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) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0... | Mathlib/Geometry/Euclidean/MongePoint.lean | 624 | 632 | theorem orthocenter_replace_orthocenter_eq_point {t₁ t₂ : Triangle ℝ P} {i₁ i₂ i₃ j₁ j₂ j₃ : Fin 3}
(hi₁₂ : i₁ ≠ i₂) (hi₁₃ : i₁ ≠ i₃) (hi₂₃ : i₂ ≠ i₃) (hj₁₂ : j₁ ≠ j₂) (hj₁₃ : j₁ ≠ j₃)
(hj₂₃ : j₂ ≠ j₃) (h₁ : t₂.points j₁ = t₁.orthocenter) (h₂ : t₂.points j₂ = t₁.points i₂)
(h₃ : t₂.points j₃ = t₁.points i₃)... |
refine (Triangle.eq_orthocenter_of_forall_mem_altitude hj₂₃ ?_ ?_).symm
· rw [altitude_replace_orthocenter_eq_affineSpan hi₁₂ hi₁₃ hi₂₃ hj₁₂ hj₁₃ hj₂₃ h₁ h₂ h₃]
exact mem_affineSpan ℝ (Set.mem_insert _ _)
· rw [altitude_replace_orthocenter_eq_affineSpan hi₁₃ hi₁₂ hi₂₃.symm hj₁₃ hj₁₂ hj₂₃.symm h₁ h₃ h₂]
e... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7... | Mathlib/Data/Set/Prod.lean | 950 | 959 | theorem update_preimage_pi [DecidableEq ι] {f : ∀ i, α i} (hi : i ∈ s)
(hf : ∀ j ∈ s, j ≠ i → f j ∈ t j) : update f i ⁻¹' s.pi t = t i := by |
ext x
refine ⟨fun h => ?_, fun hx j hj => ?_⟩
· convert h i hi
simp
· obtain rfl | h := eq_or_ne j i
· simpa
· rw [update_noteq h]
exact hf j hj h
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Data.Nat.Factorization.PrimePow
#align_import data.nat.squarefree from "leanprover-community/mathlib"@"3c1368c... | Mathlib/Data/Nat/Squarefree.lean | 76 | 91 | theorem Squarefree.ext_iff {n m : ℕ} (hn : Squarefree n) (hm : Squarefree m) :
n = m ↔ ∀ p, Prime p → (p ∣ n ↔ p ∣ m) := by |
refine ⟨by rintro rfl; simp, fun h => eq_of_factorization_eq hn.ne_zero hm.ne_zero fun p => ?_⟩
by_cases hp : p.Prime
· have h₁ := h _ hp
rw [← not_iff_not, hp.dvd_iff_one_le_factorization hn.ne_zero, not_le, lt_one_iff,
hp.dvd_iff_one_le_factorization hm.ne_zero, not_le, lt_one_iff] at h₁
have h₂ ... |
/-
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.Variables
#align_import data.mv_polynomial.comm_ring from "leanprover-community/mathlib"@"2f5b500... | Mathlib/Algebra/MvPolynomial/CommRing.lean | 96 | 97 | theorem degrees_neg (p : MvPolynomial σ R) : (-p).degrees = p.degrees := by |
rw [degrees, support_neg]; rfl
|
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 246 | 248 | theorem oneCocycles_map_mul_of_isTrivial [A.IsTrivial] (f : oneCocycles A) (g h : G) :
f.1 (g * h) = f.1 g + f.1 h := by |
rw [(mem_oneCocycles_iff f.1).1 f.2, apply_eq_self A.ρ g (f.1 h), add_comm]
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/math... | Mathlib/CategoryTheory/Iso.lean | 202 | 203 | theorem symm_self_id_assoc (α : X ≅ Y) (β : Y ≅ Z) : α.symm ≪≫ α ≪≫ β = β := by |
rw [← trans_assoc, symm_self_id, refl_trans]
|
/-
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 | 797 | 798 | theorem floor_int_add (z : ℤ) (a : α) : ⌊↑z + a⌋ = z + ⌊a⌋ := by |
simpa only [add_comm] using floor_add_int a z
|
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4... | Mathlib/Algebra/CubicDiscriminant.lean | 75 | 78 | theorem prod_X_sub_C_eq [CommRing S] {x y z : S} :
(X - C x) * (X - C y) * (X - C z) =
toPoly ⟨1, -(x + y + z), x * y + x * z + y * z, -(x * y * z)⟩ := by |
rw [← one_mul <| X - C x, ← C_1, C_mul_prod_X_sub_C_eq, one_mul, one_mul, one_mul]
|
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.init_t... | Mathlib/RingTheory/WittVector/InitTail.lean | 209 | 210 | theorem init_neg (x : 𝕎 R) (n : ℕ) : init n (-x) = init n (-init n x) := by |
init_ring using wittNeg_vars
|
/-
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.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd575... | Mathlib/Topology/Order.lean | 900 | 901 | theorem tendsto_nhds_true {l : Filter α} {p : α → Prop} :
Tendsto p l (𝓝 True) ↔ ∀ᶠ x in l, p x := by | simp
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 803 | 809 | theorem norm_mkPiAlgebra_of_empty [IsEmpty ι] :
‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = ‖(1 : A)‖ := by |
apply le_antisymm
· apply opNorm_le_bound <;> simp
· -- Porting note: have to annotate types to get mvars to unify
convert ratio_le_opNorm (ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) fun _ => (1 : A)
simp [eq_empty_of_isEmpty (univ : Finset ι)]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDo... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 614 | 617 | theorem X_sub_C_mul_divByMonic_eq_sub_modByMonic {K : Type*} [Field K] (f : K[X]) (a : K) :
(X - C a) * (f /ₘ (X - C a)) = f - f %ₘ (X - C a) := by |
rw [eq_sub_iff_add_eq, ← eq_sub_iff_add_eq', modByMonic_eq_sub_mul_div]
exact monic_X_sub_C a
|
/-
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.Algebra.Category.ModuleCat.Adjunctions
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.Algebra.Category.ModuleCat.Colimits
import Mathl... | Mathlib/RepresentationTheory/Rep.lean | 683 | 691 | theorem unit_iso_comm (V : Rep k G) (g : G) (x : V) :
unitIsoAddEquiv ((V.ρ g).toFun x) = ((ofModuleMonoidAlgebra.obj
(toModuleMonoidAlgebra.obj V)).ρ g).toFun (unitIsoAddEquiv x) := by |
dsimp [unitIsoAddEquiv, ofModuleMonoidAlgebra, toModuleMonoidAlgebra]
/- Porting note: rest of broken proof was
simp only [AddEquiv.apply_eq_iff_eq, AddEquiv.apply_symm_apply,
Representation.asModuleEquiv_symm_map_rho, Representation.ofModule_asModule_act] -/
erw [Representation.asModuleEquiv_symm_map_rho]
... |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 537 | 538 | theorem integral_fun_snd (f : β → E) : ∫ z, f z.2 ∂μ.prod ν = (μ univ).toReal • ∫ y, f y ∂ν := by |
simpa using integral_prod_smul (1 : α → ℝ) f
|
/-
Copyright (c) 2020 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.MvPolynomial.Symmetric
#align_import ring_theory.polynomial.vieta from "leanprover-community/mathlib... | Mathlib/RingTheory/Polynomial/Vieta.lean | 120 | 133 | theorem prod_X_sub_C_coeff (s : Multiset R) {k : ℕ} (h : k ≤ Multiset.card s) :
(s.map fun t => X - C t).prod.coeff k =
(-1) ^ (Multiset.card s - k) * s.esymm (Multiset.card s - k) := by |
conv_lhs =>
congr
congr
congr
ext x
rw [sub_eq_add_neg]
rw [← map_neg C x]
convert prod_X_add_C_coeff (map (fun t => -t) s) _ using 1
· rw [map_map]; rfl
· rw [esymm_neg, card_map]
· rwa [card_map]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.NormedSpace.Extend
import Mathlib.... | Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean | 44 | 59 | theorem exists_extension_norm_eq (p : Subspace ℝ E) (f : p →L[ℝ] ℝ) :
∃ g : E →L[ℝ] ℝ, (∀ x : p, g x = f x) ∧ ‖g‖ = ‖f‖ := by |
rcases exists_extension_of_le_sublinear ⟨p, f⟩ (fun x => ‖f‖ * ‖x‖)
(fun c hc x => by simp only [norm_smul c x, Real.norm_eq_abs, abs_of_pos hc, mul_left_comm])
(fun x y => by -- Porting note: placeholder filled here
rw [← left_distrib]
exact mul_le_mul_of_nonneg_left (norm_add_le x y) (@... |
/-
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 | 1,723 | 1,730 | theorem image_sUnion {f : α → β} {s : Set (Set α)} : (f '' ⋃₀ s) = ⋃₀ (image f '' s) := by |
ext b
simp only [mem_image, mem_sUnion, exists_prop, sUnion_image, mem_iUnion]
constructor
· rintro ⟨a, ⟨t, ht₁, ht₂⟩, rfl⟩
exact ⟨t, ht₁, a, ht₂, rfl⟩
· rintro ⟨t, ht₁, a, ht₂, rfl⟩
exact ⟨a, ⟨t, ht₁, ht₂⟩, rfl⟩
|
/-
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 | 1,916 | 1,917 | theorem subset_insert_diff_singleton (x : α) (s : Set α) : s ⊆ insert x (s \ {x}) := by |
rw [← diff_singleton_subset_iff]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 539 | 539 | theorem Icc_self (a : α) : Icc a a = {a} := by | rw [← coe_eq_singleton, coe_Icc, Set.Icc_self]
|
/-
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 | 339 | 339 | theorem ceil_one : ⌈(1 : α)⌉₊ = 1 := by | rw [← Nat.cast_one, ceil_natCast]
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin... | Mathlib/RingTheory/IsAdjoinRoot.lean | 345 | 347 | theorem isAdjoinRootMonic_root_eq_root (hf : Monic f) :
(AdjoinRoot.isAdjoinRootMonic f hf).root = AdjoinRoot.root f := by |
simp only [IsAdjoinRoot.root, AdjoinRoot.root, AdjoinRoot.isAdjoinRootMonic_map_eq_mk]
|
/-
Copyright (c) 2024 Lean FRO LLC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Monoidal.Braided.Opposite
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.Catego... | Mathlib/CategoryTheory/Monoidal/Comon_.lean | 73 | 74 | theorem counit_comul_hom {Z : C} (f : M.X ⟶ Z) : M.comul ≫ (M.counit ⊗ f) = f ≫ (λ_ Z).inv := by |
rw [leftUnitor_inv_naturality, tensorHom_def, counit_comul_assoc]
|
/-
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 | 143 | 144 | theorem Ico_eq_zero_iff : Ico a b = 0 ↔ ¬a < b := by |
rw [Ico, Finset.val_eq_zero, Finset.Ico_eq_empty_iff]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Operations
#align_import algebra.algebra.subalgebra.basic from "leanprover-community/mathlib"@"b915e9392ecb2a861e1e766f0e1df6ac... | Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | 1,199 | 1,200 | theorem _root_.Set.algebraMap_mem_center (r : R) : algebraMap R A r ∈ Set.center A := by |
simp only [Semigroup.mem_center_iff, commutes, forall_const]
|
/-
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.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Su... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 141 | 143 | theorem derivative_zero (n : ℕ) :
Polynomial.derivative (bernsteinPolynomial R n 0) = -n * bernsteinPolynomial R (n - 1) 0 := by |
simp [bernsteinPolynomial, Polynomial.derivative_pow]
|
/-
Copyright (c) 2023 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher
-/
import Mathlib.Topology.MetricSpace.PiNat
#align_import topology.metric_space.cantor_scheme from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb25... | Mathlib/Topology/MetricSpace/CantorScheme.lean | 168 | 194 | theorem ClosureAntitone.map_of_vanishingDiam [CompleteSpace α] (hdiam : VanishingDiam A)
(hanti : ClosureAntitone A) (hnonempty : ∀ l, (A l).Nonempty) : (inducedMap A).1 = univ := by |
rw [eq_univ_iff_forall]
intro x
choose u hu using fun n => hnonempty (res x n)
have umem : ∀ n m : ℕ, n ≤ m → u m ∈ A (res x n) := by
have : Antitone fun n : ℕ => A (res x n) := by
refine antitone_nat_of_succ_le ?_
intro n
apply hanti.antitone
intro n m hnm
exact this hnm (hu _)
... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 65 | 65 | theorem eval₂_zero : (0 : R[X]).eval₂ f x = 0 := by | simp [eval₂_eq_sum]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 377 | 381 | theorem eventually_mem_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set α} {l : Filter β}
(h : Tendsto f l (𝓝[s] a)) : ∀ᶠ i in l, f i ∈ s := by |
simp_rw [nhdsWithin_eq, tendsto_iInf, mem_setOf_eq, tendsto_principal, mem_inter_iff,
eventually_and] at h
exact (h univ ⟨mem_univ a, isOpen_univ⟩).2
|
/-
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.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Ma... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 119 | 121 | theorem rightUnitor_inv_hom {X : Action V G} : Hom.hom (ρ_ X).inv = (ρ_ X.V).inv := by |
dsimp
simp
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7... | Mathlib/Data/Set/Prod.lean | 925 | 931 | theorem pi_subset_pi_iff : pi s t₁ ⊆ pi s t₂ ↔ (∀ i ∈ s, t₁ i ⊆ t₂ i) ∨ pi s t₁ = ∅ := by |
refine
⟨fun h => or_iff_not_imp_right.2 ?_, fun h => h.elim pi_mono fun h' => h'.symm ▸ empty_subset _⟩
rw [← Ne, ← nonempty_iff_ne_empty]
intro hne i hi
simpa only [eval_image_pi hi hne, eval_image_pi hi (hne.mono h)] using
image_subset (fun f : ∀ i, α i => f i) h
|
/-
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.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | Mathlib/Order/Lattice.lean | 743 | 750 | theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by | rw [sup_inf_right, sup_comm x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := by rw [← sup_inf_left]
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.RingTheory.TensorProduct.Basic
#align_import algebra.module.bimodule from "leanprover-community/mathlib"@"58cef51f7a819e7227224461e392dee423302f2d"
/-!
# B... | Mathlib/Algebra/Module/Bimodule.lean | 90 | 92 | theorem smul_mem (p : Submodule (A ⊗[R] B) M) (a : A) {m : M} (hm : m ∈ p) : a • m ∈ p := by |
suffices a • m = a ⊗ₜ[R] (1 : B) • m by exact this.symm ▸ p.smul_mem _ hm
simp [TensorProduct.Algebra.smul_def]
|
/-
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.SuccPred.Basic
import Mathlib.Order.BoundedOrder
#align_import order.succ_pred.limit from "leanprover-community/mathlib"... | Mathlib/Order/SuccPred/Limit.lean | 205 | 213 | theorem _root_.SuccOrder.limitRecOn_succ (ha : ¬ IsMax a) :
SuccOrder.limitRecOn (succ a) H_succ H_lim
= H_succ a ha (SuccOrder.limitRecOn a H_succ H_lim) := by |
have h := not_isSuccLimit_succ_of_not_isMax ha
rw [SuccOrder.limitRecOn, WellFounded.fix_eq, dif_neg h]
have {b c hb hc} {x : ∀ a, C a} (h : b = c) :
congr_arg succ h ▸ H_succ b hb (x b) = H_succ c hc (x c) := by subst h; rfl
let x := Classical.indefiniteDescription _ (not_isSuccLimit_iff.mp h)
exact thi... |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Andrew Yang
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.sh... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | 120 | 122 | theorem PreservesPullback.iso_hom_fst :
(PreservesPullback.iso G f g).hom ≫ pullback.fst = G.map pullback.fst := by |
simp [PreservesPullback.iso]
|
/-
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 | 579 | 583 | theorem Polynomial.Separable.isIntegral {x : K} (h : (minpoly F x).Separable) : IsIntegral F x := by |
cases subsingleton_or_nontrivial F
· haveI := Module.subsingleton F K
exact ⟨1, monic_one, Subsingleton.elim _ _⟩
· exact of_not_not (h.ne_zero <| minpoly.eq_zero ·)
|
/-
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.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,296 | 1,296 | theorem sin_add_pi_div_two (x : ℂ) : sin (x + π / 2) = cos x := by | simp [sin_add]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison
-/
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
/-!
# The `abel` tactic
Evaluate expressions in the language o... | Mathlib/Tactic/Abel.lean | 210 | 211 | theorem zero_smul {α} [AddCommMonoid α] (c) : smul c (0 : α) = 0 := by |
simp [smul, nsmul_zero]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc429200506... | Mathlib/Data/Set/Image.lean | 361 | 363 | theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) :
t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by |
simp [← preimage_compl_eq_image_compl]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.... | Mathlib/Logic/Basic.lean | 1,329 | 1,331 | theorem dite_prop_iff_or {Q : P → Prop} {R : ¬P → Prop} [Decidable P] :
dite P Q R ↔ (∃ p, Q p) ∨ (∃ p, R p) := by |
by_cases h : P <;> simp [h, exists_prop_of_false, exists_prop_of_true]
|
/-
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.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
#align_import logic.equiv.set from "leanprover-... | Mathlib/Logic/Equiv/Set.lean | 163 | 165 | theorem prod_assoc_image {α β γ} {s : Set α} {t : Set β} {u : Set γ} :
Equiv.prodAssoc α β γ '' (s ×ˢ t) ×ˢ u = s ×ˢ t ×ˢ u := by |
simpa only [Equiv.image_eq_preimage] using prod_assoc_symm_preimage
|
/-
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.Algebra.Order.Module.Defs
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.order from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5... | Mathlib/Data/DFinsupp/Order.lean | 205 | 206 | theorem add_eq_zero_iff (f g : Π₀ i, α i) : f + g = 0 ↔ f = 0 ∧ g = 0 := by |
simp [DFunLike.ext_iff, forall_and]
|
/-
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.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingT... | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 66 | 70 | theorem localization_localization_map_units [IsLocalization N T]
(y : localizationLocalizationSubmodule M N) : IsUnit (algebraMap R T y) := by |
obtain ⟨y', z, eq⟩ := mem_localizationLocalizationSubmodule.mp y.prop
rw [IsScalarTower.algebraMap_apply R S T, eq, RingHom.map_mul, IsUnit.mul_iff]
exact ⟨IsLocalization.map_units T y', (IsLocalization.map_units _ z).map (algebraMap S T)⟩
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.GCongr
#align_import order.filter.archimedean fr... | Mathlib/Order/Filter/Archimedean.lean | 193 | 205 | theorem Tendsto.const_mul_atTop' (hr : 0 < r) (hf : Tendsto f l atTop) :
Tendsto (fun x => r * f x) l atTop := by |
refine tendsto_atTop.2 fun b => ?_
obtain ⟨n : ℕ, hn : 1 ≤ n • r⟩ := Archimedean.arch 1 hr
rw [nsmul_eq_mul'] at hn
filter_upwards [tendsto_atTop.1 hf (n * max b 0)] with x hx
calc
b ≤ 1 * max b 0 := by
{ rw [one_mul]
exact le_max_left _ _ }
_ ≤ r * n * max b 0 := by gcongr
_ = r * (n *... |
/-
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 | 446 | 448 | theorem neg_limZero {f : CauSeq β abv} (hf : LimZero f) : LimZero (-f) := by |
rw [← neg_one_mul f]
exact mul_limZero_right _ hf
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"... | Mathlib/Topology/Constructions.lean | 733 | 738 | theorem prod_induced_induced (f : X → Y) (g : Z → W) :
@instTopologicalSpaceProd X Z (induced f ‹_›) (induced g ‹_›) =
induced (fun p => (f p.1, g p.2)) instTopologicalSpaceProd := by |
delta instTopologicalSpaceProd
simp_rw [induced_inf, induced_compose]
rfl
|
/-
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 | 286 | 304 | theorem StrictConvexOn.secant_strict_mono_aux1 (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s)
(hz : z ∈ s) (hxy : x < y) (hyz : y < z) : (z - x) * f y < (z - y) * f x + (y - x) * f z := by |
have hxy' : 0 < y - x := by linarith
have hyz' : 0 < z - y := by linarith
have hxz' : 0 < z - x := by linarith
rw [← lt_div_iff' hxz']
have ha : 0 < (z - y) / (z - x) := by positivity
have hb : 0 < (y - x) / (z - x) := by positivity
calc
f y = f ((z - y) / (z - x) * x + (y - x) / (z - x) * z) := ?_
... |
/-
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 | 525 | 526 | theorem toIocMod_sub (a b : α) : toIocMod hp a (b - p) = toIocMod hp a b := by |
simpa only [one_zsmul] using toIocMod_sub_zsmul hp a b 1
|
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "... | Mathlib/NumberTheory/Padics/RingHoms.lean | 514 | 525 | theorem isCauSeq_nthHom (r : R) : IsCauSeq (padicNorm p) fun n => nthHom f r n := by |
intro ε hε
obtain ⟨k, hk⟩ : ∃ k : ℕ, (p : ℚ) ^ (-((k : ℕ) : ℤ)) < ε := exists_pow_neg_lt_rat p hε
use k
intro j hj
refine lt_of_le_of_lt ?_ hk
-- Need to do beta reduction first, as `norm_cast` doesn't.
-- Added to adapt to leanprover/lean4#2734.
beta_reduce
norm_cast
rw [← padicNorm.dvd_iff_norm_l... |
/-
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, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Rel
import Mathlib... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 318 | 320 | theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → SimpleGraph V} :
(⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by |
rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _), Set.forall_mem_range]
|
/-
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.Data.Fintype.Option
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Fintype.Prod
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Option
... | Mathlib/GroupTheory/Perm/Option.lean | 27 | 34 | theorem Equiv.optionCongr_swap {α : Type*} [DecidableEq α] (x y : α) :
optionCongr (swap x y) = swap (some x) (some y) := by |
ext (_ | i)
· simp [swap_apply_of_ne_of_ne]
· by_cases hx : i = x
· simp only [hx, optionCongr_apply, Option.map_some', swap_apply_left, Option.mem_def,
Option.some.injEq]
by_cases hy : i = y <;> simp [hx, hy, swap_apply_of_ne_of_ne]
|
/-
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.Hom.Basic
import Mathlib.Order.BoundedOrder
#align_import order.hom.bounded from "leanprover-community/mathlib"@"f1a2caaf51ef593799107fe9a8d5e411599... | Mathlib/Order/Hom/Bounded.lean | 146 | 149 | theorem map_eq_top_iff [LE α] [OrderTop α] [PartialOrder β] [OrderTop β] [OrderIsoClass F α β]
(f : F) {a : α} : f a = ⊤ ↔ a = ⊤ := by |
letI : TopHomClass F α β := OrderIsoClass.toTopHomClass
rw [← map_top f, (EquivLike.injective f).eq_iff]
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Fangming Li
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.List.Chain
import Mathlib.Data.List.OfFn
import Mathlib.Data.Rel
import Mathlib.Tactic.Abel
import Mathli... | Mathlib/Order/RelSeries.lean | 191 | 197 | theorem length_ne_zero (irrefl : Irreflexive r) : s.length ≠ 0 ↔ {x | x ∈ s}.Nontrivial := by |
refine ⟨fun h ↦ ⟨s 0, by simp [mem_def], s 1, by simp [mem_def], fun rid ↦ irrefl (s 0) ?_⟩,
length_ne_zero_of_nontrivial⟩
nth_rw 2 [rid]
convert s.step ⟨0, by omega⟩
ext
simpa [Nat.pos_iff_ne_zero]
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 1,069 | 1,072 | theorem epic_of_localizationMap {j k : N →* P} (h : ∀ a, j.comp f.toMap a = k.comp f.toMap a) :
j = k := by |
rw [← f.lift_of_comp j, ← f.lift_of_comp k]
congr 1 with x; exact h x
|
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Data.Part
import Mathlib.Order.Chain
import Mathlib.Order.Hom.Order
import Mathlib.Algebra.Order.Ring.Nat
#align_import o... | Mathlib/Order/OmegaCompletePartialOrder.lean | 355 | 361 | theorem eq_of_chain {c : Chain (Part α)} {a b : α} (ha : some a ∈ c) (hb : some b ∈ c) : a = b := by |
cases' ha with i ha; replace ha := ha.symm
cases' hb with j hb; replace hb := hb.symm
rw [eq_some_iff] at ha hb
rcases le_total i j with hij | hji
· have := c.monotone hij _ ha; apply mem_unique this hb
· have := c.monotone hji _ hb; apply Eq.symm; apply mem_unique this ha
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... | Mathlib/Analysis/Convex/Segment.lean | 561 | 564 | theorem Convex.mem_Icc (h : x ≤ y) :
z ∈ Icc x y ↔ ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ a * x + b * y = z := by |
rw [← segment_eq_Icc h]
rfl
|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Eric Wieser
-/
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Ta... | Mathlib/LinearAlgebra/PiTensorProduct.lean | 391 | 399 | theorem ext {φ₁ φ₂ : (⨂[R] i, s i) →ₗ[R] E}
(H : φ₁.compMultilinearMap (tprod R) = φ₂.compMultilinearMap (tprod R)) : φ₁ = φ₂ := by |
refine LinearMap.ext ?_
refine fun z ↦
PiTensorProduct.induction_on' z ?_ fun {x y} hx hy ↦ by rw [φ₁.map_add, φ₂.map_add, hx, hy]
· intro r f
rw [tprodCoeff_eq_smul_tprod, φ₁.map_smul, φ₂.map_smul]
apply _root_.congr_arg
exact MultilinearMap.congr_fun H f
|
/-
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.CharP.ExpChar
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTh... | Mathlib/RingTheory/Polynomial/Basic.lean | 502 | 506 | theorem coeff_ofSubring (p : T[X]) (n : ℕ) : coeff (ofSubring T p) n = (coeff p n : T) := by |
simp only [ofSubring, coeff_monomial, finset_sum_coeff, mem_support_iff, Finset.sum_ite_eq',
ite_eq_right_iff, Ne, ite_not, Classical.not_not, ite_eq_left_iff]
intro h
rw [h, ZeroMemClass.coe_zero]
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 511 | 520 | theorem integral_prod_smul {𝕜 : Type*} [RCLike 𝕜] [NormedSpace 𝕜 E] (f : α → 𝕜) (g : β → E) :
∫ z, f z.1 • g z.2 ∂μ.prod ν = (∫ x, f x ∂μ) • ∫ y, g y ∂ν := by |
by_cases hE : CompleteSpace E; swap; · simp [integral, hE]
by_cases h : Integrable (fun z : α × β => f z.1 • g z.2) (μ.prod ν)
· rw [integral_prod _ h]
simp_rw [integral_smul, integral_smul_const]
have H : ¬Integrable f μ ∨ ¬Integrable g ν := by
contrapose! h
exact h.1.prod_smul h.2
cases' H with... |
/-
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.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over fro... | Mathlib/RingTheory/Ideal/Over.lean | 77 | 89 | theorem injective_quotient_le_comap_map (P : Ideal R[X]) :
Function.Injective <|
Ideal.quotientMap
(Ideal.map (Polynomial.mapRingHom (Quotient.mk (P.comap (C : R →+* R[X])))) P)
(Polynomial.mapRingHom (Ideal.Quotient.mk (P.comap (C : R →+* R[X]))))
le_comap_map := by |
refine quotientMap_injective' (le_of_eq ?_)
rw [comap_map_of_surjective (mapRingHom (Ideal.Quotient.mk (P.comap (C : R →+* R[X]))))
(map_surjective (Ideal.Quotient.mk (P.comap (C : R →+* R[X]))) Ideal.Quotient.mk_surjective)]
refine le_antisymm (sup_le le_rfl ?_) (le_sup_of_le_left le_rfl)
refine fun p h... |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Pointwise.Finite
import Mathlib.Data.Set.Pointwise.SMul
... | Mathlib/Data/Finset/Pointwise.lean | 1,339 | 1,341 | theorem preimage_mul_right_singleton :
preimage {b} (· * a) (mul_left_injective _).injOn = {b * a⁻¹} := by |
classical rw [← image_mul_right', image_singleton]
|
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib... | Mathlib/Algebra/Polynomial/RingDivision.lean | 512 | 520 | theorem rootMultiplicity_mul_X_sub_C_pow {p : R[X]} {a : R} {n : ℕ} (h : p ≠ 0) :
(p * (X - C a) ^ n).rootMultiplicity a = p.rootMultiplicity a + n := by |
have h2 := monic_X_sub_C a |>.pow n |>.mul_left_ne_zero h
refine le_antisymm ?_ ?_
· rw [rootMultiplicity_le_iff h2, add_assoc, add_comm n, ← add_assoc, pow_add,
dvd_cancel_right_mem_nonZeroDivisors (monic_X_sub_C a |>.pow n |>.mem_nonZeroDivisors)]
exact pow_rootMultiplicity_not_dvd h a
· rw [le_roo... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 556 | 557 | theorem natDegree_intCast (n : ℤ) : natDegree (n : R[X]) = 0 := by |
rw [← C_eq_intCast, natDegree_C]
|
/-
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.Topology.MetricSpace.PiNat
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Gluing
import Mathlib.Topology.Sets.O... | Mathlib/Topology/MetricSpace/Polish.lean | 91 | 94 | theorem complete_polishSpaceMetric (α : Type*) [ht : TopologicalSpace α] [h : PolishSpace α] :
@CompleteSpace α (polishSpaceMetric α).toUniformSpace := by |
convert h.complete.choose_spec.2
exact MetricSpace.replaceTopology_eq _ _
|
/-
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.Data.Finset.Pointwise
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.DFinsupp.Order
import Mathlib.Order.Interval.Finset.Basic
#align_import... | Mathlib/Data/DFinsupp/Interval.lean | 189 | 190 | theorem card_Ioo : (Ioo f g).card = (∏ i ∈ f.support ∪ g.support, (Icc (f i) (g i)).card) - 2 := by |
rw [card_Ioo_eq_card_Icc_sub_two, card_Icc]
|
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanpr... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 150 | 154 | theorem Path.reverse_comp [HasReverse V] {a b c : V} (p : Path a b) (q : Path b c) :
(p.comp q).reverse = q.reverse.comp p.reverse := by |
induction' q with _ _ _ _ h
· simp
· simp [h]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Data.PFun
#align_import order.filter.partial from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56... | Mathlib/Order/Filter/Partial.lean | 278 | 283 | theorem ptendsto'_of_ptendsto {f : α →. β} {l₁ : Filter α} {l₂ : Filter β} (h : f.Dom ∈ l₁) :
PTendsto f l₁ l₂ → PTendsto' f l₁ l₂ := by |
rw [ptendsto_def, ptendsto'_def]
intro h' s sl₂
rw [PFun.preimage_eq]
exact inter_mem (h' s sl₂) h
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... | Mathlib/Analysis/Convex/Segment.lean | 617 | 619 | theorem segment_subset (x y : E × F) : segment 𝕜 x y ⊆ segment 𝕜 x.1 y.1 ×ˢ segment 𝕜 x.2 y.2 := by |
rintro z ⟨a, b, ha, hb, hab, hz⟩
exact ⟨⟨a, b, ha, hb, hab, congr_arg Prod.fst hz⟩, a, b, ha, hb, hab, congr_arg Prod.snd hz⟩
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import Mathlib.Data.Matrix.Basic
#align_import data.matrix.block from "leanprover-community/mathlib"@"c060baa79af5ca... | Mathlib/Data/Matrix/Block.lean | 247 | 254 | theorem fromBlocks_multiply [Fintype l] [Fintype m] [NonUnitalNonAssocSemiring α] (A : Matrix n l α)
(B : Matrix n m α) (C : Matrix o l α) (D : Matrix o m α) (A' : Matrix l p α) (B' : Matrix l q α)
(C' : Matrix m p α) (D' : Matrix m q α) :
fromBlocks A B C D * fromBlocks A' B' C' D' =
fromBlocks (A * ... |
ext i j
rcases i with ⟨⟩ <;> rcases j with ⟨⟩ <;> simp only [fromBlocks, mul_apply, of_apply,
Sum.elim_inr, Fintype.sum_sum_type, Sum.elim_inl, add_apply]
|
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 447 | 451 | theorem HasStrictDerivAt.clm_comp (hc : HasStrictDerivAt c c' x) (hd : HasStrictDerivAt d d' x) :
HasStrictDerivAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') x := by |
have := (hc.hasStrictFDerivAt.clm_comp hd.hasStrictFDerivAt).hasStrictDerivAt
rwa [add_apply, comp_apply, comp_apply, smulRight_apply, smulRight_apply, one_apply, one_smul,
one_smul, add_comm] at this
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalcu... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 567 | 578 | theorem integrableOn_Iic_of_intervalIntegral_norm_bounded (I b : ℝ)
(hfi : ∀ i, IntegrableOn f (Ioc (a i) b) μ) (ha : Tendsto a l atBot)
(h : ∀ᶠ i in l, (∫ x in a i..b, ‖f x‖ ∂μ) ≤ I) : IntegrableOn f (Iic b) μ := by |
have hφ : AECover (μ.restrict <| Iic b) l _ := aecover_Ioi ha
have hfi : ∀ i, IntegrableOn f (Ioi (a i)) (μ.restrict <| Iic b) := by
intro i
rw [IntegrableOn, Measure.restrict_restrict (hφ.measurableSet i)]
exact hfi i
refine hφ.integrable_of_integral_norm_bounded I hfi (h.mp ?_)
filter_upwards [ha... |
/-
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.Logic.Equiv.List
import Mathlib.Logic.Function.Iterate
#align_import computability.primrec from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3b... | Mathlib/Computability/Primrec.lean | 804 | 808 | theorem of_graph {f : α → ℕ} (h₁ : PrimrecBounded f)
(h₂ : PrimrecRel fun a b => f a = b) : Primrec f := by |
rcases h₁ with ⟨g, pg, hg : ∀ x, f x ≤ g x⟩
refine (nat_findGreatest pg h₂).of_eq fun n => ?_
exact (Nat.findGreatest_spec (P := fun b => f n = b) (hg n) rfl).symm
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathli... | Mathlib/Probability/Martingale/Basic.lean | 519 | 524 | theorem Martingale.eq_zero_of_predictable [SigmaFiniteFiltration μ 𝒢] {f : ℕ → Ω → E}
(hfmgle : Martingale f 𝒢 μ) (hfadp : Adapted 𝒢 fun n => f (n + 1)) (n : ℕ) : f n =ᵐ[μ] f 0 := by |
induction' n with k ih
· rfl
· exact ((Germ.coe_eq.mp (congr_arg Germ.ofFun <| condexp_of_stronglyMeasurable (𝒢.le _) (hfadp _)
(hfmgle.integrable _))).symm.trans (hfmgle.2 k (k + 1) k.le_succ)).trans ih
|
/-
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.MFDeriv.FDeriv
/-!
# Differentiability of specific functions
In this file, we establish differentiability r... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 357 | 360 | theorem tangentMap_prod_snd {p : TangentBundle (I.prod I') (M × M')} :
tangentMap (I.prod I') I' Prod.snd p = ⟨p.proj.2, p.2.2⟩ := by |
-- Porting note: `rfl` wasn't needed
simp [tangentMap]; rfl
|
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm
#align_import n... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 122 | 125 | theorem isIntegralClosure_adjoin_singleton_of_prime [hcycl : IsCyclotomicExtension {p} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑p) : IsIntegralClosure (adjoin ℤ ({ζ} : Set K)) ℤ K := by |
rw [← pow_one p] at hζ hcycl
exact isIntegralClosure_adjoin_singleton_of_prime_pow hζ
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
Coinductive formalization of unbounded computations.
-/
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "le... | Mathlib/Data/Seq/Computation.lean | 715 | 726 | theorem ret_bind (a) (f : α → Computation β) : bind (pure a) f = f a := by |
apply
eq_of_bisim fun c₁ c₂ => c₁ = bind (pure a) f ∧ c₂ = f a ∨ c₁ = corec (Bind.f f) (Sum.inr c₂)
· intro c₁ c₂ h
match c₁, c₂, h with
| _, _, Or.inl ⟨rfl, rfl⟩ =>
simp only [BisimO, bind, Bind.f, corec_eq, rmap, destruct_pure]
cases' destruct (f a) with b cb <;> simp [Bind.g]
| _, c,... |
/-
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.Polynomial.AlgebraMap
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 39 | 41 | theorem algebraMap_eval_T (x : R) (n : ℤ) :
algebraMap R A ((T R n).eval x) = (T A n).eval (algebraMap R A x) := by |
rw [← aeval_algebraMap_apply_eq_algebraMap_eval, aeval_T]
|
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
... | Mathlib/NumberTheory/Pell.lean | 710 | 733 | theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
∃! a₁ : Solution₁ d,
1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by |
obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
refine ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => ?_⟩
obtain ⟨Hx, Hy, H⟩ := H
obtain ⟨n₁, hn₁⟩ := H a₁
obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a
rcases hn₂ with (rfl | rfl)
· rw [← zpow_mul, eq_comm, @eq_comm... |
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.Localization.AsSubring
#align_import algebraic_geometry.prime_spec... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Maximal.lean | 65 | 69 | theorem toPrimeSpectrum_range :
Set.range (@toPrimeSpectrum R _) = { x | IsClosed ({x} : Set <| PrimeSpectrum R) } := by |
simp only [isClosed_singleton_iff_isMaximal]
ext ⟨x, _⟩
exact ⟨fun ⟨y, hy⟩ => hy ▸ y.IsMaximal, fun hx => ⟨⟨x, hx⟩, rfl⟩⟩
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 232 | 232 | theorem inner_im_symm (x y : F) : im ⟪x, y⟫ = -im ⟪y, x⟫ := by | rw [← inner_conj_symm, conj_im]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | Mathlib/Order/BooleanAlgebra.lean | 486 | 490 | theorem sup_lt_of_lt_sdiff_left (h : y < z \ x) (hxz : x ≤ z) : x ⊔ y < z := by |
rw [← sup_sdiff_cancel_right hxz]
refine (sup_le_sup_left h.le _).lt_of_not_le fun h' => h.not_le ?_
rw [← sdiff_idem]
exact (sdiff_le_sdiff_of_sup_le_sup_left h').trans sdiff_le
|
/-
Copyright (c) 2021 Ivan Sadofschi Costa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ivan Sadofschi Costa
-/
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.fin from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
/-!
# `co... | Mathlib/Data/Finsupp/Fin.lean | 60 | 64 | theorem cons_tail : cons (t 0) (tail t) = t := by |
ext a
by_cases c_a : a = 0
· rw [c_a, cons_zero]
· rw [← Fin.succ_pred a c_a, cons_succ, ← tail_apply]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Yury Kudryashov
-/
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.add_torsor from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
... | Mathlib/Algebra/AddTorsor.lean | 270 | 271 | theorem vsub_sub_vsub_comm (p₁ p₂ p₃ p₄ : P) : p₁ -ᵥ p₂ - (p₃ -ᵥ p₄) = p₁ -ᵥ p₃ - (p₂ -ᵥ p₄) := by |
rw [← vsub_vadd_eq_vsub_sub, vsub_vadd_comm, vsub_vadd_eq_vsub_sub]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Gr... | Mathlib/Algebra/Associated.lean | 1,159 | 1,168 | theorem dvdNotUnit_of_lt {a b : Associates α} (hlt : a < b) : DvdNotUnit a b := by |
constructor;
· rintro rfl
apply not_lt_of_le _ hlt
apply dvd_zero
rcases hlt with ⟨⟨x, rfl⟩, ndvd⟩
refine ⟨x, ?_, rfl⟩
contrapose! ndvd
rcases ndvd with ⟨u, rfl⟩
simp
|
/-
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 | 601 | 604 | theorem pushforward_le_bind_of_mem (S : Presieve X) (R : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S f → Sieve Y)
(f : Y ⟶ X) (h : S f) : (R h).pushforward f ≤ bind S R := by |
rintro Z _ ⟨g, rfl, hg⟩
exact ⟨_, g, f, h, hg, rfl⟩
|
/-
Copyright (c) 2021 Rémy Degenne. 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.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,037 | 1,042 | theorem setToL1_zero_left (hT : DominatedFinMeasAdditive μ (0 : Set α → E →L[ℝ] F) C)
(f : α →₁[μ] E) : setToL1 hT f = 0 := by |
suffices setToL1 hT = 0 by rw [this]; simp
refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT) _ _ _ _ ?_
ext1 f
rw [setToL1SCLM_zero_left hT f, ContinuousLinearMap.zero_comp, ContinuousLinearMap.zero_apply]
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.Matrix.AbsoluteValue
import Mathlib.NumberTheory.... | Mathlib/NumberTheory/ClassNumber/Finite.lean | 273 | 277 | theorem prod_finsetApprox_ne_zero : algebraMap R S (∏ m ∈ finsetApprox bS adm, m) ≠ 0 := by |
refine mt ((injective_iff_map_eq_zero _).mp bS.algebraMap_injective _) ?_
simp only [Finset.prod_eq_zero_iff, not_exists]
rintro x ⟨hx, rfl⟩
exact finsetApprox.zero_not_mem bS adm hx
|
/-
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.CharP.ExpChar
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTh... | Mathlib/RingTheory/Polynomial/Basic.lean | 213 | 220 | theorem exists_degree_le_of_mem_span_of_finite {s : Set R[X]} (s_fin : s.Finite) (hs : s.Nonempty) :
∃ p' ∈ s, ∀ (p : R[X]), p ∈ Submodule.span R s → degree p ≤ degree p' := by |
rcases Set.Finite.exists_maximal_wrt degree s s_fin hs with ⟨a, has, hmax⟩
refine ⟨a, has, fun p hp => ?_⟩
rcases exists_degree_le_of_mem_span hs hp with ⟨p', hp'⟩
by_cases h : degree a ≤ degree p'
· rw [← hmax p' hp'.left h] at hp'; exact hp'.right
· exact le_trans hp'.right (not_le.mp h).le
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 439 | 457 | theorem eval_monomial_one_add_sub [CommRing S] (d : ℕ) (y : S) :
eval (1 + y) (monomial d (d + 1 : S)) - eval y (monomial d (d + 1 : S)) =
∑ x_1 ∈ range (d + 1), ↑((d + 1).choose x_1) * (↑x_1 * y ^ (x_1 - 1)) := by |
have cast_succ : (d + 1 : S) = ((d.succ : ℕ) : S) := by simp only [Nat.cast_succ]
rw [cast_succ, eval_monomial, eval_monomial, add_comm, add_pow]
-- Porting note: `apply_congr` hadn't been ported yet, so `congr` & `ext` is used.
conv_lhs =>
congr
· congr
· skip
· congr
· skip
... |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 856 | 859 | theorem dotProduct_diagonal' (i : m) : (v ⬝ᵥ fun j => diagonal w j i) = v i * w i := by |
have : ∀ j ≠ i, v j * diagonal w j i = 0 := fun j hij => by
simp [diagonal_apply_ne _ hij]
convert Finset.sum_eq_single i (fun j _ => this j) _ using 1 <;> simp
|
/-
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 | 2,154 | 2,154 | theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by | rw [sInf_eq_iInf, Iic_iInf₂]
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Analysis.Complex.Polynomial
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingT... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 73 | 77 | theorem range_eval_eq_rootSet_minpoly :
(range fun φ : K →+* A => φ x) = (minpoly ℚ x).rootSet A := by |
convert (NumberField.isAlgebraic K).range_eval_eq_rootSet_minpoly A x using 1
ext a
exact ⟨fun ⟨φ, hφ⟩ => ⟨φ.toRatAlgHom, hφ⟩, fun ⟨φ, hφ⟩ => ⟨φ.toRingHom, hφ⟩⟩
|
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheor... | Mathlib/CategoryTheory/Generator.lean | 416 | 417 | theorem isSeparator_unop_iff (G : Cᵒᵖ) : IsSeparator (unop G) ↔ IsCoseparator G := by |
rw [IsSeparator, IsCoseparator, ← isSeparating_unop_iff, Set.singleton_unop]
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216... | Mathlib/Data/Bool/Basic.lean | 265 | 266 | theorem ofNat_toNat (b : Bool) : ofNat (toNat b) = b := by |
cases b <;> rfl
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,960 | 1,961 | theorem abs_le_floor_nnreal_iff (z : ℤ) (c : ℝ≥0) : |z| ≤ ⌊c⌋₊ ↔ ‖z‖₊ ≤ c := by |
rw [Int.abs_eq_natAbs, Int.ofNat_le, Nat.le_floor_iff (zero_le c), NNReal.natCast_natAbs z]
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-... | Mathlib/Order/LiminfLimsup.lean | 1,210 | 1,217 | theorem cofinite.blimsup_set_eq :
blimsup s cofinite p = { x | { n | p n ∧ x ∈ s n }.Infinite } := by |
simp only [blimsup_eq, le_eq_subset, eventually_cofinite, not_forall, sInf_eq_sInter, exists_prop]
ext x
refine ⟨fun h => ?_, fun hx t h => ?_⟩ <;> contrapose! h
· simp only [mem_sInter, mem_setOf_eq, not_forall, exists_prop]
exact ⟨{x}ᶜ, by simpa using h, by simp⟩
· exact hx.mono fun i hi => ⟨hi.1, fun ... |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra... | Mathlib/Algebra/BigOperators/Finprod.lean | 171 | 176 | theorem finprod_eq_prod_plift_of_mulSupport_toFinset_subset {f : α → M}
(hf : (mulSupport (f ∘ PLift.down)).Finite) {s : Finset (PLift α)} (hs : hf.toFinset ⊆ s) :
∏ᶠ i, f i = ∏ i ∈ s, f i.down := by |
rw [finprod, dif_pos]
refine Finset.prod_subset hs fun x _ hxf => ?_
rwa [hf.mem_toFinset, nmem_mulSupport] at hxf
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.HashMap.Basic
import Batteries.Data.Array.Lemmas
import Batteries.Data.Nat.Lemmas
namespace Batteries.HashMap
namespace Imp
attribute [-simp] ... | .lake/packages/batteries/Batteries/Data/HashMap/WF.lean | 280 | 286 | theorem modify_size [BEq α] [Hashable α] {m : Imp α β} {k}
(h : m.size = m.buckets.size) :
(modify m k f).size = (modify m k f).buckets.size := by |
dsimp [modify, cond]; rw [Buckets.update_update]
simp only [h, Buckets.size]
refine have ⟨_, _, h₁, _, eq⟩ := Buckets.exists_of_update ..; eq ▸ ?_
simp [h, h₁, Buckets.size_eq]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Finset.Update
import Mathlib.Data.Prod.TProd
import Mathlib.GroupTheory.Coset
import Mathlib.Logic.Equiv.Fin
import Mathlib.Measur... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 917 | 936 | theorem exists_measurable_piecewise {ι} [Countable ι] [Nonempty ι] (t : ι → Set α)
(t_meas : ∀ n, MeasurableSet (t n)) (g : ι → α → β) (hg : ∀ n, Measurable (g n))
(ht : Pairwise fun i j => EqOn (g i) (g j) (t i ∩ t j)) :
∃ f : α → β, Measurable f ∧ ∀ n, EqOn f (g n) (t n) := by |
inhabit ι
set g' : (i : ι) → t i → β := fun i => g i ∘ (↑)
-- see #2184
have ht' : ∀ (i j) (x : α) (hxi : x ∈ t i) (hxj : x ∈ t j), g' i ⟨x, hxi⟩ = g' j ⟨x, hxj⟩ := by
intro i j x hxi hxj
rcases eq_or_ne i j with rfl | hij
· rfl
· exact ht hij ⟨hxi, hxj⟩
set f : (⋃ i, t i) → β := iUnionLift t... |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
impo... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 115 | 127 | theorem succ_nth_stream_b_le_nth_stream_fr_inv {ifp_n ifp_succ_n : IntFractPair K}
(nth_stream_eq : IntFractPair.stream v n = some ifp_n)
(succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n) :
(ifp_succ_n.b : K) ≤ ifp_n.fr⁻¹ := by |
suffices (⌊ifp_n.fr⁻¹⌋ : K) ≤ ifp_n.fr⁻¹ by
cases' ifp_n with _ ifp_n_fr
have : ifp_n_fr ≠ 0 := by
intro h
simp [h, IntFractPair.stream, nth_stream_eq] at succ_nth_stream_eq
have : IntFractPair.of ifp_n_fr⁻¹ = ifp_succ_n := by
simpa [this, IntFractPair.stream, nth_stream_eq, Option.coe_... |
/-
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 | 718 | 726 | theorem Perm.permutations' {s t : List α} (p : s ~ t) : permutations' s ~ permutations' t := by |
induction' p with a s t _ IH a b l s t u _ _ IH₁ IH₂; · simp
· exact IH.bind_right _
· dsimp
rw [bind_assoc, bind_assoc]
apply Perm.bind_left
intro l' _
apply perm_permutations'Aux_comm
· exact IH₁.trans IH₂
|
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