Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | goals listlengths 0 224 | goals_before listlengths 0 220 |
|---|---|---|---|---|---|---|---|
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Data.Rat.Floor
#align_import algebra.continued_fractions.computation.terminates_iff_rat from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b3... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 278 | 292 | theorem stream_succ_nth_fr_num_lt_nth_fr_num_rat {ifp_n ifp_succ_n : IntFractPair ℚ}
(stream_nth_eq : IntFractPair.stream q n = some ifp_n)
(stream_succ_nth_eq : IntFractPair.stream q (n + 1) = some ifp_succ_n) :
ifp_succ_n.fr.num < ifp_n.fr.num := by |
obtain ⟨ifp_n', stream_nth_eq', ifp_n_fract_ne_zero, IntFractPair.of_eq_ifp_succ_n⟩ :
∃ ifp_n',
IntFractPair.stream q n = some ifp_n' ∧
ifp_n'.fr ≠ 0 ∧ IntFractPair.of ifp_n'.fr⁻¹ = ifp_succ_n :=
succ_nth_stream_eq_some_iff.mp stream_succ_nth_eq
have : ifp_n = ifp_n' := by injection Eq.trans ... | [
" ifp_succ_n.fr.num < ifp_n.fr.num",
" ifp_n = ifp_n'",
" (IntFractPair.of ifp_n.fr⁻¹).fr.num < ifp_n.fr.num"
] | [] |
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "leanprover-community/mathlib"@"9b2660e1b25419042c8da10bf411aa3c67f14383"
open Nat
def ack : ℕ → ℕ → ℕ
| 0, n => n + 1
| m + 1, 0 ... | Mathlib/Computability/Ackermann.lean | 70 | 70 | theorem ack_zero (n : ℕ) : ack 0 n = n + 1 := by | rw [ack]
| [
" ack 0 n = n + 1"
] | [] |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S ℕ]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 79 | 80 | theorem smeval_zero : (0 : R[X]).smeval x = 0 := by |
simp only [smeval_eq_sum, smul_pow, sum_zero_index]
| [
" p.smeval x = p.sum (smul_pow x)",
" (C r).smeval x = r • x ^ 0",
" ((monomial n) r).smeval x = r • x ^ n",
" eval r p = p.smeval r",
" (p.sum fun e a => a * r ^ e) = p.sum (smul_pow r)",
" eval₂ f x p = p.smeval x",
" (p.sum fun e a => f a * x ^ e) = p.sum (smul_pow x)",
" smeval 0 x = 0"
] | [
" p.smeval x = p.sum (smul_pow x)",
" (C r).smeval x = r • x ^ 0",
" ((monomial n) r).smeval x = r • x ^ n",
" eval r p = p.smeval r",
" (p.sum fun e a => a * r ^ e) = p.sum (smul_pow r)",
" eval₂ f x p = p.smeval x",
" (p.sum fun e a => f a * x ^ e) = p.sum (smul_pow x)"
] |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Lang... | Mathlib/ModelTheory/Definability.lean | 86 | 88 | theorem Definable.mono (hAs : A.Definable L s) (hAB : A ⊆ B) : B.Definable L s := by |
rw [definable_iff_empty_definable_with_params] at *
exact hAs.map_expansion (L.lhomWithConstantsMap (Set.inclusion hAB))
| [
" A.Definable L' s",
" A.Definable L' (setOf ψ.Realize)",
" setOf ψ.Realize = setOf ((LHom.addConstants (↑A) φ).onFormula ψ).Realize",
" x ∈ setOf ψ.Realize ↔ x ∈ setOf ((LHom.addConstants (↑A) φ).onFormula ψ).Realize",
" A.Definable L s ↔ ∃ φ, s = {v | φ.Realize (Sum.elim Subtype.val v)}",
" (∃ b, s = se... | [
" A.Definable L' s",
" A.Definable L' (setOf ψ.Realize)",
" setOf ψ.Realize = setOf ((LHom.addConstants (↑A) φ).onFormula ψ).Realize",
" x ∈ setOf ψ.Realize ↔ x ∈ setOf ((LHom.addConstants (↑A) φ).onFormula ψ).Realize",
" A.Definable L s ↔ ∃ φ, s = {v | φ.Realize (Sum.elim Subtype.val v)}",
" (∃ b, s = se... |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 178 | 179 | theorem vanishingIdeal_singleton (x : PrimeSpectrum R) :
vanishingIdeal ({x} : Set (PrimeSpectrum R)) = x.asIdeal := by | simp [vanishingIdeal]
| [
" Function.Bijective (primeSpectrumProdOfSum R S)",
" Function.Injective (primeSpectrumProdOfSum R S)",
" Sum.inl { asIdeal := I, IsPrime := hI } = Sum.inl { asIdeal := I', IsPrime := hI' }",
" Sum.inl { asIdeal := I, IsPrime := hI } = Sum.inr { asIdeal := J', IsPrime := hJ' }",
" Sum.inr { asIdeal := J, Is... | [
" Function.Bijective (primeSpectrumProdOfSum R S)",
" Function.Injective (primeSpectrumProdOfSum R S)",
" Sum.inl { asIdeal := I, IsPrime := hI } = Sum.inl { asIdeal := I', IsPrime := hI' }",
" Sum.inl { asIdeal := I, IsPrime := hI } = Sum.inr { asIdeal := J', IsPrime := hJ' }",
" Sum.inr { asIdeal := J, Is... |
import Mathlib.Data.List.Basic
open Function
open Nat hiding one_pos
assert_not_exists Set.range
namespace List
universe u v w
variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w} {l₁ l₂ : List α}
section InsertNth
variable {a : α}
@[simp]
theorem insertNth_zero (s : List α) (x : α) : insertNth 0 x s... | Mathlib/Data/List/InsertNth.lean | 103 | 112 | theorem insertNth_of_length_lt (l : List α) (x : α) (n : ℕ) (h : l.length < n) :
insertNth n x l = l := by |
induction' l with hd tl IH generalizing n
· cases n
· simp at h
· simp
· cases n
· simp at h
· simp only [Nat.succ_lt_succ_iff, length] at h
simpa using IH _ h
| [
" (insertNth n a l).eraseIdx n = l",
" modifyNthTail (tail ∘ cons a) n l = l",
" insertNth 0 a ((head✝ :: as).eraseIdx 0) = (insertNth (0 + 1) a (head✝ :: as)).eraseIdx 0",
" 0 ≤ j → j ≤ l.length → insertNth (j + 1) b (insertNth 0 a l) = insertNth 0 a (insertNth j b l)",
" i + 1 ≤ j + 1 →\n j + 1 ≤ [].le... | [
" (insertNth n a l).eraseIdx n = l",
" modifyNthTail (tail ∘ cons a) n l = l",
" insertNth 0 a ((head✝ :: as).eraseIdx 0) = (insertNth (0 + 1) a (head✝ :: as)).eraseIdx 0",
" 0 ≤ j → j ≤ l.length → insertNth (j + 1) b (insertNth 0 a l) = insertNth 0 a (insertNth j b l)",
" i + 1 ≤ j + 1 →\n j + 1 ≤ [].le... |
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Data.Rat.Floor
#align_import algebra.continued_fractions.computation.terminates_iff_rat from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b3... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 197 | 200 | theorem coe_stream'_rat_eq :
((IntFractPair.stream q).map (Option.map (mapFr (↑))) : Stream' <| Option <| IntFractPair K) =
IntFractPair.stream v := by |
funext n; exact IntFractPair.coe_stream_nth_rat_eq v_eq_q n
| [
" mapFr Rat.cast (IntFractPair.of q) = IntFractPair.of v",
" Option.map (mapFr Rat.cast) (IntFractPair.stream q n) = IntFractPair.stream v n",
" Option.map (mapFr Rat.cast) (IntFractPair.stream q 0) = IntFractPair.stream v 0",
" Option.map (mapFr Rat.cast) (IntFractPair.stream q (n + 1)) = IntFractPair.stream... | [
" mapFr Rat.cast (IntFractPair.of q) = IntFractPair.of v",
" Option.map (mapFr Rat.cast) (IntFractPair.stream q n) = IntFractPair.stream v n",
" Option.map (mapFr Rat.cast) (IntFractPair.stream q 0) = IntFractPair.stream v 0",
" Option.map (mapFr Rat.cast) (IntFractPair.stream q (n + 1)) = IntFractPair.stream... |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.fin_range from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
universe u
namespace List
variable {α : Type u}
@[simp]
theorem map_coe_finRange (n : ℕ) : ((finRange n) : List (Fin n)).map (Fin.val) = ... | Mathlib/Data/List/FinRange.lean | 64 | 72 | theorem nodup_ofFn {n} {f : Fin n → α} : Nodup (ofFn f) ↔ Function.Injective f := by |
refine ⟨?_, nodup_ofFn_ofInjective⟩
refine Fin.consInduction ?_ (fun x₀ xs ih => ?_) f
· intro _
exact Function.injective_of_subsingleton _
· intro h
rw [Fin.cons_injective_iff]
simp_rw [ofFn_succ, Fin.cons_succ, nodup_cons, Fin.cons_zero, mem_ofFn] at h
exact h.imp_right ih
| [
" map Fin.val (finRange n) = range n",
" map (fun a => a) (range n) = range n",
" finRange n.succ = 0 :: map Fin.succ (finRange n)",
" map Fin.val (finRange n.succ) = map Fin.val (0 :: map Fin.succ (finRange n))",
" 0 :: map (Nat.succ ∘ Fin.val) (finRange n) = 0 :: map (Fin.val ∘ Fin.succ) (finRange n)",
... | [
" map Fin.val (finRange n) = range n",
" map (fun a => a) (range n) = range n",
" finRange n.succ = 0 :: map Fin.succ (finRange n)",
" map Fin.val (finRange n.succ) = map Fin.val (0 :: map Fin.succ (finRange n))",
" 0 :: map (Nat.succ ∘ Fin.val) (finRange n) = 0 :: map (Fin.val ∘ Fin.succ) (finRange n)",
... |
import Mathlib.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
#align_import probability.independence.zero_one from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
open MeasureTheory MeasurableSpace
open scoped MeasureTheory ENNReal
namespace Probabili... | Mathlib/Probability/Independence/ZeroOne.lean | 109 | 115 | theorem kernel.indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
(hf : ∀ t, p t → tᶜ ∈ f) {t : Set ι} (ht : p t) :
Indep (⨆ n ∈ t, s n) (limsup s f) κ μα := by |
refine indep_of_indep_of_le_right (indep_biSup_compl h_le h_indep t) ?_
refine limsSup_le_of_le (by isBoundedDefault) ?_
simp only [Set.mem_compl_iff, eventually_map]
exact eventually_of_mem (hf t ht) le_iSup₂
| [
" ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤",
" (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤",
" μ t = 0 ∨ μ t = 1 ∨ μ t = ⊤",
" ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1",
" (κ a) t = 0 ∨ (κ a) t = 1",
" μ t = 0 ∨ μ t = 1",
" ∀ᵐ (ω : Ω) ∂μ, (μ[t.indicator fun ω => 1|m]) ω = 0 ∨ (μ[t.indicator ... | [
" ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤",
" (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤",
" μ t = 0 ∨ μ t = 1 ∨ μ t = ⊤",
" ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1",
" (κ a) t = 0 ∨ (κ a) t = 1",
" μ t = 0 ∨ μ t = 1",
" ∀ᵐ (ω : Ω) ∂μ, (μ[t.indicator fun ω => 1|m]) ω = 0 ∨ (μ[t.indicator ... |
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Nat
--@[pp_nodot] porting note: unknown attribute
def log (b : ℕ) : ℕ → ℕ
| n => i... | Mathlib/Data/Nat/Log.lean | 42 | 44 | theorem log_eq_zero_iff {b n : ℕ} : log b n = 0 ↔ n < b ∨ b ≤ 1 := by |
rw [log, dite_eq_right_iff]
simp only [Nat.add_eq_zero_iff, Nat.one_ne_zero, and_false, imp_false, not_and_or, not_le, not_lt]
| [
" (invImage (fun x => x) instWellFoundedRelationOfSizeOf).1 (n / b) a✝",
" b.log n = 0 ↔ n < b ∨ b ≤ 1",
" (∀ (h : b ≤ n ∧ 1 < b), b.log (n / b) + 1 = 0) ↔ n < b ∨ b ≤ 1"
] | [
" (invImage (fun x => x) instWellFoundedRelationOfSizeOf).1 (n / b) a✝"
] |
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 123 | 128 | theorem dvd_div_of_mul_dvd {a b c : R} (h : a * b ∣ c) : b ∣ c / a := by |
rcases eq_or_ne a 0 with (rfl | ha)
· simp only [div_zero, dvd_zero]
rcases h with ⟨d, rfl⟩
refine ⟨d, ?_⟩
rw [mul_assoc, mul_div_cancel_left₀ _ ha]
| [
" a * b / b = a",
" a - a * b / b = 0",
" False",
" b ∣ a",
" b ∣ b * (a / b)",
" a % b = 0",
" b * c = b * (b * c / b)",
" c ∣ a % b ↔ c ∣ a",
" 0 / a = 0",
" a / a = 1",
" a = c / b",
" b = c / a",
" x * y / z = x * (y / z)",
" x * y / 0 = x * (y / 0)",
" x * (z * p) / z = x * (z * p /... | [
" a * b / b = a",
" a - a * b / b = 0",
" False",
" b ∣ a",
" b ∣ b * (a / b)",
" a % b = 0",
" b * c = b * (b * c / b)",
" c ∣ a % b ↔ c ∣ a",
" 0 / a = 0",
" a / a = 1",
" a = c / b",
" b = c / a",
" x * y / z = x * (y / z)",
" x * y / 0 = x * (y / 0)",
" x * (z * p) / z = x * (z * p /... |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.Calculus.BumpFunction.Normed
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
import Mathlib.MeasureTheory.Measure.Haar.Unique
#align_import analy... | Mathlib/Analysis/Calculus/BumpFunction/Convolution.lean | 65 | 68 | theorem normed_convolution_eq_right {x₀ : G} (hg : ∀ x ∈ ball x₀ φ.rOut, g x = g x₀) :
(φ.normed μ ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀ = g x₀ := by |
rw [convolution_eq_right' _ φ.support_normed_eq.subset hg]
exact integral_normed_smul φ μ (g x₀)
| [
" (↑φ ⋆[lsmul ℝ ℝ, μ] g) x₀ = integral μ ↑φ • g x₀",
" (φ.normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀ = g x₀",
" ∫ (t : G), ((lsmul ℝ ℝ) (φ.normed μ t)) (g x₀) ∂μ = g x₀"
] | [
" (↑φ ⋆[lsmul ℝ ℝ, μ] g) x₀ = integral μ ↑φ • g x₀"
] |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 106 | 108 | theorem HasDerivAt.complexToReal_fderiv' {f : ℂ → E} {x : ℂ} {f' : E} (h : HasDerivAt f f' x) :
HasFDerivAt f (reCLM.smulRight f' + I • imCLM.smulRight f') x := by |
simpa only [Complex.restrictScalars_one_smulRight'] using h.hasFDerivAt.restrictScalars ℝ
| [
" HasStrictDerivAt (fun x => (e ↑x).re) e'.re z",
" e'.re = (reCLM.comp ((ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e')).comp ofRealCLM)) 1",
" e'.re = reCLM ((ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e')) (ofRealCLM 1))",
" HasDerivAt (fun x => (e ↑x... | [
" HasStrictDerivAt (fun x => (e ↑x).re) e'.re z",
" e'.re = (reCLM.comp ((ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e')).comp ofRealCLM)) 1",
" e'.re = reCLM ((ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e')) (ofRealCLM 1))",
" HasDerivAt (fun x => (e ↑x... |
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
namespace Relation
open Multiset Prod
variable {α : Type*}
def CutExpand (r : α → α → Prop) (s' s : Multise... | Mathlib/Logic/Hydra.lean | 138 | 146 | theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} := by |
induction' hacc with a h ih
refine Acc.intro _ fun s ↦ ?_
classical
simp only [cutExpand_iff, mem_singleton]
rintro ⟨t, a, hr, rfl, rfl⟩
refine acc_of_singleton fun a' ↦ ?_
rw [erase_singleton, zero_add]
exact ih a' ∘ hr a'
| [
" CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) ⇑toFinsupp",
" InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) (⇑toFinsupp) s t",
" (toFinsupp s) b = (toFinsupp t) b",
" (fun {i} x x_1 => x < x_1) ((toFinsupp s) a) ((toFinsupp t) a)",
" count b... | [
" CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) ⇑toFinsupp",
" InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) (⇑toFinsupp) s t",
" (toFinsupp s) b = (toFinsupp t) b",
" (fun {i} x x_1 => x < x_1) ((toFinsupp s) a) ((toFinsupp t) a)",
" count b... |
import Mathlib.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
#align_import probability.independence.zero_one from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
open MeasureTheory MeasurableSpace
open scoped MeasureTheory ENNReal
namespace Probabili... | Mathlib/Probability/Independence/ZeroOne.lean | 46 | 49 | theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
(h_indep : IndepSet t t μ) : μ t = 0 ∨ μ t = 1 ∨ μ t = ∞ := by |
simpa only [ae_dirac_eq, Filter.eventually_pure]
using kernel.measure_eq_zero_or_one_or_top_of_indepSet_self h_indep
| [
" ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤",
" (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤",
" μ t = 0 ∨ μ t = 1 ∨ μ t = ⊤"
] | [
" ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤",
" (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤"
] |
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 112 | 112 | theorem and_intro : ∀ {a b : Bool}, a → b → a && b := by | decide
| [
" b = false ∨ b = true",
" false = false ∨ false = true",
" true = false ∨ true = true",
" p x",
" p false",
" p true",
" p b ∨ p !b",
" p false ∨ p !false",
" p true ∨ p !true",
" ∃ x, p x",
" (a || b) = true",
" (false || b) = true",
" (true || b) = true",
" ∀ {a b : Bool}, (a && b) = tr... | [
" b = false ∨ b = true",
" false = false ∨ false = true",
" true = false ∨ true = true",
" p x",
" p false",
" p true",
" p b ∨ p !b",
" p false ∨ p !false",
" p true ∨ p !true",
" ∃ x, p x",
" (a || b) = true",
" (false || b) = true",
" (true || b) = true",
" ∀ {a b : Bool}, (a && b) = tr... |
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Algebra.Homology.ShortComplex.Basic
#align_import algebra.homology.homological_complex from "leanprover-community/mathlib"@"88bca0ce5d22ebfd9e73e682e51d60ea13b48347"
... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 177 | 182 | theorem next_nat_zero : (ComplexShape.down ℕ).next 0 = 0 := by |
classical
refine dif_neg ?_
push_neg
intro
apply Nat.noConfusion
| [
" (ComplexShape.down ℕ).next 0 = 0",
" ¬∃ j, (ComplexShape.down ℕ).Rel 0 j",
" ∀ (j : ℕ), ¬(ComplexShape.down ℕ).Rel 0 j",
" ¬(ComplexShape.down ℕ).Rel 0 j✝"
] | [] |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 27 | 32 | theorem add_eq_zero_iff {n m : WithBot ℕ} : n + m = 0 ↔ n = 0 ∧ m = 0 := by |
rcases n, m with ⟨_ | _, _ | _⟩
repeat (· exact ⟨fun h => Option.noConfusion h, fun h => Option.noConfusion h.1⟩)
· exact ⟨fun h => Option.noConfusion h, fun h => Option.noConfusion h.2⟩
repeat erw [WithBot.coe_eq_coe]
exact add_eq_zero_iff' (zero_le _) (zero_le _)
| [
" n + m = 0 ↔ n = 0 ∧ m = 0",
" some val✝¹ + some val✝ = 0 ↔ some val✝¹ = 0 ∧ some val✝ = 0",
" none + none = 0 ↔ none = 0 ∧ none = 0",
" none + some val✝ = 0 ↔ none = 0 ∧ some val✝ = 0",
" some val✝ + none = 0 ↔ some val✝ = 0 ∧ none = 0",
" (fun x x_1 => x + x_1) val✝¹ val✝ = 0 ↔ some val✝¹ = 0 ∧ some va... | [] |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 119 | 120 | theorem angle_const_sub (v : V) (v₁ v₂ v₃ : V) : ∠ (v - v₁) (v - v₂) (v - v₃) = ∠ v₁ v₂ v₃ := by |
simpa only [vsub_eq_sub] using angle_const_vsub v v₁ v₂ v₃
| [
" ContinuousAt (fun y => ∠ y.1 y.2.1 y.2.2) x",
" (f x).1 ≠ 0",
" (f x).2 ≠ 0",
" ∠ (f p₁) (f p₂) (f p₃) = ∠ p₁ p₂ p₃",
" ∠ (v₁ - v) (v₂ - v) (v₃ - v) = ∠ v₁ v₂ v₃",
" ∠ (v - v₁) (v - v₂) (v - v₃) = ∠ v₁ v₂ v₃"
] | [
" ContinuousAt (fun y => ∠ y.1 y.2.1 y.2.2) x",
" (f x).1 ≠ 0",
" (f x).2 ≠ 0",
" ∠ (f p₁) (f p₂) (f p₃) = ∠ p₁ p₂ p₃",
" ∠ (v₁ - v) (v₂ - v) (v₃ - v) = ∠ v₁ v₂ v₃"
] |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.Core
import Mathlib.Util.SynthesizeUsing
import Mathlib.Util.Qq
open Lean Parser Tactic Mathlib Meta NormNum Qq
initialize registerTraceClass `CancelDen... | Mathlib/Tactic/CancelDenoms/Core.lean | 105 | 109 | theorem cancel_factors_ne {α} [Field α] {a b ad bd a' b' gcd : α} (ha : ad * a = a')
(hb : bd * b = b') (had : ad ≠ 0) (hbd : bd ≠ 0) (hgcd : gcd ≠ 0) :
(a ≠ b) = (1 / gcd * (bd * a') ≠ 1 / gcd * (ad * b')) := by |
classical
rw [eq_iff_iff, not_iff_not, cancel_factors_eq ha hb had hbd hgcd]
| [
" k * (e1 * e2) = t1 * t2",
" k * (e1 / e2) = t1",
" e * n = e'",
" n * (e1 + e2) = t1 + t2",
" n * (e1 - e2) = t1 - t2",
" n * -e = -t",
" k * e1 ^ e2 = l * t1 ^ e2",
" k * e⁻¹ = n",
" (a < b) = (1 / gcd * (bd * a') < 1 / gcd * (ad * b'))",
" 0 < ad * bd",
" 0 < 1 / gcd",
" (a ≤ b) = (1 / gcd... | [
" k * (e1 * e2) = t1 * t2",
" k * (e1 / e2) = t1",
" e * n = e'",
" n * (e1 + e2) = t1 + t2",
" n * (e1 - e2) = t1 - t2",
" n * -e = -t",
" k * e1 ^ e2 = l * t1 ^ e2",
" k * e⁻¹ = n",
" (a < b) = (1 / gcd * (bd * a') < 1 / gcd * (ad * b'))",
" 0 < ad * bd",
" 0 < 1 / gcd",
" (a ≤ b) = (1 / gcd... |
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.pi from "leanprover-community/mathlib"@"e4bc74cbaf429d706cb9140902f7ca6c431e75a4"
-- Porting note: Added, since dot nota... | Mathlib/Order/Interval/Set/Pi.lean | 90 | 98 | theorem pi_univ_Ioc_update_left {x y : ∀ i, α i} {i₀ : ι} {m : α i₀} (hm : x i₀ ≤ m) :
(pi univ fun i ↦ Ioc (update x i₀ m i) (y i)) =
{ z | m < z i₀ } ∩ pi univ fun i ↦ Ioc (x i) (y i) := by |
have : Ioc m (y i₀) = Ioi m ∩ Ioc (x i₀) (y i₀) := by
rw [← Ioi_inter_Iic, ← Ioi_inter_Iic, ← inter_assoc,
inter_eq_self_of_subset_left (Ioi_subset_Ioi hm)]
simp_rw [univ_pi_update i₀ _ _ fun i z ↦ Ioc z (y i), ← pi_inter_compl ({i₀} : Set ι),
singleton_pi', ← inter_assoc, this]
rfl
| [
" (y ∈ univ.pi fun i => Ici (x i)) ↔ y ∈ Ici x",
" (y ∈ univ.pi fun i => Iic (x i)) ↔ y ∈ Iic x",
" (y ∈ univ.pi fun i => Icc (x i) (y✝ i)) ↔ y ∈ Icc x y✝",
" (univ.pi fun i => Ioc (update x i₀ m i) (y i)) = {z | m < z i₀} ∩ univ.pi fun i => Ioc (x i) (y i)",
" Ioc m (y i₀) = Ioi m ∩ Ioc (x i₀) (y i₀)",
"... | [
" (y ∈ univ.pi fun i => Ici (x i)) ↔ y ∈ Ici x",
" (y ∈ univ.pi fun i => Iic (x i)) ↔ y ∈ Iic x",
" (y ∈ univ.pi fun i => Icc (x i) (y✝ i)) ↔ y ∈ Icc x y✝"
] |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section NegOneSquare
-- This could be formulated for ... | Mathlib/NumberTheory/SumTwoSquares.lean | 108 | 120 | theorem ZMod.isSquare_neg_one_iff {n : ℕ} (hn : Squarefree n) :
IsSquare (-1 : ZMod n) ↔ ∀ {q : ℕ}, q.Prime → q ∣ n → q % 4 ≠ 3 := by |
refine ⟨fun H q hqp hqd => hqp.mod_four_ne_three_of_dvd_isSquare_neg_one hqd H, fun H => ?_⟩
induction' n using induction_on_primes with p n hpp ih
· exact False.elim (hn.ne_zero rfl)
· exact ⟨0, by simp only [mul_zero, eq_iff_true_of_subsingleton]⟩
· haveI : Fact p.Prime := ⟨hpp⟩
have hcp : p.Coprime n ... | [
" IsSquare (-1)",
" IsSquare (f (-1))",
" IsSquare (-1, -1)",
" IsSquare (x * x, y * y)",
" p % 4 ≠ 3",
" -1 = -1 ^ 2",
" IsSquare (-1) ↔ ∀ {q : ℕ}, q.Prime → q ∣ n → q % 4 ≠ 3",
" -1 = 0 * 0",
" p.Coprime n",
" False"
] | [
" IsSquare (-1)",
" IsSquare (f (-1))",
" IsSquare (-1, -1)",
" IsSquare (x * x, y * y)",
" p % 4 ≠ 3",
" -1 = -1 ^ 2"
] |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 281 | 284 | theorem normAtPlace_real (w : InfinitePlace K) (c : ℝ) :
normAtPlace w ((fun _ ↦ c, fun _ ↦ c) : (E K)) = |c| := by |
rw [show ((fun _ ↦ c, fun _ ↦ c) : (E K)) = c • 1 by ext <;> simp, normAtPlace_smul, map_one,
mul_one]
| [
" Nontrivial (({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ))",
" finrank ℝ (({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)) = finrank ℚ K",
" Function.Injective ⇑(mixedEmbedding K)",
" (fun x => if hw : w.IsReal then ‖x.1 ⟨w, hw⟩‖ else ‖x.2 ⟨w, ⋯⟩‖) 0 = 0",
" { toFun := fun x => if hw : w.IsRea... | [
" Nontrivial (({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ))",
" finrank ℝ (({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)) = finrank ℚ K",
" Function.Injective ⇑(mixedEmbedding K)",
" (fun x => if hw : w.IsReal then ‖x.1 ⟨w, hw⟩‖ else ‖x.2 ⟨w, ⋯⟩‖) 0 = 0",
" { toFun := fun x => if hw : w.IsRea... |
import Mathlib.Data.Set.Image
#align_import order.directed from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
open Function
universe u v w
variable {α : Type u} {β : Type v} {ι : Sort w} (r r' s : α → α → Prop)
local infixl:50 " ≼ " => r
def Directed (f : ι → α) :=
∀ x y, ∃ z, ... | Mathlib/Order/Directed.lean | 116 | 128 | theorem Directed.extend_bot [Preorder α] [OrderBot α] {e : ι → β} {f : ι → α}
(hf : Directed (· ≤ ·) f) (he : Function.Injective e) :
Directed (· ≤ ·) (Function.extend e f ⊥) := by |
intro a b
rcases (em (∃ i, e i = a)).symm with (ha | ⟨i, rfl⟩)
· use b
simp [Function.extend_apply' _ _ _ ha]
rcases (em (∃ i, e i = b)).symm with (hb | ⟨j, rfl⟩)
· use e i
simp [Function.extend_apply' _ _ _ hb]
rcases hf i j with ⟨k, hi, hj⟩
use e k
simp only [he.extend_apply, *, true_and_iff]... | [
" DirectedOn r s ↔ Directed r Subtype.val",
" (∀ x ∈ s, ∀ y ∈ s, ∃ z ∈ s, r x z ∧ r y z) ↔ ∀ a ∈ s, ∀ a_1 ∈ s, ∃ a_2, r a a_2 ∧ a_2 ∈ s ∧ r a_1 a_2",
" (∀ y ∈ s, ∃ z ∈ s, r x z ∧ r y z) ↔ ∀ a ∈ s, ∃ a_1, r x a_1 ∧ a_1 ∈ s ∧ r a a_1",
" Directed r f ↔ DirectedOn r (Set.range f)",
" DirectedOn r (f '' s) ↔ Di... | [
" DirectedOn r s ↔ Directed r Subtype.val",
" (∀ x ∈ s, ∀ y ∈ s, ∃ z ∈ s, r x z ∧ r y z) ↔ ∀ a ∈ s, ∀ a_1 ∈ s, ∃ a_2, r a a_2 ∧ a_2 ∈ s ∧ r a_1 a_2",
" (∀ y ∈ s, ∃ z ∈ s, r x z ∧ r y z) ↔ ∀ a ∈ s, ∃ a_1, r x a_1 ∧ a_1 ∈ s ∧ r a a_1",
" Directed r f ↔ DirectedOn r (Set.range f)",
" DirectedOn r (f '' s) ↔ Di... |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 87 | 88 | theorem fib_add_two {n : ℕ} : fib (n + 2) = fib n + fib (n + 1) := by |
simp [fib, Function.iterate_succ_apply']
| [
" (n + 2).fib = n.fib + (n + 1).fib"
] | [] |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.ord_connected_component from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Interval Function OrderDual
namespace Set
variable {α : Type*} [LinearOrder α] {s t : Set α}... | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 127 | 133 | theorem ordConnectedProj_eq {x y : s} :
ordConnectedProj s x = ordConnectedProj s y ↔ [[(x : α), y]] ⊆ s := by |
constructor <;> intro h
· rw [← mem_ordConnectedComponent, ← ordConnectedComponent_ordConnectedProj, h,
ordConnectedComponent_ordConnectedProj, self_mem_ordConnectedComponent]
exact y.2
· simp only [ordConnectedProj, ordConnectedComponent_eq h]
| [
" toDual x ∈ (⇑ofDual ⁻¹' s).ordConnectedComponent (toDual x✝) ↔ toDual x ∈ ⇑ofDual ⁻¹' s.ordConnectedComponent x✝",
" ⇑ofDual ⁻¹' [[x✝, x]] ⊆ ⇑ofDual ⁻¹' s ↔ toDual x ∈ ⇑ofDual ⁻¹' s.ordConnectedComponent x✝",
" x ∈ s.ordConnectedComponent x ↔ x ∈ s",
" s.ordConnectedComponent x = ∅ ↔ x ∉ s",
" univ.ordCon... | [
" toDual x ∈ (⇑ofDual ⁻¹' s).ordConnectedComponent (toDual x✝) ↔ toDual x ∈ ⇑ofDual ⁻¹' s.ordConnectedComponent x✝",
" ⇑ofDual ⁻¹' [[x✝, x]] ⊆ ⇑ofDual ⁻¹' s ↔ toDual x ∈ ⇑ofDual ⁻¹' s.ordConnectedComponent x✝",
" x ∈ s.ordConnectedComponent x ↔ x ∈ s",
" s.ordConnectedComponent x = ∅ ↔ x ∉ s",
" univ.ordCon... |
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Function.LocallyIntegrable
open Asymptotics MeasureTheory Set Filter
variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F]
{f : α → E} {g : α → F} {a b :... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 89 | 93 | theorem LocallyIntegrableOn.integrableOn_of_isBigO_atTop [IsMeasurablyGenerated (atTop (α := α))]
(hf : LocallyIntegrableOn f (Ici a) μ) (ho : f =O[atTop] g)
(hg : IntegrableAtFilter g atTop μ) : IntegrableOn f (Ici a) μ := by |
refine integrableOn_Ici_iff_integrableAtFilter_atTop.mpr ⟨ho.integrableAtFilter ?_ hg, hf⟩
exact ⟨Ici a, Ici_mem_atTop a, hf.aestronglyMeasurable⟩
| [
" IntegrableAtFilter f l μ",
" ∀ᵐ (a : α) ∂μ.restrict s, ‖f a‖ ≤ ‖C * ‖g a‖‖",
" ‖f x‖ ≤ ‖C * ‖g x‖‖",
" Integrable f μ",
" IntegrableAtFilter f ⊤ μ",
" StronglyMeasurableAtFilter f (cocompact α) μ",
" StronglyMeasurableAtFilter f atTop μ",
" StronglyMeasurableAtFilter f atBot μ",
" IntegrableOn f (... | [
" IntegrableAtFilter f l μ",
" ∀ᵐ (a : α) ∂μ.restrict s, ‖f a‖ ≤ ‖C * ‖g a‖‖",
" ‖f x‖ ≤ ‖C * ‖g x‖‖",
" Integrable f μ",
" IntegrableAtFilter f ⊤ μ",
" StronglyMeasurableAtFilter f (cocompact α) μ",
" StronglyMeasurableAtFilter f atTop μ",
" StronglyMeasurableAtFilter f atBot μ",
" IntegrableOn f (... |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 138 | 139 | theorem card_fintypeIic : Fintype.card (Set.Iic b) = b + 1 := by |
rw [Fintype.card_ofFinset, card_Iic]
| [
" x ∈ (fun a b => { val := ↑(List.range' a (b + 1 - a)), nodup := ⋯ }) a b ↔ a ≤ x ∧ x ≤ b",
" a ≤ x ∧ x < a + (b + 1 - a) ↔ a ≤ x ∧ x ≤ b",
" x ∈ (fun a b => { val := ↑(List.range' a (b - a)), nodup := ⋯ }) a b ↔ a ≤ x ∧ x < b",
" a ≤ x ∧ x < a + (b - a) ↔ a ≤ x ∧ x < b",
" x ∈ (fun a b => { val := ↑(List.... | [
" x ∈ (fun a b => { val := ↑(List.range' a (b + 1 - a)), nodup := ⋯ }) a b ↔ a ≤ x ∧ x ≤ b",
" a ≤ x ∧ x < a + (b + 1 - a) ↔ a ≤ x ∧ x ≤ b",
" x ∈ (fun a b => { val := ↑(List.range' a (b - a)), nodup := ⋯ }) a b ↔ a ≤ x ∧ x < b",
" a ≤ x ∧ x < a + (b - a) ↔ a ≤ x ∧ x < b",
" x ∈ (fun a b => { val := ↑(List.... |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 128 | 130 | theorem hasFiniteIntegral_iff_ofNNReal {f : α → ℝ≥0} :
HasFiniteIntegral (fun x => (f x : ℝ)) μ ↔ (∫⁻ a, f a ∂μ) < ∞ := by |
simp [hasFiniteIntegral_iff_norm]
| [
" ∫⁻ (a : α), ↑‖f a‖₊ ∂μ = ∫⁻ (a : α), edist (f a) 0 ∂μ",
" ∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ = ∫⁻ (a : α), edist (f a) 0 ∂μ",
" ∫⁻ (a : α), edist (f a) (g a) ∂μ ≤ ∫⁻ (a : α), edist (f a) (h a) ∂μ + ∫⁻ (a : α), edist (g a) (h a) ∂μ",
" ∫⁻ (a : α), edist (f a) (g a) ∂μ ≤ ∫⁻ (a : α), edist (f a) (h a) + edist... | [
" ∫⁻ (a : α), ↑‖f a‖₊ ∂μ = ∫⁻ (a : α), edist (f a) 0 ∂μ",
" ∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ = ∫⁻ (a : α), edist (f a) 0 ∂μ",
" ∫⁻ (a : α), edist (f a) (g a) ∂μ ≤ ∫⁻ (a : α), edist (f a) (h a) ∂μ + ∫⁻ (a : α), edist (g a) (h a) ∂μ",
" ∫⁻ (a : α), edist (f a) (g a) ∂μ ≤ ∫⁻ (a : α), edist (f a) (h a) + edist... |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section SigmaLift
variable {α β γ : ι → Type*} [DecidableEq ι]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 184 | 187 | theorem not_mem_sigmaLift_of_ne_left (f : ∀ ⦃i⦄, α i → β i → Finset (γ i)) (a : Sigma α)
(b : Sigma β) (x : Sigma γ) (h : a.1 ≠ x.1) : x ∉ sigmaLift f a b := by |
rw [mem_sigmaLift]
exact fun H => h H.fst
| [
" x ∈ sigmaLift f a b ↔ ∃ (ha : a.fst = x.fst) (hb : b.fst = x.fst), x.snd ∈ f (ha ▸ a.snd) (hb ▸ b.snd)",
" x ∈ sigmaLift f ⟨i, a⟩ ⟨j, b⟩ ↔\n ∃ (ha : ⟨i, a⟩.fst = x.fst) (hb : ⟨j, b⟩.fst = x.fst), x.snd ∈ f (ha ▸ ⟨i, a⟩.snd) (hb ▸ ⟨j, b⟩.snd)",
" x ∈ sigmaLift f ⟨i, a⟩ ⟨i, b⟩ ↔\n ∃ (ha : ⟨i, a⟩.fst = x.f... | [
" x ∈ sigmaLift f a b ↔ ∃ (ha : a.fst = x.fst) (hb : b.fst = x.fst), x.snd ∈ f (ha ▸ a.snd) (hb ▸ b.snd)",
" x ∈ sigmaLift f ⟨i, a⟩ ⟨j, b⟩ ↔\n ∃ (ha : ⟨i, a⟩.fst = x.fst) (hb : ⟨j, b⟩.fst = x.fst), x.snd ∈ f (ha ▸ ⟨i, a⟩.snd) (hb ▸ ⟨j, b⟩.snd)",
" x ∈ sigmaLift f ⟨i, a⟩ ⟨i, b⟩ ↔\n ∃ (ha : ⟨i, a⟩.fst = x.f... |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable s... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 158 | 167 | theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
(h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ := by |
refine ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, ?_⟩
simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal toReal_nonneg]
convert h2s.lt_top using 1
-- Porting note: was `simp_rw`
rw [prod_apply hs]
apply lintegral_congr_ae
filter_upwards [ae_measure_lt_top hs h2s] w... | [
" MeasurableSet {x | Integrable (f x) ν}",
" MeasurableSet {x | HasFiniteIntegral (f x) ν}",
" StronglyMeasurable fun x => ∫ (y : β), f x y ∂ν",
" 0 ∈ range (uncurry f) ∪ {0}",
" ∀ (n : ℕ), StronglyMeasurable (f' n)",
" StronglyMeasurable (f' n)",
" StronglyMeasurable fun x => SimpleFunc.integral ν (s' ... | [
" MeasurableSet {x | Integrable (f x) ν}",
" MeasurableSet {x | HasFiniteIntegral (f x) ν}",
" StronglyMeasurable fun x => ∫ (y : β), f x y ∂ν",
" 0 ∈ range (uncurry f) ∪ {0}",
" ∀ (n : ℕ), StronglyMeasurable (f' n)",
" StronglyMeasurable (f' n)",
" StronglyMeasurable fun x => SimpleFunc.integral ν (s' ... |
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Data.Rat.Floor
#align_import algebra.continued_fractions.computation.terminates_iff_rat from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b3... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 119 | 123 | theorem exists_rat_eq_nth_convergent : ∃ q : ℚ, (of v).convergents n = (q : K) := by |
rcases exists_rat_eq_nth_numerator v n with ⟨Aₙ, nth_num_eq⟩
rcases exists_rat_eq_nth_denominator v n with ⟨Bₙ, nth_denom_eq⟩
use Aₙ / Bₙ
simp [nth_num_eq, nth_denom_eq, convergent_eq_num_div_denom]
| [
" ∀ (n : ℕ),\n (∀ m < n, ∃ conts, (of v).continuantsAux m = Pair.map Rat.cast conts) →\n ∃ conts, (of v).continuantsAux n = Pair.map Rat.cast conts",
" ∃ conts, (of v).continuantsAux n = Pair.map Rat.cast conts",
" ∃ conts, (of v).continuantsAux 0 = Pair.map Rat.cast conts",
" ∃ gp, { a := 1, b := 0 }... | [
" ∀ (n : ℕ),\n (∀ m < n, ∃ conts, (of v).continuantsAux m = Pair.map Rat.cast conts) →\n ∃ conts, (of v).continuantsAux n = Pair.map Rat.cast conts",
" ∃ conts, (of v).continuantsAux n = Pair.map Rat.cast conts",
" ∃ conts, (of v).continuantsAux 0 = Pair.map Rat.cast conts",
" ∃ gp, { a := 1, b := 0 }... |
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import number_theory.von_mangoldt from "leanprover-community/mathlib"@"c946d6097a6925ad16d7ec55677bbc977f9846de"
namespace ArithmeticFunction
open Finset Nat
open scoped Arit... | Mathlib/NumberTheory/VonMangoldt.lean | 83 | 87 | theorem vonMangoldt_nonneg {n : ℕ} : 0 ≤ Λ n := by |
rw [vonMangoldt_apply]
split_ifs
· exact Real.log_nonneg (one_le_cast.2 (Nat.minFac_pos n))
rfl
| [
" (fun n => (↑n).log) 0 = 0",
" Λ 1 = 0",
" 0 ≤ Λ n",
" 0 ≤ if IsPrimePow n then (↑n.minFac).log else 0",
" 0 ≤ (↑n.minFac).log",
" 0 ≤ 0"
] | [
" (fun n => (↑n).log) 0 = 0",
" Λ 1 = 0"
] |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 200 | 206 | theorem norm_mk_lt' (S : AddSubgroup M) (m : M) {ε : ℝ} (hε : 0 < ε) :
∃ s ∈ S, ‖m + s‖ < ‖mk' S m‖ + ε := by |
obtain ⟨n : M, hn : mk' S n = mk' S m, hn' : ‖n‖ < ‖mk' S m‖ + ε⟩ :=
norm_mk_lt (QuotientAddGroup.mk' S m) hε
erw [eq_comm, QuotientAddGroup.eq] at hn
use -m + n, hn
rwa [add_neg_cancel_left]
| [
" ‖x‖ = infDist 0 {m | ↑m = x}",
" ‖↑x‖ = infDist x ↑S",
" infDist x (⇑(IsometryEquiv.subLeft x).symm ⁻¹' {m | ↑m = ↑x}) = infDist x ↑S",
" y ∈ ⇑(IsometryEquiv.subLeft x).symm ⁻¹' {m | ↑m = ↑x} ↔ y ∈ ↑S",
" ‖-x‖ = ‖x‖",
" sInf (norm '' {m | ↑m = -x}) = sInf (norm '' {m | ↑m = x})",
" r ∈ norm '' {m | ↑m... | [
" ‖x‖ = infDist 0 {m | ↑m = x}",
" ‖↑x‖ = infDist x ↑S",
" infDist x (⇑(IsometryEquiv.subLeft x).symm ⁻¹' {m | ↑m = ↑x}) = infDist x ↑S",
" y ∈ ⇑(IsometryEquiv.subLeft x).symm ⁻¹' {m | ↑m = ↑x} ↔ y ∈ ↑S",
" ‖-x‖ = ‖x‖",
" sInf (norm '' {m | ↑m = -x}) = sInf (norm '' {m | ↑m = x})",
" r ∈ norm '' {m | ↑m... |
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.MvPolynomial.Basic
#align_import data.mv_polynomial.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {σ R : Type*} [CommSemiring R]
namespace MvPolynomial
theorem monomial_dvd_monomial {r s : ... | Mathlib/Algebra/MvPolynomial/Division.lean | 244 | 247 | theorem monomial_one_dvd_monomial_one [Nontrivial R] {i j : σ →₀ ℕ} :
monomial i (1 : R) ∣ monomial j 1 ↔ i ≤ j := by |
rw [monomial_dvd_monomial]
simp_rw [one_ne_zero, false_or_iff, dvd_rfl, and_true_iff]
| [
" (monomial i) r ∣ (monomial j) s ↔ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
" (monomial i) r ∣ (monomial j) s → (s = 0 ∨ i ≤ j) ∧ r ∣ s",
" (s = 0 ∨ i ≤ j) ∧ r ∣ s",
" (s = 0 ∨ i ≤ j) ∧ r ∣ s → (monomial i) r ∣ (monomial j) s",
" (monomial i) r ∣ (monomial j) (r * d)",
" (monomial j) (r * d) = (monomial i) r * (monomia... | [
" (monomial i) r ∣ (monomial j) s ↔ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
" (monomial i) r ∣ (monomial j) s → (s = 0 ∨ i ≤ j) ∧ r ∣ s",
" (s = 0 ∨ i ≤ j) ∧ r ∣ s",
" (s = 0 ∨ i ≤ j) ∧ r ∣ s → (monomial i) r ∣ (monomial j) s",
" (monomial i) r ∣ (monomial j) (r * d)",
" (monomial j) (r * d) = (monomial i) r * (monomia... |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace EN... | Mathlib/MeasureTheory/Integral/Bochner.lean | 249 | 251 | theorem weightedSMul_nonneg (s : Set α) (x : ℝ) (hx : 0 ≤ x) : 0 ≤ weightedSMul μ s x := by |
simp only [weightedSMul, Algebra.id.smul_eq_mul, coe_smul', _root_.id, coe_id', Pi.smul_apply]
exact mul_nonneg toReal_nonneg hx
| [
" (weightedSMul μ s) x = (μ s).toReal • x",
" weightedSMul 0 = 0",
" weightedSMul 0 x✝ = 0 x✝",
" weightedSMul μ ∅ = 0",
" (weightedSMul μ ∅) x = 0 x",
" (μ ∅).toReal • x = 0 x",
" weightedSMul (μ + ν) s = weightedSMul μ s + weightedSMul ν s",
" (weightedSMul (μ + ν) s) x = (weightedSMul μ s + weighte... | [
" (weightedSMul μ s) x = (μ s).toReal • x",
" weightedSMul 0 = 0",
" weightedSMul 0 x✝ = 0 x✝",
" weightedSMul μ ∅ = 0",
" (weightedSMul μ ∅) x = 0 x",
" (μ ∅).toReal • x = 0 x",
" weightedSMul (μ + ν) s = weightedSMul μ s + weightedSMul ν s",
" (weightedSMul (μ + ν) s) x = (weightedSMul μ s + weighte... |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 97 | 102 | theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f 0 s x₀ x = f x₀ := by |
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
| [
" taylorWithin f (n + 1) s x₀ =\n taylorWithin f n s x₀ +\n (PolynomialModule.comp (Polynomial.X - Polynomial.C x₀))\n ((PolynomialModule.single ℝ (n + 1)) (taylorCoeffWithin f (n + 1) s x₀))",
" ∑ k ∈ Finset.range (n + 1 + 1),\n (PolynomialModule.comp (Polynomial.X - Polynomial.C x₀))\n ... | [
" taylorWithin f (n + 1) s x₀ =\n taylorWithin f n s x₀ +\n (PolynomialModule.comp (Polynomial.X - Polynomial.C x₀))\n ((PolynomialModule.single ℝ (n + 1)) (taylorCoeffWithin f (n + 1) s x₀))",
" ∑ k ∈ Finset.range (n + 1 + 1),\n (PolynomialModule.comp (Polynomial.X - Polynomial.C x₀))\n ... |
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.DifferentialObject
#align_import algebra.homology.differential_object from "leanprover-community/mathlib"@"b535c2d5d996acd9b0554b76395d9c920e186f4f"
open CategoryTheory CategoryTheory.Limits
open scoped Classical
noncomputable secti... | Mathlib/Algebra/Homology/DifferentialObject.lean | 53 | 54 | theorem objEqToHom_d {x y : β} (h : x = y) :
X.objEqToHom h ≫ X.d y = X.d x ≫ X.objEqToHom (by cases h; rfl) := by | cases h; dsimp; simp
| [
" (fun b_1 => b_1 + { as := 1 }.as • b) x = (fun b_1 => b_1 + { as := 1 }.as • b) y",
" (fun b_1 => b_1 + { as := 1 }.as • b) x = (fun b_1 => b_1 + { as := 1 }.as • b) x",
" X.objEqToHom h ≫ X.d y = X.d x ≫ X.objEqToHom ⋯",
" X.objEqToHom ⋯ ≫ X.d x = X.d x ≫ X.objEqToHom ⋯",
" 𝟙 (X.obj x) ≫ X.d x = X.d x ≫... | [] |
import Mathlib.MeasureTheory.Integral.Lebesgue
#align_import measure_theory.measure.giry_monad from "leanprover-community/mathlib"@"56f4cd1ef396e9fd389b5d8371ee9ad91d163625"
noncomputable section
open scoped Classical
open ENNReal
open scoped Classical
open Set Filter
variable {α β : Type*}
namespace MeasureT... | Mathlib/MeasureTheory/Measure/GiryMonad.lean | 91 | 96 | theorem measurable_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) :
Measurable fun μ : Measure α => ∫⁻ x, f x ∂μ := by |
simp only [lintegral_eq_iSup_eapprox_lintegral, hf, SimpleFunc.lintegral]
refine measurable_iSup fun n => Finset.measurable_sum _ fun i _ => ?_
refine Measurable.const_mul ?_ _
exact measurable_coe ((SimpleFunc.eapprox f n).measurableSet_preimage _)
| [
" borel ℝ≥0∞ ≤ MeasurableSpace.map (fun μ => μ s) (MeasurableSpace.map f inst✝)",
" borel ℝ≥0∞ ≤ MeasurableSpace.map ((fun μ => μ s) ∘ f) inst✝",
" MeasurableAdd₂ (Measure α)",
" Measurable fun b => (b.1 + b.2) s",
" Measurable fun b => b.1 s + b.2 s",
" Measurable fun b => b.1 s",
" Measurable fun b =>... | [
" borel ℝ≥0∞ ≤ MeasurableSpace.map (fun μ => μ s) (MeasurableSpace.map f inst✝)",
" borel ℝ≥0∞ ≤ MeasurableSpace.map ((fun μ => μ s) ∘ f) inst✝",
" MeasurableAdd₂ (Measure α)",
" Measurable fun b => (b.1 + b.2) s",
" Measurable fun b => b.1 s + b.2 s",
" Measurable fun b => b.1 s",
" Measurable fun b =>... |
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Init.Data.Int.Order
set_option autoImplicit true
namespace Linarith
theorem lt_irrefl {α : Type u} ... | Mathlib/Tactic/Linarith/Lemmas.lean | 36 | 37 | theorem le_of_le_of_eq {α} [OrderedSemiring α] {a b : α} (ha : a ≤ 0) (hb : b = 0) : a + b ≤ 0 := by |
simp [*]
| [
" a + b = 0",
" a + b ≤ 0",
" a + b < 0"
] | [
" a + b = 0",
" a + b ≤ 0",
" a + b < 0"
] |
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multiset.basic from "leanprover-community/mathlib"@"6c5f73fd6f6cc83122788a80a27cdd54663609f4"
assert_not_exists MonoidWithZero
variable {F ι α β γ : Type*}
names... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 105 | 106 | theorem prod_pair (a b : α) : ({a, b} : Multiset α).prod = a * b := by |
rw [insert_eq_cons, prod_cons, prod_singleton]
| [
" (fun x x_1 => x * x_1) x ((fun x x_1 => x * x_1) y z) = (fun x x_1 => x * x_1) y ((fun x x_1 => x * x_1) x z)",
" (fun x x_1 => x * x_1) ((fun x x_1 => x * x_1) x y) z = (fun x x_1 => x * x_1) ((fun x x_1 => x * x_1) x z) y",
" foldl (fun x y => y * x) ⋯ 1 s = foldl (fun x x_1 => x * x_1) ⋯ 1 s",
" s.toList... | [
" (fun x x_1 => x * x_1) x ((fun x x_1 => x * x_1) y z) = (fun x x_1 => x * x_1) y ((fun x x_1 => x * x_1) x z)",
" (fun x x_1 => x * x_1) ((fun x x_1 => x * x_1) x y) z = (fun x x_1 => x * x_1) ((fun x x_1 => x * x_1) x z) y",
" foldl (fun x y => y * x) ⋯ 1 s = foldl (fun x x_1 => x * x_1) ⋯ 1 s",
" s.toList... |
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 98 | 110 | theorem IsLindelof.image_of_continuousOn {f : X → Y} (hs : IsLindelof s) (hf : ContinuousOn f s) :
IsLindelof (f '' s) := by |
intro l lne _ ls
have : NeBot (l.comap f ⊓ 𝓟 s) :=
comap_inf_principal_neBot_of_image_mem lne (le_principal_iff.1 ls)
obtain ⟨x, hxs, hx⟩ : ∃ x ∈ s, ClusterPt x (l.comap f ⊓ 𝓟 s) := @hs _ this _ inf_le_right
haveI := hx.neBot
use f x, mem_image_of_mem f hxs
have : Tendsto f (𝓝 x ⊓ (comap f l ⊓ 𝓟 s)... | [
" sᶜ ∈ f",
" ∃ x ∈ s, sᶜ ∉ 𝓝 x ⊓ f",
" ∃ x ∈ s, (𝓝 x ⊓ (f ⊓ 𝓟 s)).NeBot",
" sᶜ ∈ 𝓝 x ⊓ f",
" ∃ i ∈ 𝓝 x ⊓ 𝓟 s, (id i)ᶜ ∈ f",
" p s",
" ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f",
" IsLindelof (s ∩ t)",
" ∃ x ∈ s ∩ t, ClusterPt x f",
" IsLindelof (f '' s)",
" ∃ x ∈ f '' s, ClusterPt x l",
" ClusterPt... | [
" sᶜ ∈ f",
" ∃ x ∈ s, sᶜ ∉ 𝓝 x ⊓ f",
" ∃ x ∈ s, (𝓝 x ⊓ (f ⊓ 𝓟 s)).NeBot",
" sᶜ ∈ 𝓝 x ⊓ f",
" ∃ i ∈ 𝓝 x ⊓ 𝓟 s, (id i)ᶜ ∈ f",
" p s",
" ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f",
" IsLindelof (s ∩ t)",
" ∃ x ∈ s ∩ t, ClusterPt x f"
] |
import Mathlib.CategoryTheory.Adjunction.Unique
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.CategoryTheory.Limits.Preserves.Finite
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open Limits
variable {C : Type u₁} [Category.{v₁} C] (J : Grothendiec... | Mathlib/CategoryTheory/Sites/Sheafification.lean | 100 | 102 | theorem sheafifyMap_comp {P Q R : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (γ : Q ⟶ R) :
sheafifyMap J (η ≫ γ) = sheafifyMap J η ≫ sheafifyMap J γ := by |
simp [sheafifyMap, sheafify]
| [
" PreservesFiniteLimits (sheafToPresheaf J A).leftAdjoint",
" sheafifyMap J (𝟙 P) = 𝟙 (sheafify J P)",
" sheafifyMap J (η ≫ γ) = sheafifyMap J η ≫ sheafifyMap J γ"
] | [
" PreservesFiniteLimits (sheafToPresheaf J A).leftAdjoint",
" sheafifyMap J (𝟙 P) = 𝟙 (sheafify J P)"
] |
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Util.AssertExists
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
namespace List
universe u v
variable {α : Type u} {β : Type v} (l :... | Mathlib/Data/List/GetD.lean | 47 | 53 | theorem getD_map {n : ℕ} (f : α → β) : (map f l).getD n (f d) = f (l.getD n d) := by |
induction l generalizing n with
| nil => rfl
| cons head tail ih =>
cases n
· rfl
· simp [ih]
| [
" l.getD n d = l.get ⟨n, hn⟩",
" [].getD n d = [].get ⟨n, hn⟩",
" (head :: tail).getD n d = (head :: tail).get ⟨n, hn⟩",
" (head :: tail).getD 0 d = (head :: tail).get ⟨0, hn⟩",
" (head :: tail).getD (n✝ + 1) d = (head :: tail).get ⟨n✝ + 1, hn⟩",
" (map f l).getD n (f d) = f (l.getD n d)",
" (map f []).... | [
" l.getD n d = l.get ⟨n, hn⟩",
" [].getD n d = [].get ⟨n, hn⟩",
" (head :: tail).getD n d = (head :: tail).get ⟨n, hn⟩",
" (head :: tail).getD 0 d = (head :: tail).get ⟨0, hn⟩",
" (head :: tail).getD (n✝ + 1) d = (head :: tail).get ⟨n✝ + 1, hn⟩"
] |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section Sigma
variable {α : ι → Type*} {β : Type*} (s s₁ s₂ : Finset ι) (... | Mathlib/Data/Finset/Sigma.lean | 99 | 104 | theorem sup_sigma [SemilatticeSup β] [OrderBot β] :
(s.sigma t).sup f = s.sup fun i => (t i).sup fun b => f ⟨i, b⟩ := by |
simp only [le_antisymm_iff, Finset.sup_le_iff, mem_sigma, and_imp, Sigma.forall]
exact
⟨fun i a hi ha => (le_sup hi).trans' <| le_sup (f := fun a => f ⟨i, a⟩) ha, fun i hi a ha =>
le_sup <| mem_sigma.2 ⟨hi, ha⟩⟩
| [
" (s.sigma t).Nonempty ↔ ∃ i ∈ s, (t i).Nonempty",
" s.sigma t = ∅ ↔ ∀ i ∈ s, t i = ∅",
" (↑s).PairwiseDisjoint fun i => map (Embedding.sigmaMk i) (t i)",
" (_root_.Disjoint on fun i => map (Embedding.sigmaMk i) (t i)) i j",
" ∀ ⦃a : (x : ι) × α x⦄, a ∈ map (Embedding.sigmaMk i) (t i) → a ∉ map (Embedding.s... | [
" (s.sigma t).Nonempty ↔ ∃ i ∈ s, (t i).Nonempty",
" s.sigma t = ∅ ↔ ∀ i ∈ s, t i = ∅",
" (↑s).PairwiseDisjoint fun i => map (Embedding.sigmaMk i) (t i)",
" (_root_.Disjoint on fun i => map (Embedding.sigmaMk i) (t i)) i j",
" ∀ ⦃a : (x : ι) × α x⦄, a ∈ map (Embedding.sigmaMk i) (t i) → a ∉ map (Embedding.s... |
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace List
variable {α β : Type*}
section FormPerm
variable [DecidableEq α] (l :... | Mathlib/GroupTheory/Perm/List.lean | 100 | 103 | theorem support_formPerm_le [Fintype α] : support (formPerm l) ≤ l.toFinset := by |
intro x hx
have hx' : x ∈ { x | formPerm l x ≠ x } := by simpa using hx
simpa using support_formPerm_le' _ hx'
| [
" (zipWith swap [] x✝¹).prod x✝ ≠ x✝ → x✝ ∈ [] ∨ x✝ ∈ x✝¹",
" (zipWith swap x✝¹ []).prod x✝ ≠ x✝ → x✝ ∈ x✝¹ ∨ x✝ ∈ []",
" (swap (?m.1920 a l b l' x hx h) (?m.1921 a l b l' x hx h)) (?m.1919 a l b l' x hx h) ≠ ?m.1919 a l b l' x hx h",
" x = a → x ∈ a :: l",
" x ∈ x :: l",
" x = b → x ∈ b :: l'",
" x ∈ x... | [
" (zipWith swap [] x✝¹).prod x✝ ≠ x✝ → x✝ ∈ [] ∨ x✝ ∈ x✝¹",
" (zipWith swap x✝¹ []).prod x✝ ≠ x✝ → x✝ ∈ x✝¹ ∨ x✝ ∈ []",
" (swap (?m.1920 a l b l' x hx h) (?m.1921 a l b l' x hx h)) (?m.1919 a l b l' x hx h) ≠ ?m.1919 a l b l' x hx h",
" x = a → x ∈ a :: l",
" x ∈ x :: l",
" x = b → x ∈ b :: l'",
" x ∈ x... |
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
#align_import algebraic_geometry.presheafed_space from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
open Opposite CategoryTheory CategoryTheory.Category CategoryTheory.Functor TopCat Topolo... | Mathlib/Geometry/RingedSpace/PresheafedSpace.lean | 112 | 121 | theorem Hom.ext {X Y : PresheafedSpace C} (α β : Hom X Y) (w : α.base = β.base)
(h : α.c ≫ whiskerRight (eqToHom (by rw [w])) _ = β.c) : α = β := by |
rcases α with ⟨base, c⟩
rcases β with ⟨base', c'⟩
dsimp at w
subst w
dsimp at h
erw [whiskerRight_id', comp_id] at h
subst h
rfl
| [
" (Opens.map α.base).op = (Opens.map β.base).op",
" α = β",
" { base := base, c := c } = β",
" { base := base, c := c } = { base := base', c := c' }",
" { base := base, c := c } = { base := base, c := c' }",
" { base := base, c := c } = { base := base, c := c }"
] | [] |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.Analytic.Basic
#align_import measure_theory.integral.circle_integral from "leanprover-communit... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 105 | 106 | theorem circleMap_sub_center (c : ℂ) (R : ℝ) (θ : ℝ) : circleMap c R θ - c = circleMap 0 R θ := by |
simp [circleMap]
| [
" circleMap c R (θ + 2 * π) = circleMap c R θ",
" circleMap c R θ - c = circleMap 0 R θ"
] | [
" circleMap c R (θ + 2 * π) = circleMap c R θ"
] |
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombination
#align_import number_theory.fermat4 from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
noncomputable section
open scope... | Mathlib/NumberTheory/FLT/Four.lean | 154 | 156 | theorem Int.coprime_of_sq_sum {r s : ℤ} (h2 : IsCoprime s r) : IsCoprime (r ^ 2 + s ^ 2) r := by |
rw [sq, sq]
exact (IsCoprime.mul_left h2 h2).mul_add_left_left r
| [
" IsCoprime (r ^ 2 + s ^ 2) r",
" IsCoprime (r * r + s * s) r"
] | [] |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Contraction
import Mathlib.RingTheory.TensorProduct.Basic
#align_import representation_... | Mathlib/RepresentationTheory/Basic.lean | 110 | 110 | theorem asAlgebraHom_single_one (g : G) : asAlgebraHom ρ (Finsupp.single g 1) = ρ g := by | simp
| [
" ρ.asAlgebraHom (Finsupp.single g r) = r • ρ g",
" ρ.asAlgebraHom (Finsupp.single g 1) = ρ g"
] | [
" ρ.asAlgebraHom (Finsupp.single g r) = r • ρ g"
] |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 83 | 86 | theorem lineMap_lt_lineMap_iff_of_lt (h : r < r') : lineMap a b r < lineMap a b r' ↔ a < b := by |
simp only [lineMap_apply_module]
rw [← lt_sub_iff_add_lt, add_sub_assoc, ← sub_lt_iff_lt_add', ← sub_smul, ← sub_smul,
sub_sub_sub_cancel_left, smul_lt_smul_iff_of_pos_left (sub_pos.2 h)]
| [
" (lineMap a b) r ≤ (lineMap a' b) r",
" (1 - r) • a + r • b ≤ (1 - r) • a' + r • b",
" (lineMap a b) r < (lineMap a' b) r",
" (1 - r) • a + r • b < (1 - r) • a' + r • b",
" (lineMap a b) r ≤ (lineMap a b') r",
" (1 - r) • a + r • b ≤ (1 - r) • a + r • b'",
" (lineMap a b) r < (lineMap a b') r",
" (1 ... | [
" (lineMap a b) r ≤ (lineMap a' b) r",
" (1 - r) • a + r • b ≤ (1 - r) • a' + r • b",
" (lineMap a b) r < (lineMap a' b) r",
" (1 - r) • a + r • b < (1 - r) • a' + r • b",
" (lineMap a b) r ≤ (lineMap a b') r",
" (1 - r) • a + r • b ≤ (1 - r) • a + r • b'",
" (lineMap a b) r < (lineMap a b') r",
" (1 ... |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
#align_import data.matrix.notation from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matrix
universe u uₘ uₙ uₒ
variable {α : Type u} {o n m... | Mathlib/Data/Matrix/Notation.lean | 174 | 175 | theorem cons_dotProduct_cons (x : α) (v : Fin n → α) (y : α) (w : Fin n → α) :
dotProduct (vecCons x v) (vecCons y w) = x * y + dotProduct v w := by | simp
| [
" vecCons v B i j = vecCons (v j) (fun i => B i j) i",
" vecCons v B 0 j = vecCons (v j) (fun i => B i j) 0",
" ∀ (i : Fin m), vecCons v B i.succ j = vecCons (v j) (fun i => B i j) i.succ",
" vecCons x v ⬝ᵥ w = x * vecHead w + v ⬝ᵥ vecTail w",
" v ⬝ᵥ vecCons x w = vecHead v * x + vecTail v ⬝ᵥ w",
" vecCon... | [
" vecCons v B i j = vecCons (v j) (fun i => B i j) i",
" vecCons v B 0 j = vecCons (v j) (fun i => B i j) 0",
" ∀ (i : Fin m), vecCons v B i.succ j = vecCons (v j) (fun i => B i j) i.succ",
" vecCons x v ⬝ᵥ w = x * vecHead w + v ⬝ᵥ vecTail w",
" v ⬝ᵥ vecCons x w = vecHead v * x + vecTail v ⬝ᵥ w"
] |
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
#align_import topology.uniform_space.uniform_embedding from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Filter Function Set Uniformity Topology
sec... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 116 | 118 | theorem UniformInducing.inducing {f : α → β} (h : UniformInducing f) : Inducing f := by |
obtain rfl := h.comap_uniformSpace
exact inducing_induced f
| [
" UniformInducing f ↔ UniformSpace.comap f inst✝¹ = inst✝²",
" (∀ (s : Set (α × α)), s ∈ comap (fun x => (f x.1, f x.2)) (𝓤 β) ↔ s ∈ 𝓤 α) ↔ ∀ (s : Set (α × α)), s ∈ 𝓤 α ↔ s ∈ 𝓤 α",
" UniformInducing f ↔ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α",
" 𝓤 α ≤ comap (fun x => (f x.1, f x.2)) (𝓤... | [
" UniformInducing f ↔ UniformSpace.comap f inst✝¹ = inst✝²",
" (∀ (s : Set (α × α)), s ∈ comap (fun x => (f x.1, f x.2)) (𝓤 β) ↔ s ∈ 𝓤 α) ↔ ∀ (s : Set (α × α)), s ∈ 𝓤 α ↔ s ∈ 𝓤 α",
" UniformInducing f ↔ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α",
" 𝓤 α ≤ comap (fun x => (f x.1, f x.2)) (𝓤... |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 79 | 81 | theorem invOf_eq [Invertible A.det] [Invertible A] : ⅟ A = ⅟ A.det • A.adjugate := by |
letI := invertibleOfDetInvertible A
convert (rfl : ⅟ A = _)
| [
" ⅟A.det • A.adjugate * A = 1",
" A * ⅟A.det • A.adjugate = 1",
" ⅟A = ⅟A.det • A.adjugate"
] | [
" ⅟A.det • A.adjugate * A = 1",
" A * ⅟A.det • A.adjugate = 1"
] |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
#align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c"
namespace Finset
variable {α : Type*}
@[simps]
protected def sym2 (s : Finset α) : Finset (Sym2 α) :... | Mathlib/Data/Finset/Sym.lean | 101 | 103 | theorem sym2_nonempty : s.sym2.Nonempty ↔ s.Nonempty := by |
rw [← not_iff_not]
simp_rw [not_nonempty_iff_eq_empty, sym2_eq_empty]
| [
" s(a, b) ∈ s.sym2 ↔ a ∈ s ∧ b ∈ s",
" m ∈ s.sym2 ↔ ∀ a ∈ m, a ∈ s",
" (∀ y ∈ m, y ∈ s.val) ↔ ∀ a ∈ m, a ∈ s",
" x ∈ univ.sym2",
" ∀ a ∈ x, a ∈ univ",
" univ.sym2 = univ",
" a✝ ∈ univ.sym2 ↔ a✝ ∈ univ",
" s.sym2 ⊆ t.sym2",
" s.val.sym2 ≤ t.val.sym2",
" s.val ≤ t.val",
" Function.Injective Finset... | [
" s(a, b) ∈ s.sym2 ↔ a ∈ s ∧ b ∈ s",
" m ∈ s.sym2 ↔ ∀ a ∈ m, a ∈ s",
" (∀ y ∈ m, y ∈ s.val) ↔ ∀ a ∈ m, a ∈ s",
" x ∈ univ.sym2",
" ∀ a ∈ x, a ∈ univ",
" univ.sym2 = univ",
" a✝ ∈ univ.sym2 ↔ a✝ ∈ univ",
" s.sym2 ⊆ t.sym2",
" s.val.sym2 ≤ t.val.sym2",
" s.val ≤ t.val",
" Function.Injective Finset... |
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional... | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 78 | 83 | theorem neg [AddGroup β] [TopologicalAddGroup β] {f : α → β} (hfm : AEStronglyMeasurable' m f μ) :
AEStronglyMeasurable' m (-f) μ := by |
rcases hfm with ⟨f', hf'_meas, hf_ae⟩
refine ⟨-f', hf'_meas.neg, hf_ae.mono fun x hx => ?_⟩
simp_rw [Pi.neg_apply]
rw [hx]
| [
" AEStronglyMeasurable' m g μ",
" AEStronglyMeasurable' m (f + g) μ",
" AEStronglyMeasurable' m (-f) μ",
" (-f) x = (-f') x",
" -f x = -f' x"
] | [
" AEStronglyMeasurable' m g μ",
" AEStronglyMeasurable' m (f + g) μ"
] |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 36 | 44 | theorem snorm'_add_le_of_le_one {f g : α → E} (hf : AEStronglyMeasurable f μ) (hq0 : 0 ≤ q)
(hq1 : q ≤ 1) : snorm' (f + g) q μ ≤ (2 : ℝ≥0∞) ^ (1 / q - 1) * (snorm' f q μ + snorm' g q μ) :=
calc
(∫⁻ a, (‖(f + g) a‖₊ : ℝ≥0∞) ^ q ∂μ) ^ (1 / q) ≤
(∫⁻ a, ((fun a => (‖f a‖₊ : ℝ≥0∞)) + fun a => (‖g a‖₊ : ℝ≥0... |
gcongr with a
simp only [Pi.add_apply, ← ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le]
_ ≤ (2 : ℝ≥0∞) ^ (1 / q - 1) * (snorm' f q μ + snorm' g q μ) :=
ENNReal.lintegral_Lp_add_le_of_le_one hf.ennnorm hq0 hq1
| [
" (∫⁻ (a : α), ↑‖(f + g) a‖₊ ^ q ∂μ) ^ (1 / q) ≤\n (∫⁻ (a : α), ((fun a => ↑‖f a‖₊) + fun a => ↑‖g a‖₊) a ^ q ∂μ) ^ (1 / q)",
" ↑‖(f + g) a‖₊ ≤ ((fun a => ↑‖f a‖₊) + fun a => ↑‖g a‖₊) a"
] | [
" (∫⁻ (a : α), ↑‖(f + g) a‖₊ ^ q ∂μ) ^ (1 / q) ≤\n (∫⁻ (a : α), ((fun a => ↑‖f a‖₊) + fun a => ↑‖g a‖₊) a ^ q ∂μ) ^ (1 / q)",
" ↑‖(f + g) a‖₊ ≤ ((fun a => ↑‖f a‖₊) + fun a => ↑‖g a‖₊) a"
] |
import Mathlib.Order.Filter.Lift
import Mathlib.Order.Filter.AtTopBot
#align_import order.filter.small_sets from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Filter
open Filter Set
variable {α β : Type*} {ι : Sort*}
namespace Filter
variable {l l' la : Filter α} {lb : Filter ... | Mathlib/Order/Filter/SmallSets.lean | 125 | 128 | theorem smallSets_comap_eq_comap_image (l : Filter β) (f : α → β) :
(comap f l).smallSets = comap (image f) l.smallSets := by |
refine (gc_map_comap _).u_comm_of_l_comm (gc_map_comap _) bind_smallSets_gc bind_smallSets_gc ?_
simp [Function.comp, map_bind, bind_map]
| [
" f.smallSets = generate (powerset '' f.sets)",
" f.lift' powerset = ⨅ b ∈ f.sets, 𝓟 (𝒫 b)",
" GaloisConnection (fun L => L.bind 𝓟) smallSets",
" (fun L => L.bind 𝓟) L ≤ l ↔ L ≤ l.smallSets",
" L.bind 𝓟 ≤ l ↔ l.sets ⊆ powerset ⁻¹' L.sets",
" Tendsto (fun x => f '' x) la.smallSets lb.smallSets ↔ Tends... | [
" f.smallSets = generate (powerset '' f.sets)",
" f.lift' powerset = ⨅ b ∈ f.sets, 𝓟 (𝒫 b)",
" GaloisConnection (fun L => L.bind 𝓟) smallSets",
" (fun L => L.bind 𝓟) L ≤ l ↔ L ≤ l.smallSets",
" L.bind 𝓟 ≤ l ↔ l.sets ⊆ powerset ⁻¹' L.sets",
" Tendsto (fun x => f '' x) la.smallSets lb.smallSets ↔ Tends... |
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 34 | 40 | theorem rank_le {n : ℕ}
(H : ∀ s : Finset M, (LinearIndependent R fun i : s => (i : M)) → s.card ≤ n) :
Module.rank R M ≤ n := by |
rw [Module.rank_def]
apply ciSup_le'
rintro ⟨s, li⟩
exact linearIndependent_bounded_of_finset_linearIndependent_bounded H _ li
| [
" Module.rank R M ≤ ↑n",
" ⨆ ι, #↑↑ι ≤ ↑n",
" ∀ (i : { s // LinearIndependent (ι := { x // x ∈ s }) R Subtype.val }), #↑↑i ≤ ↑n",
" #↑↑⟨s, li⟩ ≤ ↑n"
] | [] |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 30 | 51 | theorem TopologicalRing.of_norm {R 𝕜 : Type*} [NonUnitalNonAssocRing R] [LinearOrderedField 𝕜]
[TopologicalSpace R] [TopologicalAddGroup R] (norm : R → 𝕜)
(norm_nonneg : ∀ x, 0 ≤ norm x) (norm_mul_le : ∀ x y, norm (x * y) ≤ norm x * norm y)
(nhds_basis : (𝓝 (0 : R)).HasBasis ((0 : 𝕜) < ·) (fun ε ↦ { x ... |
have h0 : ∀ f : R → R, ∀ c ≥ (0 : 𝕜), (∀ x, norm (f x) ≤ c * norm x) →
Tendsto f (𝓝 0) (𝓝 0) := by
refine fun f c c0 hf ↦ (nhds_basis.tendsto_iff nhds_basis).2 fun ε ε0 ↦ ?_
rcases exists_pos_mul_lt ε0 c with ⟨δ, δ0, hδ⟩
refine ⟨δ, δ0, fun x hx ↦ (hf _).trans_lt ?_⟩
exact (mul_le_mul_of_nonn... | [
" TopologicalRing R",
" ∀ (f : R → R), ∀ c ≥ 0, (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)",
" ∃ ia, 0 < ia ∧ ∀ x ∈ {x | norm x < ia}, f x ∈ {x | norm x < ε}",
" c * norm x < ε",
" ∀ (x₀ : R), Tendsto (fun x => x * x₀) (𝓝 0) (𝓝 0)",
" Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 0 ×ˢ �... | [] |
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.LinearAlgebra.LinearPMap
#align_import analysis.convex.cone.basic from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
open Set LinearMap
variable {𝕜 E F G : Type*}
variable [AddCommGroup E... | Mathlib/Analysis/Convex/Cone/Extension.lean | 115 | 139 | theorem exists_top (p : E →ₗ.[ℝ] ℝ) (hp_nonneg : ∀ x : p.domain, (x : E) ∈ s → 0 ≤ p x)
(hp_dense : ∀ y, ∃ x : p.domain, (x : E) + y ∈ s) :
∃ q ≥ p, q.domain = ⊤ ∧ ∀ x : q.domain, (x : E) ∈ s → 0 ≤ q x := by |
set S := { p : E →ₗ.[ℝ] ℝ | ∀ x : p.domain, (x : E) ∈ s → 0 ≤ p x }
have hSc : ∀ c, c ⊆ S → IsChain (· ≤ ·) c → ∀ y ∈ c, ∃ ub ∈ S, ∀ z ∈ c, z ≤ ub := by
intro c hcs c_chain y hy
clear hp_nonneg hp_dense p
have cne : c.Nonempty := ⟨y, hy⟩
have hcd : DirectedOn (· ≤ ·) c := c_chain.directedOn
ref... | [
" ∃ g, f < g ∧ ∀ (x : ↥g.domain), ↑x ∈ s → 0 ≤ ↑g x",
" ∃ c, (∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c) ∧ ∀ (x : ↥f.domain), ↑x + y ∈ s → c ≤ ↑f x",
" (upperBounds Sn ∩ lowerBounds Sp).Nonempty",
" {x | -↑x - y ∈ s}.Nonempty",
" ∀ x ∈ Sn, ∀ y ∈ Sp, x ≤ y",
" ↑f xn ≤ ↑f xp",
" f < f.supSpanSingleton y (... | [
" ∃ g, f < g ∧ ∀ (x : ↥g.domain), ↑x ∈ s → 0 ≤ ↑g x",
" ∃ c, (∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c) ∧ ∀ (x : ↥f.domain), ↑x + y ∈ s → c ≤ ↑f x",
" (upperBounds Sn ∩ lowerBounds Sp).Nonempty",
" {x | -↑x - y ∈ s}.Nonempty",
" ∀ x ∈ Sn, ∀ y ∈ Sp, x ≤ y",
" ↑f xn ≤ ↑f xp",
" f < f.supSpanSingleton y (... |
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {α β γ : Type*}
class Fi... | Mathlib/Data/Fintype/Basic.lean | 113 | 114 | theorem univ_eq_empty_iff : (univ : Finset α) = ∅ ↔ IsEmpty α := by |
rw [← not_nonempty_iff, ← univ_nonempty_iff, not_nonempty_iff_eq_empty]
| [
" s = univ ↔ ∀ (x : α), x ∈ s",
" ↑univ = Set.univ",
" x✝ ∈ ↑univ ↔ x✝ ∈ Set.univ",
" ↑s = Set.univ ↔ s = univ",
" s.Nonempty → s = univ",
" s = univ",
" y ∈ s",
" univ.Nonempty ↔ Nonempty α",
" univ = ∅ ↔ IsEmpty α"
] | [
" s = univ ↔ ∀ (x : α), x ∈ s",
" ↑univ = Set.univ",
" x✝ ∈ ↑univ ↔ x✝ ∈ Set.univ",
" ↑s = Set.univ ↔ s = univ",
" s.Nonempty → s = univ",
" s = univ",
" y ∈ s",
" univ.Nonempty ↔ Nonempty α"
] |
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
#align_import topology.uniform_space.uniform_embedding from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Filter Function Set Uniformity Topology
sec... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 93 | 97 | theorem uniformInducing_of_compose {f : α → β} {g : β → γ} (hf : UniformContinuous f)
(hg : UniformContinuous g) (hgf : UniformInducing (g ∘ f)) : UniformInducing f := by |
refine ⟨le_antisymm ?_ hf.le_comap⟩
rw [← hgf.1, ← Prod.map_def, ← Prod.map_def, ← Prod.map_comp_map f f g g, ← comap_comap]
exact comap_mono hg.le_comap
| [
" UniformInducing f ↔ UniformSpace.comap f inst✝¹ = inst✝²",
" (∀ (s : Set (α × α)), s ∈ comap (fun x => (f x.1, f x.2)) (𝓤 β) ↔ s ∈ 𝓤 α) ↔ ∀ (s : Set (α × α)), s ∈ 𝓤 α ↔ s ∈ 𝓤 α",
" UniformInducing f ↔ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α",
" 𝓤 α ≤ comap (fun x => (f x.1, f x.2)) (𝓤... | [
" UniformInducing f ↔ UniformSpace.comap f inst✝¹ = inst✝²",
" (∀ (s : Set (α × α)), s ∈ comap (fun x => (f x.1, f x.2)) (𝓤 β) ↔ s ∈ 𝓤 α) ↔ ∀ (s : Set (α × α)), s ∈ 𝓤 α ↔ s ∈ 𝓤 α",
" UniformInducing f ↔ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α",
" 𝓤 α ≤ comap (fun x => (f x.1, f x.2)) (𝓤... |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 114 | 116 | theorem EuclideanSpace.norm_eq {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
(x : EuclideanSpace 𝕜 n) : ‖x‖ = √(∑ i, ‖x i‖ ^ 2) := by |
simpa only [Real.coe_sqrt, NNReal.coe_sum] using congr_arg ((↑) : ℝ≥0 → ℝ) x.nnnorm_eq
| [
" ‖x‖ ^ 2 = re ⟪x, x⟫_𝕜",
" ∀ (x y : PiLp 2 f), (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜",
" (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜",
" (starRingEnd 𝕜) ({ inner := fun x y => ∑ i : ι, InnerProductSpace.toInner.1 (x i) (y i) }.1 y x) =\n { inner := fun x y => ∑ i : ι, InnerProductSpace.toInner.1 (x i) (y i)... | [
" ‖x‖ ^ 2 = re ⟪x, x⟫_𝕜",
" ∀ (x y : PiLp 2 f), (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜",
" (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜",
" (starRingEnd 𝕜) ({ inner := fun x y => ∑ i : ι, InnerProductSpace.toInner.1 (x i) (y i) }.1 y x) =\n { inner := fun x y => ∑ i : ι, InnerProductSpace.toInner.1 (x i) (y i)... |
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
universe u v w
namespace MvPolynomial
variable {σ τ : Type*} {R : Type u} {S : Type v} {r : R} {e : ℕ} {n m : σ}
section CommSemiring
variable... | Mathlib/Algebra/MvPolynomial/Supported.lean | 91 | 92 | theorem mem_supported_vars (p : MvPolynomial σ R) : p ∈ supported R (↑p.vars : Set σ) := by |
rw [mem_supported]
| [
" supported R s = (rename Subtype.val).range",
" (aeval fun x => X ↑x).range = (aeval (X ∘ Subtype.val)).range",
" (supportedEquivMvPolynomial s).symm (C x) = (algebraMap R ↥(supported R s)) x",
" ↑((supportedEquivMvPolynomial s).symm (C x)) = ↑((algebraMap R ↥(supported R s)) x)",
" ↑((supportedEquivMvPoly... | [
" supported R s = (rename Subtype.val).range",
" (aeval fun x => X ↑x).range = (aeval (X ∘ Subtype.val)).range",
" (supportedEquivMvPolynomial s).symm (C x) = (algebraMap R ↥(supported R s)) x",
" ↑((supportedEquivMvPolynomial s).symm (C x)) = ↑((algebraMap R ↥(supported R s)) x)",
" ↑((supportedEquivMvPoly... |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 214 | 217 | theorem pnatCast_eq_natCast [Fact (∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I))] (n : ℕ+) :
((n : Rˣ) : R) = ↑n := by |
change ((PNat.isUnit_natCast (R := R) n).unit : R) = ↑n
simp only [IsUnit.unit_spec]
| [
" ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)",
" CharZero (R ⧸ I)",
" Function.Injective Nat.cast",
" a = b",
" I = ⊤",
" (algebraMap ℚ R) (↑a - ↑b) ∈ I",
" ↑a - ↑b ≠ 0",
" IsUnit ↑↑n",
" Ideal.span {↑↑n} = ⊤",
" ¬CharZero (R ⧸ Ideal.span {↑↑n})",
" False",
" ↑↑n = ↑0",
" ↑↑n ∈ Ideal.span {↑↑... | [
" ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)",
" CharZero (R ⧸ I)",
" Function.Injective Nat.cast",
" a = b",
" I = ⊤",
" (algebraMap ℚ R) (↑a - ↑b) ∈ I",
" ↑a - ↑b ≠ 0",
" IsUnit ↑↑n",
" Ideal.span {↑↑n} = ⊤",
" ¬CharZero (R ⧸ Ideal.span {↑↑n})",
" False",
" ↑↑n = ↑0",
" ↑↑n ∈ Ideal.span {↑↑... |
import Mathlib.Algebra.Group.WithOne.Defs
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.GroupWithZero.WithZero
import Mathlib.Algebra.Order.Group.Units
import Mathlib.Algebra.Order.GroupWithZero.Synonym
import Mathlib.Algebra.Order.Monoid.Basic
imp... | Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 128 | 129 | theorem le_of_le_mul_right (h : c ≠ 0) (hab : a * c ≤ b * c) : a ≤ b := by |
simpa only [mul_inv_cancel_right₀ h] using mul_le_mul_right' hab c⁻¹
| [
" a ≤ b"
] | [] |
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.PeakFunction
#align_import analysis.special_functions.trigonometric.euler_sine_prod from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open scoped Real Topology
open Real Set Filter intervalIntegra... | Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | 59 | 85 | theorem integral_cos_mul_cos_pow_aux (hn : 2 ≤ n) (hz : z ≠ 0) :
(∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ n) =
n / (2 * z) *
∫ x in (0 : ℝ)..π / 2, Complex.sin (2 * z * x) * sin x * (cos x : ℂ) ^ (n - 1) := by |
have der1 :
∀ x : ℝ,
x ∈ uIcc 0 (π / 2) →
HasDerivAt (fun y : ℝ => (cos y : ℂ) ^ n) (-n * sin x * (cos x : ℂ) ^ (n - 1)) x := by
intro x _
have b : HasDerivAt (fun y : ℝ => (cos y : ℂ)) (-sin x) x := by
simpa using (hasDerivAt_cos x).ofReal_comp
convert HasDerivAt.comp x (hasDeriv... | [
" HasDerivAt (fun y => (2 * z * ↑y).sin / (2 * z)) (2 * z * ↑x).cos x",
" HasDerivAt (fun y => -(2 * z * ↑y).cos / (2 * z)) (2 * z * ↑x).sin x",
" ∫ (x : ℝ) in 0 ..π / 2, (2 * z * ↑x).cos * ↑x.cos ^ n =\n ↑n / (2 * z) * ∫ (x : ℝ) in 0 ..π / 2, (2 * z * ↑x).sin * ↑x.sin * ↑x.cos ^ (n - 1)",
" ∀ x ∈ uIcc 0 (... | [
" HasDerivAt (fun y => (2 * z * ↑y).sin / (2 * z)) (2 * z * ↑x).cos x",
" HasDerivAt (fun y => -(2 * z * ↑y).cos / (2 * z)) (2 * z * ↑x).sin x"
] |
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α}
namespace Finset
def nonMemberSubfamily (a : α) (𝒜 : ... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 114 | 116 | theorem memberSubfamily_memberSubfamily : (𝒜.memberSubfamily a).memberSubfamily a = ∅ := by |
ext
simp
| [
" s ∈ nonMemberSubfamily a 𝒜 ↔ s ∈ 𝒜 ∧ a ∉ s",
" s ∈ memberSubfamily a 𝒜 ↔ insert a s ∈ 𝒜 ∧ a ∉ s",
" (∃ a_1, (a_1 ∈ 𝒜 ∧ a ∈ a_1) ∧ a_1.erase a = s) ↔ insert a s ∈ 𝒜 ∧ a ∉ s",
" a ∈ insert a s",
" (∃ a_1, (a_1 ∈ 𝒜 ∧ a ∈ a_1) ∧ a_1.erase a = s) → insert a s ∈ 𝒜 ∧ a ∉ s",
" insert a (s.erase a) ∈ 𝒜... | [
" s ∈ nonMemberSubfamily a 𝒜 ↔ s ∈ 𝒜 ∧ a ∉ s",
" s ∈ memberSubfamily a 𝒜 ↔ insert a s ∈ 𝒜 ∧ a ∉ s",
" (∃ a_1, (a_1 ∈ 𝒜 ∧ a ∈ a_1) ∧ a_1.erase a = s) ↔ insert a s ∈ 𝒜 ∧ a ∉ s",
" a ∈ insert a s",
" (∃ a_1, (a_1 ∈ 𝒜 ∧ a ∈ a_1) ∧ a_1.erase a = s) → insert a s ∈ 𝒜 ∧ a ∉ s",
" insert a (s.erase a) ∈ 𝒜... |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ι : Sort*} {α β X Y : Type*}... | Mathlib/Topology/Filter.lean | 125 | 125 | theorem nhds_top : 𝓝 (⊤ : Filter α) = ⊤ := by | simp [nhds_eq]
| [
" IsOpen {l | s ∈ l}",
" ∀ t₁ ∈ range (Iic ∘ 𝓟), ∀ t₂ ∈ range (Iic ∘ 𝓟), ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ range (Iic ∘ 𝓟), x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂",
" ∃ t₃ ∈ range (Iic ∘ 𝓟), l ∈ t₃ ∧ t₃ ⊆ (Iic ∘ 𝓟) s ∩ (Iic ∘ 𝓟) t",
" (Iic ∘ 𝓟) (s ∩ t) = Iic (𝓟 s) ∩ Iic (𝓟 t)",
" (∃ S ⊆ range (Iic ∘ 𝓟), s = ⋃₀ S) ↔ ∃ T, s = ⋃... | [
" IsOpen {l | s ∈ l}",
" ∀ t₁ ∈ range (Iic ∘ 𝓟), ∀ t₂ ∈ range (Iic ∘ 𝓟), ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ range (Iic ∘ 𝓟), x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂",
" ∃ t₃ ∈ range (Iic ∘ 𝓟), l ∈ t₃ ∧ t₃ ⊆ (Iic ∘ 𝓟) s ∩ (Iic ∘ 𝓟) t",
" (Iic ∘ 𝓟) (s ∩ t) = Iic (𝓟 s) ∩ Iic (𝓟 t)",
" (∃ S ⊆ range (Iic ∘ 𝓟), s = ⋃₀ S) ↔ ∃ T, s = ⋃... |
import Mathlib.Tactic.Ring.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Tactic.Conv
import Mathlib.Util.Qq
set_option autoImplicit true
-- In this file we would like to be able to use multi-character auto-implicits.
set_option relaxedAutoImplicit true
namespace Mathlib.Tactic
open Lean hiding Rat
open Qq Me... | Mathlib/Tactic/Ring/RingNF.lean | 123 | 123 | theorem int_rawCast_neg {R} [Ring R] : (Int.rawCast (.negOfNat n) : R) = -Nat.rawCast n := by | simp
| [
" a * -b = -(a * b)",
" Nat.rawCast 0 = 0",
" Nat.rawCast 1 = 1",
" (Int.negOfNat n).rawCast = -n.rawCast"
] | [
" a * -b = -(a * b)",
" Nat.rawCast 0 = 0",
" Nat.rawCast 1 = 1"
] |
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import data.zmod.coprime from "leanprover-community/mathlib"@"4b4975cf92a1ffe2ddfeff6ff91b0c46a9162bf5"
namespace ZMod
| Mathlib/Data/ZMod/Coprime.lean | 24 | 28 | theorem eq_zero_iff_gcd_ne_one {a : ℤ} {p : ℕ} [pp : Fact p.Prime] :
(a : ZMod p) = 0 ↔ a.gcd p ≠ 1 := by |
rw [Ne, Int.gcd_comm, Int.gcd_eq_one_iff_coprime,
(Nat.prime_iff_prime_int.1 pp.1).coprime_iff_not_dvd, Classical.not_not,
intCast_zmod_eq_zero_iff_dvd]
| [
" ↑a = 0 ↔ a.gcd ↑p ≠ 1"
] | [] |
import Mathlib.GroupTheory.Archimedean
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.archimedean from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set
theorem Rat.denseRange_cast {𝕜} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
... | Mathlib/Topology/Algebra/Order/Archimedean.lean | 58 | 62 | theorem dense_of_no_min (S : AddSubgroup G) (hbot : S ≠ ⊥)
(H : ¬∃ a : G, IsLeast { g : G | g ∈ S ∧ 0 < g } a) : Dense (S : Set G) := by |
refine S.dense_of_not_isolated_zero fun ε ε0 => ?_
contrapose! H
exact exists_isLeast_pos hbot ε0 (disjoint_left.2 H)
| [
" Dense ↑S",
" x ∈ ↑S",
" 0 ∈ ↑S",
" ∃ c ∈ ↑S, a < c ∧ c < b",
" a + g < b",
" ∃ g ∈ S, g ∈ Ioo 0 ε",
" ∃ a, IsLeast {g | g ∈ S ∧ 0 < g} a"
] | [
" Dense ↑S",
" x ∈ ↑S",
" 0 ∈ ↑S",
" ∃ c ∈ ↑S, a < c ∧ c < b",
" a + g < b"
] |
import Mathlib.Combinatorics.SimpleGraph.Clique
open Finset
namespace SimpleGraph
variable {V : Type*} [Fintype V] [DecidableEq V] (G H : SimpleGraph V) [DecidableRel G.Adj]
{n r : ℕ}
def IsTuranMaximal (r : ℕ) : Prop :=
G.CliqueFree (r + 1) ∧ ∀ (H : SimpleGraph V) [DecidableRel H.Adj],
H.CliqueFree (r +... | Mathlib/Combinatorics/SimpleGraph/Turan.lean | 84 | 92 | theorem not_cliqueFree_of_isTuranMaximal (hn : r ≤ Fintype.card V) (hG : G.IsTuranMaximal r) :
¬G.CliqueFree r := by |
rintro h
obtain ⟨K, _, rfl⟩ := exists_smaller_set (univ : Finset V) r hn
obtain ⟨a, -, b, -, hab, hGab⟩ : ∃ a ∈ K, ∃ b ∈ K, a ≠ b ∧ ¬ G.Adj a b := by
simpa only [isNClique_iff, IsClique, Set.Pairwise, mem_coe, ne_eq, and_true, not_forall,
exists_prop, exists_and_right] using h K
exact hGab <| le_sup_... | [
" G ≤ H ↔ G = H",
" DecidableRel (turanGraph n r).Adj",
" DecidableRel fun v w => ↑v % r ≠ ↑w % r",
" turanGraph n 0 = ⊤",
" (turanGraph n 0).Adj a b ↔ ⊤.Adj a b",
" turanGraph n r = ⊤ ↔ r = 0 ∨ n ≤ r",
" (∀ (a a_1 : Fin n), ↑a % r = ↑a_1 % r ↔ a = a_1) ↔ r = 0 ∨ n ≤ r",
" r = 0 ∨ n ≤ r",
" ∃ a a_1,... | [
" G ≤ H ↔ G = H",
" DecidableRel (turanGraph n r).Adj",
" DecidableRel fun v w => ↑v % r ≠ ↑w % r",
" turanGraph n 0 = ⊤",
" (turanGraph n 0).Adj a b ↔ ⊤.Adj a b",
" turanGraph n r = ⊤ ↔ r = 0 ∨ n ≤ r",
" (∀ (a a_1 : Fin n), ↑a % r = ↑a_1 % r ↔ a = a_1) ↔ r = 0 ∨ n ≤ r",
" r = 0 ∨ n ≤ r",
" ∃ a a_1,... |
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
universe u v w
open scoped Classical Polynomial
open Polynomial
vari... | Mathlib/FieldTheory/IsAlgClosed/Basic.lean | 68 | 69 | theorem IsAlgClosed.splits_codomain {k K : Type*} [Field k] [IsAlgClosed k] [Field K] {f : K →+* k}
(p : K[X]) : p.Splits f := by | convert IsAlgClosed.splits (p.map f); simp [splits_map_iff]
| [
" Splits f p",
" Splits f p ↔ Splits (RingHom.id k) (map f p)"
] | [] |
import Mathlib.Data.Finset.Sort
import Mathlib.Data.List.FinRange
import Mathlib.Data.Prod.Lex
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.Order.Interval.Finset.Fin
#align_import data.fin.tuple.sort from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
namespace Tuple
variable {... | Mathlib/Data/Fin/Tuple/Sort.lean | 50 | 57 | theorem graph.card (f : Fin n → α) : (graph f).card = n := by |
rw [graph, Finset.card_image_of_injective]
· exact Finset.card_fin _
· intro _ _
-- porting note (#10745): was `simp`
dsimp only
rw [Prod.ext_iff]
simp
| [
" (graph f).card = n",
" Finset.univ.card = n",
" Function.Injective fun i => (f i, i)",
" (fun i => (f i, i)) a₁✝ = (fun i => (f i, i)) a₂✝ → a₁✝ = a₂✝",
" (f a₁✝, a₁✝) = (f a₂✝, a₂✝) → a₁✝ = a₂✝",
" (f a₁✝, a₁✝).1 = (f a₂✝, a₂✝).1 ∧ (f a₁✝, a₁✝).2 = (f a₂✝, a₂✝).2 → a₁✝ = a₂✝"
] | [] |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 121 | 124 | theorem snd : IsBoundedLinearMap 𝕜 fun x : E × F => x.2 := by |
refine (LinearMap.snd 𝕜 E F).isLinear.with_bound 1 fun x => ?_
rw [one_mul]
exact le_max_right _ _
| [
" ∀ (x : E), ‖0 x‖ ≤ 0 * ‖x‖",
" ∀ (x : E), ‖LinearMap.id x‖ ≤ 1 * ‖x‖",
" IsBoundedLinearMap 𝕜 fun x => x.1",
" ‖(LinearMap.fst 𝕜 E F) x‖ ≤ 1 * ‖x‖",
" ‖(LinearMap.fst 𝕜 E F) x‖ ≤ ‖x‖",
" IsBoundedLinearMap 𝕜 fun x => x.2",
" ‖(LinearMap.snd 𝕜 E F) x‖ ≤ 1 * ‖x‖",
" ‖(LinearMap.snd 𝕜 E F) x‖ ≤ ‖... | [
" ∀ (x : E), ‖0 x‖ ≤ 0 * ‖x‖",
" ∀ (x : E), ‖LinearMap.id x‖ ≤ 1 * ‖x‖",
" IsBoundedLinearMap 𝕜 fun x => x.1",
" ‖(LinearMap.fst 𝕜 E F) x‖ ≤ 1 * ‖x‖",
" ‖(LinearMap.fst 𝕜 E F) x‖ ≤ ‖x‖"
] |
import Mathlib.Algebra.Category.ModuleCat.Abelian
import Mathlib.CategoryTheory.Limits.Shapes.Images
#align_import algebra.category.Module.images from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open CategoryTheory
open CategoryTheory.Limits
universe u v
namespace ModuleCat
set_op... | Mathlib/Algebra/Category/ModuleCat/Images.lean | 117 | 119 | theorem imageIsoRange_hom_subtype {G H : ModuleCat.{v} R} (f : G ⟶ H) :
(imageIsoRange f).hom ≫ ModuleCat.ofHom f.range.subtype = Limits.image.ι f := by |
erw [← imageIsoRange_inv_image_ι f, Iso.hom_inv_id_assoc]
| [
" (fun x => F'.e ↑(Classical.indefiniteDescription (fun x_1 => f x_1 = ↑x) ⋯)) (x + y) =\n (fun x => F'.e ↑(Classical.indefiniteDescription (fun x_1 => f x_1 = ↑x) ⋯)) x +\n (fun x => F'.e ↑(Classical.indefiniteDescription (fun x_1 => f x_1 = ↑x) ⋯)) y",
" Mono F'.m",
" F'.m ((fun x => F'.e ↑(Classical.... | [
" (fun x => F'.e ↑(Classical.indefiniteDescription (fun x_1 => f x_1 = ↑x) ⋯)) (x + y) =\n (fun x => F'.e ↑(Classical.indefiniteDescription (fun x_1 => f x_1 = ↑x) ⋯)) x +\n (fun x => F'.e ↑(Classical.indefiniteDescription (fun x_1 => f x_1 = ↑x) ⋯)) y",
" Mono F'.m",
" F'.m ((fun x => F'.e ↑(Classical.... |
import Mathlib.CategoryTheory.Galois.Basic
import Mathlib.RepresentationTheory.Action.Basic
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Logic.Equiv.... | Mathlib/CategoryTheory/Galois/Examples.lean | 127 | 145 | theorem Action.isConnected_of_transitive (X : FintypeCat) [MulAction G X]
[MulAction.IsPretransitive G X] [h : Nonempty X] :
IsConnected (Action.FintypeCat.ofMulAction G X) where
notInitial := not_initial_of_inhabited (Action.forget _ _) h.some
noTrivialComponent Y i hm hni := by |
/- We show that the induced inclusion `i.hom` of finite sets is surjective, using the
transitivity of the `G`-action. -/
obtain ⟨(y : Y.V)⟩ := (not_initial_iff_fiber_nonempty (Action.forget _ _) Y).mp hni
have : IsIso i.hom := by
refine (ConcreteCategory.isIso_iff_bijective i.hom).mpr ⟨?_, fun x'... | [
" FintypeCat",
" Y.ρ g ↑y ∈ (Set.range f.hom)ᶜ",
" False",
" ↑y ∈ Set.range f.hom",
" f.hom (X.ρ g⁻¹ x) = ↑y",
" (X.ρ g⁻¹ ≫ f.hom) x = (Y.ρ g⁻¹ * Y.ρ g) ↑y",
" Y.ρ g⁻¹ (Y.ρ g ↑y) = (Y.ρ g⁻¹ * Y.ρ g) ↑y",
" (Y.ρ g⁻¹ * Y.ρ g) ↑y = ↑y",
" (fun g y => ⟨Y.ρ g ↑y, ⋯⟩) 1 = 1",
" (fun y => ⟨𝟙 Y.V ↑y, ⋯⟩)... | [
" FintypeCat",
" Y.ρ g ↑y ∈ (Set.range f.hom)ᶜ",
" False",
" ↑y ∈ Set.range f.hom",
" f.hom (X.ρ g⁻¹ x) = ↑y",
" (X.ρ g⁻¹ ≫ f.hom) x = (Y.ρ g⁻¹ * Y.ρ g) ↑y",
" Y.ρ g⁻¹ (Y.ρ g ↑y) = (Y.ρ g⁻¹ * Y.ρ g) ↑y",
" (Y.ρ g⁻¹ * Y.ρ g) ↑y = ↑y",
" (fun g y => ⟨Y.ρ g ↑y, ⋯⟩) 1 = 1",
" (fun y => ⟨𝟙 Y.V ↑y, ⋯⟩)... |
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace List
variable {α β : Type*}
section FormPerm
variable [DecidableEq α] (l :... | Mathlib/GroupTheory/Perm/List.lean | 95 | 97 | theorem support_formPerm_le' : { x | formPerm l x ≠ x } ≤ l.toFinset := by |
refine (zipWith_swap_prod_support' l l.tail).trans ?_
simpa [Finset.subset_iff] using tail_subset l
| [
" (zipWith swap [] x✝¹).prod x✝ ≠ x✝ → x✝ ∈ [] ∨ x✝ ∈ x✝¹",
" (zipWith swap x✝¹ []).prod x✝ ≠ x✝ → x✝ ∈ x✝¹ ∨ x✝ ∈ []",
" (swap (?m.1920 a l b l' x hx h) (?m.1921 a l b l' x hx h)) (?m.1919 a l b l' x hx h) ≠ ?m.1919 a l b l' x hx h",
" x = a → x ∈ a :: l",
" x ∈ x :: l",
" x = b → x ∈ b :: l'",
" x ∈ x... | [
" (zipWith swap [] x✝¹).prod x✝ ≠ x✝ → x✝ ∈ [] ∨ x✝ ∈ x✝¹",
" (zipWith swap x✝¹ []).prod x✝ ≠ x✝ → x✝ ∈ x✝¹ ∨ x✝ ∈ []",
" (swap (?m.1920 a l b l' x hx h) (?m.1921 a l b l' x hx h)) (?m.1919 a l b l' x hx h) ≠ ?m.1919 a l b l' x hx h",
" x = a → x ∈ a :: l",
" x ∈ x :: l",
" x = b → x ∈ b :: l'",
" x ∈ x... |
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Cardinality
#align_import data.complex.cardinality from "leanprover-community/mathlib"@"1c4e18434eeb5546b212e830b2b39de6a83c473c"
-- Porting note: the lemmas `mk_complex` and `mk_univ_complex` should be in the namespace `Cardinal`
-- like their real counter... | Mathlib/Data/Complex/Cardinality.lean | 31 | 31 | theorem mk_univ_complex : #(Set.univ : Set ℂ) = 𝔠 := by | rw [mk_univ, mk_complex]
| [
" #ℂ = 𝔠",
" #↑Set.univ = 𝔠"
] | [
" #ℂ = 𝔠"
] |
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib"@"347636a7a80595d55bedf6e6fbd996a3c39da69a"
universe u v w
open Cardinal Submodule Module Function
variable {R : Type u} {M : Type v} {N : Type w}
variable... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 72 | 75 | theorem finrank_le_of_rank_le {n : ℕ} (h : Module.rank R M ≤ ↑n) : finrank R M ≤ n := by |
rwa [← Cardinal.toNat_le_iff_le_of_lt_aleph0, toNat_natCast] at h
· exact h.trans_lt (nat_lt_aleph0 n)
· exact nat_lt_aleph0 n
| [
" finrank R M = n",
" finrank R M ≤ n",
" Module.rank R M < ℵ₀",
" ↑n < ℵ₀"
] | [
" finrank R M = n"
] |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
... | Mathlib/RingTheory/PowerSeries/Basic.lean | 150 | 151 | theorem coeff_def {s : Unit →₀ ℕ} {n : ℕ} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by |
erw [coeff, ← h, ← Finsupp.unique_single s]
| [
" Inhabited R⟦X⟧",
" Inhabited (MvPowerSeries Unit R)",
" Zero R⟦X⟧",
" Zero (MvPowerSeries Unit R)",
" AddMonoid R⟦X⟧",
" AddMonoid (MvPowerSeries Unit R)",
" AddGroup R⟦X⟧",
" AddGroup (MvPowerSeries Unit R)",
" AddCommMonoid R⟦X⟧",
" AddCommMonoid (MvPowerSeries Unit R)",
" AddCommGroup R⟦X⟧"... | [
" Inhabited R⟦X⟧",
" Inhabited (MvPowerSeries Unit R)",
" Zero R⟦X⟧",
" Zero (MvPowerSeries Unit R)",
" AddMonoid R⟦X⟧",
" AddMonoid (MvPowerSeries Unit R)",
" AddGroup R⟦X⟧",
" AddGroup (MvPowerSeries Unit R)",
" AddCommMonoid R⟦X⟧",
" AddCommMonoid (MvPowerSeries Unit R)",
" AddCommGroup R⟦X⟧"... |
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc5d9d86bcd1bba10f480e948a257d65056f"
#align_import init.data.int.bitwise from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
... | Mathlib/Data/Int/Bitwise.lean | 159 | 167 | theorem bodd_neg (n : ℤ) : bodd (-n) = bodd n := by |
cases n with
| ofNat =>
rw [← negOfNat_eq, bodd_negOfNat]
simp
| negSucc n =>
rw [neg_negSucc, bodd_coe, Nat.bodd_succ]
change (!Nat.bodd n) = !(bodd n)
rw [bodd_coe]
| [
" (subNatNat m n).bodd = xor m.bodd n.bodd",
" ∀ (i n : ℕ), (↑i).bodd = xor (n + i).bodd n.bodd",
" ∀ (i m : ℕ), -[i+1].bodd = xor m.bodd (m + i + 1).bodd",
" (↑i).bodd = xor (j + i).bodd j.bodd",
" -[i+1].bodd = xor j.bodd (j + i + 1).bodd",
" i.bodd = xor (xor j.bodd i.bodd) j.bodd",
" (!i.bodd) = xor... | [
" (subNatNat m n).bodd = xor m.bodd n.bodd",
" ∀ (i n : ℕ), (↑i).bodd = xor (n + i).bodd n.bodd",
" ∀ (i m : ℕ), -[i+1].bodd = xor m.bodd (m + i + 1).bodd",
" (↑i).bodd = xor (j + i).bodd j.bodd",
" -[i+1].bodd = xor j.bodd (j + i + 1).bodd",
" i.bodd = xor (xor j.bodd i.bodd) j.bodd",
" (!i.bodd) = xor... |
import Mathlib.FieldTheory.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
open FiniteDimensional Polynomial IntermediateField Field
noncomputable section
universe u v w
variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
variable (K : Type w) [Field K] [Algebra F K]
section IsP... | Mathlib/FieldTheory/PurelyInseparable.lean | 230 | 243 | theorem isPurelyInseparable_iff_pow_mem (q : ℕ) [ExpChar F q] :
IsPurelyInseparable F E ↔ ∀ x : E, ∃ n : ℕ, x ^ q ^ n ∈ (algebraMap F E).range := by |
rw [isPurelyInseparable_iff]
refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩
· obtain ⟨g, h1, n, h2⟩ := (minpoly.irreducible (h x).1).hasSeparableContraction q
exact ⟨n, (h _).2 <| h1.of_dvd <| minpoly.dvd F _ <| by
simpa only [expand_aeval, minpoly.aeval] using congr_arg (aeval x) h2⟩
have hdeg := (minpoly.natS... | [
" IsPurelyInseparable F E",
" IsIntegral F x✝",
" IsIntegral F (e.symm x✝)",
" x ∈ (algebraMap F E).range",
" x ∈ ⊥",
" IsPurelyInseparable F E ↔ ∀ (x : E), ∃ n, x ^ q ^ n ∈ (algebraMap F E).range",
" (∀ (x : E), IsIntegral F x ∧ ((minpoly F x).Separable → x ∈ (algebraMap F E).range)) ↔\n ∀ (x : E), ... | [
" IsPurelyInseparable F E",
" IsIntegral F x✝",
" IsIntegral F (e.symm x✝)",
" x ∈ (algebraMap F E).range",
" x ∈ ⊥"
] |
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.multivariate.basic from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
universe u v
open MvFunctor
@[pp_with_univ]
structure MvPFunctor (n : ℕ) where
A : Type u
... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 160 | 170 | theorem liftP_iff {α : TypeVec n} (p : ∀ ⦃i⦄, α i → Prop) (x : P α) :
LiftP p x ↔ ∃ a f, x = ⟨a, f⟩ ∧ ∀ i j, p (f i j) := by |
constructor
· rintro ⟨y, hy⟩
cases' h : y with a f
refine ⟨a, fun i j => (f i j).val, ?_, fun i j => (f i j).property⟩
rw [← hy, h, map_eq]
rfl
rintro ⟨a, f, xeq, pf⟩
use ⟨a, fun i j => ⟨f i j, pf i j⟩⟩
rw [xeq]; rfl
| [
" get (f <$$> x) = (fun i x => f <$$> x) <$$> get x",
" get (mk x) = x",
" mk (get x) = x",
" LiftP p x ↔ ∃ a f, x = ⟨a, f⟩ ∧ ∀ (i : Fin2 n) (j : P.B a i), p (f i j)",
" LiftP p x → ∃ a f, x = ⟨a, f⟩ ∧ ∀ (i : Fin2 n) (j : P.B a i), p (f i j)",
" ∃ a f, x = ⟨a, f⟩ ∧ ∀ (i : Fin2 n) (j : P.B a i), p (f i j)"... | [
" get (f <$$> x) = (fun i x => f <$$> x) <$$> get x",
" get (mk x) = x",
" mk (get x) = x"
] |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingleton
#align_import combinatorics.composition from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open List
variable {n : ℕ}
... | Mathlib/Combinatorics/Enumerative/Composition.lean | 870 | 872 | theorem length_lt_card_boundaries : c.length < c.boundaries.card := by |
rw [c.card_boundaries_eq_succ_length]
exact lt_add_one _
| [
" 1 + ↑i < n.succ",
" 1 + (n - 1) ≤ n.succ",
" n - 1 + 1 ≤ n + 1",
" 0 ∈ {i | i = 0 ∨ i = Fin.last n ∨ ∃ j, ∃ (_ : j ∈ s), ↑i = ↑j + 1}.toFinset",
" Fin.last n ∈ {i | i = 0 ∨ i = Fin.last n ∨ ∃ j, ∃ (_ : j ∈ s), ↑i = ↑j + 1}.toFinset",
" Function.LeftInverse\n (fun s =>\n { boundaries := {i | i = ... | [
" 1 + ↑i < n.succ",
" 1 + (n - 1) ≤ n.succ",
" n - 1 + 1 ≤ n + 1",
" 0 ∈ {i | i = 0 ∨ i = Fin.last n ∨ ∃ j, ∃ (_ : j ∈ s), ↑i = ↑j + 1}.toFinset",
" Fin.last n ∈ {i | i = 0 ∨ i = Fin.last n ∨ ∃ j, ∃ (_ : j ∈ s), ↑i = ↑j + 1}.toFinset",
" Function.LeftInverse\n (fun s =>\n { boundaries := {i | i = ... |
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
#align_import combinatorics.quiver.single_obj from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
namespace Quiver
-- Porting note: Removed `deriving Unique`.
@[nolint unusedArguments]
def SingleObj ... | Mathlib/Combinatorics/Quiver/SingleObj.lean | 110 | 112 | theorem toPrefunctor_symm_comp (f : SingleObj α ⥤q SingleObj β) (g : SingleObj β ⥤q SingleObj γ) :
toPrefunctor.symm (f ⋙q g) = toPrefunctor.symm g ∘ toPrefunctor.symm f := by |
simp only [Equiv.symm_apply_eq, toPrefunctor_comp, Equiv.apply_symm_apply]
| [
" toPrefunctor.symm (f ⋙q g) = toPrefunctor.symm g ∘ toPrefunctor.symm f"
] | [] |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 140 | 145 | theorem natDegree_pow (p : R[X]) (n : ℕ) : natDegree (p ^ n) = n * natDegree p := by |
classical
obtain rfl | hp := eq_or_ne p 0
· obtain rfl | hn := eq_or_ne n 0 <;> simp [*]
exact natDegree_pow' $ by
rw [← leadingCoeff_pow, Ne, leadingCoeff_eq_zero]; exact pow_ne_zero _ hp
| [
" a✝ = 0 ∨ b✝ = 0",
" a✝.leadingCoeff = 0 ∨ b✝.leadingCoeff = 0",
" a✝.leadingCoeff * b✝.leadingCoeff = 0",
" (p * q).natDegree = p.natDegree + q.natDegree",
" (p * q).trailingDegree = p.trailingDegree + q.trailingDegree",
" ↑(p.natTrailingDegree + q.natTrailingDegree) = ↑p.natTrailingDegree + ↑q.natTrail... | [
" a✝ = 0 ∨ b✝ = 0",
" a✝.leadingCoeff = 0 ∨ b✝.leadingCoeff = 0",
" a✝.leadingCoeff * b✝.leadingCoeff = 0",
" (p * q).natDegree = p.natDegree + q.natDegree",
" (p * q).trailingDegree = p.trailingDegree + q.trailingDegree",
" ↑(p.natTrailingDegree + q.natTrailingDegree) = ↑p.natTrailingDegree + ↑q.natTrail... |
import Mathlib.Data.List.Join
#align_import data.list.permutation from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we don't import algebra
assert_not_exists Monoid
open Nat
variable {α β : Type*}
namespace List
theorem permutationsAux2_fst (t : α) (ts : List α) (r : L... | Mathlib/Data/List/Permutation.lean | 77 | 79 | theorem permutationsAux2_append (t : α) (ts : List α) (r : List β) (ys : List α) (f : List α → β) :
(permutationsAux2 t ts nil ys f).2 ++ r = (permutationsAux2 t ts r ys f).2 := by |
induction ys generalizing f <;> simp [*]
| [
" (permutationsAux2 t ts r (y :: ys) f).1 = y :: ys ++ ts",
" (permutationsAux2 t ts r (y :: ys) f).2 = f (t :: y :: ys ++ ts) :: (permutationsAux2 t ts r ys fun x => f (y :: x)).2",
" (permutationsAux2 t ts [] ys f).2 ++ r = (permutationsAux2 t ts r ys f).2",
" (permutationsAux2 t ts [] [] f).2 ++ r = (permu... | [
" (permutationsAux2 t ts r (y :: ys) f).1 = y :: ys ++ ts",
" (permutationsAux2 t ts r (y :: ys) f).2 = f (t :: y :: ys ++ ts) :: (permutationsAux2 t ts r ys fun x => f (y :: x)).2"
] |
import Mathlib.Geometry.Manifold.ContMDiff.Defs
open Set Filter Function
open scoped Topology Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
(I : Mode... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 262 | 262 | theorem smooth_one [One M'] : Smooth I I' (1 : M → M') := by | simp only [Pi.one_def, smooth_const]
| [
" ContMDiff I I' n fun x => c",
" ContMDiffAt I I' n (fun x => c) x",
" ContDiffWithinAtProp I I' n (↑(chartAt H' c) ∘ (fun x => c) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' univ)\n (↑(chartAt H x) x)",
" ContDiffWithinAt 𝕜 n (fun x => ↑I' (↑(chartAt H' c) c)) (↑I.symm ⁻¹' (↑(chartAt H x).symm ⁻¹' u... | [
" ContMDiff I I' n fun x => c",
" ContMDiffAt I I' n (fun x => c) x",
" ContDiffWithinAtProp I I' n (↑(chartAt H' c) ∘ (fun x => c) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' univ)\n (↑(chartAt H x) x)",
" ContDiffWithinAt 𝕜 n (fun x => ↑I' (↑(chartAt H' c) c)) (↑I.symm ⁻¹' (↑(chartAt H x).symm ⁻¹' u... |
import Mathlib.Analysis.InnerProductSpace.Adjoint
#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
open InnerProductSpace RCLike ContinuousLinearMap
open scoped InnerProduct ComplexConjugate
namespace ContinuousLinearMap
variable... | Mathlib/Analysis/InnerProductSpace/Positive.lean | 67 | 68 | theorem IsPositive.inner_nonneg_right {T : E →L[𝕜] E} (hT : IsPositive T) (x : E) :
0 ≤ re ⟪x, T x⟫ := by | rw [inner_re_symm]; exact hT.inner_nonneg_left x
| [
" 0 ≤ re ⟪x, T x⟫_𝕜",
" 0 ≤ re ⟪T x, x⟫_𝕜"
] | [] |
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.LinearAlgebra.Basis
#align_import linear_algebra.affine_space.basis from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
open Affine
open Set
universe u₁ u₂ u₃ u₄
structure AffineBasis (ι : Type u₁) (k : Type u₂) {V ... | Mathlib/LinearAlgebra/AffineSpace/Basis.lean | 168 | 170 | theorem coord_apply_eq (i : ι) : b.coord i (b i) = 1 := by |
simp only [coord, Basis.coe_sumCoords, LinearEquiv.map_zero, LinearEquiv.coe_coe, sub_zero,
AffineMap.coe_mk, Finsupp.sum_zero_index, vsub_self]
| [
" affineSpan k (range id) = ⊤",
" f = g",
" { toFun := toFun✝, ind' := ind'✝, tot' := tot'✝ } = g",
" { toFun := toFun✝¹, ind' := ind'✝¹, tot' := tot'✝¹ } = { toFun := toFun✝, ind' := ind'✝, tot' := tot'✝ }",
" False",
" affineSpan k (range (⇑b ∘ ⇑e.symm)) = ⊤",
" affineSpan k (range ⇑b) = ⊤",
" ⊤ ≤ S... | [
" affineSpan k (range id) = ⊤",
" f = g",
" { toFun := toFun✝, ind' := ind'✝, tot' := tot'✝ } = g",
" { toFun := toFun✝¹, ind' := ind'✝¹, tot' := tot'✝¹ } = { toFun := toFun✝, ind' := ind'✝, tot' := tot'✝ }",
" False",
" affineSpan k (range (⇑b ∘ ⇑e.symm)) = ⊤",
" affineSpan k (range ⇑b) = ⊤",
" ⊤ ≤ S... |
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 316 | 324 | theorem convexBodySumFun_eq_zero_iff (x : E K) :
convexBodySumFun x = 0 ↔ x = 0 := by |
rw [← normAtPlace_eq_zero, convexBodySumFun, Finset.sum_eq_zero_iff_of_nonneg fun _ _ =>
mul_nonneg (Nat.cast_pos.mpr mult_pos).le (normAtPlace_nonneg _ _)]
conv =>
enter [1, w, hw]
rw [mul_left_mem_nonZeroDivisors_eq_zero_iff
(mem_nonZeroDivisors_iff_ne_zero.mpr <| Nat.cast_ne_zero.mpr mult_ne_z... | [
" convexBodySumFun x = ∑ w : { w // w.IsReal }, ‖x.1 w‖ + 2 * ∑ w : { w // w.IsComplex }, ‖x.2 w‖",
" ∑ x_1 ∈ Finset.subtype (fun x => x.IsReal) Finset.univ, ↑(↑x_1).mult * (normAtPlace ↑x_1) x +\n ∑ x_1 ∈ Finset.subtype (fun x => x.IsComplex) Finset.univ, ↑(↑x_1).mult * (normAtPlace ↑x_1) x =\n ∑ x_1 ∈ F... | [
" convexBodySumFun x = ∑ w : { w // w.IsReal }, ‖x.1 w‖ + 2 * ∑ w : { w // w.IsComplex }, ‖x.2 w‖",
" ∑ x_1 ∈ Finset.subtype (fun x => x.IsReal) Finset.univ, ↑(↑x_1).mult * (normAtPlace ↑x_1) x +\n ∑ x_1 ∈ Finset.subtype (fun x => x.IsComplex) Finset.univ, ↑(↑x_1).mult * (normAtPlace ↑x_1) x =\n ∑ x_1 ∈ F... |
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.CategoryTheory.Abelian.Exact
#align_import algebra.category.Module.abelian from "leanprover-community/mathlib"@"09f981f72d43749f1fa072deade828d9c1e185bb"
open... | Mathlib/Algebra/Category/ModuleCat/Abelian.lean | 123 | 127 | theorem exact_iff : Exact f g ↔ LinearMap.range f = LinearMap.ker g := by |
rw [abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]
exact
⟨fun h => le_antisymm (range_le_ker_iff.2 h.1) (ker_le_range_iff.2 h.2), fun h =>
⟨range_le_ker_iff.1 <| le_of_eq h, ker_le_range_iff.1 <| le_of_eq h.symm⟩⟩
| [
" (((LinearMap.ker f).quotEquivOfEqBot ⋯).symm ≪≫ₗ\n (LinearMap.quotKerEquivRange f ≪≫ₗ\n LinearEquiv.ofEq (LinearMap.range f) (LinearMap.ker (LinearMap.range f).mkQ) ⋯)).toModuleIso'.hom ≫\n Fork.ι (kernelCone (LinearMap.range f).mkQ) =\n f",
" ((((LinearMap.ker f).quotEquivOfEqBo... | [
" (((LinearMap.ker f).quotEquivOfEqBot ⋯).symm ≪≫ₗ\n (LinearMap.quotKerEquivRange f ≪≫ₗ\n LinearEquiv.ofEq (LinearMap.range f) (LinearMap.ker (LinearMap.range f).mkQ) ⋯)).toModuleIso'.hom ≫\n Fork.ι (kernelCone (LinearMap.range f).mkQ) =\n f",
" ((((LinearMap.ker f).quotEquivOfEqBo... |
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.Layercake
#align_import analysis.special_functions.japanese_bracket from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
op... | Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean | 62 | 65 | theorem le_rpow_one_add_norm_iff_norm_le {r t : ℝ} (hr : 0 < r) (ht : 0 < t) (x : E) :
t ≤ (1 + ‖x‖) ^ (-r) ↔ ‖x‖ ≤ t ^ (-r⁻¹) - 1 := by |
rw [le_sub_iff_add_le', neg_inv]
exact (Real.le_rpow_inv_iff_of_neg (by positivity) ht (neg_lt_zero.mpr hr)).symm
| [
" √(1 + ‖x‖ ^ 2) ≤ 1 + ‖x‖",
" 0 ≤ 1 + ‖x‖",
" 1 + ‖x‖ ^ 2 ≤ (1 + ‖x‖) ^ 2",
" 1 + ‖x‖ ≤ √2 * √(1 + ‖x‖ ^ 2)",
" 1 + ‖x‖ ≤ √(2 * (1 + ‖x‖ ^ 2))",
" (1 + ‖x‖) ^ 2 ≤ 2 * (1 + ‖x‖ ^ 2)",
" (1 + ‖x‖ ^ 2) ^ (-r / 2) = 2 ^ (r / 2) * ((√2 * √(1 + ‖x‖ ^ 2)) ^ r)⁻¹",
" √2 ^ r ≠ 0",
" 0 ≤ √(1 + ‖x‖ ^ 2)",
"... | [
" √(1 + ‖x‖ ^ 2) ≤ 1 + ‖x‖",
" 0 ≤ 1 + ‖x‖",
" 1 + ‖x‖ ^ 2 ≤ (1 + ‖x‖) ^ 2",
" 1 + ‖x‖ ≤ √2 * √(1 + ‖x‖ ^ 2)",
" 1 + ‖x‖ ≤ √(2 * (1 + ‖x‖ ^ 2))",
" (1 + ‖x‖) ^ 2 ≤ 2 * (1 + ‖x‖ ^ 2)",
" (1 + ‖x‖ ^ 2) ^ (-r / 2) = 2 ^ (r / 2) * ((√2 * √(1 + ‖x‖ ^ 2)) ^ r)⁻¹",
" √2 ^ r ≠ 0",
" 0 ≤ √(1 + ‖x‖ ^ 2)",
"... |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Analysis.Complex.Arg
#align_import ring_theory.polynomial.cyclotomic.eval from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf16... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 36 | 37 | theorem eval₂_one_cyclotomic_prime {R S : Type*} [CommRing R] [Semiring S] (f : R →+* S) {p : ℕ}
[Fact p.Prime] : eval₂ f 1 (cyclotomic p R) = p := by | simp
| [
" eval 1 (cyclotomic p R) = ↑p",
" eval₂ f 1 (cyclotomic p R) = ↑p"
] | [
" eval 1 (cyclotomic p R) = ↑p"
] |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 125 | 125 | theorem rpow_zero (x : ℝ) : x ^ (0 : ℝ) = 1 := by | simp [rpow_def]
| [
" x ^ y = if x = 0 then if y = 0 then 1 else 0 else rexp (x.log * y)",
" (if ↑x = 0 then if ↑y = 0 then 1 else 0 else ((↑x).log * ↑y).exp).re =\n if x = 0 then if y = 0 then 1 else 0 else rexp (x.log * y)",
" Complex.re 1 = 1",
" Complex.re 1 = 0",
" Complex.re 1 = rexp (x.log * y)",
" Complex.re 0 = 1... | [
" x ^ y = if x = 0 then if y = 0 then 1 else 0 else rexp (x.log * y)",
" (if ↑x = 0 then if ↑y = 0 then 1 else 0 else ((↑x).log * ↑y).exp).re =\n if x = 0 then if y = 0 then 1 else 0 else rexp (x.log * y)",
" Complex.re 1 = 1",
" Complex.re 1 = 0",
" Complex.re 1 = rexp (x.log * y)",
" Complex.re 0 = 1... |
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Set Filter Function Metric MeasureTheory
open scoped MeasureTheory Topology Pointwise
@[... | Mathlib/NumberTheory/WellApproximable.lean | 147 | 166 | theorem smul_eq_of_mul_dvd (hn : 0 < n) (han : orderOf a ^ 2 ∣ n) :
a • approxOrderOf A n δ = approxOrderOf A n δ := by |
simp_rw [approxOrderOf, thickening_eq_biUnion_ball, ← image_smul, image_iUnion₂, image_smul,
smul_ball'', smul_eq_mul, mem_setOf_eq]
replace han : ∀ {b : A}, orderOf b = n → orderOf (a * b) = n := by
intro b hb
rw [← hb] at han hn
rw [sq] at han
rwa [(Commute.all a b).orderOf_mul_eq_right_of_fo... | [
" a ∈ approxOrderOf A n δ ↔ ∃ b, orderOf b = n ∧ a ∈ ball b δ",
" (fun y => y ^ m) '' approxOrderOf A n δ ⊆ approxOrderOf A n (↑m * δ)",
" (fun y => y ^ m) a ∈ approxOrderOf A n (↑m * δ)",
" b ^ m ∈ {u | orderOf u = n}",
" b ^ m ∈ {u | orderOf u = orderOf b}",
" (fun y => y ^ m) a ∈ ball (b ^ m) (↑m • δ)"... | [
" a ∈ approxOrderOf A n δ ↔ ∃ b, orderOf b = n ∧ a ∈ ball b δ",
" (fun y => y ^ m) '' approxOrderOf A n δ ⊆ approxOrderOf A n (↑m * δ)",
" (fun y => y ^ m) a ∈ approxOrderOf A n (↑m * δ)",
" b ^ m ∈ {u | orderOf u = n}",
" b ^ m ∈ {u | orderOf u = orderOf b}",
" (fun y => y ^ m) a ∈ ball (b ^ m) (↑m • δ)"... |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Data.Rat.Cast.Defs
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {F ι α β : Type*}
namespace Rat
open Rat
section WithDivRing
variable [DivisionRing α]
@[simp, norm_cast]
th... | Mathlib/Data/Rat/Cast/CharZero.lean | 46 | 46 | theorem cast_eq_zero [CharZero α] {n : ℚ} : (n : α) = 0 ↔ n = 0 := by | rw [← cast_zero, cast_inj]
| [
" ↑{ num := n₁, den := d₁, den_nz := d₁0, reduced := c₁ } = ↑{ num := n₂, den := d₂, den_nz := d₂0, reduced := c₂ } ↔\n { num := n₁, den := d₁, den_nz := d₁0, reduced := c₁ } = { num := n₂, den := d₂, den_nz := d₂0, reduced := c₂ }",
" { num := n₁, den := d₁, den_nz := d₁0, reduced := c₁ } = { num := n₂, den :... | [
" ↑{ num := n₁, den := d₁, den_nz := d₁0, reduced := c₁ } = ↑{ num := n₂, den := d₂, den_nz := d₂0, reduced := c₂ } ↔\n { num := n₁, den := d₁, den_nz := d₁0, reduced := c₁ } = { num := n₂, den := d₂, den_nz := d₂0, reduced := c₂ }",
" { num := n₁, den := d₁, den_nz := d₁0, reduced := c₁ } = { num := n₂, den :... |
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