import ConNF.Model.Result

/-!
# New file

In this file...

## Main declarations

* `ConNF.foo`: Something new.
-/

noncomputable section
universe u

open Cardinal Ordinal ConNF.TSet

namespace ConNF

variable [Params.{u}] {α β γ δ ε ζ : Λ} (hβ : (β : TypeIndex) < α) (hγ : (γ : TypeIndex) < β)
  (hδ : (δ : TypeIndex) < γ) (hε : (ε : TypeIndex) < δ) (hζ : (ζ : TypeIndex) < ε)

def union (x y : TSet α) : TSet α :=
  (xᶜ' ⊓' yᶜ')ᶜ'

notation:68 x:68 " ⊔[" h "] " y:68 => _root_.ConNF.union h x y
notation:68 x:68 " ⊔' " y:68 => x ⊔[by assumption] y

@[simp]
theorem mem_union_iff (x y : TSet α) :
    ∀ z : TSet β, z ∈' x ⊔' y ↔ z ∈' x ∨ z ∈' y := by
  rw [union]
  intro z
  rw [mem_compl_iff, mem_inter_iff, mem_compl_iff, mem_compl_iff, or_iff_not_and_not]

def higherIndex (α : Λ) : Λ :=
  (exists_gt α).choose

theorem lt_higherIndex {α : Λ} :
    (α : TypeIndex) < higherIndex α :=
  WithBot.coe_lt_coe.mpr (exists_gt α).choose_spec

theorem tSet_nonempty (h : ∃ β : Λ, (β : TypeIndex) < α) : Nonempty (TSet α) := by
  obtain ⟨α', hα⟩ := h
  constructor
  apply typeLower lt_higherIndex lt_higherIndex lt_higherIndex hα
  apply cardinalOne lt_higherIndex lt_higherIndex

def empty : TSet α :=
  (tSet_nonempty ⟨β, hβ⟩).some ⊓' (tSet_nonempty ⟨β, hβ⟩).someᶜ'

@[simp]
theorem mem_empty_iff :
    ∀ x : TSet β, ¬x ∈' empty hβ := by
  intro x
  rw [empty, mem_inter_iff, mem_compl_iff]
  exact and_not_self

def univ : TSet α :=
  (empty hβ)ᶜ'

@[simp]
theorem mem_univ_iff :
    ∀ x : TSet β, x ∈' univ hβ := by
  intro x
  simp only [univ, mem_compl_iff, mem_empty_iff, not_false_eq_true]

/-- The set of all ordered pairs. -/
def orderedPairs : TSet α :=
  vCross hβ hγ hδ (univ hδ)

@[simp]
theorem mem_orderedPairs_iff (x : TSet β) :
    x ∈' orderedPairs hβ hγ hδ ↔ ∃ a b, x = ⟨a, b⟩' := by
  simp only [orderedPairs, vCross_spec, mem_univ_iff, and_true]

def converse (x : TSet α) : TSet α :=
  converse' hβ hγ hδ x ⊓' orderedPairs hβ hγ hδ

@[simp]
theorem op_mem_converse_iff (x : TSet α) :
    ∀ a b, ⟨a, b⟩' ∈' converse hβ hγ hδ x ↔ ⟨b, a⟩' ∈' x := by
  intro a b
  simp only [converse, mem_inter_iff, converse'_spec, mem_orderedPairs_iff, op_inj, exists_and_left,
    exists_eq', and_true]

def cross (x y : TSet γ) : TSet α :=
  converse hβ hγ hδ (vCross hβ hγ hδ x) ⊓' vCross hβ hγ hδ y

@[simp]
theorem mem_cross_iff (x y : TSet γ) :
    ∀ a, a ∈' cross hβ hγ hδ x y ↔ ∃ b c, a = ⟨b, c⟩' ∧ b ∈' x ∧ c ∈' y := by
  intro a
  rw [cross, mem_inter_iff, vCross_spec]
  constructor
  · rintro ⟨h₁, b, c, rfl, h₂⟩
    simp only [op_mem_converse_iff, vCross_spec, op_inj] at h₁
    obtain ⟨b', c', ⟨rfl, rfl⟩, h₁⟩ := h₁
    exact ⟨b, c, rfl, h₁, h₂⟩
  · rintro ⟨b, c, rfl, h₁, h₂⟩
    simp only [op_mem_converse_iff, vCross_spec, op_inj]
    exact ⟨⟨c, b, ⟨rfl, rfl⟩, h₁⟩, ⟨b, c, ⟨rfl, rfl⟩, h₂⟩⟩

def singletonImage (x : TSet β) : TSet α :=
  singletonImage' hβ hγ hδ hε x ⊓' (cross hβ hγ hδ (cardinalOne hδ hε) (cardinalOne hδ hε))

@[simp]
theorem singletonImage_spec (x : TSet β) :
    ∀ z w,
    ⟨ {z}', {w}' ⟩' ∈' singletonImage hβ hγ hδ hε x ↔ ⟨z, w⟩' ∈' x := by
  intro z w
  rw [singletonImage, mem_inter_iff, singletonImage'_spec, and_iff_left_iff_imp]
  intro hzw
  rw [mem_cross_iff]
  refine ⟨{z}', {w}', rfl, ?_⟩
  simp only [mem_cardinalOne_iff, singleton_inj, exists_eq', and_self]

theorem exists_of_mem_singletonImage {x : TSet β} {z w : TSet δ}
    (h : ⟨z, w⟩' ∈' singletonImage hβ hγ hδ hε x) :
    ∃ a b, z = {a}' ∧ w = {b}' := by
  simp only [singletonImage, mem_inter_iff, mem_cross_iff, op_inj, mem_cardinalOne_iff] at h
  obtain ⟨-, _, _, ⟨rfl, rfl⟩, ⟨a, rfl⟩, ⟨b, rfl⟩⟩ := h
  exact ⟨a, b, rfl, rfl⟩

/-- Turn a model element encoding a relation into an actual relation. -/
def ExternalRel (r : TSet α) : Rel (TSet δ) (TSet δ) :=
  λ x y ↦ ⟨x, y⟩' ∈' r

@[simp]
theorem externalRel_converse (r : TSet α) :
    ExternalRel hβ hγ hδ (converse hβ hγ hδ r) = (ExternalRel hβ hγ hδ r).inv := by
  ext
  simp only [ExternalRel, op_mem_converse_iff, Rel.inv_apply]

/-- The codomain of a relation. -/
def codom (r : TSet α) : TSet γ :=
  (typeLower lt_higherIndex hβ hγ hδ (singletonImage lt_higherIndex hβ hγ hδ r)ᶜ[lt_higherIndex])ᶜ'

@[simp]
theorem mem_codom_iff (r : TSet α) (x : TSet δ) :
    x ∈' codom hβ hγ hδ r ↔ x ∈ (ExternalRel hβ hγ hδ r).codom := by
  simp only [codom, mem_compl_iff, mem_typeLower_iff, not_forall, not_not]
  constructor
  · rintro ⟨y, hy⟩
    obtain ⟨a, b, rfl, hb⟩ := exists_of_mem_singletonImage lt_higherIndex hβ hγ hδ hy
    rw [singleton_inj] at hb
    subst hb
    rw [singletonImage_spec] at hy
    exact ⟨a, hy⟩
  · rintro ⟨a, ha⟩
    use {a}'
    rw [singletonImage_spec]
    exact ha

/-- The domain of a relation. -/
def dom (r : TSet α) : TSet γ :=
  codom hβ hγ hδ (converse hβ hγ hδ r)

@[simp]
theorem mem_dom_iff (r : TSet α) (x : TSet δ) :
    x ∈' dom hβ hγ hδ r ↔ x ∈ (ExternalRel hβ hγ hδ r).dom := by
  rw [dom, mem_codom_iff, externalRel_converse, Rel.inv_codom]

/-- The field of a relation. -/
def field (r : TSet α) : TSet γ :=
  dom hβ hγ hδ r ⊔' codom hβ hγ hδ r

@[simp]
theorem mem_field_iff (r : TSet α) (x : TSet δ) :
    x ∈' field hβ hγ hδ r ↔ x ∈ (ExternalRel hβ hγ hδ r).field := by
  rw [field, mem_union_iff, mem_dom_iff, mem_codom_iff, Rel.field, Set.mem_union]

def subset : TSet α :=
  subset' hβ hγ hδ hε ⊓' orderedPairs hβ hγ hδ

@[simp]
theorem subset_spec :
    ∀ a b, ⟨a, b⟩' ∈' subset hβ hγ hδ hε ↔ a ⊆[TSet ε] b := by
  intro a b
  simp only [subset, mem_inter_iff, subset'_spec, mem_orderedPairs_iff, op_inj, exists_and_left,
    exists_eq', and_true]

def membership : TSet α :=
  subset hβ hγ hδ hε ⊓' cross hβ hγ hδ (cardinalOne hδ hε) (univ hδ)

@[simp]
theorem membership_spec :
    ∀ a b, ⟨{a}', b⟩' ∈' membership hβ hγ hδ hε ↔ a ∈' b := by
  intro a b
  rw [membership, mem_inter_iff, subset_spec]
  simp only [mem_cross_iff, op_inj, mem_cardinalOne_iff, mem_univ_iff, and_true, exists_and_right,
    exists_and_left, exists_eq', exists_eq_left', singleton_inj]
  constructor
  · intro h
    exact h a ((typedMem_singleton_iff' hε a a).mpr rfl)
  · intro h c hc
    simp only [typedMem_singleton_iff'] at hc
    cases hc
    exact h

def powerset (x : TSet β) : TSet α :=
  dom lt_higherIndex lt_higherIndex hβ
    (subset lt_higherIndex lt_higherIndex hβ hγ ⊓[lt_higherIndex]
      vCross lt_higherIndex lt_higherIndex hβ {x}')

@[simp]
theorem mem_powerset_iff (x y : TSet β) :
    x ∈' powerset hβ hγ y ↔ x ⊆[TSet γ] y := by
  rw [powerset, mem_dom_iff]
  constructor
  · rintro ⟨z, h⟩
    simp only [ExternalRel, mem_inter_iff, subset_spec, vCross_spec, op_inj,
      typedMem_singleton_iff', exists_eq_right, exists_and_right, exists_eq', true_and] at h
    cases h.2
    exact h.1
  · intro h
    refine ⟨y, ?_⟩
    simp only [ExternalRel, mem_inter_iff, subset_spec, h, vCross_spec, op_inj,
      typedMem_singleton_iff', exists_eq_right, and_true, exists_eq', and_self]

/-- The set `ι²''x = {{{a}} | a ∈ x}`. -/
def doubleSingleton (x : TSet γ) : TSet α :=
  cross hβ hγ hδ x x ⊓' cardinalOne hβ hγ

@[simp]
theorem mem_doubleSingleton_iff (x : TSet γ) :
    ∀ y : TSet β, y ∈' doubleSingleton hβ hγ hδ x ↔
    ∃ z : TSet δ, z ∈' x ∧ y = { {z}' }' := by
  intro y
  rw [doubleSingleton, mem_inter_iff, mem_cross_iff, mem_cardinalOne_iff]
  constructor
  · rintro ⟨⟨b, c, h₁, h₂, h₃⟩, ⟨a, rfl⟩⟩
    obtain ⟨hbc, rfl⟩ := (op_eq_singleton_iff _ _ _ _ _).mp h₁.symm
    exact ⟨c, h₃, rfl⟩
  · rintro ⟨z, h, rfl⟩
    constructor
    · refine ⟨z, z, ?_⟩
      rw [eq_comm, op_eq_singleton_iff]
      tauto
    · exact ⟨_, rfl⟩

/-- The union of a set of *singletons*: `ι⁻¹''x = {a | {a} ∈ x}`. -/
def singletonUnion (x : TSet α) : TSet β :=
  typeLower lt_higherIndex lt_higherIndex hβ hγ
    (vCross lt_higherIndex lt_higherIndex hβ x)

@[simp]
theorem mem_singletonUnion_iff (x : TSet α) :
    ∀ y : TSet γ, y ∈' singletonUnion hβ hγ x ↔ {y}' ∈' x := by
  intro y
  simp only [singletonUnion, mem_typeLower_iff, vCross_spec, op_inj]
  constructor
  · intro h
    obtain ⟨a, b, ⟨rfl, rfl⟩, hy⟩ := h {y}'
    exact hy
  · intro h b
    exact ⟨b, _, ⟨rfl, rfl⟩, h⟩

/--
The union of a set of sets.

```
singletonUnion dom {⟨{a}, b⟩ | a ∈ b} ∩ (1 × x) =
singletonUnion dom {⟨{a}, b⟩ | a ∈ b ∧ b ∈ x} =
singletonUnion {{a} | a ∈ b ∧ b ∈ x} =
{a | a ∈ b ∧ b ∈ x} =
⋃ x
```
-/
def sUnion (x : TSet α) : TSet β :=
  singletonUnion hβ hγ
    (dom lt_higherIndex lt_higherIndex hβ
      (membership lt_higherIndex lt_higherIndex hβ hγ ⊓[lt_higherIndex]
        cross lt_higherIndex lt_higherIndex hβ (cardinalOne hβ hγ) x))

@[simp]
theorem mem_sUnion_iff (x : TSet α) :
    ∀ y : TSet γ, y ∈' sUnion hβ hγ x ↔ ∃ t : TSet β, t ∈' x ∧ y ∈' t

:= by
  intro y
  simp only [sUnion, mem_singletonUnion_iff, mem_dom_iff, Rel.dom, ExternalRel, mem_inter_iff,
    mem_cross_iff, op_inj, mem_cardinalOne_iff, Set.mem_setOf_eq, membership_spec]
  constructor
  · rintro ⟨z, h₁, a, b, ⟨rfl, rfl⟩, ⟨c, h₂⟩, h₃⟩
    rw [singleton_inj] at h₂
    cases h₂
    exact ⟨z, h₃, h₁⟩
  · rintro ⟨z, h₂, h₃⟩
    exact ⟨z, h₃, _, _, ⟨rfl, rfl⟩, ⟨y, rfl⟩, h₂⟩


import ConNF.ModelData.PathEnumeration

/-!
# Supports

In this file, we define the notion of a support.

## Main declarations

* `ConNF.BaseSupport`: The type of supports of atoms.
* `ConNF.Support`: The type of supports of objects of arbitrary type indices.
-/

universe u

open Cardinal

namespace ConNF

variable [Params.{u}]

/-!
## Base supports
-/

structure BaseSupport where
  atoms : Enumeration Atom
  nearLitters : Enumeration NearLitter

namespace BaseSupport

instance : SuperA BaseSupport (Enumeration Atom) where
  superA := atoms

instance : SuperN BaseSupport (Enumeration NearLitter) where
  superN := nearLitters

@[simp]
theorem mk_atoms {a : Enumeration Atom} {N : Enumeration NearLitter} :
    (BaseSupport.mk a N)ᴬ = a :=
  rfl

@[simp]
theorem mk_nearLitters {a : Enumeration Atom} {N : Enumeration NearLitter} :
    (BaseSupport.mk a N)ᴺ = N :=
  rfl

theorem atoms_congr {S T : BaseSupport} (h : S = T) :
    Sᴬ = Tᴬ :=
  h ▸ rfl

theorem nearLitters_congr {S T : BaseSupport} (h : S = T) :
    Sᴺ = Tᴺ :=
  h ▸ rfl

@[ext]
theorem ext {S T : BaseSupport} (h₁ : Sᴬ = Tᴬ) (h₂ : Sᴺ = Tᴺ) : S = T := by
  obtain ⟨SA, SN⟩ := S
  obtain ⟨TA, TN⟩ := T
  cases h₁
  cases h₂
  rfl

instance : SMul BasePerm BaseSupport where
  smul π S := ⟨π • Sᴬ, π • Sᴺ⟩

@[simp]
theorem smul_atoms (π : BasePerm) (S : BaseSupport) :
    (π • S)ᴬ = π • Sᴬ :=
  rfl

@[simp]
theorem smul_nearLitters (π : BasePerm) (S : BaseSupport) :
    (π • S)ᴺ = π • Sᴺ :=
  rfl

@[simp]
theorem smul_atoms_eq_of_smul_eq {π : BasePerm} {S : BaseSupport}
    (h : π • S = S) :
    π • Sᴬ = Sᴬ := by
  rw [← smul_atoms, h]

@[simp]
theorem smul_nearLitters_eq_of_smul_eq {π : BasePerm} {S : BaseSupport}
    (h : π • S = S) :
    π • Sᴺ = Sᴺ := by
  rw [← smul_nearLitters, h]

instance : MulAction BasePerm BaseSupport where
  one_smul S := by
    apply ext
    · rw [smul_atoms, one_smul]
    · rw [smul_nearLitters, one_smul]
  mul_smul π₁ π₂ S := by
    apply ext
    · rw [smul_atoms, smul_atoms, smul_atoms, mul_smul]
    · rw [smul_nearLitters, smul_nearLitters, smul_nearLitters, mul_smul]

theorem smul_eq_smul_iff (π₁ π₂ : BasePerm) (S : BaseSupport) :
    π₁ • S = π₂ • S ↔ (∀ a ∈ Sᴬ, π₁ • a = π₂ • a) ∧ (∀ N ∈ Sᴺ, π₁ • N = π₂ • N) := by
  constructor
  · intro h
    constructor
    · rintro a ⟨i, ha⟩
      have := congr_arg (·ᴬ.rel i (π₁ • a)) h
      simp only [smul_atoms, Enumeration.smul_rel, inv_smul_smul, eq_iff_iff] at this
      have := Sᴬ.rel_coinjective.coinjective ha (this.mp ha)
      rw [eq_inv_smul_iff] at this
      rw [this]
    · rintro N ⟨i, hN⟩
      have := congr_arg (·ᴺ.rel i (π₁ • N)) h
      simp only [smul_nearLitters, Enumeration.smul_rel, inv_smul_smul, eq_iff_iff] at this
      have := Sᴺ.rel_coinjective.coinjective hN (this.mp hN)
      rw [eq_inv_smul_iff] at this
      rw [this]
  · intro h
    ext : 2
    · rfl
    · ext i a : 3
      rw [smul_atoms, smul_atoms, Enumeration.smul_rel, Enumeration.smul_rel]
      constructor
      · intro ha
        have := h.1 _ ⟨i, ha⟩
        rw [smul_inv_smul, ← inv_smul_eq_iff] at this
        rwa [this]
      · intro ha
        have := h.1 _ ⟨i, ha⟩
        rw [smul_inv_smul, smul_eq_iff_eq_inv_smul] at this
        rwa [← this]
    · rfl
    · ext i a : 3
      rw [smul_nearLitters, smul_nearLitters, Enumeration.smul_rel, Enumeration.smul_rel]
      constructor
      · intro hN
        have := h.2 _ ⟨i, hN⟩
        rw [smul_inv_smul, ← inv_smul_eq_iff] at this
        rwa [this]
      · intro hN
        have := h.2 _ ⟨i, hN⟩
        rw [smul_inv_smul, smul_eq_iff_eq_inv_smul] at this
        rwa [← this]

theorem smul_eq_iff (π : BasePerm) (S : BaseSupport) :
    π • S = S ↔ (∀ a ∈ Sᴬ, π • a = a) ∧ (∀ N ∈ Sᴺ, π • N = N) := by
  have := smul_eq_smul_iff π 1 S
  simp only [one_smul] at this
  exact this

noncomputable instance : Add BaseSupport where
  add S T := ⟨Sᴬ + Tᴬ, Sᴺ + Tᴺ⟩

@[simp]
theorem add_atoms (S T : BaseSupport) :
    (S + T)ᴬ = Sᴬ + Tᴬ :=
  rfl

@[simp]
theorem add_nearLitters (S T : BaseSupport) :
    (S + T)ᴺ = Sᴺ + Tᴺ :=
  rfl

end BaseSupport

def baseSupportEquiv : BaseSupport ≃ Enumeration Atom × Enumeration NearLitter where
  toFun S := (Sᴬ, Sᴺ)
  invFun S := ⟨S.1, S.2⟩
  left_inv _ := rfl
  right_inv _ := rfl

theorem card_baseSupport : #BaseSupport = #μ := by
  rw [Cardinal.eq.mpr ⟨baseSupportEquiv⟩, mk_prod, lift_id, lift_id,
    card_enumeration_eq card_atom, card_enumeration_eq card_nearLitter, mul_eq_self aleph0_lt_μ.le]

/-!
## Structural supports
-/

structure Support (α : TypeIndex) where
  atoms : Enumeration (α ↝ ⊥ × Atom)
  nearLitters : Enumeration (α ↝ ⊥ × NearLitter)

namespace Support

variable {α β : TypeIndex}

instance : SuperA (Support α) (Enumeration (α ↝ ⊥ × Atom)) where
  superA := atoms

instance : SuperN (Support α) (Enumeration (α ↝ ⊥ × NearLitter)) where
  superN := nearLitters

@[simp]
theorem mk_atoms (E : Enumeration (α ↝ ⊥ × Atom)) (F : Enumeration (α ↝ ⊥ × NearLitter)) :
    (⟨E, F⟩ : Support α)ᴬ = E :=
  rfl

@[simp]
theorem mk_nearLitters (E : Enumeration (α ↝ ⊥ × Atom)) (F : Enumeration (α ↝ ⊥ × NearLitter)) :
    (⟨E, F⟩ : Support α)ᴺ = F :=
  rfl

instance : Derivative (Support α) (Support β) α β where
  deriv S A := ⟨Sᴬ ⇘ A, Sᴺ ⇘ A⟩

instance : Coderivative (Support β) (Support α) α β where
  coderiv S A := ⟨Sᴬ ⇗ A, Sᴺ ⇗ A⟩

instance : BotDerivative (Support α) BaseSupport α where
  botDeriv S A := ⟨Sᴬ ⇘. A, Sᴺ ⇘. A⟩
  botSderiv S := ⟨Sᴬ ↘., Sᴺ ↘.⟩
  botDeriv_single S h := by dsimp only; rw [botDeriv_single, botDeriv_single]

@[simp]
theorem deriv_atoms {α β : TypeIndex} (S : Support α) (A : α ↝ β) :
    Sᴬ ⇘ A = (S ⇘ A)ᴬ :=
  rfl

@[simp]
theorem deriv_nearLitters {α β : TypeIndex} (S : Support α) (A : α ↝ β) :
    Sᴺ ⇘ A = (S ⇘ A)ᴺ :=
  rfl

@[simp]
theorem sderiv_atoms {α β : TypeIndex} (S : Support α) (h : β < α) :
    Sᴬ ↘ h = (S ↘ h)ᴬ :=
  rfl

@[simp]
theorem sderiv_nearLitters {α β : TypeIndex} (S : Support α) (h : β < α) :
    Sᴺ ↘ h = (S ↘ h)ᴺ :=
  rfl

@[simp]
theorem coderiv_atoms {α β : TypeIndex} (S : Support β) (A : α ↝ β) :
    Sᴬ ⇗ A = (S ⇗ A)ᴬ :=
  rfl

@[simp]
theorem coderiv_nearLitters {α β : TypeIndex} (S : Support β) (A : α ↝ β) :
    Sᴺ ⇗ A = (S ⇗ A)ᴺ :=
  rfl

@[simp]
theorem scoderiv_atoms {α β : TypeIndex} (S : Support β) (h : β < α) :
    Sᴬ ↗ h = (S ↗ h)ᴬ :=
  rfl

@[simp]
theorem scoderiv_nearLitters {α β : TypeIndex} (S : Support β) (h : β < α) :
    Sᴺ ↗ h = (S ↗ h)ᴺ :=
  rfl

@[simp]
theorem derivBot_atoms {α : TypeIndex} (S : Support α) (A : α ↝ ⊥) :
    Sᴬ ⇘. A = (S ⇘. A)ᴬ :=
  rfl

@[simp]
theorem derivBot_nearLitters {α : TypeIndex} (S : Support α) (A : α ↝ ⊥) :
    Sᴺ ⇘. A = (S ⇘. A)ᴺ :=
  rfl

theorem ext' {S T : Support α} (h₁ : Sᴬ = Tᴬ) (h₂ : Sᴺ = Tᴺ) : S = T := by
  obtain ⟨SA, SN⟩ := S
  obtain ⟨TA, TN⟩ := T
  cases h₁
  cases h₂
  rfl

@[ext]
theorem ext {S T : Support α} (h : ∀ A, S ⇘. A = T ⇘. A) : S = T := by
  obtain ⟨SA, SN⟩ := S
  obtain ⟨TA, TN⟩ := T
  rw [mk.injEq]
  constructor
  · apply Enumeration.ext_path
    intro A
    exact BaseSupport.atoms_congr (h A)
  · apply Enumeration.ext_path
    intro A
    exact BaseSupport.nearLitters_congr (h A)

@[simp]
theorem deriv_derivBot {α : TypeIndex} (S : Support α)
    (A : α ↝ β) (B : β ↝ ⊥) :
    S ⇘ A ⇘. B = S ⇘. (A ⇘ B) :=
  rfl

@[simp]
theorem coderiv_deriv_eq {α β : TypeIndex} (S : Support β) (A : α ↝ β) :
    S ⇗ A ⇘ A = S :=
  ext' (Sᴬ.coderiv_deriv_eq A) (Sᴺ.coderiv_deriv_eq A)

theorem eq_of_atom_mem_scoderiv_botDeriv {α β : TypeIndex} {S : Support β} {A : α ↝ ⊥}
    {h : β < α} {a : Atom} (ha : a ∈ (S ↗ h ⇘. A)ᴬ) :
    ∃ B : β ↝ ⊥, A = B ↗ h :=
  Enumeration.eq_of_mem_scoderiv_botDeriv ha

theorem eq_of_nearLitter_mem_scoderiv_botDeriv {α β : TypeIndex} {S : Support β} {A : α ↝ ⊥}
    {h : β < α} {N : NearLitter} (hN : N ∈ (S ↗ h ⇘. A)ᴺ) :
    ∃ B : β ↝ ⊥, A = B ↗ h :=
  Enumeration.eq_of_mem_scoderiv_botDeriv hN

@[simp]
theorem scoderiv_botDeriv_eq {α β : TypeIndex} (S : Support β) (A : β ↝ ⊥) (h : β < α) :
    S ↗ h ⇘. (A ↗ h) = S ⇘. A :=
  BaseSupport.ext (Enumeration.scoderiv_botDeriv_eq _ _ _) (Enumeration.scoderiv_botDeriv_eq _ _ _)

@[simp]
theorem scoderiv_deriv_eq {α β γ : TypeIndex} (S : Support β) (A : β ↝ γ) (h : β < α) :
    S ↗ h ⇘ (A ↗ h) = S ⇘ A := by
  apply ext
  intro B
  simp only [deriv_derivBot, ← scoderiv_botDeriv_eq S (A ⇘ B) h, Path.coderiv_deriv']

@[simp]
theorem coderiv_inj {α β : TypeIndex} (S T : Support β) (A : α ↝ β) :
    S ⇗ A = T ⇗ A ↔ S = T := by
  constructor
  swap
  · rintro rfl
    rfl
  intro h
  ext B : 1
  have : S ⇗ A ⇘ A ⇘. B = T ⇗ A ⇘ A ⇘. B := by rw [h]
  rwa [coderiv_deriv_eq, coderiv_deriv_eq] at this

@[simp]
theorem scoderiv_inj {α β : TypeIndex} (S T : Support β) (h : β < α) :
    S ↗ h = T ↗ h ↔ S = T :=
  coderiv_inj S T (.single h)

instance {α : TypeIndex} : SMul (StrPerm α) (Support α) where
  smul π S := ⟨π • Sᴬ, π • Sᴺ⟩

@[simp]
theorem smul_atoms {α : TypeIndex} (π : StrPerm α) (S : Support α) :
    (π • S)ᴬ = π • Sᴬ :=
  rfl

@[simp]
theorem smul_nearLitters {α : TypeIndex} (π : StrPerm α) (S : Support α) :
    (π • S)ᴺ = π • Sᴺ :=
  rfl

theorem smul_atoms_eq_of_smul_eq {α : TypeIndex} {π : StrPerm α} {S : Support α}
    (h : π • S = S) :
    π • Sᴬ = Sᴬ := by
  rw [← smul_atoms, h]

theorem smul_nearLitters_eq_of_smul_eq {α : TypeIndex} {π : StrPerm α} {S : Support α}
    (h : π • S = S) :
    π • Sᴺ = Sᴺ := by
  rw [← smul_nearLitters, h]

instance {α : TypeIndex} : MulAction (StrPerm α) (Support α) where
  one_smul S := by
    apply ext'
    · rw [smul_atoms, one_smul]
    · rw [smul_nearLitters, one_smul]
  mul_smul π₁ π₂ S := by
    apply ext'
    · rw [smul_atoms, smul_atoms, smul_atoms, mul_smul]
    · rw [smul_nearLitters, smul_nearLitters, smul_nearLitters, mul_smul]

@[simp]
theorem smul_derivBot {α : TypeIndex} (π : StrPerm α) (S : Support α) (A : α ↝ ⊥) :
    (π • S) ⇘. A = π A • (S ⇘. A) :=
  rfl

theorem smul_coderiv {α : TypeIndex} (π : StrPerm α) (S : Support β) (A : α ↝ β) :
    π • S ⇗ A = (π ⇘ A • S) ⇗ A := by
  ext B i x
  · rfl
  · constructor
    · rintro ⟨⟨C, x⟩, hS, hx⟩
      simp only [Prod.mk.injEq] at hx
      obtain ⟨rfl, rfl⟩ := hx
      exact ⟨⟨C, x⟩, hS, rfl⟩
    · rintro ⟨⟨C, x⟩, hS, hx⟩
      simp only [Prod.mk.injEq] at hx
      obtain ⟨rfl, rfl⟩ := hx
      exact ⟨⟨C, _⟩, hS, rfl⟩
  · rfl
  · constructor
    · rintro ⟨⟨C, x⟩, hS, hx⟩
      simp only [Prod.mk.injEq] at hx
      obtain ⟨rfl, rfl⟩ := hx
      exact ⟨⟨C, x⟩, hS, rfl⟩
    · rintro ⟨⟨C, a⟩, hS, hx⟩
      simp only [Prod.mk.injEq] at hx
      obtain ⟨rfl, rfl⟩ := hx
      exact ⟨⟨C, _⟩, hS, rfl⟩

theorem smul_scoderiv {α : TypeIndex} (π : StrPerm α) (S : Support β) (h : β < α) :
    π • S ↗ h = (π ↘ h • S) ↗ h :=
  smul_coderiv π S (Path.single h)

theorem smul_eq_smul_iff (π₁ π₂ : StrPerm β) (S : Support β) :
    π₁ • S = π₂ • S ↔
      ∀ A, (∀ a ∈ (S ⇘. A)ᴬ, π₁ A • a = π₂ A • a) ∧ (∀ N ∈ (S ⇘. A)ᴺ, π₁ A • N = π₂ A • N) := by
  constructor
  · intro h A
    have := congr_arg (· ⇘. A) h
    simp only [smul_derivBot, BaseSupport.smul_eq_smul_iff] at this
    exact this
  · intro h
    apply ext
    intro A
    simp only [smul_derivBot, BaseSupport.smul_eq_smul_iff]
    exact h A

theorem smul_eq_iff (π : StrPerm β) (S : Support β) :
    π • S = S ↔ ∀ A, (∀ a ∈ (S ⇘. A)ᴬ, π A • a = a) ∧ (∀ N ∈ (S ⇘. A)ᴺ, π A • N = N) := by
  have := smul_eq_smul_iff π 1 S
  simp only [one_smul, Tree.one_apply] at this
  exact this

noncomputable instance : Add (Support α) where
  add S T := ⟨Sᴬ + Tᴬ, Sᴺ + Tᴺ⟩

@[simp]
theorem add_derivBot (S T : Support α) (A : α ↝ ⊥) :
    (S + T) ⇘. A = (S ⇘. A) + (T ⇘. A) :=
  rfl

theorem smul_add (S T : Support α) (π : StrPerm α) :
    π • (S + T) = π • S + π • T :=
  rfl

theorem add_inj_of_bound_eq_bound {S T U V : Support α}
    (ha : Sᴬ.bound = Tᴬ.bound) (hN : Sᴺ.bound = Tᴺ.bound)
    (h' : S + U = T + V) : S = T ∧ U = V := by
  have ha' := Enumeration.add_inj_of_bound_eq_bound ha (congr_arg (·ᴬ) h')
  have hN' := Enumeration.add_inj_of_bound_eq_bound hN (congr_arg (·ᴺ) h')
  constructor
  · exact Support.ext' ha'.1 hN'.1
  · exact Support.ext' ha'.2 hN'.2

end Support

def supportEquiv {α : TypeIndex} : Support α ≃
    Enumeration (α ↝ ⊥ × Atom) × Enumeration (α ↝ ⊥ × NearLitter) where
  toFun S := (Sᴬ, Sᴺ)
  invFun S := ⟨S.1, S.2⟩
  left_inv _ := rfl
  right_inv _ := rfl

theorem card_support {α : TypeIndex} : #(Support α) = #μ := by
  rw [Cardinal.eq.mpr ⟨supportEquiv⟩, mk_prod, lift_id, lift_id,
    card_enumeration_eq, card_enumeration_eq, mul_eq_self aleph0_lt_μ.le]
  · rw [mk_prod, lift_id, lift_id, card_nearLitter,
      mul_eq_right aleph0_lt_μ.le (card_path_lt' α ⊥).le (card_path_ne_zero α)]
  · rw [mk_prod, lift_id, lift_id, card_atom,
      mul_eq_right aleph0_lt_μ.le (card_path_lt' α ⊥).le (card_path_ne_zero α)]

/-!
## Orders on supports
-/

-- TODO: Is this order used?
instance : LE BaseSupport where
  le S T := (∀ a ∈ Sᴬ, a ∈ Tᴬ) ∧ (∀ N ∈ Sᴺ, N ∈ Tᴺ)

instance : Preorder BaseSupport where
  le_refl S := ⟨λ _ ↦ id, λ _ ↦ id⟩
  le_trans S T U h₁ h₂ := ⟨λ a h ↦ h₂.1 _ (h₁.1 a h), λ N h ↦ h₂.2 _ (h₁.2 N h)⟩

theorem BaseSupport.smul_le_smul {S T : BaseSupport} (h : S ≤ T) (π : BasePerm) :
    π • S ≤ π • T := by
  constructor
  · intro a
    exact h.1 (π⁻¹ • a)
  · intro N
    exact h.2 (π⁻¹ • N)

theorem BaseSupport.le_add_right {S T : BaseSupport} :
    S ≤ S + T := by
  constructor
  · intro a ha
    simp only [Support.add_derivBot, BaseSupport.add_atoms, Enumeration.mem_add_iff]
    exact Or.inl ha
  · intro N hN
    simp only [Support.add_derivBot, BaseSupport.add_nearLitters, Enumeration.mem_add_iff]
    exact Or.inl hN

theorem BaseSupport.le_add_left {S T : BaseSupport} :
    S ≤ T + S := by
  constructor
  · intro a ha
    simp only [add_atoms, Enumeration.mem_add_iff]
    exact Or.inr ha
  · intro N hN
    simp only [add_nearLitters, Enumeration.mem_add_iff]
    exact Or.inr hN

def BaseSupport.Subsupport (S T : BaseSupport) : Prop :=
  Sᴬ.rel ≤ Tᴬ.rel ∧ Sᴺ.rel ≤ Tᴺ.rel

theorem BaseSupport.Subsupport.le {S T : BaseSupport}
    (h : S.Subsupport T) : S ≤ T := by
  constructor
  · rintro a ⟨i, hi⟩
    exact ⟨i, h.1 i a hi⟩
  · rintro N ⟨i, hi⟩
    exact ⟨i, h.2 i N hi⟩

theorem BaseSupport.Subsupport.trans {S T U : BaseSupport}
    (h₁ : S.Subsupport T) (h₂ : T.Subsupport U) :
    S.Subsupport U :=
  ⟨h₁.1.trans h₂.1, h₁.2.trans h₂.2⟩

theorem BaseSupport.smul_subsupport_smul {S T : BaseSupport} (h : S.Subsupport T) (π : BasePerm) :
    (π • S).Subsupport (π • T) := by
  constructor
  · intro i a ha
    exact h.1 i _ ha
  · intro i N hN
    exact h.2 i _ hN

instance {α : TypeIndex} : LE (Support α) where
  le S T := ∀ A, S ⇘. A ≤ T ⇘. A

instance {α : TypeIndex} : Preorder (Support α) where
  le_refl S := λ A ↦ le_rfl
  le_trans S T U h₁ h₂ := λ A ↦ (h₁ A).trans (h₂ A)

theorem Support.smul_le_smul {α : TypeIndex} {S T : Support α} (h : S ≤ T) (π : StrPerm α) :
    π • S ≤ π • T :=
  λ A ↦ BaseSupport.smul_le_smul (h A) (π A)

theorem Support.le_add_right {α : TypeIndex} {S T : Support α} :
    S ≤ S + T := by
  intro A
  rw [add_derivBot]
  exact BaseSupport.le_add_right

theorem Support.le_add_left {α : TypeIndex} {S T : Support α} :
    S ≤ T + S := by
  intro A
  rw [add_derivBot]
  exact BaseSupport.le_add_left

def Support.Subsupport {α : TypeIndex} (S T : Support α) : Prop :=
  ∀ A, (S ⇘. A).Subsupport (T ⇘. A)

theorem Support.Subsupport.le {α : TypeIndex} {S T : Support α}
    (h : S.Subsupport T) : S ≤ T :=
  λ A ↦ (h A).le

theorem Support.Subsupport.trans {α : TypeIndex} {S T U : Support α}
    (h₁ : S.Subsupport T) (h₂ : T.Subsupport U) :
    S.Subsupport U :=
  λ A ↦ (h₁ A).trans (h₂ A)

theorem Support.smul_subsupport_smul {α : TypeIndex} {S T : Support α}
    (h : S.Subsupport T) (π : StrPerm α) :
    (π • S).Subsupport (π • T) :=
  λ A ↦ BaseSupport.smul_subsupport_smul (h A) (π A)

theorem subsupport_add {α : TypeIndex} {S T : Support α} :
    S.Subsupport (S + T) := by
  intro A
  constructor
  · intro i a ha
    simp only [Support.add_derivBot, BaseSupport.add_atoms, Enumeration.rel_add_iff]
    exact Or.inl ha
  · intro i N hN
    simp only [Support.add_derivBot, BaseSupport.add_atoms, Enumeration.rel_add_iff]
    exact Or.inl hN

theorem smul_eq_of_subsupport {α : TypeIndex} {S T : Support α} {π : StrPerm α}
    (h₁ : S.Subsupport T) (h₂ : S.Subsupport (π • T)) :
    π • S = S := by
  rw [Support.smul_eq_iff]
  intro A
  constructor
  · rintro a ⟨i, hi⟩
    have hi₁ := (h₁ A).1 i a hi
    have hi₂ := (h₂ A).1 i a hi
    have := (T ⇘. A)ᴬ.rel_coinjective.coinjective hi₁ hi₂
    dsimp only at this
    rwa [smul_eq_iff_eq_inv_smul]
  · rintro N ⟨i, hi⟩
    have hi₁ := (h₁ A).2 i N hi
    have hi₂ := (h₂ A).2 i N hi
    have := (T ⇘. A)ᴺ.rel_coinjective.coinjective hi₁ hi₂
    dsimp only at this
    rwa [smul_eq_iff_eq_inv_smul]

theorem smul_eq_smul_of_le {α : TypeIndex} {S T : Support α} {π₁ π₂ : StrPerm α}
    (h : S ≤ T) (h₂ : π₁ • T = π₂ • T) :
    π₁ • S = π₂ • S

:= by
  rw [Support.smul_eq_smul_iff] at h₂ ⊢
  intro A
  constructor
  · intro a ha
    exact (h₂ A).1 a ((h A).1 a ha)
  · intro N hN
    exact (h₂ A).2 N ((h A).2 N hN)


import ConNF.Model.Result

/-!
# New file

In this file...

## Main declarations

* `ConNF.foo`: Something new.
-/

noncomputable section
universe u

open Cardinal Ordinal ConNF.TSet

namespace ConNF

variable [Params.{u}] {α β γ δ ε ζ : Λ} (hβ : (β : TypeIndex) < α) (hγ : (γ : TypeIndex) < β)
  (hδ : (δ : TypeIndex) < γ) (hε : (ε : TypeIndex) < δ) (hζ : (ζ : TypeIndex) < ε)

def union (x y : TSet α) : TSet α :=
  (xᶜ' ⊓' yᶜ')ᶜ'

notation:68 x:68 " ⊔[" h "] " y:68 => _root_.ConNF.union h x y
notation:68 x:68 " ⊔' " y:68 => x ⊔[by assumption] y

@[simp]
theorem mem_union_iff (x y : TSet α) :
    ∀ z : TSet β, z ∈' x ⊔' y ↔ z ∈' x ∨ z ∈' y := by
  rw [union]
  intro z
  rw [mem_compl_iff, mem_inter_iff, mem_compl_iff, mem_compl_iff, or_iff_not_and_not]

def higherIndex (α : Λ) : Λ :=
  (exists_gt α).choose

theorem lt_higherIndex {α : Λ} :
    (α : TypeIndex) < higherIndex α :=
  WithBot.coe_lt_coe.mpr (exists_gt α).choose_spec

theorem tSet_nonempty (h : ∃ β : Λ, (β : TypeIndex) < α) : Nonempty (TSet α) := by
  obtain ⟨α', hα⟩ := h
  constructor
  apply typeLower lt_higherIndex lt_higherIndex lt_higherIndex hα
  apply cardinalOne lt_higherIndex lt_higherIndex

def empty : TSet α :=
  (tSet_nonempty ⟨β, hβ⟩).some ⊓' (tSet_nonempty ⟨β, hβ⟩).someᶜ'

@[simp]
theorem mem_empty_iff :
    ∀ x : TSet β, ¬x ∈' empty hβ := by
  intro x
  rw [empty, mem_inter_iff, mem_compl_iff]
  exact and_not_self

def univ : TSet α :=
  (empty hβ)ᶜ'

@[simp]
theorem mem_univ_iff :
    ∀ x : TSet β, x ∈' univ hβ := by
  intro x
  simp only [univ, mem_compl_iff, mem_empty_iff, not_false_eq_true]

/-- The set of all ordered pairs. -/
def orderedPairs : TSet α :=
  vCross hβ hγ hδ (univ hδ)

@[simp]
theorem mem_orderedPairs_iff (x : TSet β) :
    x ∈' orderedPairs hβ hγ hδ ↔ ∃ a b, x = ⟨a, b⟩' := by
  simp only [orderedPairs, vCross_spec, mem_univ_iff, and_true]

def converse (x : TSet α) : TSet α :=
  converse' hβ hγ hδ x ⊓' orderedPairs hβ hγ hδ

@[simp]
theorem op_mem_converse_iff (x : TSet α) :
    ∀ a b, ⟨a, b⟩' ∈' converse hβ hγ hδ x ↔ ⟨b, a⟩' ∈' x := by
  intro a b
  simp only [converse, mem_inter_iff, converse'_spec, mem_orderedPairs_iff, op_inj, exists_and_left,
    exists_eq', and_true]

def cross (x y : TSet γ) : TSet α :=
  converse hβ hγ hδ (vCross hβ hγ hδ x) ⊓' vCross hβ hγ hδ y

@[simp]
theorem mem_cross_iff (x y : TSet γ) :
    ∀ a, a ∈' cross hβ hγ hδ x y ↔ ∃ b c, a = ⟨b, c⟩' ∧ b ∈' x ∧ c ∈' y := by
  intro a
  rw [cross, mem_inter_iff, vCross_spec]
  constructor
  · rintro ⟨h₁, b, c, rfl, h₂⟩
    simp only [op_mem_converse_iff, vCross_spec, op_inj] at h₁
    obtain ⟨b', c', ⟨rfl, rfl⟩, h₁⟩ := h₁
    exact ⟨b, c, rfl, h₁, h₂⟩
  · rintro ⟨b, c, rfl, h₁, h₂⟩
    simp only [op_mem_converse_iff, vCross_spec, op_inj]
    exact ⟨⟨c, b, ⟨rfl, rfl⟩, h₁⟩, ⟨b, c, ⟨rfl, rfl⟩, h₂⟩⟩

def singletonImage (x : TSet β) : TSet α :=
  singletonImage' hβ hγ hδ hε x ⊓' (cross hβ hγ hδ (cardinalOne hδ hε) (cardinalOne hδ hε))

@[simp]
theorem singletonImage_spec (x : TSet β) :
    ∀ z w,
    ⟨ {z}', {w}' ⟩' ∈' singletonImage hβ hγ hδ hε x ↔ ⟨z, w⟩' ∈' x

:= by
  intro z w
  rw [singletonImage, mem_inter_iff, singletonImage'_spec, and_iff_left_iff_imp]
  intro hzw
  rw [mem_cross_iff]
  refine ⟨{z}', {w}', rfl, ?_⟩
  simp only [mem_cardinalOne_iff, singleton_inj, exists_eq', and_self]


import ConNF.ModelData.Enumeration
import ConNF.Levels.StrPerm

/-!
# Enumerations over paths

In this file, we provide extra features to `Enumeration`s that take values of the form `α ↝ ⊥ × X`.

## Main declarations

* `ConNF.Enumeration.ext_path`: An extensionality principle for such `Enumeration`s.
-/

noncomputable section
universe u

open Cardinal Ordinal

namespace ConNF

variable [Params.{u}]

namespace Enumeration

/-- A helper function for making relations with domain and codomain of the form `α ↝ ⊥ × X`
by defining it on each branch. -/
def relWithPath {X Y : Type u} {α : TypeIndex} (f : α ↝ ⊥ → Rel X Y) :
    Rel (α ↝ ⊥ × X) (α ↝ ⊥ × Y) :=
  λ x y ↦ x.1 = y.1 ∧ f x.1 x.2 y.2

theorem relWithPath_coinjective {X Y : Type u} {α : TypeIndex} {f : α ↝ ⊥ → Rel X Y}
    (hf : ∀ A, (f A).Coinjective) :
    (relWithPath f).Coinjective := by
  constructor
  rintro ⟨_, y₁⟩ ⟨_, y₂⟩ ⟨A, x⟩ ⟨rfl, h₁⟩ ⟨rfl, h₂⟩
  cases (hf A).coinjective h₁ h₂
  rfl

instance (X : Type u) (α β : TypeIndex) :
    Derivative (Enumeration (α ↝ ⊥ × X)) (Enumeration (β ↝ ⊥ × X)) α β where
  deriv E A := E.invImage (λ x ↦ (x.1 ⇗ A, x.2))
    (λ x y h ↦ Prod.ext (Path.deriv_right_injective
      ((Prod.mk.injEq _ _ _ _).mp h).1) ((Prod.mk.injEq _ _ _ _).mp h).2)

theorem deriv_rel {X : Type _} {α β : TypeIndex} (E : Enumeration (α ↝ ⊥ × X)) (A : α ↝ β)
    (i : κ) (x : β ↝ ⊥ × X) :
    (E ⇘ A).rel i x ↔ E.rel i (x.1 ⇗ A, x.2) :=
  Iff.rfl

instance (X : Type u) (α β : TypeIndex) :
    Coderivative (Enumeration (β ↝ ⊥ × X)) (Enumeration (α ↝ ⊥ × X)) α β where
  coderiv E A := E.image (λ x ↦ (x.1 ⇗ A, x.2))

theorem coderiv_rel {X : Type _} {α β : TypeIndex} (E : Enumeration (β ↝ ⊥ × X)) (A : α ↝ β)
    (i : κ) (x : α ↝ ⊥ × X) :
    (E ⇗ A).rel i x ↔ ∃ B, x.1 = A ⇘ B ∧ E.rel i (B, x.2) := by
  constructor
  · rintro ⟨x, h, rfl⟩
    exact ⟨_, rfl, h⟩
  · rintro ⟨B, h₁, h₂⟩
    refine ⟨(B, x.2), h₂, ?_⟩
    apply Prod.ext
    · dsimp only
      exact h₁.symm
    · rfl

theorem scoderiv_rel {X : Type _} {α β : TypeIndex} (E : Enumeration (β ↝ ⊥ × X)) (h : β < α)
    (i : κ) (x : α ↝ ⊥ × X) :
    (E ↗ h).rel i x ↔ ∃ B, x.1 = B ↗ h ∧ E.rel i (B, x.2) :=
  coderiv_rel E (.single h) i x

theorem eq_of_scoderiv_mem {X : Type _} {α β γ : TypeIndex} (E : Enumeration (β ↝ ⊥ × X))
    (h : β < α) (h' : γ < α)
    (i : κ) (A : γ ↝ ⊥) (x : X) (h : (E ↗ h).rel i ⟨A ↗ h', x⟩) :
    β = γ := by
  rw [scoderiv_rel] at h
  obtain ⟨B, h₁, h₂⟩ := h
  exact Path.scoderiv_index_injective h₁.symm

instance (X : Type u) (α : TypeIndex) :
    BotDerivative (Enumeration (α ↝ ⊥ × X)) (Enumeration X) α where
  botDeriv E A := E.invImage (λ x ↦ (A, x)) (Prod.mk.inj_left A)
  botSderiv E := E.invImage (λ x ↦ (Path.nil ↘., x)) (Prod.mk.inj_left (Path.nil ↘.))
  botDeriv_single E h := by
    cases α using WithBot.recBotCoe with
    | bot => cases lt_irrefl ⊥ h
    | coe => rfl

theorem derivBot_rel {X : Type _} {α : TypeIndex} (E : Enumeration (α ↝ ⊥ × X)) (A : α ↝ ⊥)
    (i : κ) (x : X) :
    (E ⇘. A).rel i x ↔ E.rel i (A, x)


import ConNF.Model.RaiseStrong

/-!
# New file

In this file...

## Main declarations

* `ConNF.foo`: Something new.
-/

noncomputable section
universe u

open Cardinal Ordinal

open scoped Pointwise

namespace ConNF

variable [Params.{u}]

/-- A redefinition of the derivative of allowable permutations that is invariant of level,
but still has nice definitional properties. -/
@[default_instance 200]
instance {β γ : TypeIndex} : Derivative (AllPerm β) (AllPerm γ) β γ where
  deriv ρ A :=
    A.recSderiv
    (motive := λ (δ : TypeIndex) (A : β ↝ δ) ↦
      letI : Level := ⟨δ.recBotCoe (Nonempty.some inferInstance) id⟩
      letI : LeLevel δ := ⟨δ.recBotCoe (λ _ ↦ bot_le) (λ _ h ↦ WithBot.coe_le_coe.mpr h.le)
        (show δ.recBotCoe (Nonempty.some inferInstance) id = Level.α from rfl)⟩
      AllPerm δ)
    ρ (λ δ ε A h ρ ↦
      letI : Level := ⟨δ.recBotCoe (Nonempty.some inferInstance) id⟩
      letI : LeLevel δ := ⟨δ.recBotCoe (λ _ ↦ bot_le) (λ _ h ↦ WithBot.coe_le_coe.mpr h.le)
        (show δ.recBotCoe (Nonempty.some inferInstance) id = Level.α from rfl)⟩
      letI : LeLevel ε := ⟨h.le.trans LeLevel.elim⟩
      PreCoherentData.allPermSderiv h ρ)

@[simp]
theorem allPerm_deriv_nil' {β : TypeIndex}
    (ρ : AllPerm β) :
    ρ ⇘ (.nil : β ↝ β) = ρ :=
  rfl

@[simp]
theorem allPerm_deriv_sderiv' {β γ δ : TypeIndex}
    (ρ : AllPerm β) (A : β ↝ γ) (h : δ < γ) :
    ρ ⇘ (A ↘ h) = ρ ⇘ A ↘ h :=
  rfl

@[simp]
theorem allPermSderiv_forget' {β γ : TypeIndex} (h : γ < β) (ρ : AllPerm β) :
    (ρ ↘ h)ᵁ = ρᵁ ↘ h :=
  letI : Level := ⟨β.recBotCoe (Nonempty.some inferInstance) id⟩
  letI : LeLevel β := ⟨β.recBotCoe (λ _ ↦ bot_le) (λ _ h ↦ WithBot.coe_le_coe.mpr h.le)
    (show β.recBotCoe (Nonempty.some inferInstance) id = Level.α from rfl)⟩
  letI : LeLevel γ := ⟨h.le.trans LeLevel.elim⟩
  allPermSderiv_forget h ρ

@[simp]
theorem allPerm_inv_sderiv' {β γ : TypeIndex} (h : γ < β) (ρ : AllPerm β) :
    ρ⁻¹ ↘ h = (ρ ↘ h)⁻¹ := by
  apply allPermForget_injective
  rw [allPermSderiv_forget', allPermForget_inv, Tree.inv_sderiv, allPermForget_inv,
    allPermSderiv_forget']

def Symmetric {α β : Λ} (s : Set (TSet β)) (hβ : (β : TypeIndex) < α) : Prop :=
  ∃ S : Support α, ∀ ρ : AllPerm α, ρᵁ • S = S → ρ ↘ hβ • s = s

def newSetEquiv {α : Λ} :
    letI : Level := ⟨α⟩
    @TSet _ α newModelData.toPreModelData ≃ TSet α :=
  letI : Level := ⟨α⟩
  castTSet (D₁ := newModelData) (D₂ := globalModelData) rfl
    (by rw [globalModelData, motive_eq, constructMotive, globalLtData_eq])

@[simp]
theorem newSetEquiv_forget {α : Λ}
    (x : letI : Level := ⟨α⟩; @TSet _ α newModelData.toPreModelData) :
    (newSetEquiv x)ᵁ = xᵁ :=
  letI : Level := ⟨α⟩
  castTSet_forget (D₁ := newModelData) (D₂ := globalModelData) _ x

def allPermEquiv {α : Λ} :
    letI : Level := ⟨α⟩
    NewPerm ≃ AllPerm α :=
  letI : Level := ⟨α⟩
  castAllPerm (D₁ := newModelData) (D₂ := globalModelData) rfl
    (by rw [globalModelData, motive_eq, constructMotive, globalLtData_eq])

@[simp]
theorem allPermEquiv_forget {α : Λ} (ρ : letI : Level := ⟨α⟩; NewPerm) :
    (allPermEquiv ρ)ᵁ = ρᵁ :=
  letI : Level := ⟨α⟩
  castAllPerm_forget (D₁ := newModelData) (D₂ := globalModelData) _ ρ

theorem allPermEquiv_sderiv {α β : Λ}
    (ρ : letI : Level := ⟨α⟩; NewPerm) (hβ : (β : TypeIndex) < α) :
    letI : Level := ⟨α⟩
    letI : LtLevel β := ⟨hβ⟩
    allPermEquiv ρ ↘ hβ = ρ.sderiv β := by
  letI : Level := ⟨α⟩
  letI : LeLevel α := ⟨le_rfl⟩
  letI : LtLevel β := ⟨hβ⟩
  apply allPermForget_injective
  rw [allPermSderiv_forget, allPermEquiv_forget, NewPerm.forget_sderiv]

theorem TSet.exists_of_symmetric {α β : Λ} (s : Set (TSet β)) (hβ : (β : TypeIndex) < α)
    (hs : Symmetric s hβ) :
    ∃ x : TSet α, ∀ y : TSet β, y ∈[hβ] x ↔ y ∈ s := by
  letI : Level := ⟨α⟩
  letI : LtLevel β := ⟨hβ⟩
  suffices ∃ x : (@TSet _ α newModelData.toPreModelData), ∀ y : TSet β, yᵁ ∈[hβ] xᵁ ↔ y ∈ s by
    obtain ⟨x, hx⟩ := this
    use newSetEquiv x
    intro y
    rw [← hx, ← TSet.forget_mem_forget, newSetEquiv_forget]
  obtain rfl | hs' := s.eq_empty_or_nonempty
  · use none
    intro y
    simp only [Set.mem_empty_iff_false, iff_false]
    exact not_mem_none y
  · use some (Code.toSet ⟨β, s, hs'⟩ ?_)
    · intro y
      erw [mem_some_iff]
      exact Code.mem_toSet _
    · obtain ⟨S, hS⟩ := hs
      use S
      intro ρ hρS
      have := hS (allPermEquiv ρ) ?_
      · simp only [NewPerm.smul_mk, Code.mk.injEq, heq_eq_eq, true_and]
        rwa [allPermEquiv_sderiv] at this
      · rwa [allPermEquiv_forget]

theorem TSet.exists_support {α : Λ} (x : TSet α) :
    ∃ S : Support α, ∀ ρ : AllPerm α, ρᵁ • S = S → ρ • x = x := by
  letI : Level := ⟨α⟩
  obtain ⟨S, hS⟩ := NewSet.exists_support (newSetEquiv.symm x)
  use S
  intro ρ hρ
  have := @Support.Supports.supports _ _ _ newPreModelData _ _ _ hS (allPermEquiv.symm ρ) ?_
  · apply tSetForget_injective
    have := congr_arg (·ᵁ) this
    simp only at this
    erw [@smul_forget _ _ newModelData (allPermEquiv.symm ρ) (newSetEquiv.symm x),
      ← allPermEquiv_forget, ← newSetEquiv_forget, Equiv.apply_symm_apply,
      Equiv.apply_symm_apply] at this
    rwa [smul_forget]
  · rwa [← allPermEquiv_forget, Equiv.apply_symm_apply]

theorem TSet.symmetric {α β : Λ} (x : TSet α) (hβ : (β : TypeIndex) < α) :
    Symmetric {y : TSet β | y ∈[hβ] x} hβ := by
  obtain ⟨S, hS⟩ := exists_support x
  use S
  intro ρ hρ
  conv_rhs => rw [← hS ρ hρ]
  simp only [← forget_mem_forget, smul_forget, StrSet.mem_smul_iff]
  ext y
  rw [Set.mem_smul_set_iff_inv_smul_mem, Set.mem_setOf_eq, Set.mem_setOf_eq,
    smul_forget, allPermForget_inv, allPermSderiv_forget']

theorem tSet_ext' {α β : Λ} (hβ : (β : TypeIndex) < α) (x y : TSet α)
    (h : ∀ z : TSet β, z ∈[hβ] x ↔ z ∈[hβ] y) : x = y :=
  letI : Level := ⟨α⟩
  letI : LeLevel α := ⟨le_rfl⟩
  letI : LtLevel β := ⟨hβ⟩
  tSet_ext hβ x y h

@[simp]
theorem TSet.mem_smul_iff' {α β : TypeIndex}
    {x : TSet β} {y : TSet α} (h : β < α) (ρ : AllPerm α) :
    x ∈[h] ρ • y ↔ ρ⁻¹ ↘ h • x ∈[h] y := by
  letI : Level := ⟨α.recBotCoe (Nonempty.some inferInstance) id⟩
  letI : LeLevel α := ⟨α.recBotCoe (λ _ ↦ bot_le) (λ _ h ↦ WithBot.coe_le_coe.mpr h.le)
    (show α.recBotCoe (Nonempty.some inferInstance) id = Level.α from rfl)⟩
  letI : LtLevel β := ⟨h.trans_le LeLevel.elim⟩
  exact mem_smul_iff h ρ  -- For some reason, using `exact` instead of term mode speeds this up!

def singleton {α β : Λ} (hβ : (β : TypeIndex) < α) (x : TSet β) : TSet α :=
  letI : Level := ⟨α⟩
  letI : LeLevel α := ⟨le_rfl⟩
  letI : LtLevel β := ⟨hβ⟩
  PreCoherentData.singleton hβ x

@[simp]
theorem typedMem_singleton_iff' {α β : Λ} (hβ : (β : TypeIndex) < α) (x y : TSet β) :
    y ∈[hβ] singleton hβ x ↔ y = x :=
  letI : Level := ⟨α⟩
  letI : LeLevel α := ⟨le_rfl⟩
  letI : LtLevel β := ⟨hβ⟩
  typedMem_singleton_iff hβ x y

@[simp]
theorem smul_singleton {α β : Λ} (hβ : (β : TypeIndex) < α) (x : TSet β) (ρ : AllPerm α) :
    ρ • singleton hβ x = singleton hβ (ρ ↘ hβ • x) := by
  apply tSet_ext' hβ
  intro z
  rw [TSet.mem_smul_iff', allPerm_inv_sderiv', typedMem_singleton_iff', typedMem_singleton_iff',
    inv_smul_eq_iff]

theorem singleton_injective {α β : Λ} (hβ : (β : TypeIndex) < α) :
    Function.Injective (singleton hβ) := by
  intro x y hxy
  have := typedMem_singleton_iff' hβ x y
  rw [hxy, typedMem_singleton_iff'] at this
  exact (this.mp rfl).symm

@[simp]
theorem singleton_inj {α β : Λ} {hβ : (β : TypeIndex) < α} {x y : TSet β} :
    singleton hβ x = singleton hβ y ↔ x = y :=
  (singleton_injective hβ).eq_iff

theorem sUnion_singleton_symmetric_aux' {α β γ : Λ}
    (hγ : (γ : TypeIndex) < β) (hβ : (β : TypeIndex) < α) (s : Set (TSet γ)) (S : Support α)
    (hS : ∀ ρ : AllPerm α, ρᵁ • S = S → ρ ↘ hβ • singleton hγ '' s = singleton hγ '' s) :
    letI : Level := ⟨α⟩
    letI : LeLevel α := ⟨le_rfl⟩
    ∀ (ρ : AllPerm β), ρᵁ • S.strong ↘ hβ = S.strong ↘ hβ → ρ ↘ hγ • s ⊆ s := by
  letI : Level := ⟨α⟩
  letI : LeLevel α := ⟨le_rfl⟩
  letI : LtLevel β := ⟨hβ⟩
  rintro ρ hρ _ ⟨x, hx, rfl⟩
  obtain ⟨T, hT⟩ := exists_support x
  obtain ⟨ρ', hρ'₁, hρ'₂⟩ := Support.exists_allowable_of_fixes S.strong S.strong_strong T ρ hγ hρ
  have hρ's := hS ρ' (smul_eq_of_le (S.subsupport_strong.le) hρ'₁)
  have hρ'x : ρ' ↘ hβ ↘ hγ • x = ρ ↘ hγ • x := by
    apply hT.smul_eq_smul
    simp only [allPermSderiv_forget', allPermSderiv_forget, WithBot.recBotCoe_coe, id_eq, hρ'₂]
  dsimp only
  rw [← hρ'x]
  have := (Set.ext_iff.mp hρ's (ρ' ↘ hβ • singleton hγ x)).mp ⟨_, Set.mem_image_of_mem _ hx, rfl⟩
  rw [smul_singleton] at this
  rwa [(singleton_injective hγ).mem_set_image] at this

theorem sUnion_singleton_symmetric_aux {α β γ : Λ}
    (hγ : (γ : TypeIndex) < β) (hβ : (β : TypeIndex) < α) (s : Set (TSet γ)) (S : Support α)
    (hS : ∀ ρ : AllPerm α, ρᵁ • S = S → ρ ↘ hβ • singleton hγ '' s = singleton hγ '' s) :
    letI : Level := ⟨α⟩
    letI : LeLevel α := ⟨le_rfl⟩
    ∀ (ρ : AllPerm β), ρᵁ • S.strong ↘ hβ = S.strong ↘ hβ → ρ ↘ hγ • s = s := by
  intro ρ hρ
  apply subset_antisymm
  · exact sUnion_singleton_symmetric_aux' hγ hβ s S hS ρ hρ
  · have := sUnion_singleton_symmetric_aux' hγ hβ s S hS ρ⁻¹ ?_
    · rwa [allPerm_inv_sderiv', Set.set_smul_subset_iff, inv_inv] at this
    · rw [allPermForget_inv, inv_smul_eq_iff, hρ]

theorem sUnion_singleton_symmetric {α β γ : Λ} (hγ : (γ : TypeIndex) < β) (hβ : (β : TypeIndex) < α)
    (s : Set (TSet γ)) (hs : Symmetric (singleton hγ '' s) hβ) :
    Symmetric s hγ

:= by
  letI : Level := ⟨α⟩
  letI : LeLevel α := ⟨le_rfl⟩
  obtain ⟨S, hS⟩ := hs
  use S.strong ↘ hβ
  exact sUnion_singleton_symmetric_aux hγ hβ s S hS


import ConNF.Model.TTT

/-!
# New file

In this file...

## Main declarations

* `ConNF.foo`: Something new.
-/

noncomputable section
universe u

open Cardinal Ordinal

namespace ConNF

variable [Params.{u}] {α β γ δ ε ζ : Λ} (hβ : (β : TypeIndex) < α) (hγ : (γ : TypeIndex) < β)
  (hδ : (δ : TypeIndex) < γ) (hε : (ε : TypeIndex) < δ) (hζ : (ζ : TypeIndex) < ε)

namespace TSet

theorem exists_inter (x y : TSet α) :
    ∃ w : TSet α, ∀ z : TSet β, z ∈[hβ] w ↔ z ∈[hβ] x ∧ z ∈[hβ] y := by
  refine exists_of_symmetric {z | z ∈[hβ] x ∧ z ∈[hβ] y} hβ ?_
  obtain ⟨S, hS⟩ := symmetric x hβ
  obtain ⟨T, hT⟩ := symmetric y hβ
  use S + T
  intro ρ hρ
  specialize hS ρ (smul_eq_of_le Support.le_add_right hρ)
  specialize hT ρ (smul_eq_of_le Support.le_add_left hρ)
  simp [Set.ext_iff, Set.mem_smul_set_iff_inv_smul_mem] at hS hT ⊢
  aesop

theorem exists_compl (x : TSet α) :
    ∃ y : TSet α, ∀ z : TSet β, z ∈[hβ] y ↔ ¬z ∈[hβ] x := by
  refine exists_of_symmetric {z | ¬z ∈[hβ] x} hβ ?_
  obtain ⟨S, hS⟩ := symmetric x hβ
  use S
  intro ρ hρ
  specialize hS ρ hρ
  simp [Set.ext_iff, Set.mem_smul_set_iff_inv_smul_mem] at hS ⊢
  aesop

theorem exists_up (x y : TSet β) :
    ∃ w : TSet α, ∀ z : TSet β, z ∈[hβ] w ↔ z = x ∨ z = y := by
  refine exists_of_symmetric {x, y} hβ ?_
  obtain ⟨S, hS⟩ := exists_support x
  obtain ⟨T, hT⟩ := exists_support y
  use (S + T) ↗ hβ
  intro ρ hρ
  rw [Support.smul_scoderiv, Support.scoderiv_inj, ← allPermSderiv_forget'] at hρ
  specialize hS (ρ ↘ hβ) (smul_eq_of_le Support.le_add_right hρ)
  specialize hT (ρ ↘ hβ) (smul_eq_of_le Support.le_add_left hρ)
  simp only [Set.smul_set_def, Set.image, Set.mem_insert_iff, Set.mem_singleton_iff,
    exists_eq_or_imp, hS, exists_eq_left, hT]
  ext z
  rw [Set.mem_insert_iff, Set.mem_singleton_iff, Set.mem_setOf_eq]
  aesop

/-- The unordered pair. -/
def up (x y : TSet β) : TSet α :=
  (exists_up hβ x y).choose

@[simp]
theorem mem_up_iff (x y z : TSet β) :
    z ∈[hβ] up hβ x y ↔ z = x ∨ z = y :=
  (exists_up hβ x y).choose_spec z

/-- The Kuratowski ordered pair. -/
def op (x y : TSet γ) : TSet α :=
  up hβ (singleton hγ x) (up hγ x y)

theorem up_injective {x y z w : TSet β} (h : up hβ x y = up hβ z w) :
    (x = z ∧ y = w) ∨ (x = w ∧ y = z) := by
  have h₁ := mem_up_iff hβ x y z
  have h₂ := mem_up_iff hβ x y w
  have h₃ := mem_up_iff hβ z w x
  have h₄ := mem_up_iff hβ z w y
  rw [h, mem_up_iff] at h₁ h₂
  rw [← h, mem_up_iff] at h₃ h₄
  aesop

@[simp]
theorem up_inj (x y z w : TSet β) :
    up hβ x y = up hβ z w ↔ (x = z ∧ y = w) ∨ (x = w ∧ y = z) := by
  constructor
  · exact up_injective hβ
  · rintro (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
    · rfl
    · apply tSet_ext' hβ
      aesop

@[simp]
theorem up_self {x : TSet β} :
    up hβ x x = singleton hβ x := by
  apply tSet_ext' hβ
  aesop

@[simp]
theorem up_eq_singleton_iff (x y z : TSet β) :
    up hβ x y = singleton hβ z ↔ x = z ∧ y = z := by
  constructor
  · intro h
    have h₁ := typedMem_singleton_iff' hβ z x
    have h₂ := typedMem_singleton_iff' hβ z y
    rw [← h, mem_up_iff] at h₁ h₂
    aesop
  · rintro ⟨rfl, rfl⟩
    rw [up_self]

@[simp]
theorem singleton_eq_up_iff (x y z : TSet β) :
    singleton hβ z = up hβ x y ↔ x = z ∧ y = z := by
  rw [← up_eq_singleton_iff hβ x y z, eq_comm]

theorem op_injective {x y z w : TSet γ} (h : op hβ hγ x y = op hβ hγ z w) :
    x = z ∧ y = w := by
  rw [op, op] at h
  simp only [up_inj, singleton_inj, singleton_eq_up_iff, up_eq_singleton_iff] at h
  obtain (⟨rfl, ⟨h, rfl⟩ | ⟨rfl, rfl⟩⟩ | ⟨⟨rfl, rfl⟩, ⟨h, rfl⟩⟩) := h <;> simp only [and_self]

@[simp]
theorem op_inj (x y z w : TSet γ) :
    op hβ hγ x y = op hβ hγ z w ↔ x = z ∧ y = w := by
  constructor
  · exact op_injective hβ hγ
  · rintro ⟨rfl, rfl⟩
    rfl

@[simp]
theorem op_eq_singleton_iff (x y : TSet γ) (z : TSet β) :
    op hβ hγ x y = singleton hβ z ↔ singleton hγ x = z ∧ singleton hγ y = z := by
  rw [op, up_eq_singleton_iff, and_congr_right_iff]
  rintro rfl
  simp only [up_eq_singleton_iff, true_and, singleton_inj]

@[simp]
theorem smul_up (x y : TSet β) (ρ : AllPerm α) :
    ρ • up hβ x y = up hβ (ρ ↘ hβ • x) (ρ ↘ hβ • y) := by
  apply tSet_ext' hβ
  aesop

@[simp]
theorem smul_op (x y : TSet γ) (ρ : AllPerm α) :
    ρ • op hβ hγ x y = op hβ hγ (ρ ↘ hβ ↘ hγ • x) (ρ ↘ hβ ↘ hγ • y) := by
  apply tSet_ext' hβ
  simp only [op, smul_up, smul_singleton, mem_up_iff, implies_true]

theorem exists_singletonImage (x : TSet β) :
    ∃ y : TSet α, ∀ z w,
    op hγ hδ (singleton hε z) (singleton hε w) ∈[hβ] y ↔ op hδ hε z w ∈[hγ] x := by
  have := exists_of_symmetric {u | ∃ z w : TSet ε, op hδ hε z w ∈[hγ] x ∧
      u = op hγ hδ (singleton hε z) (singleton hε w)} hβ ?_
  · obtain ⟨y, hy⟩ := this
    use y
    intro z w
    rw [hy]
    simp only [Set.mem_setOf_eq, op_inj, singleton_inj, exists_eq_right_right', exists_eq_right']
  · obtain ⟨S, hS⟩ := exists_support x
    use S ↗ hβ
    intro ρ hρ
    rw [Support.smul_scoderiv, Support.scoderiv_inj, ← allPermSderiv_forget'] at hρ
    specialize hS (ρ ↘ hβ) hρ
    ext z
    constructor
    · rintro ⟨_, ⟨z, w, hab, rfl⟩, rfl⟩
      refine ⟨ρ ↘ hβ ↘ hγ ↘ hδ ↘ hε • z, ρ ↘ hβ ↘ hγ ↘ hδ ↘ hε • w, ?_, ?_⟩
      · rwa [← hS, mem_smul_iff', smul_op, allPerm_inv_sderiv', allPerm_inv_sderiv',
          allPerm_inv_sderiv', inv_smul_smul, inv_smul_smul]
      · simp only [smul_op, smul_singleton]
    · rintro ⟨z, w, hab, rfl⟩
      refine ⟨ρ⁻¹ ↘ hβ • op hγ hδ (singleton hε z) (singleton hε w), ?_, ?_⟩
      · simp only [allPerm_inv_sderiv', smul_op, smul_singleton, Set.mem_setOf_eq, op_inj,
          singleton_inj, exists_eq_right_right', exists_eq_right']
        rw [smul_eq_iff_eq_inv_smul] at hS
        rw [hS]
        simp only [mem_smul_iff', inv_inv, smul_op, smul_inv_smul]
        exact hab
      · simp only [allPerm_inv_sderiv', smul_inv_smul]

theorem exists_insertion2 (x : TSet γ) :
    ∃ y : TSet α, ∀ a b c, op hγ hδ (singleton hε (singleton hζ a)) (op hε hζ b c) ∈[hβ] y ↔
    op hε hζ a c ∈[hδ] x := by
  have := exists_of_symmetric {u | ∃ a b c : TSet ζ, op hε hζ a c ∈[hδ] x ∧
      u = op hγ hδ (singleton hε (singleton hζ a)) (op hε hζ b c)} hβ ?_
  · obtain ⟨y, hy⟩ := this
    use y
    intro a b c
    rw [hy]
    constructor
    · rintro ⟨a', b', c', h₁, h₂⟩
      simp only [op_inj, singleton_inj] at h₂
      obtain ⟨rfl, rfl, rfl⟩ := h₂
      exact h₁
    · intro h
      exact ⟨a, b, c, h, rfl⟩
  · obtain ⟨S, hS⟩ := exists_support x
    use S ↗ hγ ↗ hβ
    intro ρ hρ
    simp only [Support.smul_scoderiv, Support.scoderiv_inj, ← allPermSderiv_forget'] at hρ
    specialize hS (ρ ↘ hβ ↘ hγ) hρ
    ext z
    constructor
    · rintro ⟨_, ⟨a, b, c, hx, rfl⟩, rfl⟩
      refine ⟨ρ ↘ hβ ↘ hγ ↘ hδ ↘ hε ↘ hζ • a, ρ ↘ hβ ↘ hγ ↘ hδ ↘ hε ↘ hζ • b,
        ρ ↘ hβ ↘ hγ ↘ hδ ↘ hε ↘ hζ • c, ?_, ?_⟩
      · rw [← hS]
        simp only [mem_smul_iff', allPerm_inv_sderiv', smul_op, inv_smul_smul]
        exact hx
      · simp only [smul_op, smul_singleton]
    · rintro ⟨a, b, c, hx, rfl⟩
      rw [Set.mem_smul_set_iff_inv_smul_mem]
      refine ⟨ρ⁻¹ ↘ hβ ↘ hγ ↘ hδ ↘ hε ↘ hζ • a, ρ⁻¹ ↘ hβ ↘ hγ ↘ hδ ↘ hε ↘ hζ • b,
        ρ⁻¹ ↘ hβ ↘ hγ ↘ hδ ↘ hε ↘ hζ • c, ?_, ?_⟩
      · rw [smul_eq_iff_eq_inv_smul] at hS
        rw [hS, mem_smul_iff']
        simp only [inv_inv, allPerm_inv_sderiv', smul_op, smul_inv_smul]
        exact hx
      · simp only [smul_op, allPerm_inv_sderiv', smul_singleton]

theorem exists_insertion3 (x : TSet γ) :
    ∃ y : TSet α, ∀ a b c, op hγ hδ (singleton hε (singleton hζ a)) (op hε hζ b c) ∈[hβ] y ↔
    op hε hζ a b ∈[hδ] x := by
  have := exists_of_symmetric {u | ∃ a b c : TSet ζ, op hε hζ a b ∈[hδ] x ∧
      u = op hγ hδ (singleton hε (singleton hζ a)) (op hε hζ b c)} hβ ?_
  · obtain ⟨y, hy⟩ := this
    use y
    intro a b c
    rw [hy]
    constructor
    · rintro ⟨a', b', c', h₁, h₂⟩
      simp only [op_inj, singleton_inj] at h₂
      obtain ⟨rfl, rfl, rfl⟩ := h₂
      exact h₁
    · intro h
      exact ⟨a, b, c, h, rfl⟩
  · obtain ⟨S, hS⟩ := exists_support x
    use S ↗ hγ ↗ hβ
    intro ρ hρ
    simp only [Support.smul_scoderiv, Support.scoderiv_inj, ← allPermSderiv_forget'] at hρ
    specialize hS (ρ ↘ hβ ↘ hγ) hρ
    ext z
    constructor
    · rintro ⟨_, ⟨a, b, c, hx, rfl⟩, rfl⟩
      refine ⟨ρ ↘ hβ ↘ hγ ↘ hδ ↘ hε ↘ hζ • a, ρ ↘ hβ ↘ hγ ↘ hδ ↘ hε ↘ hζ • b,
        ρ ↘ hβ ↘ hγ ↘ hδ ↘ hε ↘ hζ • c, ?_, ?_⟩
      · rw [← hS]
        simp only [mem_smul_iff', allPerm_inv_sderiv', smul_op, inv_smul_smul]
        exact hx
      · simp only [smul_op, smul_singleton]
    · rintro ⟨a, b, c, hx, rfl⟩
      rw [Set.mem_smul_set_iff_inv_smul_mem]
      refine ⟨ρ⁻¹ ↘ hβ ↘ hγ ↘ hδ ↘ hε ↘ hζ • a, ρ⁻¹ ↘ hβ ↘ hγ ↘ hδ ↘ hε ↘ hζ • b,
        ρ⁻¹ ↘ hβ ↘ hγ ↘ hδ ↘ hε ↘ hζ • c, ?_, ?_⟩
      · rw [smul_eq_iff_eq_inv_smul] at hS
        rw [hS, mem_smul_iff']
        simp only [inv_inv, allPerm_inv_sderiv', smul_op, smul_inv_smul]
        exact hx
      · simp only [smul_op, allPerm_inv_sderiv', smul_singleton]

theorem exists_cross (x : TSet γ) :
    ∃ y : TSet α, ∀ a, a ∈[hβ] y ↔ ∃ b c, a = op hγ hδ b c ∧ c ∈[hδ] x := by
  have := exists_of_symmetric {a | ∃ b c, a = op hγ hδ b c ∧ c ∈[hδ] x} hβ ?_
  · obtain ⟨y, hy⟩ := this
    use y
    intro a
    rw [hy]
    rfl
  · obtain ⟨S, hS⟩ := exists_support x
    use S ↗ hγ ↗ hβ
    intro ρ hρ
    simp only [Support.smul_scoderiv, Support.scoderiv_inj, ← allPermSderiv_forget'] at hρ
    specialize hS (ρ ↘ hβ ↘ hγ) hρ
    ext z
    constructor
    · rintro ⟨_, ⟨a, b, rfl, hab⟩, rfl⟩
      refine ⟨ρ ↘ hβ ↘ hγ ↘ hδ • a, ρ ↘ hβ ↘ hγ ↘ hδ • b, ?_, ?_⟩
      · simp only [smul_op]
      · rw [← hS]
        simp only [mem_smul_iff', allPerm_inv_sderiv', inv_smul_smul]
        exact hab
    · rintro ⟨a, b, rfl, hab⟩
      rw [Set.mem_smul_set_iff_inv_smul_mem]
      refine ⟨ρ⁻¹ ↘ hβ ↘ hγ ↘ hδ • a, ρ⁻¹ ↘ hβ ↘ hγ ↘ hδ • b, ?_, ?_⟩
      · simp only [smul_op, allPerm_inv_sderiv']
      · rw [smul_eq_iff_eq_inv_smul] at hS
        rw [hS]
        simp only [allPerm_inv_sderiv', mem_smul_iff', inv_inv, smul_inv_smul]
        exact hab

theorem exists_typeLower (x : TSet α) :
    ∃ y : TSet δ, ∀ a, a ∈[hε] y ↔ ∀ b, op hγ hδ b (singleton hε a) ∈[hβ] x := by
  have := exists_of_symmetric {a | ∀ b, op hγ hδ b (singleton hε a) ∈[hβ] x} hε ?_
  · obtain ⟨y, hy⟩ := this
    use y
    intro a
    rw [hy]
    rfl
  · apply sUnion_singleton_symmetric hε hδ
    apply sUnion_singleton_symmetric hδ hγ
    apply sUnion_singleton_symmetric hγ hβ
    obtain ⟨S, hS⟩ := exists_support x
    use S
    intro ρ hρ
    specialize hS ρ hρ
    ext z
    constructor
    · rintro ⟨_, ⟨_, ⟨a, ⟨b, hb, rfl⟩, rfl⟩, rfl⟩, rfl⟩
      simp only [smul_singleton, Set.mem_image, Set.mem_setOf_eq, exists_exists_and_eq_and,
        singleton_inj, exists_eq_right]
      intro c
      have := hb (ρ⁻¹ ↘ hβ ↘ hγ ↘ hδ • c)
      rw [smul_eq_iff_eq_inv_smul] at hS
      rw [hS] at this
      simp only [allPerm_inv_sderiv', mem_smul_iff', inv_inv, smul_op, smul_inv_smul,
        smul_singleton] at this
      exact this
    · rintro ⟨_, ⟨a, ⟨b, hb, rfl⟩, rfl⟩, rfl⟩
      rw [Set.mem_smul_set_iff_inv_smul_mem]
      simp only [smul_singleton, allPerm_inv_sderiv', Set.mem_image, Set.mem_setOf_eq,
        exists_exists_and_eq_and, singleton_inj, exists_eq_right]
      intro c
      have := hb (ρ ↘ hβ ↘ hγ ↘ hδ • c)
      rw [← hS] at this
      simp only [mem_smul_iff', allPerm_inv_sderiv', smul_op, inv_smul_smul, smul_singleton] at this
      exact this

theorem exists_converse (x : TSet α) :
    ∃ y : TSet α, ∀ a b, op hγ hδ a b ∈[hβ] y ↔ op hγ hδ b a ∈[hβ] x := by
  have := exists_of_symmetric {a | ∃ b c, a = op hγ hδ b c ∧ op hγ hδ c b ∈[hβ] x} hβ ?_
  · obtain ⟨y, hy⟩ := this
    use y
    intro a b
    rw [hy]
    simp only [Set.mem_setOf_eq, op_inj]
    constructor
    · rintro ⟨a', b', ⟨rfl, rfl⟩, h⟩
      exact h
    · intro h
      exact ⟨a, b, ⟨rfl, rfl⟩, h⟩
  · obtain ⟨S, hS⟩ := exists_support x
    use S
    intro ρ hρ
    specialize hS ρ hρ
    ext z
    constructor
    · rintro ⟨_, ⟨a, b, rfl, hab⟩, rfl⟩
      refine ⟨ρ ↘ hβ ↘ hγ ↘ hδ • a, ρ ↘ hβ ↘ hγ ↘ hδ • b, ?_, ?_⟩
      · simp only [smul_op]
      · rw [← hS]
        simp only [mem_smul_iff', allPerm_inv_sderiv', smul_op, inv_smul_smul]
        exact hab
    · rintro ⟨a, b, rfl, hab⟩
      rw [Set.mem_smul_set_iff_inv_smul_mem]
      refine ⟨ρ⁻¹ ↘ hβ ↘ hγ ↘ hδ • a, ρ⁻¹ ↘ hβ ↘ hγ ↘ hδ • b, ?_, ?_⟩
      · simp only [smul_op, allPerm_inv_sderiv']
      · rw [smul_eq_iff_eq_inv_smul] at hS
        rw [hS]
        simp only [allPerm_inv_sderiv', mem_smul_iff', inv_inv, smul_op, smul_inv_smul]
        exact hab

theorem exists_cardinalOne :
    ∃ x : TSet α, ∀ a : TSet β, a ∈[hβ] x ↔ ∃ b, a = singleton hγ b

:= by
  have := exists_of_symmetric {a | ∃ b, a = singleton hγ b} hβ ?_
  · obtain ⟨y, hy⟩ := this
    use y
    intro a
    rw [hy]
    rfl
  · use ⟨.empty, .empty⟩
    intro ρ hρ
    ext z
    constructor
    · rintro ⟨z, ⟨a, ha⟩, rfl⟩
      refine ⟨ρ ↘ hβ ↘ hγ • a, ?_⟩
      simp only [ha, smul_singleton]
    · rintro ⟨a, ha⟩
      rw [Set.mem_smul_set_iff_inv_smul_mem]
      refine ⟨ρ⁻¹ ↘ hβ ↘ hγ • a, ?_⟩
      simp only [ha, smul_singleton, allPerm_inv_sderiv']


import ConNF.Model.Result

/-!
# New file

In this file...

## Main declarations

* `ConNF.foo`: Something new.
-/

noncomputable section
universe u

open Cardinal Ordinal ConNF.TSet

namespace ConNF

variable [Params.{u}] {α β γ δ ε ζ : Λ} (hβ : (β : TypeIndex) < α) (hγ : (γ : TypeIndex) < β)
  (hδ : (δ : TypeIndex) < γ) (hε : (ε : TypeIndex) < δ) (hζ : (ζ : TypeIndex) < ε)

def union (x y : TSet α) : TSet α :=
  (xᶜ' ⊓' yᶜ')ᶜ'

notation:68 x:68 " ⊔[" h "] " y:68 => _root_.ConNF.union h x y
notation:68 x:68 " ⊔' " y:68 => x ⊔[by assumption] y

@[simp]
theorem mem_union_iff (x y : TSet α) :
    ∀ z : TSet β, z ∈' x ⊔' y ↔ z ∈' x ∨ z ∈' y := by
  rw [union]
  intro z
  rw [mem_compl_iff, mem_inter_iff, mem_compl_iff, mem_compl_iff, or_iff_not_and_not]

def higherIndex (α : Λ) : Λ :=
  (exists_gt α).choose

theorem lt_higherIndex {α : Λ} :
    (α : TypeIndex) < higherIndex α

:=
  WithBot.coe_lt_coe.mpr (exists_gt α).choose_spec


import ConNF.External.Basic

/-!
# New file

In this file...

## Main declarations

* `ConNF.foo`: Something new.
-/

noncomputable section
universe u

open Cardinal Ordinal ConNF.TSet

namespace ConNF

variable [Params.{u}] {α β γ δ ε ζ : Λ} (hβ : (β : TypeIndex) < α) (hγ : (γ : TypeIndex) < β)
  (hδ : (δ : TypeIndex) < γ) (hε : (ε : TypeIndex) < δ) (hζ : (ζ : TypeIndex) < ε)

/-- A set in our model that is a well-order. Internal well-orders are exactly external well-orders,
so we externalise the definition for convenience. -/
def InternalWellOrder (r : TSet α) : Prop :=
  IsWellOrder (ExternalRel hβ hγ hδ r).field
    (InvImage (ExternalRel hβ hγ hδ r) Subtype.val)

def InternallyWellOrdered (x : TSet γ) : Prop :=
  {y : TSet δ | y ∈' x}.Subsingleton ∨ (∃ r, InternalWellOrder hβ hγ hδ r ∧ x = field hβ hγ hδ r)

@[simp]
theorem externalRel_smul (r : TSet α) (ρ : AllPerm α) :
    ExternalRel hβ hγ hδ (ρ • r) =
      InvImage (ExternalRel hβ hγ hδ r) ((ρ ↘ hβ ↘ hγ ↘ hδ)⁻¹ • ·) := by
  ext a b
  simp only [ExternalRel, mem_smul_iff', allPerm_inv_sderiv', smul_op, InvImage]

omit [Params] in
/-- Well-orders are rigid. -/
theorem apply_eq_of_isWellOrder {X : Type _} {r : Rel X X} {f : X → X}
    (hr : IsWellOrder X r) (hf : Function.Bijective f) (hf' : ∀ x y, r x y ↔ r (f x) (f y)) :
    ∀ x, f x = x := by
  let emb : r ≼i r := ⟨⟨⟨f, hf.injective⟩, λ {a b} ↦ (hf' a b).symm⟩, ?_⟩
  · have : emb = InitialSeg.refl r := Subsingleton.elim _ _
    intro x
    exact congr_arg (λ f ↦ f x) this
  · intro a b h
    exact hf.surjective _

omit [Params] in
theorem apply_eq_of_isWellOrder' {X : Type _} {r : Rel X X} {f : X → X}
    (hr : IsWellOrder r.field (InvImage r Subtype.val)) (hf : Function.Bijective f)
    (hf' : ∀ x y, r x y ↔ r (f x) (f y)) :
    ∀ x ∈ r.field, f x = x := by
  have : ∀ x ∈ r.field, f x ∈ r.field := by
    rintro x (⟨y, h⟩ | ⟨y, h⟩)
    · exact Or.inl ⟨f y, (hf' x y).mp h⟩
    · exact Or.inr ⟨f y, (hf' y x).mp h⟩
  have := apply_eq_of_isWellOrder (f := λ x ↦ ⟨f x.val, this x.val x.prop⟩) hr ⟨?_, ?_⟩ ?_
  · intro x hx
    exact congr_arg Subtype.val (this ⟨x, hx⟩)
  · intro x y h
    rw [Subtype.mk.injEq] at h
    exact Subtype.val_injective (hf.injective h)
  · intro x
    obtain ⟨y, hy⟩ := hf.surjective x.val
    refine ⟨⟨y, ?_⟩, ?_⟩
    · obtain (⟨z, h⟩ | ⟨z, h⟩) := x.prop <;>
          rw [← hy] at h <;>
          obtain ⟨z, rfl⟩ := hf.surjective z
      · exact Or.inl ⟨z, (hf' y z).mpr h⟩
      · exact Or.inr ⟨z, (hf' z y).mpr h⟩
    · simp only [hy]
  · intros
    apply hf'

theorem exists_common_support_of_internallyWellOrdered' {x : TSet δ}
    (h : InternallyWellOrdered hγ hδ hε x) :
    ∃ S : Support β, ∀ y, y ∈' x → S.Supports { { {y}' }' }[hγ] := by
  obtain (h | ⟨r, h, rfl⟩) := h
  · obtain (h | ⟨y, hy⟩) := h.eq_empty_or_singleton
    · use ⟨Enumeration.empty, Enumeration.empty⟩
      intro y hy
      rw [Set.eq_empty_iff_forall_not_mem] at h
      cases h y hy
    · obtain ⟨S, hS⟩ := TSet.exists_support y
      use S ↗ hε ↗ hδ ↗ hγ
      intro z hz
      rw [Set.eq_singleton_iff_unique_mem] at hy
      cases hy.2 z hz
      refine ⟨?_, λ h ↦ by cases h⟩
      intro ρ hρ
      simp only [Support.smul_scoderiv, ← allPermSderiv_forget', Support.scoderiv_inj] at hρ
      simp only [smul_singleton, singleton_inj]
      exact hS _ hρ
  obtain ⟨S, hS⟩ := TSet.exists_support r
  use S
  intro a ha
  refine ⟨?_, λ h ↦ by cases h⟩
  intro ρ hρ
  have := hS ρ hρ
  simp only [smul_singleton, singleton_inj]
  apply apply_eq_of_isWellOrder' (r := ExternalRel hγ hδ hε r)
  · exact h
  · exact MulAction.bijective (ρ ↘ hγ ↘ hδ ↘ hε)
  · intro x y
    conv_rhs => rw [← this]
    simp only [externalRel_smul, InvImage, inv_smul_smul]
  · rwa [mem_field_iff] at ha

include hγ in
theorem Support.Supports.ofSingleton {S : Support α} {x : TSet β}
    (h : S.Supports {x}') :
    letI : Level := ⟨α⟩
    letI : LeLevel α := ⟨le_rfl⟩
    (S.strong ↘ hβ).Supports x := by
  refine ⟨?_, λ h ↦ by cases h⟩
  intro ρ hρ
  open scoped Pointwise in
  have := sUnion_singleton_symmetric_aux hγ hβ {y | y ∈' x} S ?_ ρ hρ
  · apply ConNF.ext hγ
    intro z
    simp only [Set.ext_iff, Set.mem_setOf_eq, Set.mem_smul_set_iff_inv_smul_mem] at this
    rw [mem_smul_iff', allPerm_inv_sderiv', this]
  · intro ρ hρ
    ext z
    simp only [Set.mem_smul_set_iff_inv_smul_mem, Set.mem_image, Set.mem_setOf_eq]
    have := h.supports ρ hρ
    simp only [smul_singleton, singleton_inj] at this
    constructor
    · rintro ⟨y, h₁, h₂⟩
      rw [← smul_eq_iff_eq_inv_smul, smul_singleton] at h₂
      refine ⟨_, ?_, h₂⟩
      rw [← this]
      simp only [mem_smul_iff', allPerm_inv_sderiv', inv_smul_smul]
      exact h₁
    · rintro ⟨y, h, rfl⟩
      refine ⟨(ρ ↘ hβ ↘ hγ)⁻¹ • y, ?_, ?_⟩
      · rwa [← allPerm_inv_sderiv', ← mem_smul_iff', this]
      · simp only [smul_singleton, allPerm_inv_sderiv']

include hγ in
theorem supports_of_supports_singletons {S : Support α} {s : Set (TSet β)}
    (h : ∀ x ∈ s, S.Supports {x}') :
    ∃ S : Support β, ∀ x ∈ s, S.Supports x :=
  ⟨_, λ x hx ↦ (h x hx).ofSingleton hβ hγ⟩

theorem exists_common_support_of_internallyWellOrdered {x : TSet δ}
    (h : InternallyWellOrdered hγ hδ hε x) :
    ∃ S : Support δ, ∀ y, y ∈' x → S.Supports {y}' := by
  obtain ⟨S, hS⟩ := exists_common_support_of_internallyWellOrdered' hγ hδ hε h
  have := supports_of_supports_singletons (S := S)
      (s := singleton hδ '' (singleton hε '' {y | y ∈' x})) hγ hδ ?_
  swap
  · simp only [Set.mem_image, Set.mem_setOf_eq, exists_exists_and_eq_and, forall_exists_index,
      and_imp, forall_apply_eq_imp_iff₂]
    exact hS
  obtain ⟨T, hT⟩ := this
  have := supports_of_supports_singletons (S := T)
      (s := singleton hε '' {y | y ∈' x}) hδ hε ?_
  swap
  · simp only [Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp,
      forall_apply_eq_imp_iff₂] at hT ⊢
    exact hT
  simp only [Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp,
    forall_apply_eq_imp_iff₂] at this
  exact this

theorem internallyWellOrdered_of_common_support_of_nontrivial {x : TSet γ}
    (hx : {y : TSet δ | y ∈' x}.Nontrivial)
    (S : Support δ) (hS : ∀ y : TSet δ, y ∈' x → S.Supports y) :
    InternallyWellOrdered hβ hγ hδ x := by
  have := exists_of_symmetric
      {p : TSet β | ∃ a b : TSet δ, p = ⟨a, b⟩' ∧ a ∈' x ∧ b ∈' x ∧ WellOrderingRel a b} hβ ?_
  swap
  · use S ↗ hδ ↗ hγ ↗ hβ
    intro ρ hρ
    ext z
    simp only [Support.smul_scoderiv, ← allPermSderiv_forget', Support.scoderiv_inj] at hρ
    simp only [Set.mem_smul_set_iff_inv_smul_mem, Set.mem_setOf_eq]
    constructor
    · rintro ⟨a, b, h₁, h₂, h₃, h₄⟩
      refine ⟨a, b, ?_, h₂, h₃, h₄⟩
      rw [inv_smul_eq_iff] at h₁
      rw [h₁, smul_op, op_inj]
      exact ⟨(hS a h₂).supports _ hρ, (hS b h₃).supports _ hρ⟩
    · rintro ⟨a, b, h₁, h₂, h₃, h₄⟩
      refine ⟨a, b, ?_, h₂, h₃, h₄⟩
      rw [h₁, smul_op, op_inj]
      simp only [allPerm_inv_sderiv', inv_smul_eq_iff]
      rw [(hS a h₂).supports _ hρ, (hS b h₃).supports _ hρ]
      exact ⟨rfl, rfl⟩
  obtain ⟨r, hr⟩ := this
  right
  use r
  have hr' : ∀ a b, ExternalRel hβ hγ hδ r a b ↔ a ∈' x ∧ b ∈' x ∧ WellOrderingRel a b := by
    intro a b
    rw [ExternalRel, hr]
    simp only [Set.mem_setOf_eq, op_inj]
    constructor
    · rintro ⟨a, b, ⟨rfl, rfl⟩, h⟩
      exact h
    · intro h
      exact ⟨a, b, ⟨rfl, rfl⟩, h⟩
  have hrx : ∀ a, a ∈ (ExternalRel hβ hγ hδ r).field ↔ a ∈' x := by
    intro a
    constructor
    · rintro (⟨b, h⟩ | ⟨b, h⟩)
      · rw [hr'] at h
        exact h.1
      · rw [hr'] at h
        exact h.2.1
    · intro h
      obtain ⟨b, h₁, h₂⟩ := hx.exists_ne a
      obtain (h₃ | h₃ | h₃) := WellOrderingRel.isWellOrder.trichotomous a b
      · refine Or.inl ⟨b, ?_⟩
        rw [hr']
        exact ⟨h, h₁, h₃⟩
      · cases h₂ h₃.symm
      · refine Or.inr ⟨b, ?_⟩
        rw [hr']
        exact ⟨h₁, h, h₃⟩
  refine ⟨?_, ?_⟩
  swap
  · apply ext hδ
    intro z
    rw [mem_field_iff, hrx]
  refine @IsWellOrder.mk _ _ ?_ ?_ ?_
  · constructor
    intro a b
    obtain (h | h | h) := WellOrderingRel.isWellOrder.trichotomous a.val b.val
    · apply Or.inl
      rw [InvImage, hr']
      exact ⟨(hrx a).mp a.prop, (hrx b).mp b.prop, h⟩
    · exact Or.inr (Or.inl (Subtype.val_injective h))
    · apply Or.inr ∘ Or.inr
      rw [InvImage, hr']
      exact ⟨(hrx b).mp b.prop, (hrx a).mp a.prop, h⟩
  · constructor
    intro a b c h₁ h₂
    rw [InvImage, hr'] at h₁ h₂ ⊢
    exact ⟨h₁.1, h₂.2.1, WellOrderingRel.isWellOrder.trans _ _ _ h₁.2.2 h₂.2.2⟩
  · constructor
    apply InvImage.wf
    refine Subrelation.wf ?_ WellOrderingRel.isWellOrder.wf
    intro a b h
    rw [hr'] at h
    exact h.2.2

theorem internallyWellOrdered_of_common_support {x : TSet γ}
    (S : Support δ) (hS : ∀ y : TSet δ, y ∈' x → S.Supports y) :
    InternallyWellOrdered hβ hγ hδ x

:= by
  obtain (hx | hx) := Set.subsingleton_or_nontrivial {y : TSet δ | y ∈' x}
  · exact Or.inl hx
  · exact internallyWellOrdered_of_common_support_of_nontrivial hβ hγ hδ hx S hS


import ConNF.Model.Result

/-!
# New file

In this file...

## Main declarations

* `ConNF.foo`: Something new.
-/

noncomputable section
universe u

open Cardinal Ordinal ConNF.TSet

namespace ConNF

variable [Params.{u}] {α β γ δ ε ζ : Λ} (hβ : (β : TypeIndex) < α) (hγ : (γ : TypeIndex) < β)
  (hδ : (δ : TypeIndex) < γ) (hε : (ε : TypeIndex) < δ) (hζ : (ζ : TypeIndex) < ε)

def union (x y : TSet α) : TSet α :=
  (xᶜ' ⊓' yᶜ')ᶜ'

notation:68 x:68 " ⊔[" h "] " y:68 => _root_.ConNF.union h x y
notation:68 x:68 " ⊔' " y:68 => x ⊔[by assumption] y

@[simp]
theorem mem_union_iff (x y : TSet α) :
    ∀ z : TSet β, z ∈' x ⊔' y ↔ z ∈' x ∨ z ∈' y := by
  rw [union]
  intro z
  rw [mem_compl_iff, mem_inter_iff, mem_compl_iff, mem_compl_iff, or_iff_not_and_not]

def higherIndex (α : Λ) : Λ :=
  (exists_gt α).choose

theorem lt_higherIndex {α : Λ} :
    (α : TypeIndex) < higherIndex α :=
  WithBot.coe_lt_coe.mpr (exists_gt α).choose_spec

theorem tSet_nonempty (h : ∃ β : Λ, (β : TypeIndex) < α) : Nonempty (TSet α) := by
  obtain ⟨α', hα⟩ := h
  constructor
  apply typeLower lt_higherIndex lt_higherIndex lt_higherIndex hα
  apply cardinalOne lt_higherIndex lt_higherIndex

def empty : TSet α :=
  (tSet_nonempty ⟨β, hβ⟩).some ⊓' (tSet_nonempty ⟨β, hβ⟩).someᶜ'

@[simp]
theorem mem_empty_iff :
    ∀ x : TSet β, ¬x ∈' empty hβ := by
  intro x
  rw [empty, mem_inter_iff, mem_compl_iff]
  exact and_not_self

def univ : TSet α :=
  (empty hβ)ᶜ'

@[simp]
theorem mem_univ_iff :
    ∀ x : TSet β, x ∈' univ hβ := by
  intro x
  simp only [univ, mem_compl_iff, mem_empty_iff, not_false_eq_true]

/-- The set of all ordered pairs. -/
def orderedPairs : TSet α :=
  vCross hβ hγ hδ (univ hδ)

@[simp]
theorem mem_orderedPairs_iff (x : TSet β) :
    x ∈' orderedPairs hβ hγ hδ ↔ ∃ a b, x = ⟨a, b⟩' := by
  simp only [orderedPairs, vCross_spec, mem_univ_iff, and_true]

def converse (x : TSet α) : TSet α :=
  converse' hβ hγ hδ x ⊓' orderedPairs hβ hγ hδ

@[simp]
theorem op_mem_converse_iff (x : TSet α) :
    ∀ a b, ⟨a, b⟩' ∈' converse hβ hγ hδ x ↔ ⟨b, a⟩' ∈' x := by
  intro a b
  simp only [converse, mem_inter_iff, converse'_spec, mem_orderedPairs_iff, op_inj, exists_and_left,
    exists_eq', and_true]

def cross (x y : TSet γ) : TSet α :=
  converse hβ hγ hδ (vCross hβ hγ hδ x) ⊓' vCross hβ hγ hδ y

@[simp]
theorem mem_cross_iff (x y : TSet γ) :
    ∀ a, a ∈' cross hβ hγ hδ x y ↔ ∃ b c, a = ⟨b, c⟩' ∧ b ∈' x ∧ c ∈' y := by
  intro a
  rw [cross, mem_inter_iff, vCross_spec]
  constructor
  · rintro ⟨h₁, b, c, rfl, h₂⟩
    simp only [op_mem_converse_iff, vCross_spec, op_inj] at h₁
    obtain ⟨b', c', ⟨rfl, rfl⟩, h₁⟩ := h₁
    exact ⟨b, c, rfl, h₁, h₂⟩
  · rintro ⟨b, c, rfl, h₁, h₂⟩
    simp only [op_mem_converse_iff, vCross_spec, op_inj]
    exact ⟨⟨c, b, ⟨rfl, rfl⟩, h₁⟩, ⟨b, c, ⟨rfl, rfl⟩, h₂⟩⟩

def singletonImage (x : TSet β) : TSet α :=
  singletonImage' hβ hγ hδ hε x ⊓' (cross hβ hγ hδ (cardinalOne hδ hε) (cardinalOne hδ hε))

@[simp]
theorem singletonImage_spec (x : TSet β) :
    ∀ z w,
    ⟨ {z}', {w}' ⟩' ∈' singletonImage hβ hγ hδ hε x ↔ ⟨z, w⟩' ∈' x := by
  intro z w
  rw [singletonImage, mem_inter_iff, singletonImage'_spec, and_iff_left_iff_imp]
  intro hzw
  rw [mem_cross_iff]
  refine ⟨{z}', {w}', rfl, ?_⟩
  simp only [mem_cardinalOne_iff, singleton_inj, exists_eq', and_self]

theorem exists_of_mem_singletonImage {x : TSet β} {z w : TSet δ}
    (h : ⟨z, w⟩' ∈' singletonImage hβ hγ hδ hε x) :
    ∃ a b, z = {a}' ∧ w = {b}' := by
  simp only [singletonImage, mem_inter_iff, mem_cross_iff, op_inj, mem_cardinalOne_iff] at h
  obtain ⟨-, _, _, ⟨rfl, rfl⟩, ⟨a, rfl⟩, ⟨b, rfl⟩⟩ := h
  exact ⟨a, b, rfl, rfl⟩

/-- Turn a model element encoding a relation into an actual relation. -/
def ExternalRel (r : TSet α) : Rel (TSet δ) (TSet δ) :=
  λ x y ↦ ⟨x, y⟩' ∈' r

@[simp]
theorem externalRel_converse (r : TSet α) :
    ExternalRel hβ hγ hδ (converse hβ hγ hδ r) = (ExternalRel hβ hγ hδ r).inv := by
  ext
  simp only [ExternalRel, op_mem_converse_iff, Rel.inv_apply]

/-- The codomain of a relation. -/
def codom (r : TSet α) : TSet γ :=
  (typeLower lt_higherIndex hβ hγ hδ (singletonImage lt_higherIndex hβ hγ hδ r)ᶜ[lt_higherIndex])ᶜ'

@[simp]
theorem mem_codom_iff (r : TSet α) (x : TSet δ) :
    x ∈' codom hβ hγ hδ r ↔ x ∈ (ExternalRel hβ hγ hδ r).codom := by
  simp only [codom, mem_compl_iff, mem_typeLower_iff, not_forall, not_not]
  constructor
  · rintro ⟨y, hy⟩
    obtain ⟨a, b, rfl, hb⟩ := exists_of_mem_singletonImage lt_higherIndex hβ hγ hδ hy
    rw [singleton_inj] at hb
    subst hb
    rw [singletonImage_spec] at hy
    exact ⟨a, hy⟩
  · rintro ⟨a, ha⟩
    use {a}'
    rw [singletonImage_spec]
    exact ha

/-- The domain of a relation. -/
def dom (r : TSet α) : TSet γ :=
  codom hβ hγ hδ (converse hβ hγ hδ r)

@[simp]
theorem mem_dom_iff (r : TSet α) (x : TSet δ) :
    x ∈' dom hβ hγ hδ r ↔ x ∈ (ExternalRel hβ hγ hδ r).dom := by
  rw [dom, mem_codom_iff, externalRel_converse, Rel.inv_codom]

/-- The field of a relation. -/
def field (r : TSet α) : TSet γ :=
  dom hβ hγ hδ r ⊔' codom hβ hγ hδ r

@[simp]
theorem mem_field_iff (r : TSet α) (x : TSet δ) :
    x ∈' field hβ hγ hδ r ↔ x ∈ (ExternalRel hβ hγ hδ r).field := by
  rw [field, mem_union_iff, mem_dom_iff, mem_codom_iff, Rel.field, Set.mem_union]

def subset : TSet α :=
  subset' hβ hγ hδ hε ⊓' orderedPairs hβ hγ hδ

@[simp]
theorem subset_spec :
    ∀ a b, ⟨a, b⟩' ∈' subset hβ hγ hδ hε ↔ a ⊆[TSet ε] b := by
  intro a b
  simp only [subset, mem_inter_iff, subset'_spec, mem_orderedPairs_iff, op_inj, exists_and_left,
    exists_eq', and_true]

def membership : TSet α :=
  subset hβ hγ hδ hε ⊓' cross hβ hγ hδ (cardinalOne hδ hε) (univ hδ)

@[simp]
theorem membership_spec :
    ∀ a b, ⟨{a}', b⟩' ∈' membership hβ hγ hδ hε ↔ a ∈' b := by
  intro a b
  rw [membership, mem_inter_iff, subset_spec]
  simp only [mem_cross_iff, op_inj, mem_cardinalOne_iff, mem_univ_iff, and_true, exists_and_right,
    exists_and_left, exists_eq', exists_eq_left', singleton_inj]
  constructor
  · intro h
    exact h a ((typedMem_singleton_iff' hε a a).mpr rfl)
  · intro h c hc
    simp only [typedMem_singleton_iff'] at hc
    cases hc
    exact h

def powerset (x : TSet β) : TSet α :=
  dom lt_higherIndex lt_higherIndex hβ
    (subset lt_higherIndex lt_higherIndex hβ hγ ⊓[lt_higherIndex]
      vCross lt_higherIndex lt_higherIndex hβ {x}')

@[simp]
theorem mem_powerset_iff (x y : TSet β) :
    x ∈' powerset hβ hγ y ↔ x ⊆[TSet γ] y := by
  rw [powerset, mem_dom_iff]
  constructor
  · rintro ⟨z, h⟩
    simp only [ExternalRel, mem_inter_iff, subset_spec, vCross_spec, op_inj,
      typedMem_singleton_iff', exists_eq_right, exists_and_right, exists_eq', true_and] at h
    cases h.2
    exact h.1
  · intro h
    refine ⟨y, ?_⟩
    simp only [ExternalRel, mem_inter_iff, subset_spec, h, vCross_spec, op_inj,
      typedMem_singleton_iff', exists_eq_right, and_true, exists_eq', and_self]

/-- The set `ι²''x = {{{a}} | a ∈ x}`. -/
def doubleSingleton (x : TSet γ) : TSet α :=
  cross hβ hγ hδ x x ⊓' cardinalOne hβ hγ

@[simp]
theorem mem_doubleSingleton_iff (x : TSet γ) :
    ∀ y : TSet β, y ∈' doubleSingleton hβ hγ hδ x ↔
    ∃ z : TSet δ, z ∈' x ∧ y = { {z}' }' := by
  intro y
  rw [doubleSingleton, mem_inter_iff, mem_cross_iff, mem_cardinalOne_iff]
  constructor
  · rintro ⟨⟨b, c, h₁, h₂, h₃⟩, ⟨a, rfl⟩⟩
    obtain ⟨hbc, rfl⟩ := (op_eq_singleton_iff _ _ _ _ _).mp h₁.symm
    exact ⟨c, h₃, rfl⟩
  · rintro ⟨z, h, rfl⟩
    constructor
    · refine ⟨z, z, ?_⟩
      rw [eq_comm, op_eq_singleton_iff]
      tauto
    · exact ⟨_, rfl⟩

/-- The union of a set of *singletons*: `ι⁻¹''x = {a | {a} ∈ x}`. -/
def singletonUnion (x : TSet α) : TSet β :=
  typeLower lt_higherIndex lt_higherIndex hβ hγ
    (vCross lt_higherIndex lt_higherIndex hβ x)

@[simp]
theorem mem_singletonUnion_iff (x : TSet α) :
    ∀ y : TSet γ, y ∈' singletonUnion hβ hγ x ↔ {y}' ∈' x := by
  intro y
  simp only [singletonUnion, mem_typeLower_iff, vCross_spec, op_inj]
  constructor
  · intro h
    obtain ⟨a, b, ⟨rfl, rfl⟩, hy⟩ := h {y}'
    exact hy
  · intro h b
    exact ⟨b, _, ⟨rfl, rfl⟩, h⟩

/--
The union of a set of sets.

```
singletonUnion dom {⟨{a}, b⟩ | a ∈ b} ∩ (1 × x) =
singletonUnion dom {⟨{a}, b⟩ | a ∈ b ∧ b ∈ x} =
singletonUnion {{a} | a ∈ b ∧ b ∈ x} =
{a | a ∈ b ∧ b ∈ x} =
⋃ x
```
-/
def sUnion (x : TSet α) : TSet β :=
  singletonUnion hβ hγ
    (dom lt_higherIndex lt_higherIndex hβ
      (membership lt_higherIndex lt_higherIndex hβ hγ ⊓[lt_higherIndex]
        cross lt_higherIndex lt_higherIndex hβ (cardinalOne hβ hγ) x))

@[simp]
theorem mem_sUnion_iff (x : TSet α) :
    ∀ y : TSet γ, y ∈' sUnion hβ hγ x ↔ ∃ t : TSet β, t ∈' x ∧ y ∈' t := by
  intro y
  simp only [sUnion, mem_singletonUnion_iff, mem_dom_iff, Rel.dom, ExternalRel, mem_inter_iff,
    mem_cross_iff, op_inj, mem_cardinalOne_iff, Set.mem_setOf_eq, membership_spec]
  constructor
  · rintro ⟨z, h₁, a, b, ⟨rfl, rfl⟩, ⟨c, h₂⟩, h₃⟩
    rw [singleton_inj] at h₂
    cases h₂
    exact ⟨z, h₃, h₁⟩
  · rintro ⟨z, h₂, h₃⟩
    exact ⟨z, h₃, _, _, ⟨rfl, rfl⟩, ⟨y, rfl⟩, h₂⟩

theorem exists_smallUnion (s : Set (TSet α)) (hs : Small s) :
    ∃ x : TSet α, ∀ y : TSet β, y ∈' x ↔ ∃ t ∈ s, y ∈' t := by
  apply exists_of_symmetric
  have := exists_support (α := α)
  choose S hS using this
  refine ⟨⟨Enumeration.ofSet (⋃ t ∈ s, (S t)ᴬ) ?_, Enumeration.ofSet (⋃ t ∈ s, (S t)ᴺ) ?_⟩, ?_⟩
  · apply small_biUnion hs
    intros
    exact (S _)ᴬ.coe_small
  · apply small_biUnion hs
    intros
    exact (S _)ᴺ.coe_small
  intro ρ hρ
  suffices ∀ t ∈ s, ρ • t = t by
    ext y
    rw [Set.mem_smul_set_iff_inv_smul_mem]
    constructor
    · rintro ⟨t, h₁, h₂⟩
      refine ⟨t, h₁, ?_⟩
      rw [← this t h₁]
      rwa [mem_smul_iff', allPerm_inv_sderiv']
    · rintro ⟨t, h₁, h₂⟩
      refine ⟨t, h₁, ?_⟩
      have := this t h₁
      rw [smul_eq_iff_eq_inv_smul] at this
      rwa [this, mem_smul_iff', inv_inv, smul_inv_smul]
  intro t ht
  apply (hS t).supports ρ
  refine smul_eq_of_le ?_ hρ
  intro A
  constructor
  · intro a ha
    rw [← Support.derivBot_atoms, Support.mk_atoms, ← Enumeration.mem_path_iff,
      Enumeration.mem_ofSet_iff, Set.mem_iUnion]
    use t
    rw [Set.mem_iUnion]
    use ht
    exact ha
  · intro a ha
    rw [← Support.derivBot_nearLitters, Support.mk_nearLitters, ← Enumeration.mem_path_iff,
      Enumeration.mem_ofSet_iff, Set.mem_iUnion]
    use t
    rw [Set.mem_iUnion]
    use ht
    exact ha

/-- Our model is `κ`-complete; small unions exist.
In particular, the model knows the correct natural numbers. -/
def smallUnion (s : Set (TSet α)) (hs : Small s) : TSet α :=
  (exists_smallUnion hβ s hs).choose

@[simp]
theorem mem_smallUnion_iff (s : Set (TSet α)) (hs : Small s) :
    ∀ x : TSet β, x ∈' smallUnion hβ s hs ↔ ∃ t ∈ s, x ∈' t

:=
  (exists_smallUnion hβ s hs).choose_spec


import ConNF.External.Basic

/-!
# New file

In this file...

## Main declarations

* `ConNF.foo`: Something new.
-/

noncomputable section
universe u

open Cardinal Ordinal ConNF.TSet

namespace ConNF

variable [Params.{u}] {α β γ δ ε ζ : Λ} (hβ : (β : TypeIndex) < α) (hγ : (γ : TypeIndex) < β)
  (hδ : (δ : TypeIndex) < γ) (hε : (ε : TypeIndex) < δ) (hζ : (ζ : TypeIndex) < ε)

/-- A set in our model that is a well-order. Internal well-orders are exactly external well-orders,
so we externalise the definition for convenience. -/
def InternalWellOrder (r : TSet α) : Prop :=
  IsWellOrder (ExternalRel hβ hγ hδ r).field
    (InvImage (ExternalRel hβ hγ hδ r) Subtype.val)

def InternallyWellOrdered (x : TSet γ) : Prop :=
  {y : TSet δ | y ∈' x}.Subsingleton ∨ (∃ r, InternalWellOrder hβ hγ hδ r ∧ x = field hβ hγ hδ r)

@[simp]
theorem externalRel_smul (r : TSet α) (ρ : AllPerm α) :
    ExternalRel hβ hγ hδ (ρ • r) =
      InvImage (ExternalRel hβ hγ hδ r) ((ρ ↘ hβ ↘ hγ ↘ hδ)⁻¹ • ·) := by
  ext a b
  simp only [ExternalRel, mem_smul_iff', allPerm_inv_sderiv', smul_op, InvImage]

omit [Params] in
/-- Well-orders are rigid. -/
theorem apply_eq_of_isWellOrder {X : Type _} {r : Rel X X} {f : X → X}
    (hr : IsWellOrder X r) (hf : Function.Bijective f) (hf' : ∀ x y, r x y ↔ r (f x) (f y)) :
    ∀ x, f x = x := by
  let emb : r ≼i r := ⟨⟨⟨f, hf.injective⟩, λ {a b} ↦ (hf' a b).symm⟩, ?_⟩
  · have : emb = InitialSeg.refl r := Subsingleton.elim _ _
    intro x
    exact congr_arg (λ f ↦ f x) this
  · intro a b h
    exact hf.surjective _

omit [Params] in

theorem apply_eq_of_isWellOrder' {X : Type _} {r : Rel X X} {f : X → X}
    (hr : IsWellOrder r.field (InvImage r Subtype.val)) (hf : Function.Bijective f)
    (hf' : ∀ x y, r x y ↔ r (f x) (f y)) :
    ∀ x ∈ r.field, f x = x

:= by
  have : ∀ x ∈ r.field, f x ∈ r.field := by
    rintro x (⟨y, h⟩ | ⟨y, h⟩)
    · exact Or.inl ⟨f y, (hf' x y).mp h⟩
    · exact Or.inr ⟨f y, (hf' y x).mp h⟩
  have := apply_eq_of_isWellOrder (f := λ x ↦ ⟨f x.val, this x.val x.prop⟩) hr ⟨?_, ?_⟩ ?_
  · intro x hx
    exact congr_arg Subtype.val (this ⟨x, hx⟩)
  · intro x y h
    rw [Subtype.mk.injEq] at h
    exact Subtype.val_injective (hf.injective h)
  · intro x
    obtain ⟨y, hy⟩ := hf.surjective x.val
    refine ⟨⟨y, ?_⟩, ?_⟩
    · obtain (⟨z, h⟩ | ⟨z, h⟩) := x.prop <;>
          rw [← hy] at h <;>
          obtain ⟨z, rfl⟩ := hf.surjective z
      · exact Or.inl ⟨z, (hf' y z).mpr h⟩
      · exact Or.inr ⟨z, (hf' z y).mpr h⟩
    · simp only [hy]
  · intros
    apply hf'