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ea76aac | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | 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)
@[simp] theorem tail_push {B : Type*} (a : B) (L : Stack B) : tail (push a L) = L := rfl
@[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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