Theory RS_List

(***********************************************************************************
 * Copyright (c) 2025 Université Paris-Saclay
 *
 * Author: Benoît Ballenghien, Université Paris-Saclay,
           CNRS, ENS Paris-Saclay, LMF
 * Author: Benjamin Puyobro, Université Paris-Saclay,
           IRT SystemX, CNRS, ENS Paris-Saclay, LMF
 * Author: Burkhart Wolff, Université Paris-Saclay,
           CNRS, ENS Paris-Saclay, LMF
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section ‹Lists›

(*<*)
theory RS_List
  imports Restriction_Spaces "HOL-Library.Prefix_Order"
begin
  (*>*)

text ‹List is a restriction space using \<^const>‹take› as the restriction function›

instantiation list :: (type) restriction
begin

definition restriction_list :: ‹'a list ⇒ nat ⇒ 'a list› 
  where ‹L ↓ n ≡ take n L›

instance by intro_classes (simp add: restriction_list_def min.commute)

end


instance list :: (type) order_restriction_space
proof intro_classes
  show ‹L ↓ 0 ≤ M ↓ 0› for L M :: ‹'a list›
    by (simp add: restriction_list_def)
next
  show ‹L ≤ M ⟹ L ↓ n ≤ M ↓ n› for L M :: ‹'a list› and n
    unfolding restriction_list_def
    by (metis less_eq_list_def prefix_def take_append)
next
  show ‹¬ L ≤ M ⟹ ∃n. ¬ L ↓ n ≤ M ↓ n› for M L :: ‹'a list›
    unfolding restriction_list_def
    by (metis linorder_linear take_all_iff)
qed


lemma ‹OFCLASS('a list, restriction_space_class)› ..



text ‹Of course, this space is not complete. We prove this with by exhibiting a counter-example.›

notepad begin
  define σ :: ‹nat ⇒ 'a list›
    where ‹σ n = replicate n undefined› for n

  have ‹chain⇩↓ σ›
    by (intro restriction_chainI ext)
      (simp add: σ_def restriction_list_def flip: replicate_append_same)

  hence ‹∄Σ. σ ─↓→ Σ›
    by (metis σ_def convergent_restriction_chain_imp_ex1 length_replicate
        lessI nat_less_le restriction_convergentI restriction_list_def take_all)

end

(*<*)
end
  (*>*)