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🚚 Move tactic to Rubin/Tactic.lean, wrap everything in a namespace

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Shad Amethyst 12 months ago
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@ -16,122 +16,21 @@ import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.Separation import Mathlib.Topology.Separation
import Mathlib.Topology.Homeomorph import Mathlib.Topology.Homeomorph
import Lean.Meta.Tactic.Util import Rubin.Tactic
import Lean.Elab.Tactic.Basic
import Lean.Meta.Tactic.Simp.Main
import Mathlib.Tactic.Ring.Basic
#align_import rubin #align_import rubin
namespace Rubin
open Rubin.Tactic
-- TODO: remove -- TODO: remove
--@[simp] --@[simp]
theorem Rubin.GroupActionExt.smul_smul' {G α : Type _} [Group G] [MulAction G α] {g h : G} {x : α} : theorem GroupActionExt.smul_smul' {G α : Type _} [Group G] [MulAction G α] {g h : G} {x : α} :
g • h • x = (g * h) • x := g • h • x = (g * h) • x :=
smul_smul g h x smul_smul g h x
#align smul_smul' Rubin.GroupActionExt.smul_smul' #align smul_smul' Rubin.GroupActionExt.smul_smul'
--@[simp] theorem equiv_congr_ne {ι ι' : Type _} (e : ι ≃ ι') {x y : ι} : x ≠ y → e x ≠ e y :=
theorem Rubin.GroupActionExt.smul_eq_smul_inv {G α : Type _} [Group G] [MulAction G α] {g h : G}
{x y : α} : g • x = h • y ↔ (h⁻¹ * g) • x = y :=
by
constructor
· intro hyp
let res := congr_arg ((· • ·) h⁻¹) hyp
simp at res
rw [← mul_smul] at res
exact res
· intro hyp
rw [<-hyp, mul_smul]
simp
#align smul_eq_smul Rubin.GroupActionExt.smul_eq_smul_inv
theorem Rubin.GroupActionExt.smul_succ {G α : Type _} (n : ℕ) [Group G] [MulAction G α] {g : G}
{x : α} : g ^ n.succ • x = g • g ^ n • x :=
by
rw [pow_succ, mul_smul]
#align smul_succ Rubin.GroupActionExt.smul_succ
section GroupActionTactic
-- open Lean.Elab.Tactic
-- open Lean.Meta.Simp
open Lean.Parser.Tactic
syntax (name := group_action₁) "group_action₁" (location)?: tactic
macro_rules
| `(tactic| group_action₁ $[at $location]?) => `(tactic| simp only [
smul_smul,
Rubin.GroupActionExt.smul_eq_smul_inv,
Rubin.GroupActionExt.smul_succ,
one_smul,
mul_one,
one_mul,
sub_self,
add_neg_self,
neg_add_self,
neg_neg,
tsub_self,
<-mul_assoc,
one_pow,
one_zpow,
mul_zpow_neg_one,
zpow_zero,
mul_zpow,
zpow_sub,
zpow_ofNat,
zpow_neg_one,
<-zpow_mul,
zpow_add_one,
zpow_one_add,
zpow_add,
Int.ofNat_add,
Int.ofNat_mul,
Int.ofNat_zero,
Int.ofNat_one,
Int.mul_neg_eq_neg_mul_symm,
Int.neg_mul_eq_neg_mul_symm,
Mathlib.Tactic.Group.zpow_trick,
Mathlib.Tactic.Group.zpow_trick_one,
Mathlib.Tactic.Group.zpow_trick_one',
commutatorElement_def
] $[at $location]?
)
/-- Tactic for normalizing expressions in group actions, without assuming
commutativity, using only the group axioms without any information about
which group is manipulated.
Example:
```lean
example {G α : Type} [Group G] [MulAction G α] (a b : G) (x y : α) (h : a • b • x = a • y) : b⁻¹ • y = x := by
group_action at h -- normalizes `h` which becomes `h : c = d`
rw [←h] -- the goal is now `a*d*d⁻¹ = a`
group_action -- which then normalized and closed
```
-/
syntax (name := group_action) "group_action" (location)?: tactic
macro_rules
| `(tactic| group_action $[at $location]?) => `(tactic|
repeat (fail_if_no_progress (group_action₁ $[at $location]? <;> group $[at $location]?))
)
example {G α: Type _} [Group G] [MulAction G α] (g h: G) (x: α): g • h • x = (g * h) • x := by
group_action
example {G α : Type} [Group G] [MulAction G α] (a b : G) (x y : α) (h : a • b • x = a • y) : b⁻¹ • y = x := by
group_action at h -- normalizes `h` which becomes `h : c = d`
rw [←h] -- the goal is now `a*d*d⁻¹ = a`
group_action -- which then normalized and closed
example (G α : Type _) [Group G] (a b c : G) [MulAction G α] (x : α) :
⁅a * b, c⁆ • x = (a * ⁅b, c⁆ * a⁻¹ * ⁅a, c⁆) • x := by
group_action
theorem Rubin.equiv_congr_ne {ι ι' : Type _} (e : ι ≃ ι') {x y : ι} : x ≠ y → e x ≠ e y :=
by by
intro x_ne_y intro x_ne_y
by_contra h by_contra h
@ -141,7 +40,7 @@ theorem Rubin.equiv_congr_ne {ι ι' : Type _} (e : ι ≃ ι') {x y : ι} : x
-- this definitely should be added to mathlib! -- this definitely should be added to mathlib!
@[simp, to_additive] @[simp, to_additive]
theorem Rubin.GroupActionExt.subgroup_mk_smul {G α : Type _} [Group G] [MulAction G α] theorem GroupActionExt.subgroup_mk_smul {G α : Type _} [Group G] [MulAction G α]
{S : Subgroup G} {g : G} (hg : g ∈ S) (a : α) : (⟨g, hg⟩ : S) • a = g • a := {S : Subgroup G} {g : G} (hg : g ∈ S) (a : α) : (⟨g, hg⟩ : S) • a = g • a :=
rfl rfl
#align Subgroup.mk_smul Rubin.GroupActionExt.subgroup_mk_smul #align Subgroup.mk_smul Rubin.GroupActionExt.subgroup_mk_smul
@ -155,10 +54,10 @@ variable {G α β : Type _} [Group G]
---------------------------------------------------------------- ----------------------------------------------------------------
section Groups section Groups
theorem Rubin.bracket_mul {f g : G} : ⁅f, g⁆ = f * g * f⁻¹ * g⁻¹ := by tauto theorem bracket_mul {f g : G} : ⁅f, g⁆ = f * g * f⁻¹ * g⁻¹ := by tauto
#align bracket_mul Rubin.bracket_mul #align bracket_mul Rubin.bracket_mul
def Rubin.is_algebraically_disjoint (f g : G) := def is_algebraically_disjoint (f g : G) :=
∀ h : G, ∀ h : G,
¬Commute f h → ¬Commute f h →
∃ f₁ f₂ : G, Commute f₁ g ∧ Commute f₂ g ∧ Commute ⁅f₁, ⁅f₂, h⁆⁆ g ∧ ⁅f₁, ⁅f₂, h⁆⁆ ≠ 1 ∃ f₁ f₂ : G, Commute f₁ g ∧ Commute f₂ g ∧ Commute ⁅f₁, ⁅f₂, h⁆⁆ g ∧ ⁅f₁, ⁅f₂, h⁆⁆ ≠ 1
@ -172,7 +71,7 @@ section Actions
variable [MulAction G α] variable [MulAction G α]
@[simp] @[simp]
theorem Rubin.orbit_bot (G : Type _) [Group G] [MulAction G α] (p : α) : theorem orbit_bot (G : Type _) [Group G] [MulAction G α] (p : α) :
MulAction.orbit (⊥ : Subgroup G) p = {p} := MulAction.orbit (⊥ : Subgroup G) p = {p} :=
by by
ext1 ext1
@ -187,16 +86,16 @@ theorem Rubin.orbit_bot (G : Type _) [Group G] [MulAction G α] (p : α) :
-------------------------------- --------------------------------
section SmulImage section SmulImage
theorem Rubin.GroupActionExt.smul_congr (g : G) {x y : α} (h : x = y) : g • x = g • y := theorem GroupActionExt.smul_congr (g : G) {x y : α} (h : x = y) : g • x = g • y :=
congr_arg ((· • ·) g) h congr_arg ((· • ·) g) h
#align smul_congr Rubin.GroupActionExt.smul_congr #align smul_congr Rubin.GroupActionExt.smul_congr
theorem Rubin.GroupActionExt.smul_eq_iff_inv_smul_eq {x : α} {g : G} : g • x = x ↔ g⁻¹ • x = x := theorem GroupActionExt.smul_eq_iff_inv_smul_eq {x : α} {g : G} : g • x = x ↔ g⁻¹ • x = x :=
⟨fun h => (Rubin.GroupActionExt.smul_congr g⁻¹ h).symm.trans (inv_smul_smul g x), fun h => ⟨fun h => (Rubin.GroupActionExt.smul_congr g⁻¹ h).symm.trans (inv_smul_smul g x), fun h =>
(Rubin.GroupActionExt.smul_congr g h).symm.trans (smul_inv_smul g x)⟩ (Rubin.GroupActionExt.smul_congr g h).symm.trans (smul_inv_smul g x)⟩
#align smul_eq_iff_inv_smul_eq Rubin.GroupActionExt.smul_eq_iff_inv_smul_eq #align smul_eq_iff_inv_smul_eq Rubin.GroupActionExt.smul_eq_iff_inv_smul_eq
theorem Rubin.GroupActionExt.smul_pow_eq_of_smul_eq {x : α} {g : G} (n : ℕ) : theorem GroupActionExt.smul_pow_eq_of_smul_eq {x : α} {g : G} (n : ℕ) :
g • x = x → g ^ n • x = x := by g • x = x → g ^ n • x = x := by
induction n with induction n with
| zero => simp only [pow_zero, one_smul, eq_self_iff_true, imp_true_iff] | zero => simp only [pow_zero, one_smul, eq_self_iff_true, imp_true_iff]
@ -206,7 +105,7 @@ theorem Rubin.GroupActionExt.smul_pow_eq_of_smul_eq {x : α} {g : G} (n : ℕ) :
rw [← mul_smul, ← pow_succ] rw [← mul_smul, ← pow_succ]
#align smul_pow_eq_of_smul_eq Rubin.GroupActionExt.smul_pow_eq_of_smul_eq #align smul_pow_eq_of_smul_eq Rubin.GroupActionExt.smul_pow_eq_of_smul_eq
theorem Rubin.GroupActionExt.smul_zpow_eq_of_smul_eq {x : α} {g : G} (n : ℤ) : theorem GroupActionExt.smul_zpow_eq_of_smul_eq {x : α} {g : G} (n : ℤ) :
g • x = x → g ^ n • x = x := by g • x = x → g ^ n • x = x := by
intro h intro h
cases n with cases n with
@ -219,30 +118,30 @@ theorem Rubin.GroupActionExt.smul_zpow_eq_of_smul_eq {x : α} {g : G} (n : ℤ)
exact res exact res
#align smul_zpow_eq_of_smul_eq Rubin.GroupActionExt.smul_zpow_eq_of_smul_eq #align smul_zpow_eq_of_smul_eq Rubin.GroupActionExt.smul_zpow_eq_of_smul_eq
def Rubin.GroupActionExt.is_equivariant (G : Type _) {β : Type _} [Group G] [MulAction G α] def GroupActionExt.is_equivariant (G : Type _) {β : Type _} [Group G] [MulAction G α]
[MulAction G β] (f : α → β) := [MulAction G β] (f : α → β) :=
∀ g : G, ∀ x : α, f (g • x) = g • f x ∀ g : G, ∀ x : α, f (g • x) = g • f x
#align is_equivariant Rubin.GroupActionExt.is_equivariant #align is_equivariant Rubin.GroupActionExt.is_equivariant
def Rubin.SmulImage.smulImage' (g : G) (U : Set α) := def SmulImage.smulImage' (g : G) (U : Set α) :=
{x | g⁻¹ • x ∈ U} {x | g⁻¹ • x ∈ U}
#align subset_img' Rubin.SmulImage.smulImage' #align subset_img' Rubin.SmulImage.smulImage'
def Rubin.SmulImage.smul_preimage' (g : G) (U : Set α) := def SmulImage.smul_preimage' (g : G) (U : Set α) :=
{x | g • x ∈ U} {x | g • x ∈ U}
#align subset_preimg' Rubin.SmulImage.smul_preimage' #align subset_preimg' Rubin.SmulImage.smul_preimage'
def Rubin.SmulImage.SmulImage (g : G) (U : Set α) := def SmulImage.SmulImage (g : G) (U : Set α) :=
(· • ·) g '' U (· • ·) g '' U
#align subset_img Rubin.SmulImage.SmulImage #align subset_img Rubin.SmulImage.SmulImage
infixl:60 "•''" => Rubin.SmulImage.SmulImage infixl:60 "•''" => Rubin.SmulImage.SmulImage
theorem Rubin.SmulImage.smulImage_def {g : G} {U : Set α} : g•''U = (· • ·) g '' U := theorem SmulImage.smulImage_def {g : G} {U : Set α} : g•''U = (· • ·) g '' U :=
rfl rfl
#align subset_img_def Rubin.SmulImage.smulImage_def #align subset_img_def Rubin.SmulImage.smulImage_def
theorem Rubin.SmulImage.mem_smulImage {x : α} {g : G} {U : Set α} : x ∈ g•''U ↔ g⁻¹ • x ∈ U := theorem SmulImage.mem_smulImage {x : α} {g : G} {U : Set α} : x ∈ g•''U ↔ g⁻¹ • x ∈ U :=
by by
rw [Rubin.SmulImage.smulImage_def, Set.mem_image ((· • ·) g) U x] rw [Rubin.SmulImage.smulImage_def, Set.mem_image ((· • ·) g) U x]
constructor constructor
@ -253,14 +152,14 @@ theorem Rubin.SmulImage.mem_smulImage {x : α} {g : G} {U : Set α} : x ∈ g•
exact ⟨g⁻¹ • x, ⟨Set.mem_preimage.mp h, smul_inv_smul g x⟩⟩ exact ⟨g⁻¹ • x, ⟨Set.mem_preimage.mp h, smul_inv_smul g x⟩⟩
#align mem_smul'' Rubin.SmulImage.mem_smulImage #align mem_smul'' Rubin.SmulImage.mem_smulImage
theorem Rubin.SmulImage.mem_inv_smulImage {x : α} {g : G} {U : Set α} : x ∈ g⁻¹•''U ↔ g • x ∈ U := theorem SmulImage.mem_inv_smulImage {x : α} {g : G} {U : Set α} : x ∈ g⁻¹•''U ↔ g • x ∈ U :=
by by
let msi := @Rubin.SmulImage.mem_smulImage _ _ _ _ x g⁻¹ U let msi := @Rubin.SmulImage.mem_smulImage _ _ _ _ x g⁻¹ U
rw [inv_inv] at msi rw [inv_inv] at msi
exact msi exact msi
#align mem_inv_smul'' Rubin.SmulImage.mem_inv_smulImage #align mem_inv_smul'' Rubin.SmulImage.mem_inv_smulImage
theorem Rubin.SmulImage.mul_smulImage (g h : G) (U : Set α) : g * h•''U = g•''(h•''U) := theorem SmulImage.mul_smulImage (g h : G) (U : Set α) : g * h•''U = g•''(h•''U) :=
by by
ext ext
rw [Rubin.SmulImage.mem_smulImage, Rubin.SmulImage.mem_smulImage, Rubin.SmulImage.mem_smulImage, ← rw [Rubin.SmulImage.mem_smulImage, Rubin.SmulImage.mem_smulImage, Rubin.SmulImage.mem_smulImage, ←
@ -268,18 +167,18 @@ theorem Rubin.SmulImage.mul_smulImage (g h : G) (U : Set α) : g * h•''U = g
#align mul_smul'' Rubin.SmulImage.mul_smulImage #align mul_smul'' Rubin.SmulImage.mul_smulImage
@[simp] @[simp]
theorem Rubin.SmulImage.smulImage_smulImage {g h : G} {U : Set α} : g•''(h•''U) = g * h•''U := theorem SmulImage.smulImage_smulImage {g h : G} {U : Set α} : g•''(h•''U) = g * h•''U :=
(Rubin.SmulImage.mul_smulImage g h U).symm (Rubin.SmulImage.mul_smulImage g h U).symm
#align smul''_smul'' Rubin.SmulImage.smulImage_smulImage #align smul''_smul'' Rubin.SmulImage.smulImage_smulImage
@[simp] @[simp]
theorem Rubin.SmulImage.one_smulImage (U : Set α) : (1 : G)•''U = U := theorem SmulImage.one_smulImage (U : Set α) : (1 : G)•''U = U :=
by by
ext ext
rw [Rubin.SmulImage.mem_smulImage, inv_one, one_smul] rw [Rubin.SmulImage.mem_smulImage, inv_one, one_smul]
#align one_smul'' Rubin.SmulImage.one_smulImage #align one_smul'' Rubin.SmulImage.one_smulImage
theorem Rubin.SmulImage.disjoint_smulImage (g : G) {U V : Set α} : theorem SmulImage.disjoint_smulImage (g : G) {U V : Set α} :
Disjoint U V → Disjoint (g•''U) (g•''V) := Disjoint U V → Disjoint (g•''U) (g•''V) :=
by by
intro disjoint_U_V intro disjoint_U_V
@ -291,32 +190,32 @@ theorem Rubin.SmulImage.disjoint_smulImage (g : G) {U V : Set α} :
#align disjoint_smul'' Rubin.SmulImage.disjoint_smulImage #align disjoint_smul'' Rubin.SmulImage.disjoint_smulImage
-- TODO: check if this is actually needed -- TODO: check if this is actually needed
theorem Rubin.SmulImage.smulImage_congr (g : G) {U V : Set α} : U = V → g•''U = g•''V := theorem SmulImage.smulImage_congr (g : G) {U V : Set α} : U = V → g•''U = g•''V :=
congr_arg fun W : Set α => g•''W congr_arg fun W : Set α => g•''W
#align smul''_congr Rubin.SmulImage.smulImage_congr #align smul''_congr Rubin.SmulImage.smulImage_congr
theorem Rubin.SmulImage.smulImage_subset (g : G) {U V : Set α} : U ⊆ V → g•''U ⊆ g•''V := theorem SmulImage.smulImage_subset (g : G) {U V : Set α} : U ⊆ V → g•''U ⊆ g•''V :=
by by
intro h1 x intro h1 x
rw [Rubin.SmulImage.mem_smulImage, Rubin.SmulImage.mem_smulImage] rw [Rubin.SmulImage.mem_smulImage, Rubin.SmulImage.mem_smulImage]
exact fun h2 => h1 h2 exact fun h2 => h1 h2
#align smul''_subset Rubin.SmulImage.smulImage_subset #align smul''_subset Rubin.SmulImage.smulImage_subset
theorem Rubin.SmulImage.smulImage_union (g : G) {U V : Set α} : g•''U ∪ V = (g•''U) ∪ (g•''V) := theorem SmulImage.smulImage_union (g : G) {U V : Set α} : g•''U ∪ V = (g•''U) ∪ (g•''V) :=
by by
ext ext
rw [Rubin.SmulImage.mem_smulImage, Set.mem_union, Set.mem_union, Rubin.SmulImage.mem_smulImage, rw [Rubin.SmulImage.mem_smulImage, Set.mem_union, Set.mem_union, Rubin.SmulImage.mem_smulImage,
Rubin.SmulImage.mem_smulImage] Rubin.SmulImage.mem_smulImage]
#align smul''_union Rubin.SmulImage.smulImage_union #align smul''_union Rubin.SmulImage.smulImage_union
theorem Rubin.SmulImage.smulImage_inter (g : G) {U V : Set α} : g•''U ∩ V = (g•''U) ∩ (g•''V) := theorem SmulImage.smulImage_inter (g : G) {U V : Set α} : g•''U ∩ V = (g•''U) ∩ (g•''V) :=
by by
ext ext
rw [Set.mem_inter_iff, Rubin.SmulImage.mem_smulImage, Rubin.SmulImage.mem_smulImage, rw [Set.mem_inter_iff, Rubin.SmulImage.mem_smulImage, Rubin.SmulImage.mem_smulImage,
Rubin.SmulImage.mem_smulImage, Set.mem_inter_iff] Rubin.SmulImage.mem_smulImage, Set.mem_inter_iff]
#align smul''_inter Rubin.SmulImage.smulImage_inter #align smul''_inter Rubin.SmulImage.smulImage_inter
theorem Rubin.SmulImage.smulImage_eq_inv_preimage {g : G} {U : Set α} : g•''U = (· • ·) g⁻¹ ⁻¹' U := theorem SmulImage.smulImage_eq_inv_preimage {g : G} {U : Set α} : g•''U = (· • ·) g⁻¹ ⁻¹' U :=
by by
ext ext
constructor constructor
@ -324,7 +223,7 @@ theorem Rubin.SmulImage.smulImage_eq_inv_preimage {g : G} {U : Set α} : g•''U
· intro h; rw [Rubin.SmulImage.mem_smulImage]; exact Set.mem_preimage.mp h · intro h; rw [Rubin.SmulImage.mem_smulImage]; exact Set.mem_preimage.mp h
#align smul''_eq_inv_preimage Rubin.SmulImage.smulImage_eq_inv_preimage #align smul''_eq_inv_preimage Rubin.SmulImage.smulImage_eq_inv_preimage
theorem Rubin.SmulImage.smulImage_eq_of_smul_eq {g h : G} {U : Set α} : theorem SmulImage.smulImage_eq_of_smul_eq {g h : G} {U : Set α} :
(∀ x ∈ U, g • x = h • x) → g•''U = h•''U := (∀ x ∈ U, g • x = h • x) → g•''U = h•''U :=
by by
intro hU intro hU
@ -342,40 +241,40 @@ end SmulImage
-------------------------------- --------------------------------
section Support section Support
def Rubin.SmulSupport.Support (α : Type _) [MulAction G α] (g : G) := def SmulSupport.Support (α : Type _) [MulAction G α] (g : G) :=
{x : α | g • x ≠ x} {x : α | g • x ≠ x}
#align support Rubin.SmulSupport.Support #align support Rubin.SmulSupport.Support
theorem Rubin.SmulSupport.support_eq_not_fixed_by {g : G}: theorem SmulSupport.support_eq_not_fixed_by {g : G}:
Rubin.SmulSupport.Support α g = (MulAction.fixedBy α g)ᶜ := by tauto Rubin.SmulSupport.Support α g = (MulAction.fixedBy α g)ᶜ := by tauto
#align support_eq_not_fixed_by Rubin.SmulSupport.support_eq_not_fixed_by #align support_eq_not_fixed_by Rubin.SmulSupport.support_eq_not_fixed_by
theorem Rubin.SmulSupport.mem_support {x : α} {g : G} : theorem SmulSupport.mem_support {x : α} {g : G} :
x ∈ Rubin.SmulSupport.Support α g ↔ g • x ≠ x := by tauto x ∈ Rubin.SmulSupport.Support α g ↔ g • x ≠ x := by tauto
#align mem_support Rubin.SmulSupport.mem_support #align mem_support Rubin.SmulSupport.mem_support
theorem Rubin.SmulSupport.not_mem_support {x : α} {g : G} : theorem SmulSupport.not_mem_support {x : α} {g : G} :
x ∉ Rubin.SmulSupport.Support α g ↔ g • x = x := by x ∉ Rubin.SmulSupport.Support α g ↔ g • x = x := by
rw [Rubin.SmulSupport.mem_support, Classical.not_not] rw [Rubin.SmulSupport.mem_support, Classical.not_not]
#align mem_not_support Rubin.SmulSupport.not_mem_support #align mem_not_support Rubin.SmulSupport.not_mem_support
theorem Rubin.SmulSupport.smul_mem_support {g : G} {x : α} : theorem SmulSupport.smul_mem_support {g : G} {x : α} :
x ∈ Rubin.SmulSupport.Support α g → g • x ∈ Rubin.SmulSupport.Support α g := fun h => x ∈ Rubin.SmulSupport.Support α g → g • x ∈ Rubin.SmulSupport.Support α g := fun h =>
h ∘ smul_left_cancel g h ∘ smul_left_cancel g
#align smul_in_support Rubin.SmulSupport.smul_mem_support #align smul_in_support Rubin.SmulSupport.smul_mem_support
theorem Rubin.SmulSupport.inv_smul_mem_support {g : G} {x : α} : theorem SmulSupport.inv_smul_mem_support {g : G} {x : α} :
x ∈ Rubin.SmulSupport.Support α g → g⁻¹ • x ∈ Rubin.SmulSupport.Support α g := fun h k => x ∈ Rubin.SmulSupport.Support α g → g⁻¹ • x ∈ Rubin.SmulSupport.Support α g := fun h k =>
h (smul_inv_smul g x ▸ Rubin.GroupActionExt.smul_congr g k) h (smul_inv_smul g x ▸ Rubin.GroupActionExt.smul_congr g k)
#align inv_smul_in_support Rubin.SmulSupport.inv_smul_mem_support #align inv_smul_in_support Rubin.SmulSupport.inv_smul_mem_support
theorem Rubin.SmulSupport.fixed_of_disjoint {g : G} {x : α} {U : Set α} : theorem SmulSupport.fixed_of_disjoint {g : G} {x : α} {U : Set α} :
x ∈ U → Disjoint U (Rubin.SmulSupport.Support α g) → g • x = x := x ∈ U → Disjoint U (Rubin.SmulSupport.Support α g) → g • x = x :=
fun x_in_U disjoint_U_support => fun x_in_U disjoint_U_support =>
Rubin.SmulSupport.not_mem_support.mp (Set.disjoint_left.mp disjoint_U_support x_in_U) Rubin.SmulSupport.not_mem_support.mp (Set.disjoint_left.mp disjoint_U_support x_in_U)
#align fixed_of_disjoint Rubin.SmulSupport.fixed_of_disjoint #align fixed_of_disjoint Rubin.SmulSupport.fixed_of_disjoint
theorem Rubin.SmulSupport.fixed_smulImage_in_support (g : G) {U : Set α} : theorem SmulSupport.fixed_smulImage_in_support (g : G) {U : Set α} :
Rubin.SmulSupport.Support α g ⊆ U → g•''U = U := Rubin.SmulSupport.Support α g ⊆ U → g•''U = U :=
by by
intro support_in_U intro support_in_U
@ -387,13 +286,13 @@ theorem Rubin.SmulSupport.fixed_smulImage_in_support (g : G) {U : Set α} :
rw [Rubin.SmulImage.mem_smulImage, GroupActionExt.smul_eq_iff_inv_smul_eq.mp (not_mem_support.mp xfixed)] rw [Rubin.SmulImage.mem_smulImage, GroupActionExt.smul_eq_iff_inv_smul_eq.mp (not_mem_support.mp xfixed)]
#align fixes_subset_within_support Rubin.SmulSupport.fixed_smulImage_in_support #align fixes_subset_within_support Rubin.SmulSupport.fixed_smulImage_in_support
theorem Rubin.SmulSupport.smulImage_subset_in_support (g : G) (U V : Set α) : theorem SmulSupport.smulImage_subset_in_support (g : G) (U V : Set α) :
U ⊆ V → Rubin.SmulSupport.Support α g ⊆ V → g•''U ⊆ V := fun U_in_V support_in_V => U ⊆ V → Rubin.SmulSupport.Support α g ⊆ V → g•''U ⊆ V := fun U_in_V support_in_V =>
Rubin.SmulSupport.fixed_smulImage_in_support g support_in_V ▸ Rubin.SmulSupport.fixed_smulImage_in_support g support_in_V ▸
Rubin.SmulImage.smulImage_subset g U_in_V Rubin.SmulImage.smulImage_subset g U_in_V
#align moves_subset_within_support Rubin.SmulSupport.smulImage_subset_in_support #align moves_subset_within_support Rubin.SmulSupport.smulImage_subset_in_support
theorem Rubin.SmulSupport.support_mul (g h : G) (α : Type _) [MulAction G α] : theorem SmulSupport.support_mul (g h : G) (α : Type _) [MulAction G α] :
Rubin.SmulSupport.Support α (g * h) ⊆ Rubin.SmulSupport.Support α (g * h) ⊆
Rubin.SmulSupport.Support α g ∪ Rubin.SmulSupport.Support α h := Rubin.SmulSupport.Support α g ∪ Rubin.SmulSupport.Support α h :=
by by
@ -407,7 +306,7 @@ theorem Rubin.SmulSupport.support_mul (g h : G) (α : Type _) [MulAction G α] :
((congr_arg ((· • ·) g) (not_mem_support.mp res.2)).trans <| not_mem_support.mp res.1)) ((congr_arg ((· • ·) g) (not_mem_support.mp res.2)).trans <| not_mem_support.mp res.1))
#align support_mul Rubin.SmulSupport.support_mul #align support_mul Rubin.SmulSupport.support_mul
theorem Rubin.SmulSupport.support_conjugate (α : Type _) [MulAction G α] (g h : G) : theorem SmulSupport.support_conjugate (α : Type _) [MulAction G α] (g h : G) :
Rubin.SmulSupport.Support α (h * g * h⁻¹) = h•''Rubin.SmulSupport.Support α g := Rubin.SmulSupport.Support α (h * g * h⁻¹) = h•''Rubin.SmulSupport.Support α g :=
by by
ext x ext x
@ -418,7 +317,7 @@ theorem Rubin.SmulSupport.support_conjugate (α : Type _) [MulAction G α] (g h
· intro h1; by_contra h2; exact h1 (inv_smul_smul h (g • h⁻¹ • x) ▸ congr_arg ((· • ·) h⁻¹) h2) · intro h1; by_contra h2; exact h1 (inv_smul_smul h (g • h⁻¹ • x) ▸ congr_arg ((· • ·) h⁻¹) h2)
#align support_conjugate Rubin.SmulSupport.support_conjugate #align support_conjugate Rubin.SmulSupport.support_conjugate
theorem Rubin.SmulSupport.support_inv (α : Type _) [MulAction G α] (g : G) : theorem SmulSupport.support_inv (α : Type _) [MulAction G α] (g : G) :
Rubin.SmulSupport.Support α g⁻¹ = Rubin.SmulSupport.Support α g := Rubin.SmulSupport.Support α g⁻¹ = Rubin.SmulSupport.Support α g :=
by by
ext x ext x
@ -428,7 +327,7 @@ theorem Rubin.SmulSupport.support_inv (α : Type _) [MulAction G α] (g : G) :
· intro h1; by_contra h2; exact h1 (GroupActionExt.smul_eq_iff_inv_smul_eq.mpr h2) · intro h1; by_contra h2; exact h1 (GroupActionExt.smul_eq_iff_inv_smul_eq.mpr h2)
#align support_inv Rubin.SmulSupport.support_inv #align support_inv Rubin.SmulSupport.support_inv
theorem Rubin.SmulSupport.support_pow (α : Type _) [MulAction G α] (g : G) (j : ℕ) : theorem SmulSupport.support_pow (α : Type _) [MulAction G α] (g : G) (j : ℕ) :
Rubin.SmulSupport.Support α (g ^ j) ⊆ Rubin.SmulSupport.Support α g := Rubin.SmulSupport.Support α (g ^ j) ⊆ Rubin.SmulSupport.Support α g :=
by by
intro x xmoved intro x xmoved
@ -444,7 +343,7 @@ theorem Rubin.SmulSupport.support_pow (α : Type _) [MulAction G α] (g : G) (j
exact j_ih exact j_ih
#align support_pow Rubin.SmulSupport.support_pow #align support_pow Rubin.SmulSupport.support_pow
theorem Rubin.SmulSupport.support_comm (α : Type _) [MulAction G α] (g h : G) : theorem SmulSupport.support_comm (α : Type _) [MulAction G α] (g h : G) :
Rubin.SmulSupport.Support α ⁅g, h⁆ ⊆ Rubin.SmulSupport.Support α ⁅g, h⁆ ⊆
Rubin.SmulSupport.Support α h ∪ (g•''Rubin.SmulSupport.Support α h) := Rubin.SmulSupport.Support α h ∪ (g•''Rubin.SmulSupport.Support α h) :=
by by
@ -460,7 +359,7 @@ theorem Rubin.SmulSupport.support_comm (α : Type _) [MulAction G α] (g h : G)
· exact all_fixed (Or.inl xmoved) · exact all_fixed (Or.inl xmoved)
#align support_comm Rubin.SmulSupport.support_comm #align support_comm Rubin.SmulSupport.support_comm
theorem Rubin.SmulSupport.disjoint_support_comm (f g : G) {U : Set α} : theorem SmulSupport.disjoint_support_comm (f g : G) {U : Set α} :
Rubin.SmulSupport.Support α f ⊆ U → Disjoint U (g•''U) → ∀ x ∈ U, ⁅f, g⁆ • x = f • x := Rubin.SmulSupport.Support α f ⊆ U → Disjoint U (g•''U) → ∀ x ∈ U, ⁅f, g⁆ • x = f • x :=
by by
intro support_in_U disjoint_U x x_in_U intro support_in_U disjoint_U x x_in_U
@ -484,7 +383,7 @@ end Support
-- comment by Cedric: would be nicer to define just a subset, and then show it is a subgroup -- comment by Cedric: would be nicer to define just a subset, and then show it is a subgroup
def rigidStabilizer' (G : Type _) [Group G] [MulAction G α] (U : Set α) : Set G := def rigidStabilizer' (G : Type _) [Group G] [MulAction G α] (U : Set α) : Set G :=
{g : G | ∀ x : α, g • x = x ∨ x ∈ U} {g : G | ∀ x : α, g • x = x ∨ x ∈ U}
#align rigid_stabilizer' rigidStabilizer' #align rigid_stabilizer' Rubin.rigidStabilizer'
/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » U) -/ /- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » U) -/
def rigidStabilizer (G : Type _) [Group G] [MulAction G α] (U : Set α) : Subgroup G def rigidStabilizer (G : Type _) [Group G] [MulAction G α] (U : Set α) : Subgroup G
@ -493,12 +392,12 @@ def rigidStabilizer (G : Type _) [Group G] [MulAction G α] (U : Set α) : Subgr
mul_mem' ha hb x x_notin_U := by rw [mul_smul, hb x x_notin_U, ha x x_notin_U] mul_mem' ha hb x x_notin_U := by rw [mul_smul, hb x x_notin_U, ha x x_notin_U]
inv_mem' hg x x_notin_U := Rubin.GroupActionExt.smul_eq_iff_inv_smul_eq.mp (hg x x_notin_U) inv_mem' hg x x_notin_U := Rubin.GroupActionExt.smul_eq_iff_inv_smul_eq.mp (hg x x_notin_U)
one_mem' x _ := one_smul G x one_mem' x _ := one_smul G x
#align rigid_stabilizer rigidStabilizer #align rigid_stabilizer Rubin.rigidStabilizer
theorem rist_supported_in_set {g : G} {U : Set α} : theorem rist_supported_in_set {g : G} {U : Set α} :
g ∈ rigidStabilizer G U → Rubin.SmulSupport.Support α g ⊆ U := fun h x x_in_support => g ∈ rigidStabilizer G U → Rubin.SmulSupport.Support α g ⊆ U := fun h x x_in_support =>
by_contradiction (x_in_support ∘ h x) by_contradiction (x_in_support ∘ h x)
#align rist_supported_in_set rist_supported_in_set #align rist_supported_in_set Rubin.rist_supported_in_set
theorem rist_ss_rist {U V : Set α} (V_ss_U : V ⊆ U) : theorem rist_ss_rist {U V : Set α} (V_ss_U : V ⊆ U) :
(rigidStabilizer G V : Set G) ⊆ (rigidStabilizer G U : Set G) := (rigidStabilizer G V : Set G) ⊆ (rigidStabilizer G U : Set G) :=
@ -506,7 +405,7 @@ theorem rist_ss_rist {U V : Set α} (V_ss_U : V ⊆ U) :
intro g g_in_ristV x x_notin_U intro g g_in_ristV x x_notin_U
have x_notin_V : x ∉ V := by intro x_in_V; exact x_notin_U (V_ss_U x_in_V) have x_notin_V : x ∉ V := by intro x_in_V; exact x_notin_U (V_ss_U x_in_V)
exact g_in_ristV x x_notin_V exact g_in_ristV x x_notin_V
#align rist_ss_rist rist_ss_rist #align rist_ss_rist Rubin.rist_ss_rist
end Actions end Actions
@ -515,23 +414,23 @@ section TopologicalActions
variable [TopologicalSpace α] [TopologicalSpace β] variable [TopologicalSpace α] [TopologicalSpace β]
class Rubin.Topological.ContinuousMulAction (G α : Type _) [Group G] [TopologicalSpace α] extends class Topological.ContinuousMulAction (G α : Type _) [Group G] [TopologicalSpace α] extends
MulAction G α where MulAction G α where
continuous : ∀ g : G, Continuous (@SMul.smul G α _ g) continuous : ∀ g : G, Continuous (@SMul.smul G α _ g)
#align continuous_mul_action Rubin.Topological.ContinuousMulAction #align continuous_mul_action Rubin.Topological.ContinuousMulAction
structure Rubin.Topological.equivariant_homeomorph (G α β : Type _) [Group G] [TopologicalSpace α] structure Topological.equivariant_homeomorph (G α β : Type _) [Group G] [TopologicalSpace α]
[TopologicalSpace β] [MulAction G α] [MulAction G β] extends Homeomorph α β where [TopologicalSpace β] [MulAction G α] [MulAction G β] extends Homeomorph α β where
equivariant : GroupActionExt.is_equivariant G toFun equivariant : GroupActionExt.is_equivariant G toFun
#align equivariant_homeomorph Rubin.Topological.equivariant_homeomorph #align equivariant_homeomorph Rubin.Topological.equivariant_homeomorph
theorem Rubin.Topological.equivariant_fun [MulAction G α] [MulAction G β] theorem Topological.equivariant_fun [MulAction G α] [MulAction G β]
(h : Rubin.Topological.equivariant_homeomorph G α β) : (h : Rubin.Topological.equivariant_homeomorph G α β) :
Rubin.GroupActionExt.is_equivariant G h.toFun := Rubin.GroupActionExt.is_equivariant G h.toFun :=
h.equivariant h.equivariant
#align equivariant_fun Rubin.Topological.equivariant_fun #align equivariant_fun Rubin.Topological.equivariant_fun
theorem Rubin.Topological.equivariant_inv [MulAction G α] [MulAction G β] theorem Topological.equivariant_inv [MulAction G α] [MulAction G β]
(h : Rubin.Topological.equivariant_homeomorph G α β) : (h : Rubin.Topological.equivariant_homeomorph G α β) :
Rubin.GroupActionExt.is_equivariant G h.invFun := Rubin.GroupActionExt.is_equivariant G h.invFun :=
by by
@ -544,14 +443,14 @@ theorem Rubin.Topological.equivariant_inv [MulAction G α] [MulAction G β]
variable [Rubin.Topological.ContinuousMulAction G α] variable [Rubin.Topological.ContinuousMulAction G α]
theorem Rubin.Topological.img_open_open (g : G) (U : Set α) (h : IsOpen U) theorem Topological.img_open_open (g : G) (U : Set α) (h : IsOpen U)
[Rubin.Topological.ContinuousMulAction G α] : IsOpen (g•''U) := [Rubin.Topological.ContinuousMulAction G α] : IsOpen (g•''U) :=
by by
rw [Rubin.SmulImage.smulImage_eq_inv_preimage] rw [Rubin.SmulImage.smulImage_eq_inv_preimage]
exact Continuous.isOpen_preimage (Rubin.Topological.ContinuousMulAction.continuous g⁻¹) U h exact Continuous.isOpen_preimage (Rubin.Topological.ContinuousMulAction.continuous g⁻¹) U h
#align img_open_open Rubin.Topological.img_open_open #align img_open_open Rubin.Topological.img_open_open
theorem Rubin.Topological.support_open (g : G) [TopologicalSpace α] [T2Space α] theorem Topological.support_open (g : G) [TopologicalSpace α] [T2Space α]
[Rubin.Topological.ContinuousMulAction G α] : IsOpen (Rubin.SmulSupport.Support α g) := [Rubin.Topological.ContinuousMulAction G α] : IsOpen (Rubin.SmulSupport.Support α g) :=
by by
apply isOpen_iff_forall_mem_open.mpr apply isOpen_iff_forall_mem_open.mpr
@ -575,17 +474,17 @@ section FaithfulActions
variable [MulAction G α] [FaithfulSMul G α] variable [MulAction G α] [FaithfulSMul G α]
theorem Rubin.faithful_moves_point₁ {g : G} (h2 : ∀ x : α, g • x = x) : g = 1 := theorem faithful_moves_point₁ {g : G} (h2 : ∀ x : α, g • x = x) : g = 1 :=
haveI h3 : ∀ x : α, g • x = (1 : G) • x := fun x => (h2 x).symm ▸ (one_smul G x).symm haveI h3 : ∀ x : α, g • x = (1 : G) • x := fun x => (h2 x).symm ▸ (one_smul G x).symm
eq_of_smul_eq_smul h3 eq_of_smul_eq_smul h3
#align faithful_moves_point Rubin.faithful_moves_point₁ #align faithful_moves_point Rubin.faithful_moves_point₁
theorem Rubin.faithful_moves_point'₁ {g : G} (α : Type _) [MulAction G α] [FaithfulSMul G α] : theorem faithful_moves_point'₁ {g : G} (α : Type _) [MulAction G α] [FaithfulSMul G α] :
g ≠ 1 → ∃ x : α, g • x ≠ x := fun k => g ≠ 1 → ∃ x : α, g • x ≠ x := fun k =>
by_contradiction fun h => k <| Rubin.faithful_moves_point₁ <| Classical.not_exists_not.mp h by_contradiction fun h => k <| Rubin.faithful_moves_point₁ <| Classical.not_exists_not.mp h
#align faithful_moves_point' Rubin.faithful_moves_point'₁ #align faithful_moves_point' Rubin.faithful_moves_point'₁
theorem Rubin.faithful_rigid_stabilizer_moves_point {g : G} {U : Set α} : theorem faithful_rigid_stabilizer_moves_point {g : G} {U : Set α} :
g ∈ rigidStabilizer G U → g ≠ 1 → ∃ x ∈ U, g • x ≠ x := g ∈ rigidStabilizer G U → g ≠ 1 → ∃ x ∈ U, g • x ≠ x :=
by by
intro g_rigid g_ne_one intro g_rigid g_ne_one
@ -593,7 +492,7 @@ theorem Rubin.faithful_rigid_stabilizer_moves_point {g : G} {U : Set α} :
exact ⟨x, rist_supported_in_set g_rigid xmoved, xmoved⟩ exact ⟨x, rist_supported_in_set g_rigid xmoved, xmoved⟩
#align faithful_rist_moves_point Rubin.faithful_rigid_stabilizer_moves_point #align faithful_rist_moves_point Rubin.faithful_rigid_stabilizer_moves_point
theorem Rubin.ne_one_support_nonempty {g : G} : g ≠ 1 → (Rubin.SmulSupport.Support α g).Nonempty := theorem ne_one_support_nonempty {g : G} : g ≠ 1 → (Rubin.SmulSupport.Support α g).Nonempty :=
by by
intro h1 intro h1
cases' Rubin.faithful_moves_point'₁ α h1 with x h cases' Rubin.faithful_moves_point'₁ α h1 with x h
@ -602,7 +501,7 @@ theorem Rubin.ne_one_support_nonempty {g : G} : g ≠ 1 → (Rubin.SmulSupport.S
#align ne_one_support_nempty Rubin.ne_one_support_nonempty #align ne_one_support_nempty Rubin.ne_one_support_nonempty
-- FIXME: somehow clashes with another definition -- FIXME: somehow clashes with another definition
theorem Rubin.disjoint_commute₁ {f g : G} : theorem disjoint_commute₁ {f g : G} :
Disjoint (Rubin.SmulSupport.Support α f) (Rubin.SmulSupport.Support α g) → Commute f g := Disjoint (Rubin.SmulSupport.Support α f) (Rubin.SmulSupport.Support α g) → Commute f g :=
by by
intro hdisjoint intro hdisjoint
@ -638,15 +537,15 @@ section RubinActions
variable [TopologicalSpace α] [TopologicalSpace β] variable [TopologicalSpace α] [TopologicalSpace β]
def Rubin.has_no_isolated_points (α : Type _) [TopologicalSpace α] := def has_no_isolated_points (α : Type _) [TopologicalSpace α] :=
∀ x : α, (nhdsWithin x ({x}ᶜ)) ≠ ⊥ ∀ x : α, (nhdsWithin x ({x}ᶜ)) ≠ ⊥
#align has_no_isolated_points Rubin.has_no_isolated_points #align has_no_isolated_points Rubin.has_no_isolated_points
def Rubin.is_locally_dense (G α : Type _) [Group G] [TopologicalSpace α] [MulAction G α] := def is_locally_dense (G α : Type _) [Group G] [TopologicalSpace α] [MulAction G α] :=
∀ U : Set α, ∀ p ∈ U, p ∈ interior (closure (MulAction.orbit (rigidStabilizer G U) p)) ∀ U : Set α, ∀ p ∈ U, p ∈ interior (closure (MulAction.orbit (rigidStabilizer G U) p))
#align is_locally_dense Rubin.is_locally_dense #align is_locally_dense Rubin.is_locally_dense
structure Rubin.RubinAction (G α : Type _) extends Group G, TopologicalSpace α, MulAction G α, structure RubinAction (G α : Type _) extends Group G, TopologicalSpace α, MulAction G α,
FaithfulSMul G α where FaithfulSMul G α where
locally_compact : LocallyCompactSpace α locally_compact : LocallyCompactSpace α
hausdorff : T2Space α hausdorff : T2Space α
@ -661,11 +560,11 @@ section Rubin.Period.period
variable [MulAction G α] variable [MulAction G α]
noncomputable def Rubin.Period.period (p : α) (g : G) : ℕ := noncomputable def Period.period (p : α) (g : G) : ℕ :=
sInf {n : ℕ | n > 0 ∧ g ^ n • p = p} sInf {n : ℕ | n > 0 ∧ g ^ n • p = p}
#align period Rubin.Period.period #align period Rubin.Period.period
theorem Rubin.Period.period_le_fix {p : α} {g : G} {m : ℕ} (m_pos : m > 0) theorem Period.period_le_fix {p : α} {g : G} {m : ℕ} (m_pos : m > 0)
(gmp_eq_p : g ^ m • p = p) : 0 < Rubin.Period.period p g ∧ Rubin.Period.period p g ≤ m := (gmp_eq_p : g ^ m • p = p) : 0 < Rubin.Period.period p g ∧ Rubin.Period.period p g ≤ m :=
by by
constructor constructor
@ -675,7 +574,7 @@ theorem Rubin.Period.period_le_fix {p : α} {g : G} {m : ℕ} (m_pos : m > 0)
exact Nat.sInf_le ⟨m_pos, gmp_eq_p⟩ exact Nat.sInf_le ⟨m_pos, gmp_eq_p⟩
#align period_le_fix Rubin.Period.period_le_fix #align period_le_fix Rubin.Period.period_le_fix
theorem Rubin.Period.notfix_le_period {p : α} {g : G} {n : ℕ} (n_pos : n > 0) theorem Period.notfix_le_period {p : α} {g : G} {n : ℕ} (n_pos : n > 0)
(period_pos : Rubin.Period.period p g > 0) (pmoves : ∀ i : ℕ, 0 < i → i < n → g ^ i • p ≠ p) : (period_pos : Rubin.Period.period p g > 0) (pmoves : ∀ i : ℕ, 0 < i → i < n → g ^ i • p ≠ p) :
n ≤ Rubin.Period.period p g := by n ≤ Rubin.Period.period p g := by
by_contra period_le by_contra period_le
@ -684,14 +583,14 @@ theorem Rubin.Period.notfix_le_period {p : α} {g : G} {n : ℕ} (n_pos : n > 0)
(Nat.sInf_mem (Nat.nonempty_of_pos_sInf period_pos)).2 (Nat.sInf_mem (Nat.nonempty_of_pos_sInf period_pos)).2
#align notfix_le_period Rubin.Period.notfix_le_period #align notfix_le_period Rubin.Period.notfix_le_period
theorem Rubin.Period.notfix_le_period' {p : α} {g : G} {n : ℕ} (n_pos : n > 0) theorem Period.notfix_le_period' {p : α} {g : G} {n : ℕ} (n_pos : n > 0)
(period_pos : Rubin.Period.period p g > 0) (period_pos : Rubin.Period.period p g > 0)
(pmoves : ∀ i : Fin n, 0 < (i : ℕ) → g ^ (i : ℕ) • p ≠ p) : n ≤ Rubin.Period.period p g := (pmoves : ∀ i : Fin n, 0 < (i : ℕ) → g ^ (i : ℕ) • p ≠ p) : n ≤ Rubin.Period.period p g :=
Rubin.Period.notfix_le_period n_pos period_pos fun (i : ℕ) (i_pos : 0 < i) (i_lt_n : i < n) => Rubin.Period.notfix_le_period n_pos period_pos fun (i : ℕ) (i_pos : 0 < i) (i_lt_n : i < n) =>
pmoves (⟨i, i_lt_n⟩ : Fin n) i_pos pmoves (⟨i, i_lt_n⟩ : Fin n) i_pos
#align notfix_le_period' Rubin.Period.notfix_le_period' #align notfix_le_period' Rubin.Period.notfix_le_period'
theorem Rubin.Period.period_neutral_eq_one (p : α) : Rubin.Period.period p (1 : G) = 1 := theorem Period.period_neutral_eq_one (p : α) : Rubin.Period.period p (1 : G) = 1 :=
by by
have : 0 < Rubin.Period.period p (1 : G) ∧ Rubin.Period.period p (1 : G) ≤ 1 := have : 0 < Rubin.Period.period p (1 : G) ∧ Rubin.Period.period p (1 : G) ≤ 1 :=
Rubin.Period.period_le_fix (by norm_num : 1 > 0) Rubin.Period.period_le_fix (by norm_num : 1 > 0)
@ -700,12 +599,12 @@ theorem Rubin.Period.period_neutral_eq_one (p : α) : Rubin.Period.period p (1 :
linarith linarith
#align period_neutral_eq_one Rubin.Period.period_neutral_eq_one #align period_neutral_eq_one Rubin.Period.period_neutral_eq_one
def Rubin.Period.periods (U : Set α) (H : Subgroup G) : Set ℕ := def Period.periods (U : Set α) (H : Subgroup G) : Set ℕ :=
{n : ℕ | ∃ (p : α) (g : H), p ∈ U ∧ Rubin.Period.period (p : α) (g : G) = n} {n : ℕ | ∃ (p : α) (g : H), p ∈ U ∧ Rubin.Period.period (p : α) (g : G) = n}
#align periods Rubin.Period.periods #align periods Rubin.Period.periods
-- TODO: split into multiple lemmas -- TODO: split into multiple lemmas
theorem Rubin.Period.periods_lemmas {U : Set α} (U_nonempty : Set.Nonempty U) {H : Subgroup G} theorem Period.periods_lemmas {U : Set α} (U_nonempty : Set.Nonempty U) {H : Subgroup G}
(exp_ne_zero : Monoid.exponent H ≠ 0) : (exp_ne_zero : Monoid.exponent H ≠ 0) :
(Rubin.Period.periods U H).Nonempty ∧ (Rubin.Period.periods U H).Nonempty ∧
BddAbove (Rubin.Period.periods U H) ∧ BddAbove (Rubin.Period.periods U H) ∧
@ -731,7 +630,7 @@ theorem Rubin.Period.periods_lemmas {U : Set α} (U_nonempty : Set.Nonempty U) {
exact ⟨periods_nonempty, periods_bounded, m, m_pos, gmp_eq_p⟩ exact ⟨periods_nonempty, periods_bounded, m, m_pos, gmp_eq_p⟩
#align period_lemma Rubin.Period.periods_lemmas #align period_lemma Rubin.Period.periods_lemmas
theorem Rubin.Period.period_from_exponent (U : Set α) (U_nonempty : U.Nonempty) {H : Subgroup G} theorem Period.period_from_exponent (U : Set α) (U_nonempty : U.Nonempty) {H : Subgroup G}
(exp_ne_zero : Monoid.exponent H ≠ 0) : (exp_ne_zero : Monoid.exponent H ≠ 0) :
∃ (p : α) (g : H) (n : ℕ), ∃ (p : α) (g : H) (n : ℕ),
p ∈ U ∧ n > 0 ∧ Rubin.Period.period (p : α) (g : G) = n ∧ n = sSup (Rubin.Period.periods U H) := p ∈ U ∧ n > 0 ∧ Rubin.Period.period (p : α) (g : G) = n ∧ n = sSup (Rubin.Period.periods U H) :=
@ -751,7 +650,7 @@ theorem Rubin.Period.period_from_exponent (U : Set α) (U_nonempty : U.Nonempty)
⟩ ⟩
#align period_from_exponent Rubin.Period.period_from_exponent #align period_from_exponent Rubin.Period.period_from_exponent
theorem Rubin.Period.zero_lt_period_le_Sup_periods {U : Set α} (U_nonempty : U.Nonempty) theorem Period.zero_lt_period_le_Sup_periods {U : Set α} (U_nonempty : U.Nonempty)
{H : Subgroup G} (exp_ne_zero : Monoid.exponent H ≠ 0) : {H : Subgroup G} (exp_ne_zero : Monoid.exponent H ≠ 0) :
∀ (p : U) (g : H), ∀ (p : U) (g : H),
0 < Rubin.Period.period (p : α) (g : G) ∧ 0 < Rubin.Period.period (p : α) (g : G) ∧
@ -768,7 +667,7 @@ theorem Rubin.Period.zero_lt_period_le_Sup_periods {U : Set α} (U_nonempty : U.
le_csSup periods_bounded period_in_periods⟩ le_csSup periods_bounded period_in_periods⟩
#align zero_lt_period_le_Sup_periods Rubin.Period.zero_lt_period_le_Sup_periods #align zero_lt_period_le_Sup_periods Rubin.Period.zero_lt_period_le_Sup_periods
theorem Rubin.Period.pow_period_fix (p : α) (g : G) : g ^ Rubin.Period.period p g • p = p := theorem Period.pow_period_fix (p : α) (g : G) : g ^ Rubin.Period.period p g • p = p :=
by by
cases eq_zero_or_neZero (Rubin.Period.period p g) with cases eq_zero_or_neZero (Rubin.Period.period p g) with
| inl h => rw [h]; simp | inl h => rw [h]; simp
@ -786,7 +685,7 @@ section AlgebraicDisjointness
variable [TopologicalSpace α] [Rubin.Topological.ContinuousMulAction G α] [FaithfulSMul G α] variable [TopologicalSpace α] [Rubin.Topological.ContinuousMulAction G α] [FaithfulSMul G α]
def Rubin.Disjointness.IsLocallyMoving (G α : Type _) [Group G] [TopologicalSpace α] def Disjointness.IsLocallyMoving (G α : Type _) [Group G] [TopologicalSpace α]
[MulAction G α] := [MulAction G α] :=
∀ U : Set α, IsOpen U → Set.Nonempty U → rigidStabilizer G U ≠ ⊥ ∀ U : Set α, IsOpen U → Set.Nonempty U → rigidStabilizer G U ≠ ⊥
#align is_locally_moving Rubin.Disjointness.IsLocallyMoving #align is_locally_moving Rubin.Disjointness.IsLocallyMoving
@ -1059,29 +958,29 @@ section Rubin.RegularSupport.RegularSupport
variable [TopologicalSpace α] [Rubin.Topological.ContinuousMulAction G α] variable [TopologicalSpace α] [Rubin.Topological.ContinuousMulAction G α]
def Rubin.RegularSupport.InteriorClosure (U : Set α) := def RegularSupport.InteriorClosure (U : Set α) :=
interior (closure U) interior (closure U)
#align interior_closure Rubin.RegularSupport.InteriorClosure #align interior_closure Rubin.RegularSupport.InteriorClosure
theorem Rubin.RegularSupport.is_open_interiorClosure (U : Set α) : theorem RegularSupport.is_open_interiorClosure (U : Set α) :
IsOpen (Rubin.RegularSupport.InteriorClosure U) := IsOpen (Rubin.RegularSupport.InteriorClosure U) :=
isOpen_interior isOpen_interior
#align is_open_interior_closure Rubin.RegularSupport.is_open_interiorClosure #align is_open_interior_closure Rubin.RegularSupport.is_open_interiorClosure
theorem Rubin.RegularSupport.interiorClosure_mono {U V : Set α} : theorem RegularSupport.interiorClosure_mono {U V : Set α} :
U ⊆ V → Rubin.RegularSupport.InteriorClosure U ⊆ Rubin.RegularSupport.InteriorClosure V := U ⊆ V → Rubin.RegularSupport.InteriorClosure U ⊆ Rubin.RegularSupport.InteriorClosure V :=
interior_mono ∘ closure_mono interior_mono ∘ closure_mono
#align interior_closure_mono Rubin.RegularSupport.interiorClosure_mono #align interior_closure_mono Rubin.RegularSupport.interiorClosure_mono
def Set.is_regular_open (U : Set α) := def is_regular_open (U : Set α) :=
Rubin.RegularSupport.InteriorClosure U = U Rubin.RegularSupport.InteriorClosure U = U
#align set.is_regular_open Set.is_regular_open #align set.is_regular_open Rubin.is_regular_open
theorem Set.is_regular_def (U : Set α) : theorem is_regular_def (U : Set α) :
U.is_regular_open ↔ Rubin.RegularSupport.InteriorClosure U = U := by rfl is_regular_open U ↔ Rubin.RegularSupport.InteriorClosure U = U := by rfl
#align set.is_regular_def Set.is_regular_def #align set.is_regular_def Rubin.is_regular_def
theorem Rubin.RegularSupport.IsOpen.in_closure {U : Set α} : IsOpen U → U ⊆ interior (closure U) := theorem RegularSupport.IsOpen.in_closure {U : Set α} : IsOpen U → U ⊆ interior (closure U) :=
by by
intro U_open x x_in_U intro U_open x x_in_U
apply interior_mono subset_closure apply interior_mono subset_closure
@ -1089,43 +988,43 @@ theorem Rubin.RegularSupport.IsOpen.in_closure {U : Set α} : IsOpen U → U ⊆
exact x_in_U exact x_in_U
#align is_open.in_closure Rubin.RegularSupport.IsOpen.in_closure #align is_open.in_closure Rubin.RegularSupport.IsOpen.in_closure
theorem Rubin.RegularSupport.IsOpen.interiorClosure_subset {U : Set α} : theorem RegularSupport.IsOpen.interiorClosure_subset {U : Set α} :
IsOpen U → U ⊆ Rubin.RegularSupport.InteriorClosure U := fun h => IsOpen U → U ⊆ Rubin.RegularSupport.InteriorClosure U := fun h =>
(subset_interior_iff_isOpen.mpr h).trans (interior_mono subset_closure) (subset_interior_iff_isOpen.mpr h).trans (interior_mono subset_closure)
#align is_open.interior_closure_subset Rubin.RegularSupport.IsOpen.interiorClosure_subset #align is_open.interior_closure_subset Rubin.RegularSupport.IsOpen.interiorClosure_subset
theorem Rubin.RegularSupport.regular_interior_closure (U : Set α) : theorem RegularSupport.regular_interior_closure (U : Set α) :
(Rubin.RegularSupport.InteriorClosure U).is_regular_open := is_regular_open (Rubin.RegularSupport.InteriorClosure U) :=
by by
rw [Set.is_regular_def] rw [is_regular_def]
apply Set.Subset.antisymm apply Set.Subset.antisymm
exact interior_mono ((closure_mono interior_subset).trans (subset_of_eq closure_closure)) exact interior_mono ((closure_mono interior_subset).trans (subset_of_eq closure_closure))
exact (subset_of_eq interior_interior.symm).trans (interior_mono subset_closure) exact (subset_of_eq interior_interior.symm).trans (interior_mono subset_closure)
#align regular_interior_closure Rubin.RegularSupport.regular_interior_closure #align regular_interior_closure Rubin.RegularSupport.regular_interior_closure
def Rubin.RegularSupport.RegularSupport (α : Type _) [TopologicalSpace α] [MulAction G α] (g : G) := def RegularSupport.RegularSupport (α : Type _) [TopologicalSpace α] [MulAction G α] (g : G) :=
Rubin.RegularSupport.InteriorClosure (Rubin.SmulSupport.Support α g) Rubin.RegularSupport.InteriorClosure (Rubin.SmulSupport.Support α g)
#align regular_support Rubin.RegularSupport.RegularSupport #align regular_support Rubin.RegularSupport.RegularSupport
theorem Rubin.RegularSupport.regularSupport_regular {g : G} : theorem RegularSupport.regularSupport_regular {g : G} :
(Rubin.RegularSupport.RegularSupport α g).is_regular_open := is_regular_open (Rubin.RegularSupport.RegularSupport α g) :=
Rubin.RegularSupport.regular_interior_closure _ Rubin.RegularSupport.regular_interior_closure _
#align regular_regular_support Rubin.RegularSupport.regularSupport_regular #align regular_regular_support Rubin.RegularSupport.regularSupport_regular
theorem Rubin.RegularSupport.support_subset_regularSupport [T2Space α] (g : G) : theorem RegularSupport.support_subset_regularSupport [T2Space α] (g : G) :
Rubin.SmulSupport.Support α g ⊆ Rubin.RegularSupport.RegularSupport α g := Rubin.SmulSupport.Support α g ⊆ Rubin.RegularSupport.RegularSupport α g :=
Rubin.RegularSupport.IsOpen.interiorClosure_subset (Rubin.Topological.support_open g) Rubin.RegularSupport.IsOpen.interiorClosure_subset (Rubin.Topological.support_open g)
#align support_in_regular_support Rubin.RegularSupport.support_subset_regularSupport #align support_in_regular_support Rubin.RegularSupport.support_subset_regularSupport
theorem Rubin.RegularSupport.mem_regularSupport (g : G) (U : Set α) : theorem RegularSupport.mem_regularSupport (g : G) (U : Set α) :
U.is_regular_open → g ∈ rigidStabilizer G U → Rubin.RegularSupport.RegularSupport α g ⊆ U := is_regular_open U → g ∈ rigidStabilizer G U → Rubin.RegularSupport.RegularSupport α g ⊆ U :=
fun U_ro g_moves => fun U_ro g_moves =>
(Set.is_regular_def _).mp U_ro ▸ (is_regular_def _).mp U_ro ▸
Rubin.RegularSupport.interiorClosure_mono (rist_supported_in_set g_moves) Rubin.RegularSupport.interiorClosure_mono (rist_supported_in_set g_moves)
#align mem_regular_support Rubin.RegularSupport.mem_regularSupport #align mem_regular_support Rubin.RegularSupport.mem_regularSupport
-- FIXME: Weird naming? -- FIXME: Weird naming?
def Rubin.RegularSupport.AlgebraicCentralizer (f : G) : Set G := def RegularSupport.AlgebraicCentralizer (f : G) : Set G :=
{h | ∃ g, h = g ^ 12 ∧ Rubin.is_algebraically_disjoint f g} {h | ∃ g, h = g ^ 12 ∧ Rubin.is_algebraically_disjoint f g}
#align algebraic_centralizer Rubin.RegularSupport.AlgebraicCentralizer #align algebraic_centralizer Rubin.RegularSupport.AlgebraicCentralizer
@ -1218,3 +1117,5 @@ end Rubin.RegularSupport.RegularSupport
-- variables [topological_space α] [topological_space β] [continuous_mul_action G α] [continuous_mul_action G β] -- variables [topological_space α] [topological_space β] [continuous_mul_action G α] [continuous_mul_action G β]
-- noncomputable theorem rubin (hα : rubin_action G α) (hβ : rubin_action G β) : equivariant_homeomorph G α β := sorry -- noncomputable theorem rubin (hα : rubin_action G α) (hβ : rubin_action G β) : equivariant_homeomorph G α β := sorry
end Rubin end Rubin
end Rubin

113
Rubin/Tactic.lean
Unescape View File

@ -0,0 +1,113 @@
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Perm.Basic
import Lean.Meta.Tactic.Util
import Lean.Elab.Tactic.Basic
import Lean.Meta.Tactic.Simp.Main
import Mathlib.Tactic.Ring.Basic
import Mathlib.Tactic.FailIfNoProgress
import Mathlib.Tactic.Group
namespace Rubin.Tactic
open Lean.Parser.Tactic
--@[simp]
theorem smul_eq_smul_inv {G α : Type _} [Group G] [MulAction G α] {g h : G}
{x y : α} : g • x = h • y ↔ (h⁻¹ * g) • x = y :=
by
constructor
· intro hyp
let res := congr_arg ((· • ·) h⁻¹) hyp
simp at res
rw [← mul_smul] at res
exact res
· intro hyp
rw [<-hyp, mul_smul]
simp
#align smul_eq_smul Rubin.Tactic.smul_eq_smul_inv
theorem smul_succ {G α : Type _} (n : ℕ) [Group G] [MulAction G α] {g : G}
{x : α} : g ^ n.succ • x = g • g ^ n • x :=
by
rw [pow_succ, mul_smul]
#align smul_succ Rubin.Tactic.smul_succ
syntax (name := group_action₁) "group_action₁" (location)?: tactic
macro_rules
| `(tactic| group_action₁ $[at $location]?) => `(tactic| simp only [
smul_smul,
Rubin.Tactic.smul_eq_smul_inv,
Rubin.Tactic.smul_succ,
one_smul,
mul_one,
one_mul,
sub_self,
add_neg_self,
neg_add_self,
neg_neg,
tsub_self,
<-mul_assoc,
one_pow,
one_zpow,
mul_zpow_neg_one,
zpow_zero,
mul_zpow,
zpow_sub,
zpow_ofNat,
zpow_neg_one,
<-zpow_mul,
zpow_add_one,
zpow_one_add,
zpow_add,
Int.ofNat_add,
Int.ofNat_mul,
Int.ofNat_zero,
Int.ofNat_one,
Int.mul_neg_eq_neg_mul_symm,
Int.neg_mul_eq_neg_mul_symm,
Mathlib.Tactic.Group.zpow_trick,
Mathlib.Tactic.Group.zpow_trick_one,
Mathlib.Tactic.Group.zpow_trick_one',
commutatorElement_def
] $[at $location]?
)
/-- Tactic for normalizing expressions in group actions, without assuming
commutativity, using only the group axioms without any information about
which group is manipulated.
Example:
```lean
example {G α : Type} [Group G] [MulAction G α] (a b : G) (x y : α) (h : a • b • x = a • y) : b⁻¹ • y = x := by
group_action at h -- normalizes `h` which becomes `h : c = d`
rw [←h] -- the goal is now `a*d*d⁻¹ = a`
group_action -- which then normalized and closed
```
-/
syntax (name := group_action) "group_action" (location)?: tactic
macro_rules
| `(tactic| group_action $[at $location]?) => `(tactic|
repeat (fail_if_no_progress (group_action₁ $[at $location]? <;> group $[at $location]?))
)
example {G α: Type _} [Group G] [MulAction G α] (g h: G) (x: α): g • h • x = (g * h) • x := by
group_action
example {G α : Type} [Group G] [MulAction G α] (a b : G) (x y : α) (h : a • b • x = a • y) : b⁻¹ • y = x := by
group_action at h -- normalizes `h` which becomes `h : c = d`
rw [←h] -- the goal is now `a*d*d⁻¹ = a`
group_action -- which then normalized and closed
example (G α : Type _) [Group G] (a b c : G) [MulAction G α] (x : α) :
⁅a * b, c⁆ • x = (a * ⁅b, c⁆ * a⁻¹ * ⁅a, c⁆) • x := by
group_action
end Rubin.Tactic
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