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.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import ring_theory.graded_algebra.homogeneous_ideal from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
open SetLike Direc... | Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean | 102 | 107 | theorem HomogeneousIdeal.ext' {I J : HomogeneousIdeal π} (h : β i, β x β π i, x β I β x β J) :
I = J := by |
ext
rw [I.isHomogeneous.mem_iff, J.isHomogeneous.mem_iff]
apply forall_congr'
exact fun i β¦ h i _ (decompose π _ i).2
| 1,344 |
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
#align_import ring_theory.graded_algebra.radical from "leanprover-community/mathlib"@"f1944b30c97c5eb626e498307dec8b022a05bd0a"
open GradedRing DirectSum SetLike Finset
variable {ΞΉ Ο A : Type*}
variable [CommRing A]
variable [LinearOrderedCancelAddCommMono... | Mathlib/RingTheory/GradedAlgebra/Radical.lean | 47 | 136 | theorem Ideal.IsHomogeneous.isPrime_of_homogeneous_mem_or_mem {I : Ideal A} (hI : I.IsHomogeneous π)
(I_ne_top : I β β€)
(homogeneous_mem_or_mem :
β {x y : A}, Homogeneous π x β Homogeneous π y β x * y β I β x β I β¨ y β I) :
Ideal.IsPrime I :=
β¨I_ne_top, by
intro x y hxy
by_contra! rid
... |
intro x hx
rw [filter_nonempty_iff]
contrapose! hx
simp_rw [proj_apply] at hx
rw [β sum_support_decompose π x]
exact Ideal.sum_mem _ hx
set maxβ := setβ.max' (nonempty x ridβ)
set maxβ := setβ.max' (nonempty y ridβ)
have mem_maxβ : maxβ β setβ := max'_... | 1,345 |
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 81 | 83 | theorem zeroLocus_span (s : Set A) : zeroLocus π (Ideal.span s) = zeroLocus π s := by |
ext x
exact (Submodule.gi _ _).gc s x.asHomogeneousIdeal.toIdeal
| 1,346 |
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 99 | 106 | theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum π)) :
(vanishingIdeal t : Set A) =
{ f | β x : ProjectiveSpectrum π, x β t β f β x.asHomogeneousIdeal } := by |
ext f
rw [vanishingIdeal, SetLike.mem_coe, β HomogeneousIdeal.mem_iff, HomogeneousIdeal.toIdeal_iInf,
Submodule.mem_iInf]
refine forall_congr' fun x => ?_
rw [HomogeneousIdeal.toIdeal_iInf, Submodule.mem_iInf, HomogeneousIdeal.mem_iff]
| 1,346 |
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 109 | 111 | theorem mem_vanishingIdeal (t : Set (ProjectiveSpectrum π)) (f : A) :
f β vanishingIdeal t β β x : ProjectiveSpectrum π, x β t β f β x.asHomogeneousIdeal := by |
rw [β SetLike.mem_coe, coe_vanishingIdeal, Set.mem_setOf_eq]
| 1,346 |
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 115 | 117 | theorem vanishingIdeal_singleton (x : ProjectiveSpectrum π) :
vanishingIdeal ({x} : Set (ProjectiveSpectrum π)) = x.asHomogeneousIdeal := by |
simp [vanishingIdeal]
| 1,346 |
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 137 | 141 | theorem gc_set :
@GaloisConnection (Set A) (Set (ProjectiveSpectrum π))α΅α΅ _ _
(fun s => zeroLocus π s) fun t => vanishingIdeal t := by |
have ideal_gc : GaloisConnection Ideal.span _ := (Submodule.gi A _).gc
simpa [zeroLocus_span, Function.comp] using GaloisConnection.compose ideal_gc (gc_ideal π)
| 1,346 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 51 | 53 | theorem univ_fin2 : (univ : Finset (Fin 2)) = {0, 1} := by |
ext x
fin_cases x <;> simp
| 1,347 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 72 | 74 | theorem weightedVSubOfPoint_apply (w : ΞΉ β k) (p : ΞΉ β P) (b : P) :
s.weightedVSubOfPoint p b w = β i β s, w i β’ (p i -α΅₯ b) := by |
simp [weightedVSubOfPoint, LinearMap.sum_apply]
| 1,347 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 79 | 81 | theorem weightedVSubOfPoint_apply_const (w : ΞΉ β k) (p : P) (b : P) :
s.weightedVSubOfPoint (fun _ => p) b w = (β i β s, w i) β’ (p -α΅₯ b) := by |
rw [weightedVSubOfPoint_apply, sum_smul]
| 1,347 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 86 | 91 | theorem weightedVSubOfPoint_congr {wβ wβ : ΞΉ β k} (hw : β i β s, wβ i = wβ i) {pβ pβ : ΞΉ β P}
(hp : β i β s, pβ i = pβ i) (b : P) :
s.weightedVSubOfPoint pβ b wβ = s.weightedVSubOfPoint pβ b wβ := by |
simp_rw [weightedVSubOfPoint_apply]
refine sum_congr rfl fun i hi => ?_
rw [hw i hi, hp i hi]
| 1,347 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 96 | 104 | theorem weightedVSubOfPoint_eq_of_weights_eq (p : ΞΉ β P) (j : ΞΉ) (wβ wβ : ΞΉ β k)
(hw : β i, i β j β wβ i = wβ i) :
s.weightedVSubOfPoint p (p j) wβ = s.weightedVSubOfPoint p (p j) wβ := by |
simp only [Finset.weightedVSubOfPoint_apply]
congr
ext i
rcases eq_or_ne i j with h | h
Β· simp [h]
Β· simp [hw i h]
| 1,347 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 109 | 118 | theorem weightedVSubOfPoint_eq_of_sum_eq_zero (w : ΞΉ β k) (p : ΞΉ β P) (h : β i β s, w i = 0)
(bβ bβ : P) : s.weightedVSubOfPoint p bβ w = s.weightedVSubOfPoint p bβ w := by |
apply eq_of_sub_eq_zero
rw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply, β sum_sub_distrib]
conv_lhs =>
congr
Β· skip
Β· ext
rw [β smul_sub, vsub_sub_vsub_cancel_left]
rw [β sum_smul, h, zero_smul]
| 1,347 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 123 | 135 | theorem weightedVSubOfPoint_vadd_eq_of_sum_eq_one (w : ΞΉ β k) (p : ΞΉ β P) (h : β i β s, w i = 1)
(bβ bβ : P) : s.weightedVSubOfPoint p bβ w +α΅₯ bβ = s.weightedVSubOfPoint p bβ w +α΅₯ bβ := by |
erw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply, β @vsub_eq_zero_iff_eq V,
vadd_vsub_assoc, vsub_vadd_eq_vsub_sub, β add_sub_assoc, add_comm, add_sub_assoc, β
sum_sub_distrib]
conv_lhs =>
congr
Β· skip
Β· congr
Β· skip
Β· ext
rw [β smul_sub, vsub_sub_vsub_cancel_left]
... | 1,347 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 141 | 145 | theorem weightedVSubOfPoint_erase [DecidableEq ΞΉ] (w : ΞΉ β k) (p : ΞΉ β P) (i : ΞΉ) :
(s.erase i).weightedVSubOfPoint p (p i) w = s.weightedVSubOfPoint p (p i) w := by |
rw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply]
apply sum_erase
rw [vsub_self, smul_zero]
| 1,347 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 151 | 155 | theorem weightedVSubOfPoint_insert [DecidableEq ΞΉ] (w : ΞΉ β k) (p : ΞΉ β P) (i : ΞΉ) :
(insert i s).weightedVSubOfPoint p (p i) w = s.weightedVSubOfPoint p (p i) w := by |
rw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply]
apply sum_insert_zero
rw [vsub_self, smul_zero]
| 1,347 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 160 | 165 | theorem weightedVSubOfPoint_indicator_subset (w : ΞΉ β k) (p : ΞΉ β P) (b : P) {sβ sβ : Finset ΞΉ}
(h : sβ β sβ) :
sβ.weightedVSubOfPoint p b w = sβ.weightedVSubOfPoint p b (Set.indicator (βsβ) w) := by |
rw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply]
exact Eq.symm <|
sum_indicator_subset_of_eq_zero w (fun i wi => wi β’ (p i -α΅₯ b : V)) h fun i => zero_smul k _
| 1,347 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 796 | 797 | theorem sum_centroidWeights_eq_one_of_cast_card_ne_zero (h : (card s : k) β 0) :
β i β s, s.centroidWeights k i = 1 := by | simp [h]
| 1,347 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 804 | 809 | theorem sum_centroidWeights_eq_one_of_card_ne_zero [CharZero k] (h : card s β 0) :
β i β s, s.centroidWeights k i = 1 := by |
-- Porting note: `simp` cannot find `mul_inv_cancel` and does not use `norm_cast`
simp only [centroidWeights_apply, sum_const, nsmul_eq_mul, ne_eq, Nat.cast_eq_zero, card_eq_zero]
refine mul_inv_cancel ?_
norm_cast
| 1,347 |
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Order.OrderIsoNat
#align_import data.nat.nth from "leanprover-community/mathlib"@"7fdd4f3746cb059edfdb5d52cba98f66fce418c0"
open Finset
namespace Nat
variable (p : β β Prop)
noncomputable d... | Mathlib/Data/Nat/Nth.lean | 62 | 63 | theorem nth_of_card_le (hf : (setOf p).Finite) {n : β} (hn : hf.toFinset.card β€ n) :
nth p n = 0 := by | rw [nth, dif_pos hf, List.getD_eq_default]; rwa [Finset.length_sort]
| 1,348 |
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Order.OrderIsoNat
#align_import data.nat.nth from "leanprover-community/mathlib"@"7fdd4f3746cb059edfdb5d52cba98f66fce418c0"
open Finset
namespace Nat
variable (p : β β Prop)
noncomputable d... | Mathlib/Data/Nat/Nth.lean | 71 | 73 | theorem nth_eq_orderEmbOfFin (hf : (setOf p).Finite) {n : β} (hn : n < hf.toFinset.card) :
nth p n = hf.toFinset.orderEmbOfFin rfl β¨n, hnβ© := by |
rw [nth_eq_getD_sort hf, Finset.orderEmbOfFin_apply, List.getD_eq_get]
| 1,348 |
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Order.OrderIsoNat
#align_import data.nat.nth from "leanprover-community/mathlib"@"7fdd4f3746cb059edfdb5d52cba98f66fce418c0"
open Finset
namespace Nat
variable (p : β β Prop)
noncomputable d... | Mathlib/Data/Nat/Nth.lean | 76 | 80 | theorem nth_strictMonoOn (hf : (setOf p).Finite) :
StrictMonoOn (nth p) (Set.Iio hf.toFinset.card) := by |
rintro m (hm : m < _) n (hn : n < _) h
simp only [nth_eq_orderEmbOfFin, *]
exact OrderEmbedding.strictMono _ h
| 1,348 |
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Order.OrderIsoNat
#align_import data.nat.nth from "leanprover-community/mathlib"@"7fdd4f3746cb059edfdb5d52cba98f66fce418c0"
open Finset
namespace Nat
variable (p : β β Prop)
noncomputable d... | Mathlib/Data/Nat/Nth.lean | 113 | 119 | theorem image_nth_Iio_card (hf : (setOf p).Finite) : nth p '' Set.Iio hf.toFinset.card = setOf p :=
calc
nth p '' Set.Iio hf.toFinset.card = Set.range (hf.toFinset.orderEmbOfFin rfl) := by |
ext x
simp only [Set.mem_image, Set.mem_range, Fin.exists_iff, β nth_eq_orderEmbOfFin hf,
Set.mem_Iio, exists_prop]
_ = setOf p := by rw [range_orderEmbOfFin, Set.Finite.coe_toFinset]
| 1,348 |
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Order.OrderIsoNat
#align_import data.nat.nth from "leanprover-community/mathlib"@"7fdd4f3746cb059edfdb5d52cba98f66fce418c0"
open Finset
namespace Nat
variable (p : β β Prop)
noncomputable d... | Mathlib/Data/Nat/Nth.lean | 127 | 129 | theorem exists_lt_card_finite_nth_eq (hf : (setOf p).Finite) {x} (h : p x) :
β n, n < hf.toFinset.card β§ nth p n = x := by |
rwa [β @Set.mem_setOf_eq _ _ p, β image_nth_Iio_card hf] at h
| 1,348 |
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Order.OrderIsoNat
#align_import data.nat.nth from "leanprover-community/mathlib"@"7fdd4f3746cb059edfdb5d52cba98f66fce418c0"
open Finset
namespace Nat
variable (p : β β Prop)
noncomputable d... | Mathlib/Data/Nat/Nth.lean | 137 | 138 | theorem nth_apply_eq_orderIsoOfNat (hf : (setOf p).Infinite) (n : β) :
nth p n = @Nat.Subtype.orderIsoOfNat (setOf p) hf.to_subtype n := by | rw [nth, dif_neg hf]
| 1,348 |
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 76 | 88 | theorem wellFoundedOn_iff :
s.WellFoundedOn r β WellFounded fun a b : Ξ± => r a b β§ a β s β§ b β s := by |
have f : RelEmbedding (fun (a : s) (b : s) => r a b) fun a b : Ξ± => r a b β§ a β s β§ b β s :=
β¨β¨(β), Subtype.coe_injectiveβ©, by simpβ©
refine β¨fun h => ?_, f.wellFoundedβ©
rw [WellFounded.wellFounded_iff_has_min]
intro t ht
by_cases hst : (s β© t).Nonempty
Β· rw [β Subtype.preimage_coe_nonempty] at hst
... | 1,349 |
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 92 | 93 | theorem wellFoundedOn_univ : (univ : Set Ξ±).WellFoundedOn r β WellFounded r := by |
simp [wellFoundedOn_iff]
| 1,349 |
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 101 | 108 | theorem wellFoundedOn_range : (range f).WellFoundedOn r β WellFounded (r on f) := by |
let f' : Ξ² β range f := fun c => β¨f c, c, rflβ©
refine β¨fun h => (InvImage.wf f' h).mono fun c c' => id, fun h => β¨?_β©β©
rintro β¨_, c, rflβ©
refine Acc.of_downward_closed f' ?_ _ ?_
Β· rintro _ β¨_, c', rflβ© -
exact β¨c', rflβ©
Β· exact h.apply _
| 1,349 |
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 112 | 113 | theorem wellFoundedOn_image {s : Set Ξ²} : (f '' s).WellFoundedOn r β s.WellFoundedOn (r on f) := by |
rw [image_eq_range]; exact wellFoundedOn_range
| 1,349 |
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 146 | 161 | theorem acc_iff_wellFoundedOn {Ξ±} {r : Ξ± β Ξ± β Prop} {a : Ξ±} :
TFAE [Acc r a,
WellFoundedOn { b | ReflTransGen r b a } r,
WellFoundedOn { b | TransGen r b a } r] := by |
tfae_have 1 β 2
Β· refine fun h => β¨fun b => InvImage.accessible _ ?_β©
rw [β acc_transGen_iff] at h β’
obtain h' | h' := reflTransGen_iff_eq_or_transGen.1 b.2
Β· rwa [h'] at h
Β· exact h.inv h'
tfae_have 2 β 3
Β· exact fun h => h.subset fun _ => TransGen.to_reflTransGen
tfae_have 3 β 1
Β· refine ... | 1,349 |
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 286 | 293 | theorem PartiallyWellOrderedOn.union (hs : s.PartiallyWellOrderedOn r)
(ht : t.PartiallyWellOrderedOn r) : (s βͺ t).PartiallyWellOrderedOn r := by |
rintro f hf
rcases Nat.exists_subseq_of_forall_mem_union f hf with β¨g, hgs | hgtβ©
Β· rcases hs _ hgs with β¨m, n, hlt, hrβ©
exact β¨g m, g n, g.strictMono hlt, hrβ©
Β· rcases ht _ hgt with β¨m, n, hlt, hrβ©
exact β¨g m, g n, g.strictMono hlt, hrβ©
| 1,349 |
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 303 | 309 | theorem PartiallyWellOrderedOn.image_of_monotone_on (hs : s.PartiallyWellOrderedOn r)
(hf : β aβ β s, β aβ β s, r aβ aβ β r' (f aβ) (f aβ)) : (f '' s).PartiallyWellOrderedOn r' := by |
intro g' hg'
choose g hgs heq using hg'
obtain rfl : f β g = g' := funext heq
obtain β¨m, n, hlt, hmnβ© := hs g hgs
exact β¨m, n, hlt, hf _ (hgs m) _ (hgs n) hmnβ©
| 1,349 |
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 312 | 317 | theorem _root_.IsAntichain.finite_of_partiallyWellOrderedOn (ha : IsAntichain r s)
(hp : s.PartiallyWellOrderedOn r) : s.Finite := by |
refine not_infinite.1 fun hi => ?_
obtain β¨m, n, hmn, hβ© := hp (fun n => hi.natEmbedding _ n) fun n => (hi.natEmbedding _ n).2
exact hmn.ne ((hi.natEmbedding _).injective <| Subtype.val_injective <|
ha.eq (hi.natEmbedding _ m).2 (hi.natEmbedding _ n).2 h)
| 1,349 |
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 345 | 348 | theorem partiallyWellOrderedOn_insert :
PartiallyWellOrderedOn (insert a s) r β PartiallyWellOrderedOn s r := by |
simp only [β singleton_union, partiallyWellOrderedOn_union,
partiallyWellOrderedOn_singleton, true_and_iff]
| 1,349 |
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 356 | 373 | theorem partiallyWellOrderedOn_iff_finite_antichains [IsSymm Ξ± r] :
s.PartiallyWellOrderedOn r β β t, t β s β IsAntichain r t β t.Finite := by |
refine β¨fun h t ht hrt => hrt.finite_of_partiallyWellOrderedOn (h.mono ht), ?_β©
rintro hs f hf
by_contra! H
refine infinite_range_of_injective (fun m n hmn => ?_) (hs _ (range_subset_iff.2 hf) ?_)
Β· obtain h | h | h := lt_trichotomy m n
Β· refine (H _ _ h ?_).elim
rw [hmn]
exact refl _
Β· e... | 1,349 |
import Mathlib.Algebra.Order.Group.PiLex
import Mathlib.Data.DFinsupp.Order
import Mathlib.Data.DFinsupp.NeLocus
import Mathlib.Order.WellFoundedSet
#align_import data.dfinsupp.lex from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
variable {ΞΉ : Type*} {Ξ± : ΞΉ β Type*}
namespace DFinsu... | Mathlib/Data/DFinsupp/Lex.lean | 51 | 58 | theorem lex_lt_of_lt_of_preorder [β i, Preorder (Ξ± i)] (r) [IsStrictOrder ΞΉ r] {x y : Ξ β i, Ξ± i}
(hlt : x < y) : β i, (β j, r j i β x j β€ y j β§ y j β€ x j) β§ x i < y i := by |
obtain β¨hle, j, hltβ© := Pi.lt_def.1 hlt
classical
have : (x.neLocus y : Set ΞΉ).WellFoundedOn r := (x.neLocus y).finite_toSet.wellFoundedOn
obtain β¨i, hi, hlβ© := this.has_min { i | x i < y i } β¨β¨j, mem_neLocus.2 hlt.neβ©, hltβ©
refine β¨i, fun k hk β¦ β¨hle k, ?_β©, hiβ©
exact of_not_not fun h β¦ hl β¨k, mem_neLocus... | 1,350 |
import Mathlib.Algebra.Order.Group.PiLex
import Mathlib.Data.DFinsupp.Order
import Mathlib.Data.DFinsupp.NeLocus
import Mathlib.Order.WellFoundedSet
#align_import data.dfinsupp.lex from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
variable {ΞΉ : Type*} {Ξ± : ΞΉ β Type*}
namespace DFinsu... | Mathlib/Data/DFinsupp/Lex.lean | 61 | 64 | theorem lex_lt_of_lt [β i, PartialOrder (Ξ± i)] (r) [IsStrictOrder ΞΉ r] {x y : Ξ β i, Ξ± i}
(hlt : x < y) : Pi.Lex r (Β· < Β·) x y := by |
simp_rw [Pi.Lex, le_antisymm_iff]
exact lex_lt_of_lt_of_preorder r hlt
| 1,350 |
import Mathlib.Algebra.Order.Group.PiLex
import Mathlib.Data.DFinsupp.Order
import Mathlib.Data.DFinsupp.NeLocus
import Mathlib.Order.WellFoundedSet
#align_import data.dfinsupp.lex from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
variable {ΞΉ : Type*} {Ξ± : ΞΉ β Type*}
namespace DFinsu... | Mathlib/Data/DFinsupp/Lex.lean | 133 | 139 | theorem toLex_monotone : Monotone (@toLex (Ξ β i, Ξ± i)) := by |
intro a b h
refine le_of_lt_or_eq (or_iff_not_imp_right.2 fun hne β¦ ?_)
classical
exact β¨Finset.min' _ (nonempty_neLocus_iff.2 hne),
fun j hj β¦ not_mem_neLocus.1 fun h β¦ (Finset.min'_le _ _ h).not_lt hj,
(h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)β©
| 1,350 |
import Mathlib.Data.DFinsupp.Lex
import Mathlib.Order.GameAdd
import Mathlib.Order.Antisymmetrization
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Tactic.AdaptationNote
#align_import data.dfinsupp.well_founded from "leanprover-community/mathlib"@"e9b8651eb1ad354f4de6be35a38ef31efcd2cfaa"
variable {ΞΉ : Ty... | Mathlib/Data/DFinsupp/WellFounded.lean | 69 | 98 | theorem lex_fibration [β (i) (s : Set ΞΉ), Decidable (i β s)] :
Fibration (InvImage (GameAdd (DFinsupp.Lex r s) (DFinsupp.Lex r s)) snd) (DFinsupp.Lex r s)
fun x => piecewise x.2.1 x.2.2 x.1 := by |
rintro β¨p, xβ, xββ© x β¨i, hr, hsβ©
simp_rw [piecewise_apply] at hs hr
split_ifs at hs with hp
Β· refine β¨β¨{ j | r j i β j β p }, piecewise xβ x { j | r j i }, xββ©,
.fst β¨i, fun j hj β¦ ?_, ?_β©, ?_β© <;> simp only [piecewise_apply, Set.mem_setOf_eq]
Β· simp only [if_pos hj]
Β· split_ifs with hi
Β· r... | 1,351 |
import Mathlib.Data.DFinsupp.Lex
import Mathlib.Order.GameAdd
import Mathlib.Order.Antisymmetrization
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Tactic.AdaptationNote
#align_import data.dfinsupp.well_founded from "leanprover-community/mathlib"@"e9b8651eb1ad354f4de6be35a38ef31efcd2cfaa"
variable {ΞΉ : Ty... | Mathlib/Data/DFinsupp/WellFounded.lean | 103 | 109 | theorem Lex.acc_of_single_erase [DecidableEq ΞΉ] {x : Ξ β i, Ξ± i} (i : ΞΉ)
(hs : Acc (DFinsupp.Lex r s) <| single i (x i)) (hu : Acc (DFinsupp.Lex r s) <| x.erase i) :
Acc (DFinsupp.Lex r s) x := by |
classical
convert β @Acc.of_fibration _ _ _ _ _ (lex_fibration r s) β¨{i}, _β©
(InvImage.accessible snd <| hs.prod_gameAdd hu)
convert piecewise_single_erase x i
| 1,351 |
import Mathlib.Data.DFinsupp.Lex
import Mathlib.Order.GameAdd
import Mathlib.Order.Antisymmetrization
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Tactic.AdaptationNote
#align_import data.dfinsupp.well_founded from "leanprover-community/mathlib"@"e9b8651eb1ad354f4de6be35a38ef31efcd2cfaa"
variable {ΞΉ : Ty... | Mathlib/Data/DFinsupp/WellFounded.lean | 118 | 129 | theorem Lex.acc_of_single [DecidableEq ΞΉ] [β (i) (x : Ξ± i), Decidable (x β 0)] (x : Ξ β i, Ξ± i) :
(β i β x.support, Acc (DFinsupp.Lex r s) <| single i (x i)) β Acc (DFinsupp.Lex r s) x := by |
generalize ht : x.support = t; revert x
classical
induction' t using Finset.induction with b t hb ih
Β· intro x ht
rw [support_eq_empty.1 ht]
exact fun _ => Lex.acc_zero hbot
refine fun x ht h => Lex.acc_of_single_erase b (h b <| t.mem_insert_self b) ?_
refine ih _ (by rw [support_erase,... | 1,351 |
import Mathlib.Data.DFinsupp.Lex
import Mathlib.Order.GameAdd
import Mathlib.Order.Antisymmetrization
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Tactic.AdaptationNote
#align_import data.dfinsupp.well_founded from "leanprover-community/mathlib"@"e9b8651eb1ad354f4de6be35a38ef31efcd2cfaa"
variable {ΞΉ : Ty... | Mathlib/Data/DFinsupp/WellFounded.lean | 134 | 153 | theorem Lex.acc_single [DecidableEq ΞΉ] {i : ΞΉ} (hi : Acc (rαΆ β (Β· β Β·)) i) :
β a, Acc (DFinsupp.Lex r s) (single i a) := by |
induction' hi with i _ ih
refine fun a => WellFounded.induction (hs i)
(C := fun x β¦ Acc (DFinsupp.Lex r s) (single i x)) a fun a ha β¦ ?_
refine Acc.intro _ fun x β¦ ?_
rintro β¨k, hr, hsβ©
rw [single_apply] at hs
split_ifs at hs with hik
swap
Β· exact (hbot hs).elim
subst hik
classical
refine ... | 1,351 |
import Mathlib.Data.DFinsupp.WellFounded
import Mathlib.Data.Finsupp.Lex
#align_import data.finsupp.well_founded from "leanprover-community/mathlib"@"5fd3186f1ec30a75d5f65732e3ce5e623382556f"
variable {Ξ± N : Type*}
namespace Finsupp
variable [Zero N] {r : Ξ± β Ξ± β Prop} {s : N β N β Prop} (hbot : β β¦nβ¦, Β¬s n 0)
... | Mathlib/Data/Finsupp/WellFounded.lean | 37 | 42 | theorem Lex.acc (x : Ξ± ββ N) (h : β a β x.support, Acc (rαΆ β (Β· β Β·)) a) :
Acc (Finsupp.Lex r s) x := by |
rw [lex_eq_invImage_dfinsupp_lex]
classical
refine InvImage.accessible toDFinsupp (DFinsupp.Lex.acc (fun _ => hbot) (fun _ => hs) _ ?_)
simpa only [toDFinsupp_support] using h
| 1,352 |
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.WellFoundedSet
#align_import group_theory.submonoid.pointwise from "leanprover-community/mathlib"@"2bbc7e3884ba234309d2a43b19144105a753292e"
open Set P... | Mathlib/Algebra/Group/Submonoid/Pointwise.lean | 72 | 76 | theorem coe_mul_self_eq (s : Submonoid M) : (s : Set M) * s = s := by |
ext x
refine β¨?_, fun h => β¨x, h, 1, s.one_mem, mul_one xβ©β©
rintro β¨a, ha, b, hb, rflβ©
exact s.mul_mem ha hb
| 1,353 |
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.WellFoundedSet
#align_import group_theory.submonoid.pointwise from "leanprover-community/mathlib"@"2bbc7e3884ba234309d2a43b19144105a753292e"
open Set P... | Mathlib/Algebra/Group/Submonoid/Pointwise.lean | 98 | 107 | theorem pow_smul_mem_closure_smul {N : Type*} [CommMonoid N] [MulAction M N] [IsScalarTower M N N]
(r : M) (s : Set N) {x : N} (hx : x β closure s) : β n : β, r ^ n β’ x β closure (r β’ s) := by |
refine @closure_induction N _ s (fun x : N => β n : β, r ^ n β’ x β closure (r β’ s)) _ hx ?_ ?_ ?_
Β· intro x hx
exact β¨1, subset_closure β¨_, hx, by rw [pow_one]β©β©
Β· exact β¨0, by simpa using one_mem _β©
Β· rintro x y β¨nx, hxβ© β¨ny, hyβ©
use ny + nx
rw [pow_add, mul_smul, β smul_mul_assoc, mul_comm, β smu... | 1,353 |
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.WellFoundedSet
#align_import group_theory.submonoid.pointwise from "leanprover-community/mathlib"@"2bbc7e3884ba234309d2a43b19144105a753292e"
open Set P... | Mathlib/Algebra/Group/Submonoid/Pointwise.lean | 700 | 704 | theorem submonoid_closure (hpos : β x : Ξ±, x β s β 1 β€ x) (h : s.IsPWO) :
IsPWO (Submonoid.closure s : Set Ξ±) := by |
rw [Submonoid.closure_eq_image_prod]
refine (h.partiallyWellOrderedOn_sublistForallβ (Β· β€ Β·)).image_of_monotone_on ?_
exact fun l1 _ l2 hl2 h12 => h12.prod_le_prod' fun x hx => hpos x <| hl2 x hx
| 1,353 |
import Mathlib.Algebra.Group.Submonoid.Pointwise
#align_import group_theory.submonoid.inverses from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {M : Type*}
namespace Submonoid
@[to_additive]
noncomputable instance [Monoid M] : Group (IsUnit.submonoid M) :=
{ inferInstanc... | Mathlib/GroupTheory/Submonoid/Inverses.lean | 73 | 76 | theorem leftInv_leftInv_le : S.leftInv.leftInv β€ S := by |
rintro x β¨β¨y, z, hββ©, hβ : x * y = 1β©
convert z.prop
rw [β mul_one x, β hβ, β mul_assoc, hβ, one_mul]
| 1,354 |
import Mathlib.Algebra.Group.Submonoid.Pointwise
#align_import group_theory.submonoid.inverses from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {M : Type*}
namespace Submonoid
@[to_additive]
noncomputable instance [Monoid M] : Group (IsUnit.submonoid M) :=
{ inferInstanc... | Mathlib/GroupTheory/Submonoid/Inverses.lean | 87 | 94 | theorem leftInv_leftInv_eq (hS : S β€ IsUnit.submonoid M) : S.leftInv.leftInv = S := by |
refine le_antisymm S.leftInv_leftInv_le ?_
intro x hx
have : x = ((hS hx).unitβ»ΒΉβ»ΒΉ : MΛ£) := by
rw [inv_inv (hS hx).unit]
rfl
rw [this]
exact S.leftInv.unit_mem_leftInv _ (S.unit_mem_leftInv _ hx)
| 1,354 |
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
#align_import group_theory.subgroup.pointwise from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
open Set
open Pointwise
variable {Ξ± G A S... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 73 | 81 | theorem closure_toSubmonoid (S : Set G) :
(closure S).toSubmonoid = Submonoid.closure (S βͺ Sβ»ΒΉ) := by |
refine le_antisymm (fun x hx => ?_) (Submonoid.closure_le.2 ?_)
Β· refine
closure_induction hx
(fun x hx => Submonoid.closure_mono subset_union_left (Submonoid.subset_closure hx))
(Submonoid.one_mem _) (fun x y hx hy => Submonoid.mul_mem _ hx hy) fun x hx => ?_
rwa [β Submonoid.mem_closure... | 1,355 |
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
#align_import group_theory.subgroup.pointwise from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
open Set
open Pointwise
variable {Ξ± G A S... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 89 | 102 | theorem closure_induction_left {p : (x : G) β x β closure s β Prop} (one : p 1 (one_mem _))
(mul_left : β x (hx : x β s), β (y) hy, p y hy β p (x * y) (mul_mem (subset_closure hx) hy))
(mul_left_inv : β x (hx : x β s), β (y) hy, p y hy β
p (xβ»ΒΉ * y) (mul_mem (inv_mem (subset_closure hx)) hy))
{x : G} ... |
revert h
simp_rw [β mem_toSubmonoid, closure_toSubmonoid] at *
intro h
induction h using Submonoid.closure_induction_left with
| one => exact one
| mul_left x hx y hy ih =>
cases hx with
| inl hx => exact mul_left _ hx _ hy ih
| inr hx => simpa only [inv_inv] using mul_left_inv _ hx _ hy ih
| 1,355 |
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
#align_import group_theory.subgroup.pointwise from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
open Set
open Pointwise
variable {Ξ± G A S... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 125 | 126 | theorem closure_inv (s : Set G) : closure sβ»ΒΉ = closure s := by |
simp only [β toSubmonoid_eq, closure_toSubmonoid, inv_inv, union_comm]
| 1,355 |
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.quotient_group from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
open scope... | Mathlib/GroupTheory/QuotientGroup.lean | 108 | 113 | theorem sound (U : Set (G β§Έ N)) (g : N.op) :
g β’ (mk' N) β»ΒΉ' U = (mk' N) β»ΒΉ' U := by |
ext x
simp only [Set.mem_preimage, Set.mem_smul_set_iff_inv_smul_mem]
congr! 1
exact Quotient.sound β¨gβ»ΒΉ, rflβ©
| 1,356 |
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.quotient_group from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
open scope... | Mathlib/GroupTheory/QuotientGroup.lean | 129 | 131 | theorem eq_one_iff {N : Subgroup G} [nN : N.Normal] (x : G) : (x : G β§Έ N) = 1 β x β N := by |
refine QuotientGroup.eq.trans ?_
rw [mul_one, Subgroup.inv_mem_iff]
| 1,356 |
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.quotient_group from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
open scope... | Mathlib/GroupTheory/QuotientGroup.lean | 149 | 152 | theorem eq_iff_div_mem {N : Subgroup G} [nN : N.Normal] {x y : G} :
(x : G β§Έ N) = y β x / y β N := by |
refine eq_comm.trans (QuotientGroup.eq.trans ?_)
rw [nN.mem_comm_iff, div_eq_mul_inv]
| 1,356 |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ΞΉ : Type*} {G : ΞΉ β Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [β i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 88 | 93 | theorem of_comp_eq_base (i : ΞΉ) : (of i).comp (Ο i) = (base Ο) := by |
ext x
apply (Con.eq _).2
refine ConGen.Rel.of _ _ ?_
simp only [MonoidHom.comp_apply, Set.mem_iUnion, Set.mem_range]
exact β¨_, _, rfl, rflβ©
| 1,357 |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ΞΉ : Type*} {G : ΞΉ β Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [β i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 96 | 97 | theorem of_apply_eq_base (i : ΞΉ) (x : H) : of i (Ο i x) = base Ο x := by |
rw [β MonoidHom.comp_apply, of_comp_eq_base]
| 1,357 |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ΞΉ : Type*} {G : ΞΉ β Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [β i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 111 | 116 | theorem lift_of (f : β i, G i β* K) (k : H β* K)
(hf : β i, (f i).comp (Ο i) = k)
{i : ΞΉ} (g : G i) : (lift f k hf) (of i g : PushoutI Ο) = f i g := by |
delta PushoutI lift of
simp only [MonoidHom.coe_comp, Con.coe_mk', comp_apply, Con.lift_coe,
lift_apply_inl, CoprodI.lift_of]
| 1,357 |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ΞΉ : Type*} {G : ΞΉ β Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [β i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 119 | 123 | theorem lift_base (f : β i, G i β* K) (k : H β* K)
(hf : β i, (f i).comp (Ο i) = k)
(g : H) : (lift f k hf) (base Ο g : PushoutI Ο) = k g := by |
delta PushoutI lift base
simp only [MonoidHom.coe_comp, Con.coe_mk', comp_apply, Con.lift_coe, lift_apply_inr]
| 1,357 |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ΞΉ : Type*} {G : ΞΉ β Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [β i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 163 | 165 | theorem ofCoprodI_of (i : ΞΉ) (g : G i) :
(ofCoprodI (CoprodI.of g) : PushoutI Ο) = of i g := by |
simp [ofCoprodI]
| 1,357 |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ΞΉ : Type*} {G : ΞΉ β Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [β i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 167 | 184 | theorem induction_on {motive : PushoutI Ο β Prop}
(x : PushoutI Ο)
(of : β (i : ΞΉ) (g : G i), motive (of i g))
(base : β h, motive (base Ο h))
(mul : β x y, motive x β motive y β motive (x * y)) : motive x := by |
delta PushoutI PushoutI.of PushoutI.base at *
induction x using Con.induction_on with
| H x =>
induction x using Coprod.induction_on with
| inl g =>
induction g using CoprodI.induction_on with
| h_of i g => exact of i g
| h_mul x y ihx ihy =>
rw [map_mul]
exact mul _ _ i... | 1,357 |
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.char_zero.quotient from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
variable {R : Type*} [DivisionRing R] [CharZero R] {p : R}
namespace AddSubgroup
| Mathlib/Algebra/CharZero/Quotient.lean | 20 | 39 | theorem zsmul_mem_zmultiples_iff_exists_sub_div {r : R} {z : β€} (hz : z β 0) :
z β’ r β AddSubgroup.zmultiples p β
β k : Fin z.natAbs, r - (k : β) β’ (p / z : R) β AddSubgroup.zmultiples p := by |
rw [AddSubgroup.mem_zmultiples_iff]
simp_rw [AddSubgroup.mem_zmultiples_iff, div_eq_mul_inv, β smul_mul_assoc, eq_sub_iff_add_eq]
have hz' : (z : R) β 0 := Int.cast_ne_zero.mpr hz
conv_rhs => simp (config := { singlePass := true }) only [β (mul_right_injectiveβ hz').eq_iff]
simp_rw [β zsmul_eq_mul, smul_add,... | 1,358 |
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.char_zero.quotient from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
variable {R : Type*} [DivisionRing R] [CharZero R] {p : R}
namespace AddSubgroup
theorem zsmul_mem_zmultiples_iff_exists_sub_div {r : R} {z : β€} (hz :... | Mathlib/Algebra/CharZero/Quotient.lean | 42 | 47 | theorem nsmul_mem_zmultiples_iff_exists_sub_div {r : R} {n : β} (hn : n β 0) :
n β’ r β AddSubgroup.zmultiples p β
β k : Fin n, r - (k : β) β’ (p / n : R) β AddSubgroup.zmultiples p := by |
rw [β natCast_zsmul r, zsmul_mem_zmultiples_iff_exists_sub_div (Int.natCast_ne_zero.mpr hn),
Int.cast_natCast]
rfl
| 1,358 |
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Constructions
#align_import topology.algebra.group.basic from "leanprover-community/mathlib"@"3b1890e71632be9e3... | Mathlib/Topology/Algebra/Group/Basic.lean | 71 | 73 | theorem Homeomorph.mulLeft_symm (a : G) : (Homeomorph.mulLeft a).symm = Homeomorph.mulLeft aβ»ΒΉ := by |
ext
rfl
| 1,359 |
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Constructions
#align_import topology.algebra.group.basic from "leanprover-community/mathlib"@"3b1890e71632be9e3... | Mathlib/Topology/Algebra/Group/Basic.lean | 114 | 117 | theorem Homeomorph.mulRight_symm (a : G) :
(Homeomorph.mulRight a).symm = Homeomorph.mulRight aβ»ΒΉ := by |
ext
rfl
| 1,359 |
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Constructions
#align_import topology.algebra.group.basic from "leanprover-community/mathlib"@"3b1890e71632be9e3... | Mathlib/Topology/Algebra/Group/Basic.lean | 146 | 154 | theorem discreteTopology_of_isOpen_singleton_one (h : IsOpen ({1} : Set G)) :
DiscreteTopology G := by |
rw [β singletons_open_iff_discrete]
intro g
suffices {g} = (gβ»ΒΉ * Β·) β»ΒΉ' {1} by
rw [this]
exact (continuous_mul_left gβ»ΒΉ).isOpen_preimage _ h
simp only [mul_one, Set.preimage_mul_left_singleton, eq_self_iff_true, inv_inv,
Set.singleton_eq_singleton_iff]
| 1,359 |
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Ring.Prod
import Mathlib.Algebra.Ring.Subring.Basic
import Mathlib.Topology.Algebra.Group.Basic
#align_import topology.algebra.ring.basic from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpac... | Mathlib/Topology/Algebra/Ring/Basic.lean | 63 | 66 | theorem TopologicalSemiring.continuousNeg_of_mul [TopologicalSpace Ξ±] [NonAssocRing Ξ±]
[ContinuousMul Ξ±] : ContinuousNeg Ξ± where
continuous_neg := by |
simpa using (continuous_const.mul continuous_id : Continuous fun x : Ξ± => -1 * x)
| 1,360 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.Ring.Basic
#align_import topology.algebra.nonarchimedean.basic from "leanprover-community/mathlib"@"83f81aea33931a1edb94ce0f32b9a5d484de6978"
open scoped Pointwise Topology
class Nonarchimede... | Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean | 69 | 75 | theorem nonarchimedean_of_emb (f : G β* H) (emb : OpenEmbedding f) : NonarchimedeanGroup H :=
{ is_nonarchimedean := fun U hU =>
have hβ : f β»ΒΉ' U β π (1 : G) := by |
apply emb.continuous.tendsto
rwa [f.map_one]
let β¨V, hVβ© := is_nonarchimedean (f β»ΒΉ' U) hβ
β¨{ Subgroup.map f V with isOpen' := emb.isOpenMap _ V.isOpen }, Set.image_subset_iff.2 hVβ© }
| 1,361 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.Ring.Basic
#align_import topology.algebra.nonarchimedean.basic from "leanprover-community/mathlib"@"83f81aea33931a1edb94ce0f32b9a5d484de6978"
open scoped Pointwise Topology
class Nonarchimede... | Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean | 84 | 93 | theorem prod_subset {U} (hU : U β π (1 : G Γ K)) :
β (V : OpenSubgroup G) (W : OpenSubgroup K), (V : Set G) ΓΛ’ (W : Set K) β U := by |
erw [nhds_prod_eq, Filter.mem_prod_iff] at hU
rcases hU with β¨Uβ, hUβ, Uβ, hUβ, hβ©
cases' is_nonarchimedean _ hUβ with V hV
cases' is_nonarchimedean _ hUβ with W hW
use V; use W
rw [Set.prod_subset_iff]
intro x hX y hY
exact Set.Subset.trans (Set.prod_mono hV hW) h (Set.mem_sep hX hY)
| 1,361 |
import Mathlib.Topology.Algebra.Nonarchimedean.Basic
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Algebra.Module.Submodule.Pointwise
#align_import topology.algebra.nonarchimedean.bases from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Function Lattice
ope... | Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean | 339 | 345 | theorem nonarchimedean (hB : SubmodulesBasis B) : @NonarchimedeanAddGroup M _ hB.topology := by |
letI := hB.topology
constructor
intro U hU
obtain β¨-, β¨i, rflβ©, hi : (B i : Set M) β Uβ© :=
hB.toModuleFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff.mp hU
exact β¨hB.openAddSubgroup i, hiβ©
| 1,362 |
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.nonarchimedean.adic_topology from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R : Type*} [CommRing R]
open S... | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 54 | 73 | theorem adic_basis (I : Ideal R) : SubmodulesRingBasis fun n : β => (I ^ n β’ β€ : Ideal R) :=
{ inter := by |
suffices β i j : β, β k, I ^ k β€ I ^ i β§ I ^ k β€ I ^ j by
simpa only [smul_eq_mul, mul_top, Algebra.id.map_eq_id, map_id, le_inf_iff] using this
intro i j
exact β¨max i j, pow_le_pow_right (le_max_left i j), pow_le_pow_right (le_max_right i j)β©
leftMul := by
suffices β (a : R) (i : β... | 1,363 |
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.nonarchimedean.adic_topology from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R : Type*} [CommRing R]
open S... | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 92 | 103 | theorem hasBasis_nhds_zero_adic (I : Ideal R) :
HasBasis (@nhds R I.adicTopology (0 : R)) (fun _n : β => True) fun n =>
((I ^ n : Ideal R) : Set R) :=
β¨by
intro U
rw [I.ringFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff]
constructor
Β· rintro β¨-, β¨i, rflβ©, hβ©
replace h : β... | simpa using h
exact β¨i, trivial, hβ©
Β· rintro β¨i, -, hβ©
exact β¨(I ^ i : Ideal R), β¨i, by simpβ©, hβ©β©
| 1,363 |
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.nonarchimedean.adic_topology from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R : Type*} [CommRing R]
open S... | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 106 | 111 | theorem hasBasis_nhds_adic (I : Ideal R) (x : R) :
HasBasis (@nhds R I.adicTopology x) (fun _n : β => True) fun n =>
(fun y => x + y) '' (I ^ n : Ideal R) := by |
letI := I.adicTopology
have := I.hasBasis_nhds_zero_adic.map fun y => x + y
rwa [map_add_left_nhds_zero x] at this
| 1,363 |
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.nonarchimedean.adic_topology from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R : Type*} [CommRing R]
open S... | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 116 | 126 | theorem adic_module_basis :
I.ringFilterBasis.SubmodulesBasis fun n : β => I ^ n β’ (β€ : Submodule R M) :=
{ inter := fun i j =>
β¨max i j,
le_inf_iff.mpr
β¨smul_mono_left <| pow_le_pow_right (le_max_left i j),
smul_mono_left <| pow_le_pow_right (le_max_right i j)β©β©
smul := fu... | simpa [(I ^ i).mul_top] using a_in
exact smul_mem_smul a_in mem_topβ© }
| 1,363 |
import Mathlib.Algebra.Field.Subfield
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Order.LocalExtr
#align_import topology.algebra.field from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862... | Mathlib/Topology/Algebra/Field.lean | 112 | 114 | theorem IsLocalMin.inv {f : Ξ± β Ξ²} {a : Ξ±} (h1 : IsLocalMin f a) (h2 : βαΆ z in π a, 0 < f z) :
IsLocalMax fβ»ΒΉ a := by |
filter_upwards [h1, h2] with z h3 h4 using(inv_le_inv h4 h2.self_of_nhds).mpr h3
| 1,364 |
import Mathlib.Algebra.Field.Subfield
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Order.LocalExtr
#align_import topology.algebra.field from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862... | Mathlib/Topology/Algebra/Field.lean | 130 | 136 | theorem IsPreconnected.eq_one_or_eq_neg_one_of_sq_eq [Ring π] [NoZeroDivisors π]
(hS : IsPreconnected S) (hf : ContinuousOn f S) (hsq : EqOn (f ^ 2) 1 S) :
EqOn f 1 S β¨ EqOn f (-1) S := by |
have : DiscreteTopology ({1, -1} : Set π) := discrete_of_t1_of_finite
have hmaps : MapsTo f S {1, -1} := by
simpa only [EqOn, Pi.one_apply, Pi.pow_apply, sq_eq_one_iff] using hsq
simpa using hS.eqOn_const_of_mapsTo hf hmaps
| 1,364 |
import Mathlib.Algebra.Field.Subfield
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Order.LocalExtr
#align_import topology.algebra.field from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862... | Mathlib/Topology/Algebra/Field.lean | 142 | 149 | theorem IsPreconnected.eq_or_eq_neg_of_sq_eq [Field π] [HasContinuousInvβ π] [ContinuousMul π]
(hS : IsPreconnected S) (hf : ContinuousOn f S) (hg : ContinuousOn g S)
(hsq : EqOn (f ^ 2) (g ^ 2) S) (hg_ne : β {x : Ξ±}, x β S β g x β 0) :
EqOn f g S β¨ EqOn f (-g) S := by |
have hsq : EqOn ((f / g) ^ 2) 1 S := fun x hx => by
simpa [div_eq_one_iff_eq (pow_ne_zero _ (hg_ne hx))] using hsq hx
simpa (config := { contextual := true }) [EqOn, div_eq_iff (hg_ne _)]
using hS.eq_one_or_eq_neg_one_of_sq_eq (hf.div hg fun z => hg_ne) hsq
| 1,364 |
import Mathlib.Algebra.Field.Subfield
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.uniform_field from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open uniformity Topology
... | Mathlib/Topology/Algebra/UniformField.lean | 72 | 93 | theorem continuous_hatInv [CompletableTopField K] {x : hat K} (h : x β 0) :
ContinuousAt hatInv x := by |
refine denseInducing_coe.continuousAt_extend ?_
apply mem_of_superset (compl_singleton_mem_nhds h)
intro y y_ne
rw [mem_compl_singleton_iff] at y_ne
apply CompleteSpace.complete
have : (fun (x : K) => (βxβ»ΒΉ: hat K)) =
((fun (y : K) => (βy: hat K))β(fun (x : K) => (xβ»ΒΉ : K))) := by
unfold Function... | 1,365 |
import Mathlib.Algebra.Field.Subfield
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.uniform_field from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open uniformity Topology
... | Mathlib/Topology/Algebra/UniformField.lean | 112 | 121 | theorem coe_inv (x : K) : (x : hat K)β»ΒΉ = ((xβ»ΒΉ : K) : hat K) := by |
by_cases h : x = 0
Β· rw [h, inv_zero]
dsimp [Inv.inv]
norm_cast
simp
Β· conv_lhs => dsimp [Inv.inv]
rw [if_neg]
Β· exact hatInv_extends h
Β· exact fun H => h (denseEmbedding_coe.inj H)
| 1,365 |
import Mathlib.Algebra.Field.Subfield
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.uniform_field from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open uniformity Topology
... | Mathlib/Topology/Algebra/UniformField.lean | 126 | 153 | theorem mul_hatInv_cancel {x : hat K} (x_ne : x β 0) : x * hatInv x = 1 := by |
haveI : T1Space (hat K) := T2Space.t1Space
let f := fun x : hat K => x * hatInv x
let c := (fun (x : K) => (x : hat K))
change f x = 1
have cont : ContinuousAt f x := by
letI : TopologicalSpace (hat K Γ hat K) := instTopologicalSpaceProd
have : ContinuousAt (fun y : hat K => ((y, hatInv y) : hat K Γ ... | 1,365 |
import Mathlib.Topology.Algebra.Valuation
import Mathlib.Topology.Algebra.WithZeroTopology
import Mathlib.Topology.Algebra.UniformField
#align_import topology.algebra.valued_field from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Filter Set
open Topology
section DivisionRing
v... | Mathlib/Topology/Algebra/ValuedField.lean | 51 | 72 | theorem Valuation.inversion_estimate {x y : K} {Ξ³ : ΞβΛ£} (y_ne : y β 0)
(h : v (x - y) < min (Ξ³ * (v y * v y)) (v y)) : v (xβ»ΒΉ - yβ»ΒΉ) < Ξ³ := by |
have hyp1 : v (x - y) < Ξ³ * (v y * v y) := lt_of_lt_of_le h (min_le_left _ _)
have hyp1' : v (x - y) * (v y * v y)β»ΒΉ < Ξ³ := mul_inv_lt_of_lt_mulβ hyp1
have hyp2 : v (x - y) < v y := lt_of_lt_of_le h (min_le_right _ _)
have key : v x = v y := Valuation.map_eq_of_sub_lt v hyp2
have x_ne : x β 0 := by
intro... | 1,366 |
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.DiscreteSubset
import Mathlib.Tactic.Abel... | Mathlib/Topology/Algebra/UniformGroup.lean | 89 | 92 | theorem UniformContinuous.inv [UniformSpace Ξ²] {f : Ξ² β Ξ±} (hf : UniformContinuous f) :
UniformContinuous fun x => (f x)β»ΒΉ := by |
have : UniformContinuous fun x => 1 / f x := uniformContinuous_const.div hf
simp_all
| 1,367 |
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.DiscreteSubset
import Mathlib.Tactic.Abel... | Mathlib/Topology/Algebra/UniformGroup.lean | 103 | 106 | theorem UniformContinuous.mul [UniformSpace Ξ²] {f : Ξ² β Ξ±} {g : Ξ² β Ξ±} (hf : UniformContinuous f)
(hg : UniformContinuous g) : UniformContinuous fun x => f x * g x := by |
have : UniformContinuous fun x => f x / (g x)β»ΒΉ := hf.div hg.inv
simp_all
| 1,367 |
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.Algebra.UniformGroup
import Mathlib.Topology.UniformSpace.Cauchy
import Ma... | Mathlib/Analysis/LocallyConvex/Bounded.lean | 80 | 84 | theorem _root_.Filter.HasBasis.isVonNBounded_iff {q : ΞΉ β Prop} {s : ΞΉ β Set E} {A : Set E}
(h : (π (0 : E)).HasBasis q s) : IsVonNBounded π A β β i, q i β Absorbs π (s i) A := by |
refine β¨fun hA i hi => hA (h.mem_of_mem hi), fun hA V hV => ?_β©
rcases h.mem_iff.mp hV with β¨i, hi, hVβ©
exact (hA i hi).mono_left hV
| 1,368 |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.Multilinear.Basic
open Bornology Filter Set Function
open scoped Topology
namespace Bornology.IsVonNBounded
variable {ΞΉ π F : Type*} {E : ΞΉ β Type*} [NormedField π]
[β i, AddCommGroup (E i)] [β i, Module π (E i)] [β i, Topol... | Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean | 44 | 83 | theorem image_multilinear' [Nonempty ΞΉ] {s : Set (β i, E i)} (hs : IsVonNBounded π s)
(f : ContinuousMultilinearMap π E F) : IsVonNBounded π (f '' s) := fun V hV β¦ by
classical
if hβ : β c : π, βcβ β€ 1 then
exact absorbs_iff_norm.2 β¨2, fun c hc β¦ by linarith [hβ c]β©
else
let _ : NontriviallyNormed... |
have hfV : f β»ΒΉ' V β π 0 := (map_continuous f).tendsto' _ _ f.map_zero hV
rwa [nhds_pi, Filter.mem_pi, exists_finite_iff_finset] at hfV
have : β i, β c : π, c β 0 β§ β c' : π, βc'β β€ βcβ β β x β s, c' β’ x i β t i := fun i β¦ by
rw [isVonNBounded_pi_iff] at hs
have := (hs i).tendsto_smallSe... | 1,369 |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.Multilinear.Basic
open Bornology Filter Set Function
open scoped Topology
namespace Bornology.IsVonNBounded
variable {ΞΉ π F : Type*} {E : ΞΉ β Type*} [NormedField π]
[β i, AddCommGroup (E i)] [β i, Module π (E i)] [β i, Topol... | Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean | 90 | 96 | theorem image_multilinear [ContinuousSMul π F] {s : Set (β i, E i)} (hs : IsVonNBounded π s)
(f : ContinuousMultilinearMap π E F) : IsVonNBounded π (f '' s) := by |
cases isEmpty_or_nonempty ΞΉ with
| inl h =>
exact (isBounded_iff_isVonNBounded _).1 <|
@Set.Finite.isBounded _ (vonNBornology π F) _ (s.toFinite.image _)
| inr h => exact hs.image_multilinear' f
| 1,369 |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.StrongTopology
#align_import analysis.normed_space.compact_operator from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Function Set Filter Bornology Metric Pointwise Topology
def IsCompactOperat... | Mathlib/Analysis/NormedSpace/CompactOperator.lean | 84 | 89 | theorem isCompactOperator_iff_exists_mem_nhds_isCompact_closure_image [T2Space Mβ] (f : Mβ β Mβ) :
IsCompactOperator f β β V β (π 0 : Filter Mβ), IsCompact (closure <| f '' V) := by |
rw [isCompactOperator_iff_exists_mem_nhds_image_subset_compact]
exact
β¨fun β¨V, hV, K, hK, hKVβ© => β¨V, hV, hK.closure_of_subset hKVβ©,
fun β¨V, hV, hVcβ© => β¨V, hV, closure (f '' V), hVc, subset_closureβ©β©
| 1,370 |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.StrongTopology
#align_import analysis.normed_space.compact_operator from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Function Set Filter Bornology Metric Pointwise Topology
def IsCompactOperat... | Mathlib/Analysis/NormedSpace/CompactOperator.lean | 228 | 230 | theorem IsCompactOperator.sub [TopologicalAddGroup Mβ] {f g : Mβ β Mβ} (hf : IsCompactOperator f)
(hg : IsCompactOperator g) : IsCompactOperator (f - g) := by |
rw [sub_eq_add_neg]; exact hf.add hg.neg
| 1,370 |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.StrongTopology
#align_import analysis.normed_space.compact_operator from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Function Set Filter Bornology Metric Pointwise Topology
def IsCompactOperat... | Mathlib/Analysis/NormedSpace/CompactOperator.lean | 252 | 257 | theorem IsCompactOperator.comp_clm [AddCommMonoid Mβ] [Module Rβ Mβ] {f : Mβ β Mβ}
(hf : IsCompactOperator f) (g : Mβ βSL[Οββ] Mβ) : IsCompactOperator (f β g) := by |
have := g.continuous.tendsto 0
rw [map_zero] at this
rcases hf with β¨K, hK, hKfβ©
exact β¨K, hK, this hKfβ©
| 1,370 |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.StrongTopology
#align_import analysis.normed_space.compact_operator from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Function Set Filter Bornology Metric Pointwise Topology
def IsCompactOperat... | Mathlib/Analysis/NormedSpace/CompactOperator.lean | 260 | 265 | theorem IsCompactOperator.continuous_comp {f : Mβ β Mβ} (hf : IsCompactOperator f) {g : Mβ β Mβ}
(hg : Continuous g) : IsCompactOperator (g β f) := by |
rcases hf with β¨K, hK, hKfβ©
refine β¨g '' K, hK.image hg, mem_of_superset hKf ?_β©
rw [preimage_comp]
exact preimage_mono (subset_preimage_image _ _)
| 1,370 |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.StrongTopology
#align_import analysis.normed_space.compact_operator from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Function Set Filter Bornology Metric Pointwise Topology
def IsCompactOperat... | Mathlib/Analysis/NormedSpace/CompactOperator.lean | 336 | 365 | theorem IsCompactOperator.continuous {f : Mβ βββ[Οββ] Mβ} (hf : IsCompactOperator f) :
Continuous f := by |
letI : UniformSpace Mβ := TopologicalAddGroup.toUniformSpace _
haveI : UniformAddGroup Mβ := comm_topologicalAddGroup_is_uniform
-- Since `f` is linear, we only need to show that it is continuous at zero.
-- Let `U` be a neighborhood of `0` in `Mβ`.
refine continuous_of_continuousAt_zero f fun U hU => ?_
r... | 1,370 |
import Mathlib.Topology.Algebra.UniformGroup
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Data.Matrix.Basic
#align_import topology.uniform_space.matrix from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Uniformity Topology
variable (m n π : Type*) [UniformSpace π]
na... | Mathlib/Topology/UniformSpace/Matrix.lean | 30 | 34 | theorem uniformity :
π€ (Matrix m n π) = β¨
(i : m) (j : n), (π€ π).comap fun a => (a.1 i j, a.2 i j) := by |
erw [Pi.uniformity]
simp_rw [Pi.uniformity, Filter.comap_iInf, Filter.comap_comap]
rfl
| 1,371 |
import Mathlib.Topology.Algebra.UniformGroup
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Data.Matrix.Basic
#align_import topology.uniform_space.matrix from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Uniformity Topology
variable (m n π : Type*) [UniformSpace π]
na... | Mathlib/Topology/UniformSpace/Matrix.lean | 37 | 40 | theorem uniformContinuous {Ξ² : Type*} [UniformSpace Ξ²] {f : Ξ² β Matrix m n π} :
UniformContinuous f β β i j, UniformContinuous fun x => f x i j := by |
simp only [UniformContinuous, Matrix.uniformity, Filter.tendsto_iInf, Filter.tendsto_comap_iff]
apply Iff.intro <;> intro a <;> apply a
| 1,371 |
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.UniformGroup
#align_import topology.algebra.uniform_filter_basis from "leanprover-community/mathlib"@"531db2ef0fdddf8b3c8dcdcd87138fe969e1a81a"
open uniformity Filter
open Filter
namespace AddGroupFilterBasis
variable {G : Type*} [AddC... | Mathlib/Topology/Algebra/UniformFilterBasis.lean | 42 | 51 | theorem cauchy_iff {F : Filter G} :
@Cauchy G B.uniformSpace F β
F.NeBot β§ β U β B, β M β F, βα΅ (x β M) (y β M), y - x β U := by |
letI := B.uniformSpace
haveI := B.uniformAddGroup
suffices F ΓΛ’ F β€ uniformity G β β U β B, β M β F, βα΅ (x β M) (y β M), y - x β U by
constructor <;> rintro β¨h', hβ© <;> refine β¨h', ?_β© <;> [rwa [β this]; rwa [this]]
rw [uniformity_eq_comap_nhds_zero G, β map_le_iff_le_comap]
change Tendsto _ _ _ β _
si... | 1,372 |
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
noncomputable section
open Filter Finset Function
open scoped Topology
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*}
section TopologicalGroup
variable [CommGroup Ξ±] [TopologicalSpace Ξ±] [TopologicalGroup Ξ±]
variable {f g : Ξ² β Ξ±} {a aβ... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 30 | 31 | theorem HasProd.inv (h : HasProd f a) : HasProd (fun b β¦ (f b)β»ΒΉ) aβ»ΒΉ := by |
simpa only using h.map (MonoidHom.id Ξ±)β»ΒΉ continuous_inv
| 1,373 |
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
noncomputable section
open Filter Finset Function
open scoped Topology
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*}
section TopologicalGroup
variable [CommGroup Ξ±] [TopologicalSpace Ξ±] [TopologicalGroup Ξ±]
variable {f g : Ξ² β Ξ±} {a aβ... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 40 | 41 | theorem Multipliable.of_inv (hf : Multipliable fun b β¦ (f b)β»ΒΉ) : Multipliable f := by |
simpa only [inv_inv] using hf.inv
| 1,373 |
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
noncomputable section
open Filter Finset Function
open scoped Topology
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*}
section TopologicalGroup
variable [CommGroup Ξ±] [TopologicalSpace Ξ±] [TopologicalGroup Ξ±]
variable {f g : Ξ² β Ξ±} {a aβ... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 50 | 53 | theorem HasProd.div (hf : HasProd f aβ) (hg : HasProd g aβ) :
HasProd (fun b β¦ f b / g b) (aβ / aβ) := by |
simp only [div_eq_mul_inv]
exact hf.mul hg.inv
| 1,373 |
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
noncomputable section
open Filter Finset Function
open scoped Topology
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*}
section TopologicalGroup
variable [CommGroup Ξ±] [TopologicalSpace Ξ±] [TopologicalGroup Ξ±]
variable {f g : Ξ² β Ξ±} {a aβ... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 63 | 65 | theorem Multipliable.trans_div (hg : Multipliable g) (hfg : Multipliable fun b β¦ f b / g b) :
Multipliable f := by |
simpa only [div_mul_cancel] using hfg.mul hg
| 1,373 |
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
noncomputable section
open Filter Finset Function
open scoped Topology
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*}
section TopologicalGroup
variable [CommGroup Ξ±] [TopologicalSpace Ξ±] [TopologicalGroup Ξ±]
variable {f g : Ξ² β Ξ±} {a aβ... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 75 | 81 | theorem HasProd.update (hf : HasProd f aβ) (b : Ξ²) [DecidableEq Ξ²] (a : Ξ±) :
HasProd (update f b a) (a / f b * aβ) := by |
convert (hasProd_ite_eq b (a / f b)).mul hf with b'
by_cases h : b' = b
Β· rw [h, update_same]
simp [eq_self_iff_true, if_true, sub_add_cancel]
Β· simp only [h, update_noteq, if_false, Ne, one_mul, not_false_iff]
| 1,373 |
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
noncomputable section
open Filter Finset Function
open scoped Topology
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*}
section TopologicalGroup
variable [CommGroup Ξ±] [TopologicalSpace Ξ±] [TopologicalGroup Ξ±]
variable {f g : Ξ² β Ξ±} {a aβ... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 91 | 96 | theorem HasProd.hasProd_compl_iff {s : Set Ξ²} (hf : HasProd (f β (β) : s β Ξ±) aβ) :
HasProd (f β (β) : βsαΆ β Ξ±) aβ β HasProd f (aβ * aβ) := by |
refine β¨fun h β¦ hf.mul_compl h, fun h β¦ ?_β©
rw [hasProd_subtype_iff_mulIndicator] at hf β’
rw [Set.mulIndicator_compl]
simpa only [div_eq_mul_inv, mul_inv_cancel_comm] using h.div hf
| 1,373 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Nonarchimedean.Basic
open Filter Topology
namespace NonarchimedeanGroup
variable {Ξ± G : Type*}
variable [CommGroup G] [UniformSpace G] [UniformGroup G] [NonarchimedeanGroup G]
@[to_additive "Let `G` be a nonarchimedean additive ab... | Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean | 31 | 48 | theorem cauchySeq_prod_of_tendsto_cofinite_one {f : Ξ± β G} (hf : Tendsto f cofinite (π 1)) :
CauchySeq (fun s β¦ β i β s, f i) := by |
/- Let `U` be a neighborhood of `1`. It suffices to show that there exists `s : Finset Ξ±` such
that for any `t : Finset Ξ±` disjoint from `s`, we have `β i β t, f i β U`. -/
apply cauchySeq_finset_iff_prod_vanishing.mpr
intro U hU
-- Since `G` is nonarchimedean, `U` contains an open subgroup `V`.
rcases is_... | 1,374 |
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