import ConNF.Model.Externalise

/-!
# New file

In this file...

## Main declarations

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

noncomputable section
universe u

open Cardinal Ordinal

namespace ConNF

variable [Params.{u}] {β γ : Λ} {hγ : (γ : TypeIndex) < β}

namespace Support

theorem not_mem_scoderiv_botDeriv (S : Support γ) (N : NearLitter) :
    N ∉ (S ↗ hγ ⇘. (Path.nil ↘.))ᴺ := by
  rintro ⟨i, ⟨A, N'⟩, h₁, h₂⟩
  simp only [Prod.mk.injEq] at h₂
  cases A
  case sderiv δ A hδ _ =>
    simp only [Path.deriv_sderiv] at h₂
    cases A
    case nil => cases h₂.1
    case sderiv ζ A hζ _ =>
      simp only [Path.deriv_sderiv] at h₂
      cases h₂.1

variable [Level] [LtLevel β]

theorem not_mem_strong_botDeriv (S : Support γ) (N : NearLitter) :
    N ∉ ((S ↗ hγ).strong ⇘. (Path.nil ↘.))ᴺ := by
  rintro h
  rw [strong, close_nearLitters, preStrong_nearLitters, Enumeration.mem_add_iff] at h
  obtain h | h := h
  · exact not_mem_scoderiv_botDeriv S N h
  · rw [mem_constrainsNearLitters_nearLitters] at h
    obtain ⟨B, N', hN', h⟩ := h
    cases h using Relation.ReflTransGen.head_induction_on
    case refl => exact not_mem_scoderiv_botDeriv S N hN'
    case head x hx₁ hx₂ _ =>
      obtain ⟨⟨γ, δ, ε, hδ, hε, hδε, A⟩, t, B, hB, hN, ht⟩ := hx₂
      simp only at hB
      cases B
      case nil =>
        cases hB
        obtain ⟨C, N''⟩ := x
        simp only at ht
        cases ht.1
        change _ ∈ t.supportᴺ at hN
        rw [t.support_supports.2 rfl] at hN
        obtain ⟨i, hN⟩ := hN
        cases hN
      case sderiv δ B hδ _ _ =>
        cases B
        case nil => cases hB
        case sderiv ζ B hζ _ _ => cases hB

theorem raise_preStrong' (S : Support α) (hS : S.Strong) (T : Support γ) (ρ : AllPerm β)
    (hγ : (γ : TypeIndex) < β) :
    (S + (ρᵁ • ((T ↗ hγ).strong +
      (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim).PreStrong := by
  apply hS.toPreStrong.add
  constructor
  intro A N hN P t hA ht
  obtain ⟨A, rfl⟩ := eq_of_nearLitter_mem_scoderiv_botDeriv hN
  simp only [scoderiv_botDeriv_eq, add_derivBot, smul_derivBot,
    BaseSupport.add_nearLitters, BaseSupport.smul_nearLitters, interferenceSupport_nearLitters,
    Enumeration.mem_add_iff, Enumeration.mem_smul, Enumeration.not_mem_empty, or_false] at hN
  obtain ⟨δ, ε, ζ, hε, hζ, hεζ, B⟩ := P
  dsimp only at *
  cases A
  case sderiv ζ' A hζ' _ =>
    rw [← Path.coderiv_deriv] at hA
    cases Path.sderiv_index_injective hA
    apply Path.sderiv_left_inj.mp at hA
    cases A
    case nil =>
      cases hA
      cases not_mem_strong_botDeriv T _ hN
    case sderiv ι A hι _ _ =>
      rw [← Path.coderiv_deriv] at hA
      cases Path.sderiv_index_injective hA
      cases hA
      haveI : LtLevel δ := ⟨A.le.trans_lt LtLevel.elim⟩
      haveI : LtLevel ε := ⟨hε.trans LtLevel.elim⟩
      haveI : LtLevel ζ := ⟨hζ.trans LtLevel.elim⟩
      have := (T ↗ hγ).strong_strong.support_le hN ⟨δ, ε, ζ, hε, hζ, hεζ, A⟩
          (ρ⁻¹ ⇘ A ↘ hε • t) rfl ?_
      · simp only [Tangle.smul_support, allPermSderiv_forget, allPermDeriv_forget,
          allPermForget_inv, Tree.inv_deriv, Tree.inv_sderiv] at this
        have := smul_le_smul this (ρᵁ ⇘ A ↘ hε)
        simp only [smul_inv_smul] at this
        apply le_trans this
        intro B
        constructor
        · intro a ha
          simp only [smul_derivBot, Tree.sderiv_apply, Tree.deriv_apply, Path.deriv_scoderiv,
            deriv_derivBot, Enumeration.mem_smul] at ha
          rw [deriv_derivBot, ← Path.deriv_scoderiv, Path.coderiv_deriv', scoderiv_botDeriv_eq,]
          simp only [Path.deriv_scoderiv, add_derivBot, smul_derivBot,
            BaseSupport.add_atoms, BaseSupport.smul_atoms, Enumeration.mem_add_iff,
            Enumeration.mem_smul]
          exact Or.inl ha
        · intro N hN
          simp only [smul_derivBot, Tree.sderiv_apply, Tree.deriv_apply, Path.deriv_scoderiv,
            deriv_derivBot, Enumeration.mem_smul] at hN
          rw [deriv_derivBot, ← Path.deriv_scoderiv, Path.coderiv_deriv', scoderiv_botDeriv_eq,]
          simp only [Path.deriv_scoderiv, add_derivBot, smul_derivBot,
            BaseSupport.add_nearLitters, BaseSupport.smul_nearLitters, Enumeration.mem_add_iff,
            Enumeration.mem_smul]
          exact Or.inl hN
      · rw [← smul_fuzz hε hζ hεζ, ← ht]
        simp only [Path.botSderiv_coe_eq, BasePerm.smul_nearLitter_litter, allPermDeriv_forget,
          allPermForget_inv, Tree.inv_deriv, Tree.inv_sderiv, Tree.inv_sderivBot]
        rfl

theorem raise_closed' (S : Support α) (hS : S.Strong) (T : Support γ) (ρ : AllPerm β)
    (hγ : (γ : TypeIndex) < β)
    (hρ : ρᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim) :
    (S + (ρᵁ • ((T ↗ hγ).strong +
      (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim).Closed := by
  constructor
  intro A
  constructor
  intro N₁ N₂ hN₁ hN₂ a ha
  simp only [add_derivBot, BaseSupport.add_nearLitters, Enumeration.mem_add_iff,
    BaseSupport.add_atoms] at hN₁ hN₂ ⊢
  obtain hN₁ | hN₁ := hN₁
  · obtain hN₂ | hN₂ := hN₂
    · exact Or.inl ((hS.closed A).interference_subset hN₁ hN₂ a ha)
    · obtain ⟨B, rfl⟩ := eq_of_nearLitter_mem_scoderiv_botDeriv hN₂
      simp only [smul_add, scoderiv_botDeriv_eq, add_derivBot, smul_derivBot,
        BaseSupport.add_nearLitters, BaseSupport.smul_nearLitters, Enumeration.mem_add_iff,
        Enumeration.mem_smul, BaseSupport.add_atoms, BaseSupport.smul_atoms] at hN₁ hN₂ ⊢
      refine Or.inr (Or.inr ?_)
      rw [mem_interferenceSupport_atoms]
      refine ⟨(ρᵁ B)⁻¹ • N₁, ?_, (ρᵁ B)⁻¹ • N₂, ?_, ?_⟩
      · simp only [add_derivBot, BaseSupport.add_nearLitters, Enumeration.mem_add_iff]
        rw [← Enumeration.mem_smul, ← BaseSupport.smul_nearLitters, ← smul_derivBot, hρ]
        exact Or.inl hN₁
      · simp only [add_derivBot, BaseSupport.add_nearLitters, Enumeration.mem_add_iff]
        simp only [interferenceSupport_nearLitters, Enumeration.not_mem_empty, or_false] at hN₂
        exact Or.inr hN₂
      · rw [← BasePerm.smul_interference]
        exact Set.smul_mem_smul_set ha
  · obtain ⟨B, rfl⟩ := eq_of_nearLitter_mem_scoderiv_botDeriv hN₁
    simp only [smul_add, scoderiv_botDeriv_eq, add_derivBot, smul_derivBot,
      BaseSupport.add_nearLitters, BaseSupport.smul_nearLitters, Enumeration.mem_add_iff,
      Enumeration.mem_smul, BaseSupport.add_atoms, BaseSupport.smul_atoms] at hN₁ hN₂ ⊢
    refine Or.inr (Or.inr ?_)
    rw [mem_interferenceSupport_atoms]
    refine ⟨(ρᵁ B)⁻¹ • N₁, ?_, (ρᵁ B)⁻¹ • N₂, ?_, ?_⟩
    · simp only [add_derivBot, BaseSupport.add_nearLitters, Enumeration.mem_add_iff]
      simp only [interferenceSupport_nearLitters, Enumeration.not_mem_empty, or_false] at hN₁
      exact Or.inr hN₁
    · simp only [add_derivBot, BaseSupport.add_nearLitters, Enumeration.mem_add_iff]
      simp only [interferenceSupport_nearLitters, Enumeration.not_mem_empty, or_false] at hN₂
      obtain hN₂ | hN₂ := hN₂
      · rw [← Enumeration.mem_smul, ← BaseSupport.smul_nearLitters, ← smul_derivBot, hρ]
        exact Or.inl hN₂
      · exact Or.inr hN₂
    · rw [← BasePerm.smul_interference]
      exact Set.smul_mem_smul_set ha

theorem raise_strong' (S : Support α) (hS : S.Strong) (T : Support γ) (ρ : AllPerm β)
    (hγ : (γ : TypeIndex) < β)
    (hρ : ρᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim) :
    (S + (ρᵁ • ((T ↗ hγ).strong +
      (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim).Strong :=
  ⟨raise_preStrong' S hS T ρ hγ, raise_closed' S hS T ρ hγ hρ⟩

theorem convAtoms_injective_of_fixes {S : Support α} {T : Support γ}
    {ρ₁ ρ₂ : AllPerm β} {hγ : (γ : TypeIndex) < β}
    (hρ₁ : ρ₁ᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim)
    (hρ₂ : ρ₂ᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim)
    (A : ↑α ↝ ⊥) :
    (convAtoms
      (S + (ρ₁ᵁ • ((T ↗ hγ).strong + (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗
        LtLevel.elim)
      (S + (ρ₂ᵁ • ((T ↗ hγ).strong + (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗
        LtLevel.elim) A).Injective := by
  rw [Support.smul_eq_iff] at hρ₁ hρ₂
  constructor
  rintro a₁ a₂ a₃ ⟨i, hi₁, hi₂⟩ ⟨j, hj₁, hj₂⟩
  simp only [add_derivBot, BaseSupport.add_atoms, Rel.inv_apply,
    Enumeration.rel_add_iff] at hi₁ hi₂ hj₁ hj₂
  obtain hi₁ | ⟨i, rfl, hi₁⟩ := hi₁
  · obtain hi₂ | ⟨i', rfl, _⟩ := hi₂
    swap
    · have := Enumeration.lt_bound _ _ ⟨_, hi₁⟩
      simp only [add_lt_iff_neg_left] at this
      cases (κ_zero_le i').not_lt this
    cases (Enumeration.rel_coinjective _).coinjective hi₁ hi₂
    obtain hj₁ | ⟨j, rfl, hj₁⟩ := hj₁
    · obtain hj₂ | ⟨j', rfl, _⟩ := hj₂
      · exact (Enumeration.rel_coinjective _).coinjective hj₂ hj₁
      · have := Enumeration.lt_bound _ _ ⟨_, hj₁⟩
        simp only [add_lt_iff_neg_left] at this
        cases (κ_zero_le j').not_lt this
    · obtain hj₂ | hj₂ := hj₂
      · have := Enumeration.lt_bound _ _ ⟨_, hj₂⟩
        simp only [add_lt_iff_neg_left] at this
        cases (κ_zero_le j).not_lt this
      · simp only [add_right_inj, exists_eq_left] at hj₂
        obtain ⟨B, rfl⟩ := eq_of_atom_mem_scoderiv_botDeriv ⟨j, hj₁⟩
        simp only [scoderiv_botDeriv_eq, smul_derivBot, add_derivBot, BaseSupport.smul_atoms,
          BaseSupport.add_atoms, Enumeration.smul_rel, add_right_inj, exists_eq_left] at hj₁ hj₂
        have := (Enumeration.rel_coinjective _).coinjective hj₁ hj₂
        rw [← (hρ₂ B).1 a₁ ⟨_, hi₁⟩, inv_smul_smul, inv_smul_eq_iff, (hρ₁ B).1 a₁ ⟨_, hi₁⟩] at this
        exact this.symm
  · obtain ⟨B, rfl⟩ := eq_of_atom_mem_scoderiv_botDeriv ⟨i, hi₁⟩
    simp only [scoderiv_botDeriv_eq, smul_derivBot, add_derivBot, BaseSupport.smul_atoms,
      BaseSupport.add_atoms, Enumeration.smul_rel, add_right_inj, exists_eq_left] at hi₁ hi₂ hj₁ hj₂
    obtain hi₂ | hi₂ := hi₂
    · have := Enumeration.lt_bound _ _ ⟨_, hi₂⟩
      simp only [add_lt_iff_neg_left] at this
      cases (κ_zero_le i).not_lt this
    have hi := (Enumeration.rel_coinjective _).coinjective hi₁ hi₂
    suffices hj : (ρ₁ᵁ B)⁻¹ • a₂ = (ρ₂ᵁ B)⁻¹ • a₃ by
      rwa [← hj, smul_left_cancel_iff] at hi
    obtain hj₁ | ⟨j, rfl, hj₁⟩ := hj₁
    · obtain hj₂ | ⟨j', rfl, _⟩ := hj₂
      · rw [← (hρ₁ B).1 a₂ ⟨_, hj₁⟩, ← (hρ₂ B).1 a₃ ⟨_, hj₂⟩, inv_smul_smul, inv_smul_smul]
        exact (Enumeration.rel_coinjective _).coinjective hj₁ hj₂
      · have := Enumeration.lt_bound _ _ ⟨_, hj₁⟩
        simp only [add_lt_iff_neg_left] at this
        cases (κ_zero_le j').not_lt this
    · obtain hj₂ | hj₂ := hj₂
      · have := Enumeration.lt_bound _ _ ⟨_, hj₂⟩
        simp only [add_lt_iff_neg_left] at this
        cases (κ_zero_le j).not_lt this
      · simp only [add_right_inj, exists_eq_left] at hj₂
        exact (Enumeration.rel_coinjective _).coinjective hj₁ hj₂

theorem atomMemRel_le_of_fixes {S : Support α} {T : Support γ}
    {ρ₁ ρ₂ : AllPerm β} {hγ : (γ : TypeIndex) < β}
    (hρ₁ : ρ₁ᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim)
    (hρ₂ : ρ₂ᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim)
    (A : ↑α ↝ ⊥) :
    atomMemRel (S + (ρ₁ᵁ • ((T ↗ hγ).strong +
      (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim) A ≤
    atomMemRel (S + (ρ₂ᵁ • ((T ↗ hγ).strong +
      (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim) A := by
  rw [Support.smul_eq_iff] at hρ₁ hρ₂
  rintro i j ⟨N, hN, a, haN, ha⟩
  simp only [add_derivBot, BaseSupport.add_atoms, Rel.inv_apply, Enumeration.rel_add_iff,
    BaseSupport.add_nearLitters] at ha hN
  obtain hN | ⟨i, rfl, hi⟩ := hN
  · obtain ha | ⟨j, rfl, hj⟩ := ha
    · exact ⟨N, Or.inl hN, a, haN, Or.inl ha⟩
    · obtain ⟨B, rfl⟩ := eq_of_atom_mem_scoderiv_botDeriv ⟨j, hj⟩
      simp only [scoderiv_botDeriv_eq, smul_derivBot, add_derivBot, BaseSupport.smul_atoms,
        BaseSupport.add_atoms, Enumeration.smul_rel] at hj hN
      refine ⟨N, Or.inl hN, ρ₂ᵁ B • (ρ₁ᵁ B)⁻¹ • a, ?_, ?_⟩
      · dsimp only
        rw [← (hρ₂ B).2 N ⟨_, hN⟩, BasePerm.smul_nearLitter_atoms, Set.smul_mem_smul_set_iff]
        have := (hρ₁ B).2 N ⟨_, hN⟩
        rw [smul_eq_iff_eq_inv_smul] at this
        rwa [this, BasePerm.smul_nearLitter_atoms, Set.smul_mem_smul_set_iff]
      · rw [Rel.inv_apply, add_derivBot, BaseSupport.add_atoms, Enumeration.rel_add_iff]
        simp only [add_right_inj, scoderiv_botDeriv_eq, smul_derivBot, add_derivBot,
          BaseSupport.smul_atoms, BaseSupport.add_atoms, Enumeration.smul_rel, inv_smul_smul,
          exists_eq_left]
        exact Or.inr hj
  · obtain ⟨B, rfl⟩ := eq_of_nearLitter_mem_scoderiv_botDeriv ⟨i, hi⟩
    simp only [scoderiv_botDeriv_eq, smul_derivBot, add_derivBot, BaseSupport.smul_atoms,
      BaseSupport.add_atoms, Enumeration.smul_rel] at hi ha
    obtain ha | ⟨j, rfl, hj⟩ := ha
    · refine ⟨ρ₂ᵁ B • (ρ₁ᵁ B)⁻¹ • N, ?_, a, ?_, Or.inl ha⟩
      · rw [add_derivBot, BaseSupport.add_nearLitters, Enumeration.rel_add_iff]
        simp only [add_right_inj, scoderiv_botDeriv_eq, smul_derivBot, add_derivBot,
          BaseSupport.smul_nearLitters, BaseSupport.add_nearLitters, Enumeration.smul_rel,
          inv_smul_smul, exists_eq_left]
        exact Or.inr hi
      · dsimp only
        rw [← (hρ₂ B).1 a ⟨_, ha⟩, BasePerm.smul_nearLitter_atoms, Set.smul_mem_smul_set_iff]
        have := (hρ₁ B).1 a ⟨_, ha⟩
        rw [smul_eq_iff_eq_inv_smul] at this
        rwa [this, BasePerm.smul_nearLitter_atoms, Set.smul_mem_smul_set_iff]
    · refine ⟨ρ₂ᵁ B • (ρ₁ᵁ B)⁻¹ • N, ?_, ρ₂ᵁ B • (ρ₁ᵁ B)⁻¹ • a, ?_, ?_⟩
      · rw [add_derivBot, BaseSupport.add_nearLitters, Enumeration.rel_add_iff]
        simp only [add_right_inj, scoderiv_botDeriv_eq, smul_derivBot, add_derivBot,
          BaseSupport.smul_nearLitters, BaseSupport.add_nearLitters, Enumeration.smul_rel,
          inv_smul_smul, exists_eq_left]
        exact Or.inr hi
      · simp only [BasePerm.smul_nearLitter_atoms, Set.smul_mem_smul_set_iff]
        exact haN
      · rw [Rel.inv_apply, add_derivBot, BaseSupport.add_atoms, Enumeration.rel_add_iff]
        simp only [add_right_inj, scoderiv_botDeriv_eq, smul_derivBot, add_derivBot,
          BaseSupport.smul_atoms, BaseSupport.add_atoms, Enumeration.smul_rel, inv_smul_smul,
          exists_eq_left]
        exact Or.inr hj

theorem convNearLitters_cases {S : Support α} {T : Support γ}
    {ρ₁ ρ₂ : AllPerm β} {hγ : (γ : TypeIndex) < β}
    {A : α ↝ ⊥} {N₁ N₂ : NearLitter} :
  convNearLitters
    (S + (ρ₁ᵁ • ((T ↗ hγ).strong +
      (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim)
    (S + (ρ₂ᵁ • ((T ↗ hγ).strong +
      (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim) A N₁ N₂ →
    N₁ = N₂ ∧ N₁ ∈ (S ⇘. A)ᴺ ∨
    ∃ B : β ↝ ⊥, A = B ↗ LtLevel.elim ∧ (ρ₁ᵁ B)⁻¹ • N₁ = (ρ₂ᵁ B)⁻¹ • N₂ ∧
      (ρ₁ᵁ B)⁻¹ • N₁ ∈ (((T ↗ hγ).strong +
        (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport) ⇘. B)ᴺ := by
  rintro ⟨i, hN₁, hN₂⟩
  simp only [add_derivBot, BaseSupport.add_nearLitters, Rel.inv_apply,
    Enumeration.rel_add_iff] at hN₁ hN₂
  obtain hN₁ | ⟨i, rfl, hN₁⟩ := hN₁
  · obtain hN₂ | ⟨i, rfl, hN₂⟩ := hN₂
    swap
    · have := Enumeration.lt_bound _ _ ⟨_, hN₁⟩
      simp only [add_lt_iff_neg_left] at this
      cases (κ_zero_le i).not_lt this
    exact Or.inl ⟨(Enumeration.rel_coinjective _).coinjective hN₁ hN₂, _, hN₁⟩
  · obtain ⟨B, rfl⟩ := eq_of_nearLitter_mem_scoderiv_botDeriv ⟨i, hN₁⟩
    simp only [scoderiv_botDeriv_eq, smul_derivBot, add_derivBot, BaseSupport.smul_nearLitters,
      BaseSupport.add_nearLitters, Enumeration.smul_rel, add_right_inj, exists_eq_left] at hN₁ hN₂
    obtain hN₂ | hN₂ := hN₂
    · have := Enumeration.lt_bound _ _ ⟨_, hN₂⟩
      simp only [add_lt_iff_neg_left] at this
      cases (κ_zero_le i).not_lt this
    exact Or.inr ⟨B, rfl, (Enumeration.rel_coinjective _).coinjective hN₁ hN₂, _, hN₁⟩

theorem inflexible_of_inflexible_of_fixes {S : Support α} {T : Support γ}
    {ρ₁ ρ₂ : AllPerm β} {hγ : (γ : TypeIndex) < β}
    (hρ₁ : ρ₁ᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim)
    (hρ₂ : ρ₂ᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim)
    {A : α ↝ ⊥} {N₁ N₂ : NearLitter} :
    convNearLitters
      (S + (ρ₁ᵁ • ((T ↗ hγ).strong +
        (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim)
      (S + (ρ₂ᵁ • ((T ↗ hγ).strong +
        (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim) A N₁ N₂ →
      ∀ (P : InflexiblePath ↑α) (t : Tangle P.δ), A = P.A ↘ P.hε ↘. → N₁ᴸ = fuzz P.hδε t →
      ∃ ρ : AllPerm P.δ, N₂ᴸ = fuzz P.hδε (ρ • t) := by
  rintro hN ⟨γ, δ, ε, hδ, hε, hδε, A⟩ t hA ht
  haveI : LeLevel γ := ⟨A.le⟩
  haveI : LtLevel δ := ⟨hδ.trans_le LeLevel.elim⟩
  haveI : LtLevel ε := ⟨hε.trans_le LeLevel.elim⟩
  obtain ⟨rfl, _⟩ | ⟨B, rfl, hN'⟩ := convNearLitters_cases hN
  · use 1
    rw [one_smul, ht]
  · clear hN
    cases B
    case sderiv ε B hε' _ =>
      rw [← Path.coderiv_deriv] at hA
      cases Path.sderiv_index_injective hA
      apply Path.sderiv_path_injective at hA
      cases B
      case nil =>
        simp only [Path.botSderiv_coe_eq, add_derivBot, BaseSupport.add_nearLitters,
          interferenceSupport_nearLitters, Enumeration.add_empty] at hN'
        cases not_mem_strong_botDeriv _ _ hN'.2
      case sderiv ζ B hζ _ _ =>
        rw [← Path.coderiv_deriv] at hA
        cases Path.sderiv_index_injective hA
        apply Path.sderiv_path_injective at hA
        dsimp only at hA hζ hε' B t
        cases hA
        use (ρ₂ * ρ₁⁻¹) ⇘ B ↘ hδ
        rw [inv_smul_eq_iff] at hN'
        rw [← smul_fuzz hδ hε hδε, ← ht, hN'.1]
        simp only [allPermDeriv_forget, allPermForget_mul, allPermForget_inv, Tree.mul_deriv,
          Tree.inv_deriv, Tree.mul_sderiv, Tree.inv_sderiv, Tree.mul_sderivBot, Tree.inv_sderivBot,
          Path.botSderiv_coe_eq, BasePerm.smul_nearLitter_litter, mul_smul]
        erw [inv_smul_smul, smul_inv_smul]

theorem atoms_of_inflexible_of_fixes {S : Support α} (hS : S.Strong) {T : Support γ}
    {ρ₁ ρ₂ : AllPerm β} {hγ : (γ : TypeIndex) < β}
    (hρ₁ : ρ₁ᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim)
    (hρ₂ : ρ₂ᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim)
    (A : α ↝ ⊥) (N₁ N₂ : NearLitter) (P : InflexiblePath ↑α) (t : Tangle P.δ) (ρ : AllPerm P.δ) :
    A = P.A ↘ P.hε ↘. → N₁ᴸ = fuzz P.hδε t → N₂ᴸ = fuzz P.hδε (ρ • t) →
    convNearLitters
      (S + (ρ₁ᵁ • ((T ↗ hγ).strong +
        (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim)
      (S + (ρ₂ᵁ • ((T ↗ hγ).strong +
        (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim) A N₁ N₂ →
    ∀ (B : P.δ ↝ ⊥), ∀ a ∈ (t.support ⇘. B)ᴬ, ∀ (i : κ),
      ((S + (ρ₁ᵁ • ((T ↗ hγ).strong + (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗
        LtLevel.elim) ⇘. (P.A ↘ P.hδ ⇘ B))ᴬ.rel i a →
      ((S + (ρ₂ᵁ • ((T ↗ hγ).strong + (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗
        LtLevel.elim) ⇘. (P.A ↘ P.hδ ⇘ B))ᴬ.rel i (ρᵁ B • a) := by
  rw [Support.smul_eq_iff] at hρ₁ hρ₂
  obtain ⟨γ, δ, ε, hδ, hε, hδε, B⟩ := P
  haveI : LeLevel γ := ⟨B.le⟩
  haveI : LtLevel δ := ⟨hδ.trans_le LeLevel.elim⟩
  haveI : LtLevel ε := ⟨hε.trans_le LeLevel.elim⟩
  dsimp only at t ρ ⊢
  intro hA hN₁ hN₂ hN C a ha i hi
  obtain ⟨rfl, hN'⟩ | ⟨A, rfl, hN₁', hN₂'⟩ := convNearLitters_cases hN
  · have haS := (hS.support_le hN' ⟨γ, δ, ε, hδ, hε, hδε, _⟩ t hA hN₁ _).1 a ha
    rw [hN₂] at hN₁
    have hρt := congr_arg Tangle.support (fuzz_injective hN₁)
    rw [Tangle.smul_support, Support.smul_eq_iff] at hρt
    simp only [add_derivBot, BaseSupport.add_atoms, Enumeration.rel_add_iff] at hi ⊢
    rw [(hρt C).1 a ha]
    obtain hi | ⟨i, rfl, hi⟩ := hi
    · exact Or.inl hi
    · simp only [add_right_inj, exists_eq_left]
      obtain ⟨D, hD⟩ := eq_of_atom_mem_scoderiv_botDeriv ⟨i, hi⟩
      cases B using Path.recScoderiv
      case nil =>
        cases Path.scoderiv_index_injective hD
        cases Path.scoderiv_left_inj.mp hD
        simp only [hD, Path.coderiv_deriv, Path.coderiv_deriv', scoderiv_botDeriv_eq, smul_derivBot,
          add_derivBot, BaseSupport.smul_atoms, BaseSupport.add_atoms, Enumeration.smul_rel] at hi ⊢
        rw [deriv_derivBot, hD] at haS
        rw [← (hρ₂ _).1 a haS, inv_smul_smul]
        rw [← (hρ₁ _).1 a haS, inv_smul_smul] at hi
        exact Or.inr hi
      case scoderiv ζ B hζ' _ =>
        rw [Path.coderiv_deriv, Path.coderiv_deriv'] at hD
        cases Path.scoderiv_index_injective hD
        rw [Path.scoderiv_left_inj] at hD
        cases hD
        simp only [Path.coderiv_deriv, Path.coderiv_deriv', scoderiv_botDeriv_eq, smul_derivBot,
          add_derivBot, BaseSupport.smul_atoms, BaseSupport.add_atoms, Enumeration.smul_rel] at hi ⊢
        rw [deriv_derivBot, Path.coderiv_deriv, Path.coderiv_deriv'] at haS
        rw [← (hρ₂ _).1 a haS, inv_smul_smul]
        rw [← (hρ₁ _).1 a haS, inv_smul_smul] at hi
        exact Or.inr hi
  · simp only [add_derivBot, BaseSupport.add_nearLitters, interferenceSupport_nearLitters,
      Enumeration.add_empty] at hN₂'
    cases A
    case sderiv ζ A hζ' _ =>
      rw [← Path.coderiv_deriv] at hA
      cases Path.sderiv_index_injective hA
      apply Path.sderiv_path_injective at hA
      cases A
      case nil =>
        cases hA
        cases not_mem_strong_botDeriv _ _ hN₂'
      case sderiv ζ A hζ _ _ =>
        rw [← Path.coderiv_deriv] at hA
        cases Path.sderiv_index_injective hA
        apply Path.sderiv_path_injective at hA
        cases hA
        simp only [Path.coderiv_deriv, Path.coderiv_deriv', add_derivBot, scoderiv_botDeriv_eq,
          smul_derivBot, BaseSupport.add_atoms, BaseSupport.smul_atoms] at hi ⊢
        have : N₂ᴸ = (ρ₂ ⇘ A)ᵁ ↘ hζ ↘. • (ρ₁⁻¹ ⇘ A)ᵁ ↘ hζ ↘. • fuzz hδε t := by
          rw [inv_smul_eq_iff] at hN₁'
          rw [hN₁', Path.botSderiv_coe_eq, BasePerm.smul_nearLitter_litter,
            BasePerm.smul_nearLitter_litter, smul_smul, smul_eq_iff_eq_inv_smul,
            mul_inv_rev, inv_inv, mul_smul, ← Tree.inv_apply, ← allPermForget_inv] at hN₁
          rw [hN₁]
          simp only [allPermForget_inv, Tree.inv_apply, allPermDeriv_forget, Tree.inv_deriv,
            Tree.inv_sderiv, Tree.inv_sderivBot]
          rfl
        rw [smul_fuzz hδ hε hδε, smul_fuzz hδ hε hδε] at this
        have := fuzz_injective (hN₂.symm.trans this)
        rw [smul_smul] at this
        rw [t.smul_atom_eq_of_mem_support this ha]
        rw [Enumeration.rel_add_iff] at hi ⊢
        obtain hi | ⟨i, rfl, hi⟩ := hi
        · left
          simp only [allPermForget_mul, allPermSderiv_forget, allPermDeriv_forget,
            allPermForget_inv, Tree.inv_deriv, Tree.inv_sderiv, Tree.mul_apply, Tree.sderiv_apply,
            Tree.deriv_apply, Path.deriv_scoderiv, Tree.inv_apply, mul_smul]
          rwa [← (hρ₁ _).1 a ⟨i, hi⟩, inv_smul_smul, (hρ₂ _).1 a ⟨i, hi⟩]
        · refine Or.inr ⟨i, rfl, ?_⟩
          simp only [allPermForget_mul, allPermSderiv_forget, allPermDeriv_forget,
            allPermForget_inv, Tree.inv_deriv, Tree.inv_sderiv, Tree.mul_apply, Tree.sderiv_apply,
            Tree.deriv_apply, Path.deriv_scoderiv, Tree.inv_apply, mul_smul, Enumeration.smul_rel,
            inv_smul_smul]
          exact hi

theorem nearLitters_of_inflexible_of_fixes {S : Support α} (hS : S.Strong) {T : Support γ}
    {ρ₁ ρ₂ : AllPerm β} {hγ : (γ : TypeIndex) < β}
    (hρ₁ : ρ₁ᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim)
    (hρ₂ : ρ₂ᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim)
    (A : α ↝ ⊥) (N₁ N₂ : NearLitter) (P : InflexiblePath ↑α) (t : Tangle P.δ) (ρ : AllPerm P.δ) :
    A = P.A ↘ P.hε ↘. → N₁ᴸ = fuzz P.hδε t → N₂ᴸ = fuzz P.hδε (ρ • t) →
    convNearLitters
      (S + (ρ₁ᵁ • ((T ↗ hγ).strong +
        (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim)
      (S + (ρ₂ᵁ • ((T ↗ hγ).strong +
        (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim) A N₁ N₂ →
    ∀ (B : P.δ ↝ ⊥), ∀ N ∈ (t.support ⇘. B)ᴺ, ∀ (i : κ),
      ((S + (ρ₁ᵁ • ((T ↗ hγ).strong + (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗
        LtLevel.elim) ⇘. (P.A ↘ P.hδ ⇘ B))ᴺ.rel i N →
      ((S + (ρ₂ᵁ • ((T ↗ hγ).strong + (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗
        LtLevel.elim) ⇘. (P.A ↘ P.hδ ⇘ B))ᴺ.rel i (ρᵁ B • N) := by
  rw [Support.smul_eq_iff] at hρ₁ hρ₂
  obtain ⟨γ, δ, ε, hδ, hε, hδε, B⟩ := P
  haveI : LeLevel γ := ⟨B.le⟩
  haveI : LtLevel δ := ⟨hδ.trans_le LeLevel.elim⟩
  haveI : LtLevel ε := ⟨hε.trans_le LeLevel.elim⟩
  dsimp only at t ρ ⊢
  intro hA hN₁ hN₂ hN C N₀ hN₀ i hi
  obtain ⟨rfl, hN'⟩ | ⟨A, rfl, hN₁', hN₂'⟩ := convNearLitters_cases hN
  · have haS := (hS.support_le hN' ⟨γ, δ, ε, hδ, hε, hδε, _⟩ t hA hN₁ _).2 N₀ hN₀
    rw [hN₂] at hN₁
    have hρt := congr_arg Tangle.support (fuzz_injective hN₁)
    rw [Tangle.smul_support, Support.smul_eq_iff] at hρt
    simp only [add_derivBot, BaseSupport.add_nearLitters, Enumeration.rel_add_iff] at hi ⊢
    rw [(hρt C).2 N₀ hN₀]
    obtain hi | ⟨i, rfl, hi⟩ := hi
    · exact Or.inl hi
    · simp only [add_right_inj, exists_eq_left]
      obtain ⟨D, hD⟩ := eq_of_nearLitter_mem_scoderiv_botDeriv ⟨i, hi⟩
      cases B using Path.recScoderiv
      case nil =>
        cases Path.scoderiv_index_injective hD
        cases Path.scoderiv_left_inj.mp hD
        simp only [hD, Path.coderiv_deriv, Path.coderiv_deriv', scoderiv_botDeriv_eq, smul_derivBot,
          add_derivBot, BaseSupport.smul_nearLitters, BaseSupport.add_nearLitters, Enumeration.smul_rel] at hi ⊢
        rw [deriv_derivBot, hD] at haS
        rw [← (hρ₂ _).2 N₀ haS, inv_smul_smul]
        rw [← (hρ₁ _).2 N₀ haS, inv_smul_smul] at hi
        exact Or.inr hi
      case scoderiv ζ B hζ' _ =>
        rw [Path.coderiv_deriv, Path.coderiv_deriv'] at hD
        cases Path.scoderiv_index_injective hD
        rw [Path.scoderiv_left_inj] at hD
        cases hD
        simp only [Path.coderiv_deriv, Path.coderiv_deriv', scoderiv_botDeriv_eq, smul_derivBot,
          add_derivBot, BaseSupport.smul_nearLitters, BaseSupport.add_nearLitters, Enumeration.smul_rel] at hi ⊢
        rw [deriv_derivBot, Path.coderiv_deriv, Path.coderiv_deriv'] at haS
        rw [← (hρ₂ _).2 N₀ haS, inv_smul_smul]
        rw [← (hρ₁ _).2 N₀ haS, inv_smul_smul] at hi
        exact Or.inr hi
  · simp only [add_derivBot, BaseSupport.add_nearLitters, interferenceSupport_nearLitters,
      Enumeration.add_empty] at hN₂'
    cases A
    case sderiv ζ A hζ' _ =>
      rw [← Path.coderiv_deriv] at hA
      cases Path.sderiv_index_injective hA
      apply Path.sderiv_path_injective at hA
      cases A
      case nil =>
        cases hA
        cases not_mem_strong_botDeriv _ _ hN₂'
      case sderiv ζ A hζ _ _ =>
        rw [← Path.coderiv_deriv] at hA
        cases Path.sderiv_index_injective hA
        apply Path.sderiv_path_injective at hA
        cases hA
        simp only [Path.coderiv_deriv, Path.coderiv_deriv', add_derivBot, scoderiv_botDeriv_eq,
          smul_derivBot, BaseSupport.add_nearLitters, BaseSupport.smul_nearLitters] at hi ⊢
        have : N₂ᴸ = (ρ₂ ⇘ A)ᵁ ↘ hζ ↘. • (ρ₁⁻¹ ⇘ A)ᵁ ↘ hζ ↘. • fuzz hδε t := by
          rw [inv_smul_eq_iff] at hN₁'
          rw [hN₁', Path.botSderiv_coe_eq, BasePerm.smul_nearLitter_litter,
            BasePerm.smul_nearLitter_litter, smul_smul, smul_eq_iff_eq_inv_smul,
            mul_inv_rev, inv_inv, mul_smul, ← Tree.inv_apply, ← allPermForget_inv] at hN₁
          rw [hN₁]
          simp only [allPermForget_inv, Tree.inv_apply, allPermDeriv_forget, Tree.inv_deriv,
            Tree.inv_sderiv, Tree.inv_sderivBot]
          rfl
        rw [smul_fuzz hδ hε hδε, smul_fuzz hδ hε hδε] at this
        have := fuzz_injective (hN₂.symm.trans this)
        rw [smul_smul] at this
        rw [t.smul_nearLitter_eq_of_mem_support this hN₀]
        rw [Enumeration.rel_add_iff] at hi ⊢
        obtain hi | ⟨i, rfl, hi⟩ := hi
        · left
          simp only [allPermForget_mul, allPermSderiv_forget, allPermDeriv_forget,
            allPermForget_inv, Tree.inv_deriv, Tree.inv_sderiv, Tree.mul_apply, Tree.sderiv_apply,
            Tree.deriv_apply, Path.deriv_scoderiv, Tree.inv_apply, mul_smul]
          rwa [← (hρ₁ _).2 N₀ ⟨i, hi⟩, inv_smul_smul, (hρ₂ _).2 N₀ ⟨i, hi⟩]
        · refine Or.inr ⟨i, rfl, ?_⟩
          simp only [allPermForget_mul, allPermSderiv_forget, allPermDeriv_forget,
            allPermForget_inv, Tree.inv_deriv, Tree.inv_sderiv, Tree.mul_apply, Tree.sderiv_apply,
            Tree.deriv_apply, Path.deriv_scoderiv, Tree.inv_apply, mul_smul, Enumeration.smul_rel,
            inv_smul_smul]
          exact hi

theorem litter_eq_of_flexible_of_fixes {S : Support α} {T : Support γ}
    {ρ₁ ρ₂ : AllPerm β} {hγ : (γ : TypeIndex) < β}
    (hρ₁ : ρ₁ᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim)
    (hρ₂ : ρ₂ᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim)
    {A : ↑α ↝ ⊥} {N₁ N₂ N₃ N₄ : NearLitter} :
    convNearLitters
      (S + (ρ₁ᵁ • ((T ↗ hγ).strong +
        (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim)
      (S + (ρ₂ᵁ • ((T ↗ hγ).strong +
        (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim) A N₁ N₂ →
    convNearLitters
      (S + (ρ₁ᵁ • ((T ↗ hγ).strong +
        (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim)
      (S + (ρ₂ᵁ • ((T ↗ hγ).strong +
        (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim) A N₃ N₄ →
      ¬Inflexible A N₁ᴸ → ¬Inflexible A N₂ᴸ → ¬Inflexible A N₃ᴸ → ¬Inflexible A N₄ᴸ →
      N₁ᴸ = N₃ᴸ → N₂ᴸ = N₄ᴸ := by
  rw [Support.smul_eq_iff] at hρ₁ hρ₂
  rintro ⟨i, hi₁, hi₂⟩ ⟨j, hj₁, hj₂⟩ hN₁ hN₂ hN₃ hN₄ hN₁₃
  simp only [add_derivBot, BaseSupport.add_nearLitters, Rel.inv_apply,
    Enumeration.rel_add_iff] at hi₁ hi₂ hj₁ hj₂
  obtain hi₁ | ⟨i, rfl, hi₁⟩ := hi₁
  · obtain hi₂ | ⟨i, rfl, hi₂⟩ := hi₂
    swap
    · have := Enumeration.lt_bound _ _ ⟨_, hi₁⟩
      simp only [add_lt_iff_neg_left] at this
      cases (κ_zero_le i).not_lt this
    cases (Enumeration.rel_coinjective _).coinjective hi₁ hi₂
    obtain hj₁ | ⟨j, rfl, hj₁⟩ := hj₁
    · obtain hj₂ | ⟨j, rfl, hj₂⟩ := hj₂
      swap
      · have := Enumeration.lt_bound _ _ ⟨_, hj₁⟩
        simp only [add_lt_iff_neg_left] at this
        cases (κ_zero_le j).not_lt this
      cases (Enumeration.rel_coinjective _).coinjective hj₁ hj₂
      exact hN₁₃
    · simp only [add_right_inj, exists_eq_left] at hj₂
      obtain hj₂ | hj₂ := hj₂
      · have := Enumeration.lt_bound _ _ ⟨_, hj₂⟩
        simp only [add_lt_iff_neg_left] at this
        cases (κ_zero_le j).not_lt this
      obtain ⟨A, rfl⟩ := eq_of_nearLitter_mem_scoderiv_botDeriv ⟨j, hj₁⟩
      simp only [scoderiv_botDeriv_eq, smul_derivBot, add_derivBot, BaseSupport.smul_nearLitters,
        BaseSupport.add_nearLitters, Enumeration.smul_rel] at hj₁ hj₂
      have := congr_arg (·ᴸ) ((Enumeration.rel_coinjective _).coinjective hj₁ hj₂)
      simp only [BasePerm.smul_nearLitter_litter] at this
      rw [← hN₁₃, ← (hρ₁ A).2 N₁ ⟨i, hi₁⟩, BasePerm.smul_nearLitter_litter, inv_smul_smul] at this
      have hN₁' := (hρ₂ A).2 N₁ ⟨i, hi₁⟩
      rw [smul_eq_iff_eq_inv_smul] at hN₁'
      rwa [hN₁', BasePerm.smul_nearLitter_litter, smul_left_cancel_iff] at this
  · obtain hi₂ | hi₂ := hi₂
    · have := Enumeration.lt_bound _ _ ⟨_, hi₂⟩
      simp only [add_lt_iff_neg_left] at this
      cases (κ_zero_le i).not_lt this
    simp only [add_right_inj, exists_eq_left] at hi₂
    obtain ⟨A, rfl⟩ := eq_of_nearLitter_mem_scoderiv_botDeriv ⟨i, hi₁⟩
    simp only [scoderiv_botDeriv_eq, smul_derivBot, add_derivBot, BaseSupport.smul_nearLitters,
      BaseSupport.add_nearLitters, Enumeration.smul_rel] at hi₁ hi₂ hj₁ hj₂
    have hN₁₂ := congr_arg (·ᴸ) ((Enumeration.rel_coinjective _).coinjective hi₁ hi₂)
    obtain hj₁ | ⟨j, rfl, hj₁⟩ := hj₁
    · obtain hj₂ | ⟨j, rfl, hj₂⟩ := hj₂
      swap
      · have := Enumeration.lt_bound _ _ ⟨_, hj₁⟩
        simp only [add_lt_iff_neg_left] at this
        cases (κ_zero_le j).not_lt this
      cases (Enumeration.rel_coinjective _).coinjective hj₁ hj₂
      simp only [BasePerm.smul_nearLitter_litter] at hN₁₂
      rw [hN₁₃, ← (hρ₁ A).2 N₃ ⟨j, hj₁⟩, BasePerm.smul_nearLitter_litter, inv_smul_smul,
        eq_inv_smul_iff, ← BasePerm.smul_nearLitter_litter, (hρ₂ A).2 N₃ ⟨j, hj₁⟩] at hN₁₂
      rw [hN₁₂]
    · simp only [add_right_inj, exists_eq_left] at hj₂
      obtain hj₂ | hj₂ := hj₂
      · have := Enumeration.lt_bound _ _ ⟨_, hj₂⟩
        simp only [add_lt_iff_neg_left] at this
        cases (κ_zero_le j).not_lt this
      have hN₃₄ := congr_arg (·ᴸ) ((Enumeration.rel_coinjective _).coinjective hj₁ hj₂)
      simp only [BasePerm.smul_nearLitter_litter] at hN₁₂ hN₃₄
      rw [hN₁₃] at hN₁₂
      rwa [hN₁₂, smul_left_cancel_iff] at hN₃₄

theorem sameSpecLe_of_fixes (S : Support α) (hS : S.Strong) (T : Support γ) (ρ₁ ρ₂ : AllPerm β)
    (hγ : (γ : TypeIndex) < β)
    (hρ₁ : ρ₁ᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim)
    (hρ₂ : ρ₂ᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim) :
    SameSpecLE
    (S + (ρ₁ᵁ • ((T ↗ hγ).strong +
      (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim)
    (S + (ρ₂ᵁ • ((T ↗ hγ).strong +
      (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim)

:= by
  constructor
  case atoms_bound_eq => intro; rfl
  case nearLitters_bound_eq => intro; rfl
  case atoms_dom_subset =>
    simp only [add_derivBot, BaseSupport.add_atoms, Enumeration.add_rel_dom,
      Set.union_subset_iff, Set.subset_union_left, true_and]
    rintro A _ ⟨i, ⟨a, ⟨A, a⟩, h₁, h₂⟩, rfl⟩
    cases h₂
    right
    apply Set.mem_image_of_mem
    refine ⟨ρ₂ᵁ A • (ρ₁ᵁ A)⁻¹ • a, ⟨A, ρ₂ᵁ A • (ρ₁ᵁ A)⁻¹ • a⟩, ?_, rfl⟩
    rw [smul_atoms, Enumeration.smulPath_rel] at h₁ ⊢
    simp only [inv_smul_smul]
    exact h₁
  case nearLitters_dom_subset =>
    simp only [add_derivBot, BaseSupport.add_nearLitters, Enumeration.add_rel_dom,
      Set.union_subset_iff, Set.subset_union_left, true_and]
    rintro A _ ⟨i, ⟨N, ⟨A, N⟩, h₁, h₂⟩, rfl⟩
    cases h₂
    right
    apply Set.mem_image_of_mem
    refine ⟨ρ₂ᵁ A • (ρ₁ᵁ A)⁻¹ • N, ⟨A, ρ₂ᵁ A • (ρ₁ᵁ A)⁻¹ • N⟩, ?_, rfl⟩
    rw [smul_nearLitters, Enumeration.smulPath_rel] at h₁ ⊢
    simp only [inv_smul_smul]
    exact h₁
  case convAtoms_injective => exact convAtoms_injective_of_fixes hρ₁ hρ₂
  case atomMemRel_le => exact atomMemRel_le_of_fixes hρ₁ hρ₂
  case inflexible_of_inflexible => exact inflexible_of_inflexible_of_fixes hρ₁ hρ₂
  case atoms_of_inflexible => exact atoms_of_inflexible_of_fixes hS hρ₁ hρ₂
  case nearLitters_of_inflexible => exact nearLitters_of_inflexible_of_fixes hS hρ₁ hρ₂
  case litter_eq_of_flexible => exact litter_eq_of_flexible_of_fixes hρ₁ hρ₂


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


import ConNF.Model.Externalise

/-!
# New file

In this file...

## Main declarations

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

noncomputable section
universe u

open Cardinal Ordinal

namespace ConNF

variable [Params.{u}] {β γ : Λ} {hγ : (γ : TypeIndex) < β}

namespace Support

theorem not_mem_scoderiv_botDeriv (S : Support γ) (N : NearLitter) :
    N ∉ (S ↗ hγ ⇘. (Path.nil ↘.))ᴺ := by
  rintro ⟨i, ⟨A, N'⟩, h₁, h₂⟩
  simp only [Prod.mk.injEq] at h₂
  cases A
  case sderiv δ A hδ _ =>
    simp only [Path.deriv_sderiv] at h₂
    cases A
    case nil => cases h₂.1
    case sderiv ζ A hζ _ =>
      simp only [Path.deriv_sderiv] at h₂
      cases h₂.1

variable [Level] [LtLevel β]

theorem not_mem_strong_botDeriv (S : Support γ) (N : NearLitter) :
    N ∉ ((S ↗ hγ).strong ⇘. (Path.nil ↘.))ᴺ := by
  rintro h
  rw [strong, close_nearLitters, preStrong_nearLitters, Enumeration.mem_add_iff] at h
  obtain h | h := h
  · exact not_mem_scoderiv_botDeriv S N h
  · rw [mem_constrainsNearLitters_nearLitters] at h
    obtain ⟨B, N', hN', h⟩ := h
    cases h using Relation.ReflTransGen.head_induction_on
    case refl => exact not_mem_scoderiv_botDeriv S N hN'
    case head x hx₁ hx₂ _ =>
      obtain ⟨⟨γ, δ, ε, hδ, hε, hδε, A⟩, t, B, hB, hN, ht⟩ := hx₂
      simp only at hB
      cases B
      case nil =>
        cases hB
        obtain ⟨C, N''⟩ := x
        simp only at ht
        cases ht.1
        change _ ∈ t.supportᴺ at hN
        rw [t.support_supports.2 rfl] at hN
        obtain ⟨i, hN⟩ := hN
        cases hN
      case sderiv δ B hδ _ _ =>
        cases B
        case nil => cases hB
        case sderiv ζ B hζ _ _ => cases hB

theorem raise_preStrong' (S : Support α) (hS : S.Strong) (T : Support γ) (ρ : AllPerm β)
    (hγ : (γ : TypeIndex) < β) :
    (S + (ρᵁ • ((T ↗ hγ).strong +
      (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim).PreStrong

:= by
  apply hS.toPreStrong.add
  constructor
  intro A N hN P t hA ht
  obtain ⟨A, rfl⟩ := eq_of_nearLitter_mem_scoderiv_botDeriv hN
  simp only [scoderiv_botDeriv_eq, add_derivBot, smul_derivBot,
    BaseSupport.add_nearLitters, BaseSupport.smul_nearLitters, interferenceSupport_nearLitters,
    Enumeration.mem_add_iff, Enumeration.mem_smul, Enumeration.not_mem_empty, or_false] at hN
  obtain ⟨δ, ε, ζ, hε, hζ, hεζ, B⟩ := P
  dsimp only at *
  cases A
  case sderiv ζ' A hζ' _ =>
    rw [← Path.coderiv_deriv] at hA
    cases Path.sderiv_index_injective hA
    apply Path.sderiv_left_inj.mp at hA
    cases A
    case nil =>
      cases hA
      cases not_mem_strong_botDeriv T _ hN
    case sderiv ι A hι _ _ =>
      rw [← Path.coderiv_deriv] at hA
      cases Path.sderiv_index_injective hA
      cases hA
      haveI : LtLevel δ := ⟨A.le.trans_lt LtLevel.elim⟩
      haveI : LtLevel ε := ⟨hε.trans LtLevel.elim⟩
      haveI : LtLevel ζ := ⟨hζ.trans LtLevel.elim⟩
      have := (T ↗ hγ).strong_strong.support_le hN ⟨δ, ε, ζ, hε, hζ, hεζ, A⟩
          (ρ⁻¹ ⇘ A ↘ hε • t) rfl ?_
      · simp only [Tangle.smul_support, allPermSderiv_forget, allPermDeriv_forget,
          allPermForget_inv, Tree.inv_deriv, Tree.inv_sderiv] at this
        have := smul_le_smul this (ρᵁ ⇘ A ↘ hε)
        simp only [smul_inv_smul] at this
        apply le_trans this
        intro B
        constructor
        · intro a ha
          simp only [smul_derivBot, Tree.sderiv_apply, Tree.deriv_apply, Path.deriv_scoderiv,
            deriv_derivBot, Enumeration.mem_smul] at ha
          rw [deriv_derivBot, ← Path.deriv_scoderiv, Path.coderiv_deriv', scoderiv_botDeriv_eq,]
          simp only [Path.deriv_scoderiv, add_derivBot, smul_derivBot,
            BaseSupport.add_atoms, BaseSupport.smul_atoms, Enumeration.mem_add_iff,
            Enumeration.mem_smul]
          exact Or.inl ha
        · intro N hN
          simp only [smul_derivBot, Tree.sderiv_apply, Tree.deriv_apply, Path.deriv_scoderiv,
            deriv_derivBot, Enumeration.mem_smul] at hN
          rw [deriv_derivBot, ← Path.deriv_scoderiv, Path.coderiv_deriv', scoderiv_botDeriv_eq,]
          simp only [Path.deriv_scoderiv, add_derivBot, smul_derivBot,
            BaseSupport.add_nearLitters, BaseSupport.smul_nearLitters, Enumeration.mem_add_iff,
            Enumeration.mem_smul]
          exact Or.inl hN
      · rw [← smul_fuzz hε hζ hεζ, ← ht]
        simp only [Path.botSderiv_coe_eq, BasePerm.smul_nearLitter_litter, allPermDeriv_forget,
          allPermForget_inv, Tree.inv_deriv, Tree.inv_sderiv, Tree.inv_sderivBot]
        rfl


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]


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]


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


import ConNF.Model.RunInduction

/-!
# Externalisation

In this file, we convert many of our *internal* results (i.e. inside the induction) to *external*
ones (i.e. defined using the global `TSet`/`AllPerm` definitions).

## Main declarations

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

noncomputable section
universe u

open Cardinal Ordinal WithBot

namespace ConNF

variable [Params.{u}]

instance globalModelData : {α : TypeIndex} → ModelData α
  | (α : Λ) => (motive α).data
  | ⊥ => botModelData

instance globalPosition : {α : TypeIndex} → Position (Tangle α)
  | (α : Λ) => (motive α).pos
  | ⊥ => botPosition

instance globalTypedNearLitters {α : Λ} : TypedNearLitters α :=
  (motive α).typed

instance globalLtData [Level] : LtData where

instance globalLeData [Level] : LeData where

omit [Params] in
theorem heq_funext {α : Sort _} {β γ : α → Sort _} {f : (x : α) → β x} {g : (x : α) → γ x}
    (h : ∀ x, HEq (f x) (g x)) : HEq f g := by
  cases funext λ x ↦ type_eq_of_heq (h x)
  simp only [heq_eq_eq] at h ⊢
  exact funext h

theorem globalLtData_eq [Level] :
    globalLtData = ltData (λ β _ ↦ motive β) := by
  apply LtData.ext
  · ext β hβ
    induction β using recBotCoe
    case bot => rfl
    case coe β => rfl
  · apply heq_funext
    intro β
    induction β using recBotCoe
    case bot => rfl
    case coe β => rfl
  · rfl

theorem globalLeData_eq [Level] :
    globalLeData = leData (λ β _ ↦ motive β)

:= by
  apply LeData.ext
  · ext β hβ
    induction β using recBotCoe
    case bot => rfl
    case coe β =>
      by_cases h : (β : TypeIndex) = α
      · cases coe_injective h
        rw [leData_data_eq]
        unfold globalLeData globalModelData
        dsimp only
        rw [motive_eq]
        rfl
      · rw [leData_data_lt _ (hβ.elim.lt_of_ne h)]
        rfl
  · apply heq_funext
    intro β
    apply heq_funext
    intro hβ
    induction β using recBotCoe
    case bot => rfl
    case coe β =>
      rw [leData]
      simp only [coe_inj, id_eq, eq_mpr_eq_cast, recBotCoe_bot, recBotCoe_coe, LtLevel.elim.ne]
      exact HEq.symm (cast_heq _ _)


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']


import ConNF.Base.TypeIndex

/-!
# Paths of type indices

In this file, we define the notion of a *path*, and the derivative and coderivative operations.

## Main declarations

* `ConNF.Path`: A path of type indices.
* `ConNF.Path.recSderiv`, `ConNF.Path.recSderivLe`, `ConNF.Path.recSderivGlobal`:
    Downwards induction principles for paths.
* `ConNF.Path.recScoderiv`: An upwards induction principle for paths.
-/

universe u

open Cardinal WithBot

namespace ConNF

variable [Params.{u}] {α β γ δ : TypeIndex}

/-- A path of type indices starting at `α` and ending at `β` is a finite sequence of type indices
`α > ... > β`. -/
inductive Path (α : TypeIndex) : TypeIndex → Type u
  | nil : Path α α
  | cons {β γ : TypeIndex} : Path α β → γ < β → Path α γ

@[inherit_doc] infix:70 " ↝ "  => Path

def Path.single {α β : TypeIndex} (h : β < α) : α ↝ β :=
  .cons .nil h

/-- Typeclass for the `↘` notation. -/
class SingleDerivative (X : Type _) (Y : outParam <| Type _)
    (β : outParam TypeIndex) (γ : TypeIndex) where
  sderiv : X → γ < β → Y

/-- Typeclass for the `⇘` notation. -/
class Derivative (X : Type _) (Y : outParam <| Type _)
    (β : outParam TypeIndex) (γ : TypeIndex) extends SingleDerivative X Y β γ where
  deriv : X → β ↝ γ → Y
  sderiv x h := deriv x (.single h)
  deriv_single : ∀ x : X, ∀ h : γ < β, deriv x (.single h) = sderiv x h := by intros; rfl

/-- Typeclass for the `↘.` notation. -/
class BotSingleDerivative (X : Type _) (Y : outParam <| Type _) where
  botSderiv : X → Y

/-- Typeclass for the `⇘.` notation. -/
class BotDerivative (X : Type _) (Y : outParam <| Type _) (β : outParam TypeIndex)
    extends BotSingleDerivative X Y where
  botDeriv : X → β ↝ ⊥ → Y
  /-- We often need to do case analysis on `β` to show that it's a proper type index here.
  This case check doesn't need to be done in most actual use cases of the notation. -/
  botDeriv_single : ∀ x : X, ∀ h : ⊥ < β, botDeriv x (.single h) = botSderiv x

/-- Typeclass for the `↗` notation. -/
class SingleCoderivative (X : Type _) (Y : outParam <| Type _)
    (β : TypeIndex) (γ : outParam TypeIndex) where
  scoderiv : X → γ < β → Y

/-- Typeclass for the `⇗` notation. -/
class Coderivative (X : Type _) (Y : outParam <| Type _)
    (β : TypeIndex) (γ : outParam TypeIndex) extends SingleCoderivative X Y β γ where
  coderiv : X → β ↝ γ → Y
  scoderiv x h := coderiv x (.single h)
  coderiv_single : ∀ x : X, ∀ h : γ < β, coderiv x (.single h) = scoderiv x h := by intros; rfl

infixl:75 " ↘ " => SingleDerivative.sderiv
infixl:75 " ⇘ " => Derivative.deriv
postfix:75 " ↘." => BotSingleDerivative.botSderiv
infixl:75 " ⇘. " => BotDerivative.botDeriv
infixl:75 " ↗ " => SingleCoderivative.scoderiv
infixl:75 " ⇗ " => Coderivative.coderiv

@[simp]
theorem deriv_single {X Y : Type _} [Derivative X Y β γ] (x : X) (h : γ < β) :
    x ⇘ .single h = x ↘ h :=
  Derivative.deriv_single x h

@[simp]
theorem coderiv_single {X Y : Type _} [Coderivative X Y β γ] (x : X) (h : γ < β) :
    x ⇗ .single h = x ↗ h :=
  Coderivative.coderiv_single x h

@[simp]
theorem botDeriv_single {X Y : Type _} [BotDerivative X Y β] (x : X) (h : ⊥ < β) :
    x ⇘. .single h = x ↘. :=
  BotDerivative.botDeriv_single x h

/-!
## Downwards recursion along paths
-/

instance : SingleDerivative (α ↝ β) (α ↝ γ) β γ where
  sderiv := .cons

/-- The downwards recursion principle for paths. -/
@[elab_as_elim, induction_eliminator, cases_eliminator]
def Path.recSderiv {motive : ∀ β, α ↝ β → Sort _}
    (nil : motive α .nil)
    (sderiv : ∀ β γ (A : α ↝ β) (h : γ < β), motive β A → motive γ (A ↘ h)) :
    {β : TypeIndex} → (A : α ↝ β) → motive β A
  | _, .nil => nil
  | _, .cons A h => sderiv _ _ A h (recSderiv nil sderiv A)

@[simp]
theorem Path.recSderiv_nil {motive : ∀ β, α ↝ β → Sort _}
    (nil : motive α .nil)
    (sderiv : ∀ β γ (A : α ↝ β) (h : γ < β), motive β A → motive γ (A ↘ h)) :
    recSderiv (motive := motive) nil sderiv .nil = nil :=
  rfl

@[simp]
theorem Path.recSderiv_sderiv {motive : ∀ β, α ↝ β → Sort _}
    (nil : motive α .nil)
    (sderiv : ∀ β γ (A : α ↝ β) (h : γ < β), motive β A → motive γ (A ↘ h))
    {β γ : TypeIndex} (A : α ↝ β) (h : γ < β) :
    recSderiv (motive := motive) nil sderiv (A ↘ h) = sderiv β γ A h (recSderiv nil sderiv A) :=
  rfl

theorem Path.le (A : α ↝ β) : β ≤ α := by
  induction A with
  | nil => exact le_rfl
  | sderiv β γ _A h h' => exact h.le.trans h'

/-- The downwards recursion principle for paths, specialised to the case where the motive at `β`
only depends on the fact that `β ≤ α`. -/
def Path.recSderivLe {motive : ∀ β ≤ α, Sort _}
    (nil : motive α le_rfl)
    (sderiv : ∀ β γ, ∀ (A : α ↝ β) (h : γ < β), motive β A.le → motive γ (h.le.trans A.le)) :
    {β : TypeIndex} → (A : α ↝ β) → motive β A.le :=
  Path.recSderiv (motive := λ β A ↦ motive β A.le) nil sderiv

@[simp]
theorem Path.recSderivLe_nil {motive : ∀ β ≤ α, Sort _}
    (nil : motive α le_rfl)
    (sderiv : ∀ β γ (A : α ↝ β) (h : γ < β), motive β A.le → motive γ (h.le.trans A.le)) :
    recSderivLe (motive := motive) nil sderiv .nil = nil :=
  rfl

@[simp]
theorem Path.recSderivLe_sderiv {motive : ∀ β ≤ α, Sort _}
    (nil : motive α le_rfl)
    (sderiv : ∀ β γ (A : α ↝ β) (h : γ < β), motive β A.le → motive γ (h.le.trans A.le))
    {β γ : TypeIndex} (A : α ↝ β) (h : γ < β) :
    recSderivLe (motive := motive) nil sderiv (A ↘ h) = sderiv β γ A h (recSderiv nil sderiv A) :=
  rfl

/-- The downwards recursion principle for paths, specialised to the case where the motive is not
dependent on the relation of `β` to `α`. -/
@[elab_as_elim]
def Path.recSderivGlobal {motive : TypeIndex → Sort _}
    (nil : motive α)
    (sderiv : ∀ β γ, α ↝ β → γ < β → motive β → motive γ) :
    {β : TypeIndex} → α ↝ β → motive β :=
  Path.recSderiv (motive := λ β _ ↦ motive β) nil sderiv

@[simp]
theorem Path.recSderivGlobal_nil {motive : TypeIndex → Sort _}
    (nil : motive α)
    (sderiv : ∀ β γ, α ↝ β → γ < β → motive β → motive γ) :
    recSderivGlobal (motive := motive) nil sderiv .nil = nil :=
  rfl

@[simp]
theorem Path.recSderivGlobal_sderiv {motive : TypeIndex → Sort _}
    (nil : motive α)
    (sderiv : ∀ β γ, α ↝ β → γ < β → motive β → motive γ)
    {β γ : TypeIndex} (A : α ↝ β) (h : γ < β) :
    recSderivGlobal (motive := motive) nil sderiv (A ↘ h) =
      sderiv β γ A h (recSderiv nil sderiv A) :=
  rfl

/-!
## Derivatives of paths
-/

instance : Derivative (α ↝ β) (α ↝ γ) β γ where
  deriv A := Path.recSderivGlobal A (λ _ _ _ h B ↦ B ↘ h)

instance : BotDerivative (α ↝ β) (α ↝ ⊥) β where
  botDeriv A B := A ⇘ B
  botSderiv A :=
    match β with
      | ⊥ => A
      | (β : Λ) => A ↘ bot_lt_coe β
  botDeriv_single A h := by
    cases β using WithBot.recBotCoe with
    | bot => cases lt_irrefl ⊥ h
    | coe => rfl

instance : Coderivative (β ↝ γ) (α ↝ γ) α β where
  coderiv A B := B ⇘ A

@[simp]
theorem Path.deriv_nil (A : α ↝ β) :
    A ⇘ .nil = A :=
  rfl

@[simp]
theorem Path.deriv_sderiv (A : α ↝ β) (B : β ↝ γ) (h : δ < γ) :
    A ⇘ (B ↘ h) = A ⇘ B ↘ h :=
  rfl

@[simp]
theorem Path.nil_deriv (A : α ↝ β) :
    (.nil : α ↝ α) ⇘ A = A := by
  induction A with
  | nil => rfl
  | sderiv γ δ A h ih => rw [deriv_sderiv, ih]

@[simp]
theorem Path.deriv_sderivBot (A : α ↝ β) (B : β ↝ γ) :
    A ⇘ (B ↘.) = A ⇘ B ↘. := by
  cases γ using WithBot.recBotCoe with
  | bot => rfl
  | coe => rfl

@[simp]
theorem Path.botSderiv_bot_eq (A : α ↝ ⊥) :
    A ↘. = A :=
  rfl

@[simp]
theorem Path.botSderiv_coe_eq {β : Λ} (A : α ↝ β) :
    A ↘ bot_lt_coe β = A ↘. :=
  rfl

@[simp]
theorem Path.deriv_assoc (A : α ↝ β) (B : β ↝ γ) (C : γ ↝ δ) :
    A ⇘ (B ⇘ C) = A ⇘ B ⇘ C := by
  induction C with
  | nil => rfl
  | sderiv ε ζ C h ih => simp only [deriv_sderiv, ih]

@[simp]
theorem Path.deriv_sderiv_assoc (A : α ↝ β) (B : β ↝ γ) (h : δ < γ) :
    A ⇘ (B ↘ h) = A ⇘ B ↘ h :=
  rfl

@[simp]
theorem Path.deriv_scoderiv (A : α ↝ β) (B : γ ↝ δ) (h : γ < β) :
    A ⇘ (B ↗ h) = A ↘ h ⇘ B := by
  induction B with
  | nil => rfl
  | sderiv ε ζ B h' ih =>
    rw [deriv_sderiv, ← ih]
    rfl

@[simp]
theorem Path.botDeriv_scoderiv (A : α ↝ β) (B : γ ↝ ⊥) (h : γ < β) :
    A ⇘. (B ↗ h) = A ↘ h ⇘. B


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]


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


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


import ConNF.Model.Externalise

/-!
# New file

In this file...

## Main declarations

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

noncomputable section
universe u

open Cardinal Ordinal

namespace ConNF

variable [Params.{u}] {β γ : Λ} {hγ : (γ : TypeIndex) < β}

namespace Support

theorem not_mem_scoderiv_botDeriv (S : Support γ) (N : NearLitter) :
    N ∉ (S ↗ hγ ⇘. (Path.nil ↘.))ᴺ := by
  rintro ⟨i, ⟨A, N'⟩, h₁, h₂⟩
  simp only [Prod.mk.injEq] at h₂
  cases A
  case sderiv δ A hδ _ =>
    simp only [Path.deriv_sderiv] at h₂
    cases A
    case nil => cases h₂.1
    case sderiv ζ A hζ _ =>
      simp only [Path.deriv_sderiv] at h₂
      cases h₂.1

variable [Level] [LtLevel β]

theorem not_mem_strong_botDeriv (S : Support γ) (N : NearLitter) :
    N ∉ ((S ↗ hγ).strong ⇘. (Path.nil ↘.))ᴺ := by
  rintro h
  rw [strong, close_nearLitters, preStrong_nearLitters, Enumeration.mem_add_iff] at h
  obtain h | h := h
  · exact not_mem_scoderiv_botDeriv S N h
  · rw [mem_constrainsNearLitters_nearLitters] at h
    obtain ⟨B, N', hN', h⟩ := h
    cases h using Relation.ReflTransGen.head_induction_on
    case refl => exact not_mem_scoderiv_botDeriv S N hN'
    case head x hx₁ hx₂ _ =>
      obtain ⟨⟨γ, δ, ε, hδ, hε, hδε, A⟩, t, B, hB, hN, ht⟩ := hx₂
      simp only at hB
      cases B
      case nil =>
        cases hB
        obtain ⟨C, N''⟩ := x
        simp only at ht
        cases ht.1
        change _ ∈ t.supportᴺ at hN
        rw [t.support_supports.2 rfl] at hN
        obtain ⟨i, hN⟩ := hN
        cases hN
      case sderiv δ B hδ _ _ =>
        cases B
        case nil => cases hB
        case sderiv ζ B hζ _ _ => cases hB

theorem raise_preStrong' (S : Support α) (hS : S.Strong) (T : Support γ) (ρ : AllPerm β)
    (hγ : (γ : TypeIndex) < β) :
    (S + (ρᵁ • ((T ↗ hγ).strong +
      (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim).PreStrong := by
  apply hS.toPreStrong.add
  constructor
  intro A N hN P t hA ht
  obtain ⟨A, rfl⟩ := eq_of_nearLitter_mem_scoderiv_botDeriv hN
  simp only [scoderiv_botDeriv_eq, add_derivBot, smul_derivBot,
    BaseSupport.add_nearLitters, BaseSupport.smul_nearLitters, interferenceSupport_nearLitters,
    Enumeration.mem_add_iff, Enumeration.mem_smul, Enumeration.not_mem_empty, or_false] at hN
  obtain ⟨δ, ε, ζ, hε, hζ, hεζ, B⟩ := P
  dsimp only at *
  cases A
  case sderiv ζ' A hζ' _ =>
    rw [← Path.coderiv_deriv] at hA
    cases Path.sderiv_index_injective hA
    apply Path.sderiv_left_inj.mp at hA
    cases A
    case nil =>
      cases hA
      cases not_mem_strong_botDeriv T _ hN
    case sderiv ι A hι _ _ =>
      rw [← Path.coderiv_deriv] at hA
      cases Path.sderiv_index_injective hA
      cases hA
      haveI : LtLevel δ := ⟨A.le.trans_lt LtLevel.elim⟩
      haveI : LtLevel ε := ⟨hε.trans LtLevel.elim⟩
      haveI : LtLevel ζ := ⟨hζ.trans LtLevel.elim⟩
      have := (T ↗ hγ).strong_strong.support_le hN ⟨δ, ε, ζ, hε, hζ, hεζ, A⟩
          (ρ⁻¹ ⇘ A ↘ hε • t) rfl ?_
      · simp only [Tangle.smul_support, allPermSderiv_forget, allPermDeriv_forget,
          allPermForget_inv, Tree.inv_deriv, Tree.inv_sderiv] at this
        have := smul_le_smul this (ρᵁ ⇘ A ↘ hε)
        simp only [smul_inv_smul] at this
        apply le_trans this
        intro B
        constructor
        · intro a ha
          simp only [smul_derivBot, Tree.sderiv_apply, Tree.deriv_apply, Path.deriv_scoderiv,
            deriv_derivBot, Enumeration.mem_smul] at ha
          rw [deriv_derivBot, ← Path.deriv_scoderiv, Path.coderiv_deriv', scoderiv_botDeriv_eq,]
          simp only [Path.deriv_scoderiv, add_derivBot, smul_derivBot,
            BaseSupport.add_atoms, BaseSupport.smul_atoms, Enumeration.mem_add_iff,
            Enumeration.mem_smul]
          exact Or.inl ha
        · intro N hN
          simp only [smul_derivBot, Tree.sderiv_apply, Tree.deriv_apply, Path.deriv_scoderiv,
            deriv_derivBot, Enumeration.mem_smul] at hN
          rw [deriv_derivBot, ← Path.deriv_scoderiv, Path.coderiv_deriv', scoderiv_botDeriv_eq,]
          simp only [Path.deriv_scoderiv, add_derivBot, smul_derivBot,
            BaseSupport.add_nearLitters, BaseSupport.smul_nearLitters, Enumeration.mem_add_iff,
            Enumeration.mem_smul]
          exact Or.inl hN
      · rw [← smul_fuzz hε hζ hεζ, ← ht]
        simp only [Path.botSderiv_coe_eq, BasePerm.smul_nearLitter_litter, allPermDeriv_forget,
          allPermForget_inv, Tree.inv_deriv, Tree.inv_sderiv, Tree.inv_sderivBot]
        rfl

theorem raise_closed' (S : Support α) (hS : S.Strong) (T : Support γ) (ρ : AllPerm β)
    (hγ : (γ : TypeIndex) < β)
    (hρ : ρᵁ • (S ↘ LtLevel.elim : Support β) = S ↘ LtLevel.elim) :
    (S + (ρᵁ • ((T ↗ hγ).strong +
      (S ↘ LtLevel.elim + (T ↗ hγ).strong).interferenceSupport)) ↗ LtLevel.elim).Closed

:= by
  constructor
  intro A
  constructor
  intro N₁ N₂ hN₁ hN₂ a ha
  simp only [add_derivBot, BaseSupport.add_nearLitters, Enumeration.mem_add_iff,
    BaseSupport.add_atoms] at hN₁ hN₂ ⊢
  obtain hN₁ | hN₁ := hN₁
  · obtain hN₂ | hN₂ := hN₂
    · exact Or.inl ((hS.closed A).interference_subset hN₁ hN₂ a ha)
    · obtain ⟨B, rfl⟩ := eq_of_nearLitter_mem_scoderiv_botDeriv hN₂
      simp only [smul_add, scoderiv_botDeriv_eq, add_derivBot, smul_derivBot,
        BaseSupport.add_nearLitters, BaseSupport.smul_nearLitters, Enumeration.mem_add_iff,
        Enumeration.mem_smul, BaseSupport.add_atoms, BaseSupport.smul_atoms] at hN₁ hN₂ ⊢
      refine Or.inr (Or.inr ?_)
      rw [mem_interferenceSupport_atoms]
      refine ⟨(ρᵁ B)⁻¹ • N₁, ?_, (ρᵁ B)⁻¹ • N₂, ?_, ?_⟩
      · simp only [add_derivBot, BaseSupport.add_nearLitters, Enumeration.mem_add_iff]
        rw [← Enumeration.mem_smul, ← BaseSupport.smul_nearLitters, ← smul_derivBot, hρ]
        exact Or.inl hN₁
      · simp only [add_derivBot, BaseSupport.add_nearLitters, Enumeration.mem_add_iff]
        simp only [interferenceSupport_nearLitters, Enumeration.not_mem_empty, or_false] at hN₂
        exact Or.inr hN₂
      · rw [← BasePerm.smul_interference]
        exact Set.smul_mem_smul_set ha
  · obtain ⟨B, rfl⟩ := eq_of_nearLitter_mem_scoderiv_botDeriv hN₁
    simp only [smul_add, scoderiv_botDeriv_eq, add_derivBot, smul_derivBot,
      BaseSupport.add_nearLitters, BaseSupport.smul_nearLitters, Enumeration.mem_add_iff,
      Enumeration.mem_smul, BaseSupport.add_atoms, BaseSupport.smul_atoms] at hN₁ hN₂ ⊢
    refine Or.inr (Or.inr ?_)
    rw [mem_interferenceSupport_atoms]
    refine ⟨(ρᵁ B)⁻¹ • N₁, ?_, (ρᵁ B)⁻¹ • N₂, ?_, ?_⟩
    · simp only [add_derivBot, BaseSupport.add_nearLitters, Enumeration.mem_add_iff]
      simp only [interferenceSupport_nearLitters, Enumeration.not_mem_empty, or_false] at hN₁
      exact Or.inr hN₁
    · simp only [add_derivBot, BaseSupport.add_nearLitters, Enumeration.mem_add_iff]
      simp only [interferenceSupport_nearLitters, Enumeration.not_mem_empty, or_false] at hN₂
      obtain hN₂ | hN₂ := hN₂
      · rw [← Enumeration.mem_smul, ← BaseSupport.smul_nearLitters, ← smul_derivBot, hρ]
        exact Or.inl hN₂
      · exact Or.inr hN₂
    · rw [← BasePerm.smul_interference]
      exact Set.smul_mem_smul_set ha


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]


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