| import Mathlib |
|
|
| namespace OAI |
|
|
| /-! Infinite stacks, finite prefixes and tails. -/ |
|
|
| noncomputable section |
| open scoped ContDiff |
| namespace PrefixFlows |
| namespace Interspersed |
| abbrev Stack (A : Type*) := ℕ → A |
|
|
| def push {A : Type*} (a : A) (L : Stack A) : Stack A |
| | 0 => a |
| | n + 1 => L n |
|
|
| @[simp] theorem push_zero {A : Type*} (a : A) (L : Stack A) : push a L 0 = a := rfl |
|
|
| @[simp] theorem push_succ {A : Type*} (a : A) (L : Stack A) (n : ℕ) : |
| push a L (n + 1) = L n := rfl |
|
|
| @[simp] theorem push_inj {A : Type*} {a b : A} {L R : Stack A} : |
| push a L = push b R ↔ a = b ∧ L = R := by |
| constructor |
| · intro h |
| exact ⟨congrFun h 0, funext (fun n => congrFun h (n + 1))⟩ |
| · rintro ⟨rfl, rfl⟩ |
| rfl |
|
|
| def prepend {B : Type*} : List B → Stack B → Stack B |
| | [], L => L |
| | a :: w, L => push a (prepend w L) |
|
|
| @[simp] theorem prepend_nil {B : Type*} (L : Stack B) : prepend [] L = L := rfl |
|
|
| @[simp] theorem prepend_cons {B : Type*} (a : B) (w : List B) (L : Stack B) : |
| prepend (a :: w) L = push a (prepend w L) := rfl |
|
|
| theorem prepend_append {B : Type*} (v w : List B) (L : Stack B) : |
| prepend (v ++ w) L = prepend v (prepend w L) := by |
| induction v with |
| | nil => rfl |
| | cons a v ih => simp [ih] |
|
|
| def tail {B : Type*} (L : Stack B) : Stack B := fun n => L (n + 1) |
|
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| @[simp] theorem tail_push {B : Type*} (a : B) (L : Stack B) : tail (push a L) = L := rfl |
|
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| @[simp] theorem push_constant {B : Type*} (b : B) : |
| push b (fun _ => b) = (fun _ => b) := by |
| funext n |
| cases n <;> rfl |
| end Interspersed |
| end PrefixFlows |
| end |
|
|
| end OAI |
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|
