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.