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 | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 52 | 60 | theorem hasStrictDerivAt_const_cpow {x y : β} (h : x β 0 β¨ y β 0) :
HasStrictDerivAt (fun y => x ^ y) (x ^ y * log x) y := by |
rcases em (x = 0) with (rfl | hx)
Β· replace h := h.neg_resolve_left rfl
rw [log_zero, mul_zero]
refine (hasStrictDerivAt_const _ 0).congr_of_eventuallyEq ?_
exact (isOpen_ne.eventually_mem h).mono fun y hy => (zero_cpow hy).symm
Β· simpa only [cpow_def_of_ne_zero hx, mul_one] using
((hasStrictDe... | 2,159 |
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 276 | 285 | theorem hasStrictFDerivAt_rpow_of_pos (p : β Γ β) (hp : 0 < p.1) :
HasStrictFDerivAt (fun x : β Γ β => x.1 ^ x.2)
((p.2 * p.1 ^ (p.2 - 1)) β’ ContinuousLinearMap.fst β β β +
(p.1 ^ p.2 * log p.1) β’ ContinuousLinearMap.snd β β β) p := by |
have : (fun x : β Γ β => x.1 ^ x.2) =αΆ [π p] fun x => exp (log x.1 * x.2) :=
(continuousAt_fst.eventually (lt_mem_nhds hp)).mono fun p hp => rpow_def_of_pos hp _
refine HasStrictFDerivAt.congr_of_eventuallyEq ?_ this.symm
convert ((hasStrictFDerivAt_fst.log hp.ne').mul hasStrictFDerivAt_snd).exp using 1
rw... | 2,159 |
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 289 | 301 | theorem hasStrictFDerivAt_rpow_of_neg (p : β Γ β) (hp : p.1 < 0) :
HasStrictFDerivAt (fun x : β Γ β => x.1 ^ x.2)
((p.2 * p.1 ^ (p.2 - 1)) β’ ContinuousLinearMap.fst β β β +
(p.1 ^ p.2 * log p.1 - exp (log p.1 * p.2) * sin (p.2 * Ο) * Ο) β’
ContinuousLinearMap.snd β β β) p := by |
have : (fun x : β Γ β => x.1 ^ x.2) =αΆ [π p] fun x => exp (log x.1 * x.2) * cos (x.2 * Ο) :=
(continuousAt_fst.eventually (gt_mem_nhds hp)).mono fun p hp => rpow_def_of_neg hp _
refine HasStrictFDerivAt.congr_of_eventuallyEq ?_ this.symm
convert ((hasStrictFDerivAt_fst.log hp.ne).mul hasStrictFDerivAt_snd).e... | 2,159 |
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 305 | 313 | theorem contDiffAt_rpow_of_ne (p : β Γ β) (hp : p.1 β 0) {n : ββ} :
ContDiffAt β n (fun p : β Γ β => p.1 ^ p.2) p := by |
cases' hp.lt_or_lt with hneg hpos
exacts
[(((contDiffAt_fst.log hneg.ne).mul contDiffAt_snd).exp.mul
(contDiffAt_snd.mul contDiffAt_const).cos).congr_of_eventuallyEq
((continuousAt_fst.eventually (gt_mem_nhds hneg)).mono fun p hp => rpow_def_of_neg hp _),
((contDiffAt_fst.log hpos.ne').mul ... | 2,159 |
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 321 | 326 | theorem _root_.HasStrictDerivAt.rpow {f g : β β β} {f' g' : β} (hf : HasStrictDerivAt f f' x)
(hg : HasStrictDerivAt g g' x) (h : 0 < f x) : HasStrictDerivAt (fun x => f x ^ g x)
(f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * Real.log (f x)) x := by |
convert (hasStrictFDerivAt_rpow_of_pos ((fun x => (f x, g x)) x) h).comp_hasStrictDerivAt x
(hf.prod hg) using 1
simp [mul_assoc, mul_comm, mul_left_comm]
| 2,159 |
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 329 | 335 | theorem hasStrictDerivAt_rpow_const_of_ne {x : β} (hx : x β 0) (p : β) :
HasStrictDerivAt (fun x => x ^ p) (p * x ^ (p - 1)) x := by |
cases' hx.lt_or_lt with hx hx
Β· have := (hasStrictFDerivAt_rpow_of_neg (x, p) hx).comp_hasStrictDerivAt x
((hasStrictDerivAt_id x).prod (hasStrictDerivAt_const _ _))
convert this using 1; simp
Β· simpa using (hasStrictDerivAt_id x).rpow (hasStrictDerivAt_const x p) hx
| 2,159 |
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 338 | 340 | theorem hasStrictDerivAt_const_rpow {a : β} (ha : 0 < a) (x : β) :
HasStrictDerivAt (fun x => a ^ x) (a ^ x * log a) x := by |
simpa using (hasStrictDerivAt_const _ _).rpow (hasStrictDerivAt_id x) ha
| 2,159 |
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 648 | 656 | theorem tendsto_one_plus_div_rpow_exp (t : β) :
Tendsto (fun x : β => (1 + t / x) ^ x) atTop (π (exp t)) := by |
apply ((Real.continuous_exp.tendsto _).comp (tendsto_mul_log_one_plus_div_atTop t)).congr' _
have hβ : (1 : β) / 2 < 1 := by linarith
have hβ : Tendsto (fun x : β => 1 + t / x) atTop (π 1) := by
simpa using (tendsto_inv_atTop_zero.const_mul t).const_add 1
refine (eventually_ge_of_tendsto_gt hβ hβ).mono fu... | 2,159 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.caratheodory from "leanprover-community/mathlib"@"e6fab1dc073396d45da082c644642c4f8bff2264"
open Set Finset
universe u
variable {π : Type*} {E : Type u} ... | Mathlib/Analysis/Convex/Caratheodory.lean | 52 | 98 | theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : Β¬AffineIndependent π ((β) : t β E))
{x : E} (m : x β convexHull π (βt : Set E)) :
β y : (βt : Set E), x β convexHull π (β(t.erase y) : Set E) := by |
simp only [Finset.convexHull_eq, mem_setOf_eq] at m β’
obtain β¨f, fpos, fsum, rflβ© := m
obtain β¨g, gcombo, gsum, gposβ© := exists_nontrivial_relation_sum_zero_of_not_affine_ind h
replace gpos := exists_pos_of_sum_zero_of_exists_nonzero g gsum gpos
clear h
let s := @Finset.filter _ (fun z => 0 < g z) (fun _ =... | 2,160 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.caratheodory from "leanprover-community/mathlib"@"e6fab1dc073396d45da082c644642c4f8bff2264"
open Set Finset
universe u
variable {π : Type*} {E : Type u} ... | Mathlib/Analysis/Convex/Caratheodory.lean | 119 | 121 | theorem minCardFinsetOfMemConvexHull_nonempty : (minCardFinsetOfMemConvexHull hx).Nonempty := by |
rw [β Finset.coe_nonempty, β @convexHull_nonempty_iff π]
exact β¨x, mem_minCardFinsetOfMemConvexHull hxβ©
| 2,160 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprover-community/mathlib"@"ea050b44c0f9aba9d16a948c7cc7d2e7c8493567"
namespace Cardinal
open Cardinal
universe u
variable {a b : Cardinal.{u}} {n m : β... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 43 | 58 | theorem isUnit_iff : IsUnit a β a = 1 := by |
refine
β¨fun h => ?_, by
rintro rfl
exact isUnit_oneβ©
rcases eq_or_ne a 0 with (rfl | ha)
Β· exact (not_isUnit_zero h).elim
rw [isUnit_iff_forall_dvd] at h
cases' h 1 with t ht
rw [eq_comm, mul_eq_one_iff'] at ht
Β· exact ht.1
Β· exact one_le_iff_ne_zero.mpr ha
Β· apply one_le_iff_ne_zero.... | 2,161 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprover-community/mathlib"@"ea050b44c0f9aba9d16a948c7cc7d2e7c8493567"
namespace Cardinal
open Cardinal
universe u
variable {a b : Cardinal.{u}} {n m : β... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 76 | 89 | theorem prime_of_aleph0_le (ha : β΅β β€ a) : Prime a := by |
refine β¨(aleph0_pos.trans_le ha).ne', ?_, fun b c hbc => ?_β©
Β· rw [isUnit_iff]
exact (one_lt_aleph0.trans_le ha).ne'
rcases eq_or_ne (b * c) 0 with hz | hz
Β· rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
wlog h : c β€ b
Β· cases le_total c b <;> [solve_by_elim; rw [or_comm]]
apply_assumption
... | 2,161 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprover-community/mathlib"@"ea050b44c0f9aba9d16a948c7cc7d2e7c8493567"
namespace Cardinal
open Cardinal
universe u
variable {a b : Cardinal.{u}} {n m : β... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 92 | 96 | theorem not_irreducible_of_aleph0_le (ha : β΅β β€ a) : Β¬Irreducible a := by |
rw [irreducible_iff, not_and_or]
refine Or.inr fun h => ?_
simpa [mul_aleph0_eq ha, isUnit_iff, (one_lt_aleph0.trans_le ha).ne', one_lt_aleph0.ne'] using
h a β΅β
| 2,161 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprover-community/mathlib"@"ea050b44c0f9aba9d16a948c7cc7d2e7c8493567"
namespace Cardinal
open Cardinal
universe u
variable {a b : Cardinal.{u}} {n m : β... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 100 | 108 | theorem nat_coe_dvd_iff : (n : Cardinal) β£ m β n β£ m := by |
refine β¨?_, fun β¨h, htβ© => β¨h, mod_cast htβ©β©
rintro β¨k, hkβ©
have : βm < β΅β := nat_lt_aleph0 m
rw [hk, mul_lt_aleph0_iff] at this
rcases this with (h | h | β¨-, hk'β©)
iterate 2 simp only [h, mul_zero, zero_mul, Nat.cast_eq_zero] at hk; simp [hk]
lift k to β using hk'
exact β¨k, mod_cast hkβ©
| 2,161 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprover-community/mathlib"@"ea050b44c0f9aba9d16a948c7cc7d2e7c8493567"
namespace Cardinal
open Cardinal
universe u
variable {a b : Cardinal.{u}} {n m : β... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 112 | 134 | theorem nat_is_prime_iff : Prime (n : Cardinal) β n.Prime := by |
simp only [Prime, Nat.prime_iff]
refine and_congr (by simp) (and_congr ?_ β¨fun h b c hbc => ?_, fun h b c hbc => ?_β©)
Β· simp only [isUnit_iff, Nat.isUnit_iff]
exact mod_cast Iff.rfl
Β· exact mod_cast h b c (mod_cast hbc)
cases' lt_or_le (b * c) β΅β with h' h'
Β· rcases mul_lt_aleph0_iff.mp h' with (rfl | ... | 2,161 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprover-community/mathlib"@"ea050b44c0f9aba9d16a948c7cc7d2e7c8493567"
namespace Cardinal
open Cardinal
universe u
variable {a b : Cardinal.{u}} {n m : β... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 137 | 141 | theorem is_prime_iff {a : Cardinal} : Prime a β β΅β β€ a β¨ β p : β, a = p β§ p.Prime := by |
rcases le_or_lt β΅β a with h | h
Β· simp [h]
lift a to β using id h
simp [not_le.mpr h]
| 2,161 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprover-community/mathlib"@"ea050b44c0f9aba9d16a948c7cc7d2e7c8493567"
namespace Cardinal
open Cardinal
universe u
variable {a b : Cardinal.{u}} {n m : β... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 144 | 158 | theorem isPrimePow_iff {a : Cardinal} : IsPrimePow a β β΅β β€ a β¨ β n : β, a = n β§ IsPrimePow n := by |
by_cases h : β΅β β€ a
Β· simp [h, (prime_of_aleph0_le h).isPrimePow]
simp only [h, Nat.cast_inj, exists_eq_left', false_or_iff, isPrimePow_nat_iff]
lift a to β using not_le.mp h
rw [isPrimePow_def]
refine
β¨?_, fun β¨n, han, p, k, hp, hk, hβ© =>
β¨p, k, nat_is_prime_iff.2 hp, hk, by rw [han]; exact ... | 2,161 |
import Mathlib.Algebra.Field.ULift
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.Data.Rat.Denumerable
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.RingTheory.Localization.Cardinality
import Mathlib.... | Mathlib/FieldTheory/Cardinality.lean | 40 | 49 | theorem Fintype.isPrimePow_card_of_field {Ξ±} [Fintype Ξ±] [Field Ξ±] : IsPrimePow βΞ±β := by |
-- TODO: `Algebra` version of `CharP.exists`, of type `β p, Algebra (ZMod p) Ξ±`
cases' CharP.exists Ξ± with p _
haveI hp := Fact.mk (CharP.char_is_prime Ξ± p)
letI : Algebra (ZMod p) Ξ± := ZMod.algebra _ _
let b := IsNoetherian.finsetBasis (ZMod p) Ξ±
rw [Module.card_fintype b, ZMod.card, isPrimePow_pow_iff]
... | 2,162 |
import Mathlib.Algebra.Field.ULift
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.Data.Rat.Denumerable
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.RingTheory.Localization.Cardinality
import Mathlib.... | Mathlib/FieldTheory/Cardinality.lean | 53 | 57 | theorem Fintype.nonempty_field_iff {Ξ±} [Fintype Ξ±] : Nonempty (Field Ξ±) β IsPrimePow βΞ±β := by |
refine β¨fun β¨hβ© => Fintype.isPrimePow_card_of_field, ?_β©
rintro β¨p, n, hp, hn, hΞ±β©
haveI := Fact.mk hp.nat_prime
exact β¨(Fintype.equivOfCardEq ((GaloisField.card p n hn.ne').trans hΞ±)).symm.fieldβ©
| 2,162 |
import Mathlib.Algebra.Field.ULift
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.Data.Rat.Denumerable
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.RingTheory.Localization.Cardinality
import Mathlib.... | Mathlib/FieldTheory/Cardinality.lean | 66 | 76 | theorem Infinite.nonempty_field {Ξ± : Type u} [Infinite Ξ±] : Nonempty (Field Ξ±) := by |
letI K := FractionRing (MvPolynomial Ξ± <| ULift.{u} β)
suffices #Ξ± = #K by
obtain β¨eβ© := Cardinal.eq.1 this
exact β¨e.fieldβ©
rw [β IsLocalization.card (MvPolynomial Ξ± <| ULift.{u} β)β° K le_rfl]
apply le_antisymm
Β· refine
β¨β¨fun a => MvPolynomial.monomial (Finsupp.single a 1) (1 : ULift.{u} β), fu... | 2,162 |
import Mathlib.Algebra.Field.ULift
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.Data.Rat.Denumerable
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.RingTheory.Localization.Cardinality
import Mathlib.... | Mathlib/FieldTheory/Cardinality.lean | 80 | 85 | theorem Field.nonempty_iff {Ξ± : Type u} : Nonempty (Field Ξ±) β IsPrimePow #Ξ± := by |
rw [Cardinal.isPrimePow_iff]
cases' fintypeOrInfinite Ξ± with h h
Β· simpa only [Cardinal.mk_fintype, Nat.cast_inj, exists_eq_left',
(Cardinal.nat_lt_aleph0 _).not_le, false_or_iff] using Fintype.nonempty_field_iff
Β· simpa only [β Cardinal.infinite_iff, h, true_or_iff, iff_true_iff] using Infinite.nonempty... | 2,162 |
import Mathlib.Computability.Encoding
import Mathlib.Logic.Small.List
import Mathlib.ModelTheory.Syntax
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import model_theory.encoding from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u v w u' v'
namespace FirstOrder
namespace... | Mathlib/ModelTheory/Encoding.lean | 67 | 98 | theorem listDecode_encode_list (l : List (L.Term Ξ±)) :
listDecode (l.bind listEncode) = l.map Option.some := by |
suffices h : β (t : L.Term Ξ±) (l : List (Sum Ξ± (Ξ£i, L.Functions i))),
listDecode (t.listEncode ++ l) = some t::listDecode l by
induction' l with t l lih
Β· rfl
Β· rw [cons_bind, h t (l.bind listEncode), lih, List.map]
intro t
induction' t with a n f ts ih <;> intro l
Β· rw [listEncode, singleton... | 2,163 |
import Mathlib.Computability.Encoding
import Mathlib.Logic.Small.List
import Mathlib.ModelTheory.Syntax
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import model_theory.encoding from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u v w u' v'
namespace FirstOrder
namespace... | Mathlib/ModelTheory/Encoding.lean | 122 | 151 | theorem card_sigma : #(Ξ£n, L.Term (Sum Ξ± (Fin n))) = max β΅β #(Sum Ξ± (Ξ£i, L.Functions i)) := by |
refine le_antisymm ?_ ?_
Β· rw [mk_sigma]
refine (sum_le_iSup_lift _).trans ?_
rw [mk_nat, lift_aleph0, mul_eq_max_of_aleph0_le_left le_rfl, max_le_iff,
ciSup_le_iff' (bddAbove_range _)]
Β· refine β¨le_max_left _ _, fun i => card_le.trans ?_β©
refine max_le (le_max_left _ _) ?_
rw [β add_... | 2,163 |
import Mathlib.Computability.Encoding
import Mathlib.Logic.Small.List
import Mathlib.ModelTheory.Syntax
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import model_theory.encoding from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u v w u' v'
namespace FirstOrder
namespace... | Mathlib/ModelTheory/Encoding.lean | 235 | 287 | theorem listDecode_encode_list (l : List (Ξ£n, L.BoundedFormula Ξ± n)) :
(listDecode (l.bind fun Ο => Ο.2.listEncode)).1 = l.headI := by |
suffices h : β (Ο : Ξ£n, L.BoundedFormula Ξ± n) (l),
(listDecode (listEncode Ο.2 ++ l)).1 = Ο β§ (listDecode (listEncode Ο.2 ++ l)).2.1 = l by
induction' l with Ο l _
Β· rw [List.nil_bind]
simp [listDecode]
Β· rw [cons_bind, (h Ο _).1, headI_cons]
rintro β¨n, Οβ©
induction' Ο with _ _ _ _ Ο_n Ο_... | 2,163 |
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import data.polynomial.cardinal from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
universe u
open Cardinal Polynomial
open Cardinal
namespace Polynomial
@[simp]
theorem cardinal_mk_eq_max {R :... | Mathlib/Algebra/Polynomial/Cardinal.lean | 34 | 37 | theorem cardinal_mk_le_max {R : Type u} [Semiring R] : #(R[X]) β€ max #R β΅β := by |
cases subsingleton_or_nontrivial R
Β· exact (mk_eq_one _).trans_le (le_max_of_le_right one_le_aleph0)
Β· exact cardinal_mk_eq_max.le
| 2,164 |
import Mathlib.Algebra.Polynomial.Cardinal
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.ZMod.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.AlgebraicIndependent
#align_import field_theory.is_alg_closed.classification from "leanprover-community/mathlib"@"0723536a0522... | Mathlib/FieldTheory/IsAlgClosed/Classification.lean | 41 | 59 | theorem cardinal_mk_le_sigma_polynomial :
#L β€ #(Ξ£ p : R[X], { x : L // x β p.aroots L }) :=
@mk_le_of_injective L (Ξ£ p : R[X], {x : L | x β p.aroots L})
(fun x : L =>
let p := Classical.indefiniteDescription _ (Algebra.IsAlgebraic.isAlgebraic x)
β¨p.1, x, by
dsimp
have h : p.1.map ... |
rw [Ne, β Polynomial.degree_eq_bot,
Polynomial.degree_map_eq_of_injective (NoZeroSMulDivisors.algebraMap_injective R L),
Polynomial.degree_eq_bot]
exact p.2.1
erw [Polynomial.mem_roots h, Polynomial.IsRoot, Polynomial.eval_map, β Polynomial.aeval_def,
p.2.2... | 2,165 |
import Mathlib.Algebra.Polynomial.Cardinal
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.ZMod.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.AlgebraicIndependent
#align_import field_theory.is_alg_closed.classification from "leanprover-community/mathlib"@"0723536a0522... | Mathlib/FieldTheory/IsAlgClosed/Classification.lean | 64 | 76 | theorem cardinal_mk_le_max : #L β€ max #R β΅β :=
calc
#L β€ #(Ξ£ p : R[X], { x : L // x β p.aroots L }) :=
cardinal_mk_le_sigma_polynomial R L
_ = Cardinal.sum fun p : R[X] => #{x : L | x β p.aroots L} := by |
rw [β mk_sigma]; rfl
_ β€ Cardinal.sum.{u, u} fun _ : R[X] => β΅β :=
(sum_le_sum _ _ fun p => (Multiset.finite_toSet _).lt_aleph0.le)
_ = #(R[X]) * β΅β := sum_const' _ _
_ β€ max (max #(R[X]) β΅β) β΅β := mul_le_max _ _
_ β€ max (max (max #R β΅β) β΅β) β΅β :=
(max_le_max (max_le_max Polynomial.ca... | 2,165 |
import Mathlib.Algebra.Polynomial.Cardinal
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.ZMod.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.AlgebraicIndependent
#align_import field_theory.is_alg_closed.classification from "leanprover-community/mathlib"@"0723536a0522... | Mathlib/FieldTheory/IsAlgClosed/Classification.lean | 96 | 100 | theorem isAlgClosure_of_transcendence_basis [IsAlgClosed K] (hv : IsTranscendenceBasis R v) :
IsAlgClosure (Algebra.adjoin R (Set.range v)) K :=
letI := RingHom.domain_nontrivial (algebraMap R K)
{ alg_closed := by | infer_instance
algebraic := hv.isAlgebraic }
| 2,165 |
import Mathlib.Algebra.Polynomial.Cardinal
import Mathlib.RingTheory.Algebraic
#align_import algebra.algebraic_card from "leanprover-community/mathlib"@"40494fe75ecbd6d2ec61711baa630cf0a7b7d064"
universe u v
open Cardinal Polynomial Set
open Cardinal Polynomial
namespace Algebraic
theorem infinite_of_charZero... | Mathlib/Algebra/AlgebraicCard.lean | 45 | 54 | theorem cardinal_mk_lift_le_mul :
Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } β€ Cardinal.lift.{v} #R[X] * β΅β := by |
rw [β mk_uLift, β mk_uLift]
choose g hgβ hgβ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop
refine lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => ?_
rw [lift_le_aleph0, le_aleph0_iff_set_countable]
suffices MapsTo (β) (g β»ΒΉ' {f}) (f.rootSet A) from
this.countable_of_injOn Subtype.coe_i... | 2,166 |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ΞΉ : Type u} {Ξ± Ξ² : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter Ξ±) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 52 | 55 | theorem _root_.Filter.cardinalInterFilter_aleph0 (l : Filter Ξ±) : CardinalInterFilter l aleph0 where
cardinal_sInter_mem := by |
simp_all only [aleph_zero, lt_aleph0_iff_subtype_finite, setOf_mem_eq, sInter_mem,
implies_true, forall_const]
| 2,167 |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ΞΉ : Type u} {Ξ± Ξ² : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter Ξ±) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 90 | 94 | theorem cardinal_iInter_mem {s : ΞΉ β Set Ξ±} (hic : #ΞΉ < c) :
(β i, s i) β l β β i, s i β l := by |
rw [β sInter_range _]
apply (cardinal_sInter_mem (lt_of_le_of_lt Cardinal.mk_range_le hic)).trans
exact forall_mem_range
| 2,167 |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ΞΉ : Type u} {Ξ± Ξ² : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter Ξ±) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 96 | 100 | theorem cardinal_bInter_mem {S : Set ΞΉ} (hS : #S < c)
{s : β i β S, Set Ξ±} :
(β i, β hi : i β S, s i βΉ_βΊ) β l β β i, β hi : i β S, s i βΉ_βΊ β l := by |
rw [biInter_eq_iInter]
exact (cardinal_iInter_mem hS).trans Subtype.forall
| 2,167 |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ΞΉ : Type u} {Ξ± Ξ² : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter Ξ±) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 102 | 105 | theorem eventually_cardinal_forall {p : Ξ± β ΞΉ β Prop} (hic : #ΞΉ < c) :
(βαΆ x in l, β i, p x i) β β i, βαΆ x in l, p x i := by |
simp only [Filter.Eventually, setOf_forall]
exact cardinal_iInter_mem hic
| 2,167 |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ΞΉ : Type u} {Ξ± Ξ² : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter Ξ±) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 107 | 111 | theorem eventually_cardinal_ball {S : Set ΞΉ} (hS : #S < c)
{p : Ξ± β β i β S, Prop} :
(βαΆ x in l, β i hi, p x i hi) β β i hi, βαΆ x in l, p x i hi := by |
simp only [Filter.Eventually, setOf_forall]
exact cardinal_bInter_mem hS
| 2,167 |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ΞΉ : Type u} {Ξ± Ξ² : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter Ξ±) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 123 | 127 | theorem EventuallyLE.cardinal_bUnion {S : Set ΞΉ} (hS : #S < c)
{s t : β i β S, Set Ξ±} (h : β i hi, s i hi β€αΆ [l] t i hi) :
β i β S, s i βΉ_βΊ β€αΆ [l] β i β S, t i βΉ_βΊ := by |
simp only [biUnion_eq_iUnion]
exact EventuallyLE.cardinal_iUnion hS fun i => h i i.2
| 2,167 |
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Filter.CountableInter
import Mathlib.Order.Filter.CardinalInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.Order.Filter.Bases
open Set Filter Cardinal
universe u
variable {ΞΉ : Type u} {Ξ± Ξ² : Type u}... | Mathlib/Order/Filter/Cocardinal.lean | 61 | 68 | theorem hasBasis_cocardinal : HasBasis (cocardinal Ξ± hreg) {s : Set Ξ± | #s < c} compl :=
β¨fun s =>
β¨fun h => β¨sαΆ, h, (compl_compl s).subsetβ©, fun β¨_t, htf, htsβ© => by
have : #βsαΆ < c := by |
apply lt_of_le_of_lt _ htf
rw [compl_subset_comm] at hts
apply Cardinal.mk_le_mk_of_subset hts
simp_all only [mem_cocardinal] β©β©
| 2,168 |
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Filter.CountableInter
import Mathlib.Order.Filter.CardinalInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.Order.Filter.Bases
open Set Filter Cardinal
universe u
variable {ΞΉ : Type u} {Ξ± Ξ² : Type u}... | Mathlib/Order/Filter/Cocardinal.lean | 70 | 72 | theorem frequently_cocardinal {p : Ξ± β Prop} :
(βαΆ x in cocardinal Ξ± hreg, p x) β c β€ # { x | p x } := by |
simp only [Filter.Frequently, eventually_cocardinal, not_not,coe_setOf, not_lt]
| 2,168 |
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Filter.CountableInter
import Mathlib.Order.Filter.CardinalInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.Order.Filter.Bases
open Set Filter Cardinal
universe u
variable {ΞΉ : Type u} {Ξ± Ξ² : Type u}... | Mathlib/Order/Filter/Cocardinal.lean | 98 | 100 | theorem eventually_cocardinal_ne (x : Ξ±) : βαΆ a in cocardinal Ξ± hreg, a β x := by |
simp [Set.finite_singleton x]
exact hreg.nat_lt 1
| 2,168 |
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Filter.CountableInter
import Mathlib.Order.Filter.CardinalInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.Order.Filter.Bases
open Set Filter Cardinal
universe u
variable {ΞΉ : Type u} {Ξ± Ξ² : Type u}... | Mathlib/Order/Filter/Cocardinal.lean | 105 | 107 | theorem mem_cocountable {s : Set Ξ±} :
s β cocountable β (sαΆ : Set Ξ±).Countable := by |
rw [Cardinal.countable_iff_lt_aleph_one, mem_cocardinal]
| 2,168 |
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ β΅β
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 41 | 42 | theorem lift_continuum : lift.{v} π = π := by |
rw [β two_power_aleph0, lift_two_power, lift_aleph0, two_power_aleph0]
| 2,169 |
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ β΅β
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 46 | 48 | theorem continuum_le_lift {c : Cardinal.{u}} : π β€ lift.{v} c β π β€ c := by |
-- Porting note: added explicit universes
rw [β lift_continuum.{u,v}, lift_le]
| 2,169 |
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ β΅β
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 52 | 54 | theorem lift_le_continuum {c : Cardinal.{u}} : lift.{v} c β€ π β c β€ π := by |
-- Porting note: added explicit universes
rw [β lift_continuum.{u,v}, lift_le]
| 2,169 |
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ β΅β
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 58 | 60 | theorem continuum_lt_lift {c : Cardinal.{u}} : π < lift.{v} c β π < c := by |
-- Porting note: added explicit universes
rw [β lift_continuum.{u,v}, lift_lt]
| 2,169 |
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ β΅β
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 64 | 66 | theorem lift_lt_continuum {c : Cardinal.{u}} : lift.{v} c < π β c < π := by |
-- Porting note: added explicit universes
rw [β lift_continuum.{u,v}, lift_lt]
| 2,169 |
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ β΅β
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 83 | 83 | theorem beth_one : beth 1 = π := by | simpa using beth_succ 0
| 2,169 |
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ β΅β
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 90 | 90 | theorem mk_set_nat : #(Set β) = π := by | simp
| 2,169 |
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ β΅β
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 101 | 103 | theorem aleph_one_le_continuum : aleph 1 β€ π := by |
rw [β succ_aleph0]
exact Order.succ_le_of_lt aleph0_lt_continuum
| 2,169 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {Ξ± : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 55 | 59 | theorem self_subset_generateMeasurableRec (s : Set (Set Ξ±)) (i : Οβ) :
s β generateMeasurableRec s i := by |
unfold generateMeasurableRec
apply_rules [subset_union_of_subset_left]
exact subset_rfl
| 2,170 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {Ξ± : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 62 | 65 | theorem empty_mem_generateMeasurableRec (s : Set (Set Ξ±)) (i : Οβ) :
β
β generateMeasurableRec s i := by |
unfold generateMeasurableRec
exact mem_union_left _ (mem_union_left _ (mem_union_right _ (mem_singleton β
)))
| 2,170 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {Ξ± : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 68 | 71 | theorem compl_mem_generateMeasurableRec {s : Set (Set Ξ±)} {i j : Οβ} (h : j < i) {t : Set Ξ±}
(ht : t β generateMeasurableRec s j) : tαΆ β generateMeasurableRec s i := by |
unfold generateMeasurableRec
exact mem_union_left _ (mem_union_right _ β¨t, mem_iUnion.2 β¨β¨j, hβ©, htβ©, rflβ©)
| 2,170 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {Ξ± : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 74 | 78 | theorem iUnion_mem_generateMeasurableRec {s : Set (Set Ξ±)} {i : Οβ} {f : β β Set Ξ±}
(hf : β n, β j < i, f n β generateMeasurableRec s j) :
(β n, f n) β generateMeasurableRec s i := by |
unfold generateMeasurableRec
exact mem_union_right _ β¨fun n => β¨f n, let β¨j, hj, hfβ© := hf n; mem_iUnion.2 β¨β¨j, hjβ©, hfβ©β©, rflβ©
| 2,170 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {Ξ± : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 91 | 113 | theorem cardinal_generateMeasurableRec_le (s : Set (Set Ξ±)) (i : Οβ) :
#(generateMeasurableRec s i) β€ max #s 2 ^ aleph0.{u} := by |
apply (aleph 1).ord.out.wo.wf.induction i
intro i IH
have A := aleph0_le_aleph 1
have B : aleph 1 β€ max #s 2 ^ aleph0.{u} :=
aleph_one_le_continuum.trans (power_le_power_right (le_max_right _ _))
have C : β΅β β€ max #s 2 ^ aleph0.{u} := A.trans B
have J : #(β j : Iio i, generateMeasurableRec s j.1) β€ max... | 2,170 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {Ξ± : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 117 | 151 | theorem generateMeasurable_eq_rec (s : Set (Set Ξ±)) :
{ t | GenerateMeasurable s t } =
β (i : (Quotient.out (aleph 1).ord).Ξ±), generateMeasurableRec s i := by |
ext t; refine β¨fun ht => ?_, fun ht => ?_β©
Β· inhabit Οβ
induction' ht with u hu u _ IH f _ IH
Β· exact mem_iUnion.2 β¨default, self_subset_generateMeasurableRec s _ huβ©
Β· exact mem_iUnion.2 β¨default, empty_mem_generateMeasurableRec s _β©
Β· rcases mem_iUnion.1 IH with β¨i, hiβ©
obtain β¨j, hjβ© := ex... | 2,170 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 64 | 65 | theorem cantorFunctionAux_true (h : f n = true) : cantorFunctionAux c f n = c ^ n := by |
simp [cantorFunctionAux, h]
| 2,171 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 69 | 70 | theorem cantorFunctionAux_false (h : f n = false) : cantorFunctionAux c f n = 0 := by |
simp [cantorFunctionAux, h]
| 2,171 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 73 | 75 | theorem cantorFunctionAux_nonneg (h : 0 β€ c) : 0 β€ cantorFunctionAux c f n := by |
cases h' : f n <;> simp [h']
apply pow_nonneg h
| 2,171 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 78 | 79 | theorem cantorFunctionAux_eq (h : f n = g n) :
cantorFunctionAux c f n = cantorFunctionAux c g n := by | simp [cantorFunctionAux, h]
| 2,171 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 82 | 83 | theorem cantorFunctionAux_zero (f : β β Bool) : cantorFunctionAux c f 0 = cond (f 0) 1 0 := by |
cases h : f 0 <;> simp [h]
| 2,171 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 86 | 90 | theorem cantorFunctionAux_succ (f : β β Bool) :
(fun n => cantorFunctionAux c f (n + 1)) = fun n =>
c * cantorFunctionAux c (fun n => f (n + 1)) n := by |
ext n
cases h : f (n + 1) <;> simp [h, _root_.pow_succ']
| 2,171 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 93 | 96 | theorem summable_cantor_function (f : β β Bool) (h1 : 0 β€ c) (h2 : c < 1) :
Summable (cantorFunctionAux c f) := by |
apply (summable_geometric_of_lt_one h1 h2).summable_of_eq_zero_or_self
intro n; cases h : f n <;> simp [h]
| 2,171 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 105 | 110 | theorem cantorFunction_le (h1 : 0 β€ c) (h2 : c < 1) (h3 : β n, f n β g n) :
cantorFunction c f β€ cantorFunction c g := by |
apply tsum_le_tsum _ (summable_cantor_function f h1 h2) (summable_cantor_function g h1 h2)
intro n; cases h : f n
Β· simp [h, cantorFunctionAux_nonneg h1]
replace h3 : g n = true := h3 n h; simp [h, h3]
| 2,171 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 113 | 117 | theorem cantorFunction_succ (f : β β Bool) (h1 : 0 β€ c) (h2 : c < 1) :
cantorFunction c f = cond (f 0) 1 0 + c * cantorFunction c fun n => f (n + 1) := by |
rw [cantorFunction, tsum_eq_zero_add (summable_cantor_function f h1 h2)]
rw [cantorFunctionAux_succ, tsum_mul_left, cantorFunctionAux, _root_.pow_zero]
rfl
| 2,171 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 123 | 164 | theorem increasing_cantorFunction (h1 : 0 < c) (h2 : c < 1 / 2) {n : β} {f g : β β Bool}
(hn : β k < n, f k = g k) (fn : f n = false) (gn : g n = true) :
cantorFunction c f < cantorFunction c g := by |
have h3 : c < 1 := by
apply h2.trans
norm_num
induction' n with n ih generalizing f g
Β· let f_max : β β Bool := fun n => Nat.rec false (fun _ _ => true) n
have hf_max : β n, f n β f_max n := by
intro n hn
cases n
Β· rw [fn] at hn
contradiction
apply rfl
let g_min : ... | 2,171 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 168 | 197 | theorem cantorFunction_injective (h1 : 0 < c) (h2 : c < 1 / 2) :
Function.Injective (cantorFunction c) := by |
intro f g hfg
classical
by_contra h
revert hfg
have : β n, f n β g n := by
rw [β not_forall]
intro h'
apply h
ext
apply h'
let n := Nat.find this
have hn : β k : β, k < n β f k = g k := by
intro k hk
apply of_not_not
exact Nat.find_min this hk
... | 2,171 |
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 | 25 | 26 | theorem mk_complex : #β = π := by |
rw [mk_congr Complex.equivRealProd, mk_prod, lift_id, mk_real, continuum_mul_self]
| 2,172 |
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 β) = π := by | rw [mk_univ, mk_complex]
| 2,172 |
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 | 35 | 37 | theorem not_countable_complex : Β¬(Set.univ : Set β).Countable := by |
rw [β le_aleph0_iff_set_countable, not_le, mk_univ_complex]
apply cantor
| 2,172 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Data.Complex.Cardinality
import Mathlib.Data.Fin.VecNotation
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import data.complex.module from "leanprover-community/mathlib"@"c7bce2818663f456335892ddbdd1809f111a... | Mathlib/Data/Complex/Module.lean | 125 | 127 | theorem algHom_ext β¦f g : β ββ[β] Aβ¦ (h : f I = g I) : f = g := by |
ext β¨x, yβ©
simp only [mk_eq_add_mul_I, AlgHom.map_add, AlgHom.map_coe_real_complex, AlgHom.map_mul, h]
| 2,173 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Data.Complex.Cardinality
import Mathlib.Data.Fin.VecNotation
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import data.complex.module from "leanprover-community/mathlib"@"c7bce2818663f456335892ddbdd1809f111a... | Mathlib/Data/Complex/Module.lean | 171 | 172 | theorem finrank_real_complex : FiniteDimensional.finrank β β = 2 := by |
rw [finrank_eq_card_basis basisOneI, Fintype.card_fin]
| 2,173 |
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.Determinant
#align_import data.complex.determinant from "leanprover-community/mathlib"@"65ec59902eb17e4ab7da8d7e3d0bd9774d1b8b99"
namespace Complex
@[simp]
| Mathlib/Data/Complex/Determinant.lean | 24 | 26 | theorem det_conjAe : LinearMap.det conjAe.toLinearMap = -1 := by |
rw [β LinearMap.det_toMatrix basisOneI, toMatrix_conjAe, Matrix.det_fin_two_of]
simp
| 2,174 |
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.Determinant
#align_import data.complex.determinant from "leanprover-community/mathlib"@"65ec59902eb17e4ab7da8d7e3d0bd9774d1b8b99"
namespace Complex
@[simp]
theorem det_conjAe : LinearMap.det conjAe.toLinearMap = -1 := by
rw [β LinearMap.det_toMat... | Mathlib/Data/Complex/Determinant.lean | 31 | 33 | theorem linearEquiv_det_conjAe : LinearEquiv.det conjAe.toLinearEquiv = -1 := by |
rw [β Units.eq_iff, LinearEquiv.coe_det, AlgEquiv.toLinearEquiv_toLinearMap, det_conjAe,
Units.coe_neg_one]
| 2,174 |
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Trace
#align_import ring_theory.complex from "leanprover-community/mathlib"@"9015c511549dc77a0f8d6eba021d8ac4bba20c82"
open Complex
| Mathlib/RingTheory/Complex.lean | 17 | 28 | theorem Algebra.leftMulMatrix_complex (z : β) :
Algebra.leftMulMatrix Complex.basisOneI z = !![z.re, -z.im; z.im, z.re] := by |
ext i j
rw [Algebra.leftMulMatrix_eq_repr_mul, Complex.coe_basisOneI_repr, Complex.coe_basisOneI, mul_re,
mul_im, Matrix.of_apply]
fin_cases j
Β· simp only [Fin.mk_zero, Matrix.cons_val_zero, one_re, mul_one, one_im, mul_zero, sub_zero,
zero_add]
fin_cases i <;> rfl
Β· simp only [Fin.mk_one, Matr... | 2,175 |
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Trace
#align_import ring_theory.complex from "leanprover-community/mathlib"@"9015c511549dc77a0f8d6eba021d8ac4bba20c82"
open Complex
theorem Algebra.leftMulMatrix_complex (z : β) :
Algebra.leftMulMatrix Complex.basisOn... | Mathlib/RingTheory/Complex.lean | 31 | 34 | theorem Algebra.trace_complex_apply (z : β) : Algebra.trace β β z = 2 * z.re := by |
rw [Algebra.trace_eq_matrix_trace Complex.basisOneI, Algebra.leftMulMatrix_complex,
Matrix.trace_fin_two]
exact (two_mul _).symm
| 2,175 |
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Trace
#align_import ring_theory.complex from "leanprover-community/mathlib"@"9015c511549dc77a0f8d6eba021d8ac4bba20c82"
open Complex
theorem Algebra.leftMulMatrix_complex (z : β) :
Algebra.leftMulMatrix Complex.basisOn... | Mathlib/RingTheory/Complex.lean | 37 | 40 | theorem Algebra.norm_complex_apply (z : β) : Algebra.norm β z = Complex.normSq z := by |
rw [Algebra.norm_eq_matrix_det Complex.basisOneI, Algebra.leftMulMatrix_complex,
Matrix.det_fin_two, normSq_apply]
simp
| 2,175 |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 90 | 96 | theorem reverse_apply (x : CliffordAlgebra (0 : QuadraticForm R Unit)) :
reverse (R := R) x = x := by |
induction x using CliffordAlgebra.induction with
| algebraMap r => exact reverse.commutes _
| ΞΉ x => rw [ΞΉ_eq_zero, LinearMap.zero_apply, reverse.map_zero]
| mul xβ xβ hxβ hxβ => rw [reverse.map_mul, mul_comm, hxβ, hxβ]
| add xβ xβ hxβ hxβ => rw [reverse.map_add, hxβ, hxβ]
| 2,176 |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 106 | 107 | theorem involute_eq_id :
(involute : CliffordAlgebra (0 : QuadraticForm R Unit) ββ[R] _) = AlgHom.id R _ := by | ext; simp
| 2,176 |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 157 | 164 | theorem toComplex_involute (c : CliffordAlgebra Q) :
toComplex (involute c) = conj (toComplex c) := by |
have : toComplex (involute (ΞΉ Q 1)) = conj (toComplex (ΞΉ Q 1)) := by
simp only [involute_ΞΉ, toComplex_ΞΉ, AlgHom.map_neg, one_smul, Complex.conj_I]
suffices toComplex.comp involute = Complex.conjAe.toAlgHom.comp toComplex by
exact AlgHom.congr_fun this c
ext : 2
exact this
| 2,176 |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 182 | 185 | theorem toComplex_comp_ofComplex : toComplex.comp ofComplex = AlgHom.id β β := by |
ext1
dsimp only [AlgHom.comp_apply, Subtype.coe_mk, AlgHom.id_apply]
rw [ofComplex_I, toComplex_ΞΉ, one_smul]
| 2,176 |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 194 | 198 | theorem ofComplex_comp_toComplex : ofComplex.comp toComplex = AlgHom.id β (CliffordAlgebra Q) := by |
ext
dsimp only [LinearMap.comp_apply, Subtype.coe_mk, AlgHom.id_apply, AlgHom.toLinearMap_apply,
AlgHom.comp_apply]
rw [toComplex_ΞΉ, one_smul, ofComplex_I]
| 2,176 |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 223 | 228 | theorem reverse_apply (x : CliffordAlgebra Q) : reverse (R := β) x = x := by |
induction x using CliffordAlgebra.induction with
| algebraMap r => exact reverse.commutes _
| ΞΉ x => rw [reverse_ΞΉ]
| mul xβ xβ hxβ hxβ => rw [reverse.map_mul, mul_comm, hxβ, hxβ]
| add xβ xβ hxβ hxβ => rw [reverse.map_add, hxβ, hxβ]
| 2,176 |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 311 | 322 | theorem toQuaternion_star (c : CliffordAlgebra (Q cβ cβ)) :
toQuaternion (star c) = star (toQuaternion c) := by |
simp only [CliffordAlgebra.star_def']
induction c using CliffordAlgebra.induction with
| algebraMap r =>
simp only [reverse.commutes, AlgHom.commutes, QuaternionAlgebra.coe_algebraMap,
QuaternionAlgebra.star_coe]
| ΞΉ x =>
rw [reverse_ΞΉ, involute_ΞΉ, toQuaternion_ΞΉ, AlgHom.map_neg, toQuaternion_ΞΉ,
... | 2,176 |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 339 | 348 | theorem ofQuaternion_comp_toQuaternion :
ofQuaternion.comp toQuaternion = AlgHom.id R (CliffordAlgebra (Q cβ cβ)) := by |
ext : 1
dsimp -- before we end up with two goals and have to do this twice
ext
all_goals
dsimp
rw [toQuaternion_ΞΉ]
dsimp
simp only [toQuaternion_ΞΉ, zero_smul, one_smul, zero_add, add_zero, RingHom.map_zero]
| 2,176 |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 358 | 360 | theorem toQuaternion_comp_ofQuaternion :
toQuaternion.comp ofQuaternion = AlgHom.id R β[R,cβ,cβ] := by |
ext : 1 <;> simp
| 2,176 |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 400 | 403 | theorem ΞΉ_mul_ΞΉ (rβ rβ) : ΞΉ (0 : QuadraticForm R R) rβ * ΞΉ (0 : QuadraticForm R R) rβ = 0 := by |
rw [β mul_one rβ, β mul_one rβ, β smul_eq_mul R, β smul_eq_mul R, LinearMap.map_smul,
LinearMap.map_smul, smul_mul_smul, ΞΉ_sq_scalar, QuadraticForm.zero_apply, RingHom.map_zero,
smul_zero]
| 2,176 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
#align_import analysis.special_functions.trigonometric.chebyshev from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
set_... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 29 | 30 | theorem aeval_T (x : A) (n : β€) : aeval x (T R n) = (T A n).eval x := by |
rw [aeval_def, evalβ_eq_eval_map, map_T]
| 2,177 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
#align_import analysis.special_functions.trigonometric.chebyshev from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
set_... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 34 | 35 | theorem aeval_U (x : A) (n : β€) : aeval x (U R n) = (U A n).eval x := by |
rw [aeval_def, evalβ_eq_eval_map, map_U]
| 2,177 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
#align_import analysis.special_functions.trigonometric.chebyshev from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
set_... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 39 | 41 | theorem algebraMap_eval_T (x : R) (n : β€) :
algebraMap R A ((T R n).eval x) = (T A n).eval (algebraMap R A x) := by |
rw [β aeval_algebraMap_apply_eq_algebraMap_eval, aeval_T]
| 2,177 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
#align_import analysis.special_functions.trigonometric.chebyshev from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
set_... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 45 | 47 | theorem algebraMap_eval_U (x : R) (n : β€) :
algebraMap R A ((U R n).eval x) = (U A n).eval (algebraMap R A x) := by |
rw [β aeval_algebraMap_apply_eq_algebraMap_eval, aeval_U]
| 2,177 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
#align_import analysis.special_functions.trigonometric.chebyshev from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
set_... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 73 | 86 | theorem T_complex_cos (n : β€) : (T β n).eval (cos ΞΈ) = cos (n * ΞΈ) := by |
induction n using Polynomial.Chebyshev.induct with
| zero => simp
| one => simp
| add_two n ih1 ih2 =>
simp only [T_add_two, eval_sub, eval_mul, eval_X, eval_ofNat, ih1, ih2, sub_eq_iff_eq_add,
cos_add_cos]
push_cast
ring_nf
| neg_add_one n ih1 ih2 =>
simp only [T_sub_one, eval_sub, eva... | 2,177 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
#align_import analysis.special_functions.trigonometric.chebyshev from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
set_... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 92 | 105 | theorem U_complex_cos (n : β€) : (U β n).eval (cos ΞΈ) * sin ΞΈ = sin ((n + 1) * ΞΈ) := by |
induction n using Polynomial.Chebyshev.induct with
| zero => simp
| one => simp [one_add_one_eq_two, sin_two_mul]; ring
| add_two n ih1 ih2 =>
simp only [U_add_two, add_sub_cancel_right, eval_sub, eval_mul, eval_X, eval_ofNat, sub_mul,
mul_assoc, ih1, ih2, sub_eq_iff_eq_add, sin_add_sin]
push_cas... | 2,177 |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 58 | 59 | theorem norm_exp_ofReal_mul_I (t : β) : βexp (t * I)β = 1 := by |
simp only [norm_eq_abs, abs_exp_ofReal_mul_I]
| 2,178 |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 102 | 104 | theorem dist_eq_re_im (z w : β) : dist z w = β((z.re - w.re) ^ 2 + (z.im - w.im) ^ 2) := by |
rw [sq, sq]
rfl
| 2,178 |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 113 | 114 | theorem dist_of_re_eq {z w : β} (h : z.re = w.re) : dist z w = dist z.im w.im := by |
rw [dist_eq_re_im, h, sub_self, zero_pow two_ne_zero, zero_add, Real.sqrt_sq_eq_abs, Real.dist_eq]
| 2,178 |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 121 | 122 | theorem edist_of_re_eq {z w : β} (h : z.re = w.re) : edist z w = edist z.im w.im := by |
rw [edist_nndist, edist_nndist, nndist_of_re_eq h]
| 2,178 |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 125 | 126 | theorem dist_of_im_eq {z w : β} (h : z.im = w.im) : dist z w = dist z.re w.re := by |
rw [dist_eq_re_im, h, sub_self, zero_pow two_ne_zero, add_zero, Real.sqrt_sq_eq_abs, Real.dist_eq]
| 2,178 |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 133 | 134 | theorem edist_of_im_eq {z w : β} (h : z.im = w.im) : edist z w = edist z.re w.re := by |
rw [edist_nndist, edist_nndist, nndist_of_im_eq h]
| 2,178 |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 137 | 139 | theorem dist_conj_self (z : β) : dist (conj z) z = 2 * |z.im| := by |
rw [dist_of_re_eq (conj_re z), conj_im, dist_comm, Real.dist_eq, sub_neg_eq_add, β two_mul,
_root_.abs_mul, abs_of_pos (zero_lt_two' β)]
| 2,178 |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 146 | 146 | theorem dist_self_conj (z : β) : dist z (conj z) = 2 * |z.im| := by | rw [dist_comm, dist_conj_self]
| 2,178 |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 149 | 150 | theorem nndist_self_conj (z : β) : nndist z (conj z) = 2 * Real.nnabs z.im := by |
rw [nndist_comm, nndist_conj_self]
| 2,178 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.Topology.TietzeExtension
import Mathlib.Analysis.NormedSpace.HomeomorphBall
import Mathlib.Analysis.NormedSpace.RCLike
universe u uβ v w
-- this is not an instance because Lean cannot determine `π`.
theorem TietzeExtension.o... | Mathlib/Analysis/Complex/Tietze.lean | 105 | 118 | theorem exists_norm_eq_restrict_eq (f : s βα΅ E) :
β g : X βα΅ E, βgβ = βfβ β§ g.restrict s = f := by |
by_cases hf : βfβ = 0; Β· exact β¨0, by aesopβ©
have := Metric.instTietzeExtensionClosedBall.{u, v} π (0 : E) (by aesop : 0 < βfβ)
have hf' x : f x β Metric.closedBall 0 βfβ := by simpa using f.norm_coe_le_norm x
obtain β¨g, hg_mem, hgβ© := (f : C(s, E)).exists_forall_mem_restrict_eq hs hf'
simp only [Metric.mem... | 2,179 |
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