Theory RS_Int

(***********************************************************************************
 * 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 ‹Integers›

(*<*)
theory RS_Int
  imports Restriction_Spaces
begin
  (*>*)

instantiation int :: restriction
begin

definition restriction_int :: ‹int ⇒ nat ⇒ int›
  where ‹x ↓ n ≡ if ¦x¦ ≤ int n then x else if 0 ≤ x then int n else - int n›

instance by intro_classes (simp add: restriction_int_def min_def)

end


instance int :: restriction_space
  by (intro_classes, simp_all add: restriction_int_def)
    (metis le_eq_less_or_eq linorder_not_less nat_le_iff)



lemma restriction_int_0_is_0 [simp] : ‹x ↓ 0 = (0 :: int)›
  by (simp add: restriction_int_def)


text ‹Restriction shift plus›

lemma restriction_shift_on_pos_plus : ‹restriction_shift_on (λx. x + k) k {x. 0 ≤ x}›
  by (intro restriction_shift_onI)
    (simp add: restriction_int_def split: if_split_asm)

lemma restriction_shift_on_neg_minus : ‹restriction_shift_on (λx. x - k) k {x. x ≤ 0}›
  by (intro restriction_shift_onI)
    (simp add: restriction_int_def split: if_split_asm)



(*<*)
end
  (*>*)