lean/OAI/GroupTheory/UniversalGroup/GraphActions.lean

import OAI.GroupTheory.UniversalGroup.GraphBundles

namespace OAI

noncomputable section

open Set Metric Topology BigOperators
open Set Metric Topology
open Set Topology
open Set Metric Topology
open scoped NNReal
open Set Metric Topology
open scoped NNReal ENNReal
open Set Metric Topology
open scoped TensorProduct

namespace UniversalFInfinity.GraphBundle

section
open Set Topology
open scoped unitInterval
universe u v w
variable {C : Type u} {A : Type v} [TopologicalSpace A] {B : Type w} [TopologicalSpace B]
variable (τ : C → C) (L R : C → C(B,A))

lemma edgeImage_edge_slice [DecidableEq C] (c d : C) (S : Set (I × B)) :
    (fun z : I × B => edge τ L R d z.1 z.2) ⁻¹'
      ((fun z : I × B => edge τ L R c z.1 z.2) '' S) =
      (if d = c then S else ∅) ∪
      ({0} ×ˢ ((L d) ⁻¹' ((vertex τ L R d) ⁻¹'
        ((fun z : I × B => edge τ L R c z.1 z.2) '' S)))) ∪
      ({1} ×ˢ ((R d) ⁻¹' ((vertex τ L R (τ d)) ⁻¹'
        ((fun z : I × B => edge τ L R c z.1 z.2) '' S)))) := by
  classical
  ext ⟨t,x⟩
  constructor
  · intro hz
    by_cases h0 : t = 0
    · subst t
      change edge τ L R d 0 x ∈ ((fun z : I × B => edge τ L R c z.1 z.2) '' S) at hz
      exact Or.inl (Or.inr ⟨rfl,by
        change vertex τ L R d (L d x) ∈ ((fun z : I × B => edge τ L R c z.1 z.2) '' S)
        rw [edge_zero] at hz; exact hz⟩)
    · by_cases h1 : t = 1
      · subst t
        change edge τ L R d 1 x ∈ ((fun z : I × B => edge τ L R c z.1 z.2) '' S) at hz
        exact Or.inr ⟨rfl,by
          change vertex τ L R (τ d) (R d x) ∈ ((fun z : I × B => edge τ L R c z.1 z.2) '' S)
          rw [edge_one] at hz; exact hz⟩
      · let s : Inner := ⟨t.1,lt_of_le_of_ne t.2.1 (fun h => h0 (Subtype.ext h.symm)),
          lt_of_le_of_ne t.2.2 (fun h => h1 (Subtype.ext h))⟩
        have he : edge τ L R d t x = interior τ L R d s x := edge_inner τ L R d s x
        obtain ⟨z,hz,hze⟩ := hz
        have hm : z ∈ (fun z : I × B => edge τ L R c z.1 z.2) ⁻¹' {interior τ L R d s x} := hze.trans he
        rw [interior_preimage] at hm
        split_ifs at hm with hcd
        · change z = (t,x) at hm
          exact Or.inl (Or.inl (by rw [ite_eq_left hcd.symm]; exact hm ▸ hz))
        · exact False.elim hm
  · rintro ((hz|hz)|hz)
    · split_ifs at hz with hdc
      · subst d; exact ⟨(t,x),hz,rfl⟩
      · exact False.elim hz
    · rcases hz with ⟨h0,hx⟩
      change t = 0 at h0
      subst t
      change edge τ L R d 0 x ∈ ((fun z : I × B => edge τ L R c z.1 z.2) '' S)
      rw [edge_zero]
      exact hx
    · rcases hz with ⟨h1,hx⟩
      change t = 1 at h1
      subst t
      change edge τ L R d 1 x ∈ ((fun z : I × B => edge τ L R c z.1 z.2) '' S)
      rw [edge_one]
      exact hx

lemma edge_injective (c : C) (hτ : τ c ≠ c) (hL : Function.Injective (L c))
    (hR : Function.Injective (R c)) : Function.Injective (fun z : I × B => edge τ L R c z.1 z.2) := by
  classical
  rintro ⟨t,x⟩ ⟨s,y⟩ he
  by_cases h0 : t = 0
  · subst t
    have he' : edge τ L R c s y = vertex τ L R c (L c x) := he.symm.trans (edge_zero τ L R c x)
    rcases (edge_eq_vertex τ L R c c s y (L c x)).mp he' with ⟨rfl,_,hy⟩|⟨_,ht,_⟩
    · exact Prod.ext rfl (hL hy).symm
    · exact False.elim (hτ ht)
  · by_cases h1 : t = 1
    · subst t
      have he' : edge τ L R c s y = vertex τ L R (τ c) (R c x) := he.symm.trans (edge_one τ L R c x)
      rcases (edge_eq_vertex τ L R c (τ c) s y (R c x)).mp he' with ⟨_,ht,_⟩|⟨rfl,_,hy⟩
      · exact False.elim (hτ ht.symm)
      · exact Prod.ext rfl (hR hy).symm
    · let r : Inner := ⟨t.1,lt_of_le_of_ne t.2.1 (fun h => h0 (Subtype.ext h.symm)),
        lt_of_le_of_ne t.2.2 (fun h => h1 (Subtype.ext h))⟩
      have ht : edge τ L R c t x = interior τ L R c r x := edge_inner τ L R c r x
      have hm : (s,y) ∈ (fun z : I × B => edge τ L R c z.1 z.2) ⁻¹' {interior τ L R c r x} := he.symm.trans ht
      rw [interior_preimage,ite_eq_left rfl] at hm
      exact (show (s,y) = (t,x) from hm).symm

variable [TopologicalSpace C] [DiscreteTopology C]
lemma edge_closedMap (c : C) (hL : IsClosedMap (L c)) (hR : IsClosedMap (R c)) :
    IsClosedMap (fun z : I × B => edge τ L R c z.1 z.2) := by
  classical
  intro S hS
  apply (isClosed_iff τ L R _).mpr
  refine ⟨fun d => edgeImage_vertex_closed τ L R c d S hS hL hR,fun d => ?_⟩
  rw [edgeImage_edge_slice]
  apply IsClosed.union
  · apply IsClosed.union
    · split_ifs; exact hS; exact isClosed_empty
    · exact isClosed_singleton.prod ((edgeImage_vertex_closed τ L R c d S hS hL hR).preimage (L d).continuous)
  · exact isClosed_singleton.prod ((edgeImage_vertex_closed τ L R c (τ d) S hS hL hR).preimage (R d).continuous)

lemma edge_closedEmbedding (c : C) (hτ : τ c ≠ c) (hL : IsClosedEmbedding (L c))
    (hR : IsClosedEmbedding (R c)) : IsClosedEmbedding (fun z : I × B => edge τ L R c z.1 z.2) :=
  IsClosedEmbedding.of_continuous_injective_isClosedMap (edgeMap τ L R c).continuous
    (edge_injective τ L R c hτ hL.injective hR.injective)
    (edge_closedMap τ L R c hL.isClosedMap hR.isClosedMap)
end

open Set Topology
open scoped unitInterval
universe u v w z
variable {C : Type u} [TopologicalSpace C] [DiscreteTopology C]
variable {A : Type v} [TopologicalSpace A] {B : Type w} [TopologicalSpace B]
variable (τ : C → C) (L R : C → C(B,A))
variable {Y : Type z} [TopologicalSpace Y]

omit [DiscreteTopology C] in
lemma hom_ext {f g : C(Space τ L R,Y)}
    (hv : ∀ c x,f (vertex τ L R c x) = g (vertex τ L R c x))
    (he : ∀ c (z : I × B),f (edge τ L R c z.1 z.2) = g (edge τ L R c z.1 z.2)) : f = g := by
  apply ContinuousMap.ext
  intro z
  change Core (C := C) (A := A) (B := B) at z
  cases z with
  | inl z => exact hv z.1 z.2
  | inr z =>
    have hh := he z.1 (⟨z.2.1.1,z.2.1.2.1.le,z.2.1.2.2.le⟩,z.2.2)
    rw [edge_inner] at hh
    exact hh

variable (f g : C(Space τ L R,Y))
variable (V : ∀ c,(f.comp (vertexMap τ L R c)).Homotopy (g.comp (vertexMap τ L R c)))
variable (E : ∀ c,(f.comp (edgeMap τ L R c)).Homotopy (g.comp (edgeMap τ L R c)))
variable (hE0 : ∀ c t x,E c (t,(0,x)) = V c (t,L c x))
variable (hE1 : ∀ c t x,E c (t,(1,x)) = V (τ c) (t,R c x))

def homotopyPaths : C(Space τ L R,C(I,Y)) :=
  lift τ L R (fun c => ContinuousMap.curry ((V c).toContinuousMap.comp ⟨Prod.swap,continuous_swap⟩))
    (fun c => ContinuousMap.curry ((E c).toContinuousMap.comp ⟨Prod.swap,continuous_swap⟩)) (by
      intro c x; apply ContinuousMap.ext; intro t; exact hE0 c t x) (by
      intro c x; apply ContinuousMap.ext; intro t; exact hE1 c t x)

def homotopy : f.Homotopy g where
  toFun z := homotopyPaths τ L R f g V E hE0 hE1 z.2 z.1
  continuous_toFun := (homotopyPaths τ L R f g V E hE0 hE1).uncurry.continuous.comp continuous_swap
  map_zero_left z := by
    change Core (C := C) (A := A) (B := B) at z
    cases z with
    | inl z => exact (V z.1).apply_zero z.2
    | inr z =>
      have he := (E z.1).apply_zero (⟨z.2.1.1,z.2.1.2.1.le,z.2.1.2.2.le⟩,z.2.2)
      change E z.1 (0,(⟨z.2.1.1,z.2.1.2.1.le,z.2.1.2.2.le⟩,z.2.2)) = f (edge τ L R z.1 ⟨z.2.1.1,z.2.1.2.1.le,z.2.1.2.2.le⟩ z.2.2) at he
      rw [edge_inner] at he
      exact he
  map_one_left z := by
    change Core (C := C) (A := A) (B := B) at z
    cases z with
    | inl z => exact (V z.1).apply_one z.2
    | inr z =>
      have he := (E z.1).apply_one (⟨z.2.1.1,z.2.1.2.1.le,z.2.1.2.2.le⟩,z.2.2)
      change E z.1 (1,(⟨z.2.1.1,z.2.1.2.1.le,z.2.1.2.2.le⟩,z.2.2)) = g (edge τ L R z.1 ⟨z.2.1.1,z.2.1.2.1.le,z.2.1.2.2.le⟩ z.2.2) at he
      rw [edge_inner] at he
      exact he
end UniversalFInfinity.GraphBundle

namespace UniversalFInfinity.PathFamilies
open scoped unitInterval
universe u v
variable {J : Type u} [TopologicalSpace J] {Y : Type v} [TopologicalSpace Y]
def intervalPath : Path (0:I) 1 where
  toFun := id
  continuous_toFun := continuous_id
  source' := rfl
  target' := rfl

def unpause : C(I × I,I) := (Path.Homotopy.transRefl intervalPath).toContinuousMap

lemma transRefl_natural {x y : Y} (p : Path x y) (z : I × I) :
    Path.Homotopy.transRefl p z = p (unpause z) := rfl

lemma continuous_transRefl {a b : J → Y} (p : ∀ j,Path (a j) (b j))
    (hp : Continuous (fun z : J × I => p z.1 z.2)) :
    Continuous (fun z : J × (I × I) => Path.Homotopy.transRefl (p z.1) z.2) := by
  change Continuous (fun z : J × (I × I) => p z.1 (unpause z.2))
  exact hp.comp (continuous_fst.prodMk (unpause.continuous.comp continuous_snd))
end UniversalFInfinity.PathFamilies

namespace UniversalFInfinity.GraphBundle

section
open Set Topology
open scoped unitInterval
universe u v w
variable {C : Type u} [TopologicalSpace C] [DiscreteTopology C]
variable {A : Type v} [TopologicalSpace A] {B : Type w} [TopologicalSpace B]
variable (τ : C → C) (L R : C → C(B,A))

def edgePath (c : C) (x : B) : Path (vertex τ L R c (L c x)) (vertex τ L R (τ c) (R c x)) where
  toFun t := edge τ L R c t x
  continuous_toFun := (edgeMap τ L R c).continuous.comp (continuous_id.prodMk continuous_const)
  source' := edge_zero τ L R c x
  target' := edge_one τ L R c x

omit [DiscreteTopology C] in
lemma edgePath_continuous (c : C) : Continuous (fun z : B × I => edgePath τ L R c z.1 z.2) :=
  (edgeMap τ L R c).continuous.comp continuous_swap

def pausedEdge (c : C) (x : B) := (edgePath τ L R c x).trans (Path.refl _)
omit [DiscreteTopology C] in
lemma pausedEdge_continuous (c : C) : Continuous (fun z : B × I => pausedEdge τ L R c z.1 z.2) :=
  Path.trans_continuous_family (edgePath τ L R c) (edgePath_continuous τ L R c)
    (fun x => Path.refl (vertex τ L R (τ c) (R c x)))
    ((vertexMap τ L R (τ c)).continuous.comp ((R c).continuous.comp continuous_fst))

def pause : C(Space τ L R,Space τ L R) :=
  lift τ L R (vertexMap τ L R)
    (fun c => ⟨fun z => pausedEdge τ L R c z.2 z.1,(pausedEdge_continuous τ L R c).comp continuous_swap⟩)
    (fun c x => (pausedEdge τ L R c x).source)
    (fun c x => (pausedEdge τ L R c x).target)
lemma pause_vertex (c : C) (x : A) : pause τ L R (vertex τ L R c x) = vertex τ L R c x := rfl
lemma pause_edge (c : C) (z : I × B) : pause τ L R (edge τ L R c z.1 z.2) = pausedEdge τ L R c z.2 z.1 :=
  lift_edge τ L R _ _ _ _ c z

def pauseVertexHomotopy (c : C) :
    ((pause τ L R).comp (vertexMap τ L R c)).Homotopy ((ContinuousMap.id _).comp (vertexMap τ L R c)) :=
  ContinuousMap.Homotopy.refl (vertexMap τ L R c)

omit [DiscreteTopology C] in
lemma pauseEdgeHomotopy_continuous (c : C) :
    Continuous (fun z : I × (I × B) =>
      Path.Homotopy.transRefl (edgePath τ L R c z.2.2) (z.1,z.2.1)) := by
  change Continuous (fun z : I × (I × B) =>
    edge τ L R c (PathFamilies.unpause (z.1,z.2.1)) z.2.2)
  exact (edgeMap τ L R c).continuous.comp
    ((PathFamilies.unpause.continuous.comp (continuous_fst.prodMk continuous_snd.fst)).prodMk
      continuous_snd.snd)

def pauseEdgeHomotopy (c : C) :
    ((pause τ L R).comp (edgeMap τ L R c)).Homotopy ((ContinuousMap.id _).comp (edgeMap τ L R c)) where
  toFun z := Path.Homotopy.transRefl (edgePath τ L R c z.2.2) (z.1,z.2.1)
  continuous_toFun := pauseEdgeHomotopy_continuous τ L R c
  map_zero_left z := by
    change Path.Homotopy.transRefl (edgePath τ L R c z.2) (0,z.1) = pause τ L R (edge τ L R c z.1 z.2)
    rw [pause_edge]
    exact (Path.Homotopy.transRefl (edgePath τ L R c z.2)).apply_zero z.1
  map_one_left z := (Path.Homotopy.transRefl (edgePath τ L R c z.2)).apply_one z.1

def pauseHomotopy : (pause τ L R).Homotopy (ContinuousMap.id _) :=
  homotopy τ L R (pause τ L R) (ContinuousMap.id _) (pauseVertexHomotopy τ L R)
    (pauseEdgeHomotopy τ L R) (fun c t x => (Path.Homotopy.transRefl (edgePath τ L R c x)).source t)
    (fun c t x => (Path.Homotopy.transRefl (edgePath τ L R c x)).target t)
end

section
open Set Topology
open scoped unitInterval
universe u v w z
variable {C : Type u} [TopologicalSpace C] [DiscreteTopology C]
variable {D : Type v} [TopologicalSpace D] [DiscreteTopology D]
variable {A : Type w} [TopologicalSpace A] {B : Type z} [TopologicalSpace B]
variable (τ : C → C) (F : C → C(A,A)) (ν : D → D) (K : D → C(B,B))
variable (sig : C → D) (hsig : ∀ c,sig (τ c) = ν (sig c)) (a : C → C(A,B))
variable (ha : ∀ c x,a (τ c) (F c x) = K (sig c) (a c x))

def homogeneousMap : C(Space τ (fun _ => .id A) F,Space ν (fun _ => .id B) K) :=
  lift τ (fun _ => .id A) F
    (fun c => (vertexMap ν (fun _ => .id B) K (sig c)).comp (a c))
    (fun c => (edgeMap ν (fun _ => .id B) K (sig c)).comp
      ⟨fun z => (z.1,a c z.2), continuous_fst.prodMk ((a c).continuous.comp continuous_snd)⟩)
    (fun c x => edge_zero ν _ K (sig c) (a c x))
    (fun c x => by change edge ν _ K (sig c) 1 (a c x) = vertex ν _ K (sig (τ c)) (a (τ c) (F c x)); rw [edge_one,hsig,ha])
omit [DiscreteTopology D] in
lemma homogeneousMap_vertex (c : C) (x : A) :
    homogeneousMap τ F ν K sig hsig a ha (vertex τ (fun _ => .id A) F c x) =
      vertex ν (fun _ => .id B) K (sig c) (a c x) := rfl
omit [DiscreteTopology D] in
lemma homogeneousMap_edge (c : C) (t : I) (x : A) :
    homogeneousMap τ F ν K sig hsig a ha (edge τ (fun _ => .id A) F c t x) =
      edge ν (fun _ => .id B) K (sig c) t (a c x) := lift_edge τ _ F _ _ _ _ c (t,x)
end

open Set Topology
open scoped unitInterval
universe u v w
variable {G : Type u} [Group G]
variable {C : Type v} [TopologicalSpace C] [DiscreteTopology C] [MulAction G C]
variable {A : Type w} [TopologicalSpace A]
variable (τ : C → C) (F : C → C(A,A)) (hτ : ∀ (g : G) c,τ (g • c) = g • τ c)
variable (a : G → C → C(A,A))
variable (ha : ∀ (g : G) c x,a g (τ c) (F c x) = F (g • c) (a g c x))
variable (a_one : ∀ c x,a 1 c x = x)
variable (a_mul : ∀ (g h : G) c x,a (g*h) c x = a g (h • c) (a h c x))

def actionMap (g : G) : C(Space τ (fun _ => .id A) F,Space τ (fun _ => .id A) F) :=
  homogeneousMap τ F τ F (g • ·) (fun c => (hτ g c).symm) (a g) (ha g)
lemma actionMap_vertex (g : G) (c : C) (x : A) :
    actionMap τ F hτ a ha g (vertex τ (fun _ => .id A) F c x) =
      vertex τ (fun _ => .id A) F (g • c) (a g c x) := rfl
lemma actionMap_edge (g : G) (c : C) (t : I) (x : A) :
    actionMap τ F hτ a ha g (edge τ (fun _ => .id A) F c t x) =
      edge τ (fun _ => .id A) F (g • c) t (a g c x) :=
  homogeneousMap_edge τ F τ F _ _ _ _ c t x

@[instance_reducible]
def graphAction : MulAction G (Space τ (fun _ => .id A) F) where
  smul g x := actionMap τ F hτ a ha g x
  one_smul x := by
    have he : actionMap τ F hτ a ha 1 = ContinuousMap.id _ := by
      apply hom_ext τ (fun _ => .id A) F
      · intro c y; rw [actionMap_vertex,one_smul,a_one]; rfl
      · intro c z; rw [actionMap_edge,one_smul,a_one]; rfl
    exact ContinuousMap.congr_fun he x
  mul_smul g h x := by
    have he : actionMap τ F hτ a ha (g*h) = (actionMap τ F hτ a ha g).comp (actionMap τ F hτ a ha h) := by
      apply hom_ext τ (fun _ => .id A) F
      · intro c y; simp only [ContinuousMap.comp_apply,actionMap_vertex,mul_smul,a_mul]
      · intro c z; simp only [ContinuousMap.comp_apply,actionMap_edge,mul_smul,a_mul]
    exact ContinuousMap.congr_fun he x

lemma graphAction_continuous : @ContinuousConstSMul G (Space τ (fun _ => .id A) F) _
    (graphAction τ F hτ a ha a_one a_mul).toSMul :=
by
  let := graphAction τ F hτ a ha a_one a_mul
  exact ⟨fun g => (actionMap τ F hτ a ha g).continuous⟩
end UniversalFInfinity.GraphBundle

end

end OAI