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| import Discretion.Wk.List | ||
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| /-- The function `ρ` sends `Γ` to `Δ` -/ | ||
| def List.NEWkn (Γ Δ : List α) (ρ : ℕ → ℕ) : Prop | ||
| := ∀n, (hΔ : n < Δ.length) → ∃hΓ : ρ n < Γ.length, Γ[ρ n] = Δ[n] | ||
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| theorem List.NEWkn.bounded {ρ : ℕ → ℕ} (h : List.NEWkn Γ Δ ρ) (n : ℕ) (hΔ : n < Δ.length) | ||
| : ρ n < Γ.length := match h n hΔ with | ⟨hΓ, _⟩ => hΓ | ||
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| def List.NEWkn.toFinWk {ρ : ℕ → ℕ} (h : List.NEWkn Γ Δ ρ) : Fin (Δ.length) → Fin (Γ.length) | ||
| := Fin.wkOfBounded ρ h.bounded | ||
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| -- theorem List.NEWkn.toIsFWk (Γ Δ : List α) (ρ : ℕ → ℕ) | ||
| -- (h : List.NEWkn Γ Δ ρ) : List.IsFWk Γ Δ (List.NEWkn.toFinWk h) | ||
| -- := funext λ⟨i, hi⟩ => have ⟨_, h⟩ := h i hi; h | ||
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| -- ... TODO: NEWkns | ||
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| @[simp] | ||
| theorem List.NEWkn.id (Γ : List α) : List.NEWkn Γ Γ id | ||
| := λ_ hΓ => ⟨hΓ, rfl⟩ | ||
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| -- ... TODO: len_le | ||
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| @[simp] | ||
| theorem List.NEWkn.drop_all (Γ : List α) (ρ) : List.NEWkn Γ [] ρ | ||
| := λi h => by cases h | ||
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| theorem List.NEWkn.comp {ρ : ℕ → ℕ} {σ : ℕ → ℕ} (hρ : List.NEWkn Γ Δ ρ) (hσ : List.NEWkn Δ Ξ σ) | ||
| : List.NEWkn Γ Ξ (ρ ∘ σ) := λn hΞ => | ||
| have ⟨hΔ, hσ⟩ := hσ n hΞ; | ||
| have ⟨hΓ, hρ⟩ := hρ _ hΔ; | ||
| ⟨hΓ, hρ ▸ hσ⟩ | ||
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| theorem List.NEWkn.lift {ρ : ℕ → ℕ} (hρ : List.NEWkn Γ Δ ρ) | ||
| : List.NEWkn (A :: Γ) (A :: Δ) (Nat.liftWk ρ) := λn hΔ => match n with | ||
| | 0 => ⟨Nat.zero_lt_succ _, rfl⟩ | ||
| | n+1 => have ⟨hΔ, hρ⟩ := hρ n (Nat.lt_of_succ_lt_succ hΔ); ⟨Nat.succ_lt_succ hΔ, hρ⟩ | ||
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| theorem List.NEWkn.lift_tail {ρ : ℕ → ℕ} (h : List.NEWkn (A :: Γ) (B :: Δ) (Nat.liftWk ρ)) | ||
| : List.NEWkn Γ Δ ρ | ||
| := λi hΔ => have ⟨hΔ, hρ⟩ := h i.succ (Nat.succ_lt_succ hΔ); ⟨Nat.lt_of_succ_lt_succ hΔ, hρ⟩ | ||
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| theorem List.NEWkn.lift_head {ρ : ℕ → ℕ} (h : List.NEWkn (A :: Γ) (B :: Δ) (Nat.liftWk ρ)) : A = B | ||
| := (h 0 (Nat.zero_lt_succ _)).2 | ||
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| theorem List.NEWkn.lift_iff (A B) (Γ Δ : List α) (ρ : ℕ → ℕ) | ||
| : List.NEWkn (A :: Γ) (B :: Δ) (Nat.liftWk ρ) ↔ A = B ∧ List.NEWkn Γ Δ ρ | ||
| := ⟨ | ||
| λh => ⟨h.lift_head, List.NEWkn.lift_tail h⟩, | ||
| λ⟨rfl, hρ⟩ => List.NEWkn.lift hρ | ||
| ⟩ | ||
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| theorem List.NEWkn.lift_id (hρ : List.NEWkn Γ Δ _root_.id) | ||
| : List.NEWkn (A :: Γ) (A :: Δ) _root_.id := Nat.liftWk_id ▸ hρ.lift | ||
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| theorem List.NEWkn.lift_id_tail (h : List.NEWkn (left :: Γ) (right :: Δ) _root_.id) | ||
| : List.NEWkn Γ Δ _root_.id | ||
| := (Nat.liftWk_id ▸ h).lift_tail | ||
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| theorem List.NEWkn.lift_id_head (h : List.NEWkn (left :: Γ) (right :: Δ) _root_.id) | ||
| : left = right | ||
| := (Nat.liftWk_id ▸ h).lift_head | ||
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| theorem List.NEWkn.lift_id_iff (h : List.NEWkn (left :: Γ) (right :: Δ) _root_.id) | ||
| : left = right ∧ List.NEWkn Γ Δ _root_.id | ||
| := ⟨h.lift_id_head, h.lift_id_tail⟩ | ||
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| theorem List.NEWkn.lift₂ {ρ : ℕ → ℕ} (hρ : List.NEWkn Γ Δ ρ) | ||
| : List.NEWkn (A₁ :: A₂ :: Γ) (A₁ :: A₂ :: Δ) (Nat.liftWk (Nat.liftWk ρ)) | ||
| := hρ.lift.lift | ||
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| theorem List.NEWkn.liftn₂ {ρ : ℕ → ℕ} (hρ : List.NEWkn Γ Δ ρ) | ||
| : List.NEWkn (A₁ :: A₂ :: Γ) (A₁ :: A₂ :: Δ) (Nat.liftnWk 2 ρ) | ||
| := by rw [Nat.liftnWk_two]; exact hρ.lift₂ | ||
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| theorem List.NEWkn.liftn_append {ρ : ℕ → ℕ} (Ξ : List α) (hρ : List.NEWkn Γ Δ ρ) | ||
| : List.NEWkn (Ξ ++ Γ) (Ξ ++ Δ) (Nat.liftnWk Ξ.length ρ) := by | ||
| induction Ξ with | ||
| | nil => exact hρ | ||
| | cons A Ξ I => | ||
| rw [List.length, Nat.liftnWk_succ'] | ||
| exact I.lift | ||
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| theorem List.NEWkn.liftn_append' {ρ : ℕ → ℕ} (Ξ : List α) (hΞ : Ξ.length = n) | ||
| (hρ : List.NEWkn Γ Δ ρ) : List.NEWkn (Ξ ++ Γ) (Ξ ++ Δ) (Nat.liftnWk n ρ) := hΞ ▸ hρ.liftn_append Ξ | ||
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| theorem List.NEWkn.step {ρ : ℕ → ℕ} (A : α) (hρ : List.NEWkn Γ Δ ρ) | ||
| : List.NEWkn (A :: Γ) Δ (Nat.succ ∘ ρ) | ||
| := λn hΔ => have ⟨hΔ, hρ⟩ := hρ n hΔ; ⟨Nat.succ_lt_succ hΔ, hρ⟩ | ||
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| @[simp] | ||
| theorem List.NEWkn.succ (A : α) : List.NEWkn (A :: Γ) Γ .succ := step (ρ := _root_.id) A (id _) | ||
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| theorem List.NEWkn.step_tail {ρ : ℕ → ℕ} (h : List.NEWkn (A :: Γ) Δ (Nat.succ ∘ ρ)) : List.NEWkn Γ Δ ρ | ||
| := λi hΔ => have ⟨hΔ, hρ⟩ := h i hΔ; ⟨Nat.lt_of_succ_lt_succ hΔ, hρ⟩ | ||
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| theorem List.NEWkn.step_iff (A) (Γ Δ : List α) (ρ : ℕ → ℕ) | ||
| : List.NEWkn (A :: Γ) Δ (Nat.succ ∘ ρ) ↔ List.NEWkn Γ Δ ρ | ||
| := ⟨ | ||
| List.NEWkn.step_tail, | ||
| List.NEWkn.step A | ||
| ⟩ | ||
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| theorem List.NEWkn.stepn_append {ρ : ℕ → ℕ} (Ξ : List α) (hρ : List.NEWkn Γ Δ ρ) | ||
| : List.NEWkn (Ξ ++ Γ) Δ (Nat.stepnWk Ξ.length ρ) | ||
| := by induction Ξ with | ||
| | nil => exact hρ | ||
| | cons A Ξ I => | ||
| rw [List.length, Nat.stepnWk_succ'] | ||
| exact I.step _ | ||
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| theorem List.NEWkn.stepn_append' {ρ : ℕ → ℕ} (Ξ : List α) (hΞ : Ξ.length = n) | ||
| (hρ : List.NEWkn Γ Δ ρ) : List.NEWkn (Ξ ++ Γ) Δ (Nat.stepnWk n ρ) := hΞ ▸ hρ.stepn_append Ξ | ||
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| theorem List.NEWkn.toNWkn [PartialOrder α] (Γ Δ : List α) (ρ : ℕ → ℕ) | ||
| (h : List.NEWkn Γ Δ ρ) : List.NWkn Γ Δ ρ | ||
| := λn hΔ => match h n hΔ with | ⟨hΓ, h⟩ => ⟨hΓ, le_of_eq h⟩ |