Theory Arrays

(*
 * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
 * Copyright (c) 2022 Apple Inc. All rights reserved.
 *
 * SPDX-License-Identifier: BSD-2-Clause
 *)

(* Port of Anthony Fox's HOL4 realisation of John Harrison's paper
   in TPHOLs 2005 on finite cartesian products *)

theory Arrays
imports
  "HOL-Library.Numeral_Type"
begin

definition has_size :: "'a set ⇒ nat ⇒ bool" where
   "has_size s n = (finite s ∧ card s = n)"

― ‹If @{typ 'a} is not finite, there is no @{term "n < CARD('a)"}›
definition finite_index :: "nat ⇒ 'a::finite" where
  "finite_index = (SOME f. ∀x. ∃!n. n < CARD('a) ∧ f n = x)"

lemma card_image_inj:
  "⟦ finite S; ⋀x y. ⟦ x ∈ S; y ∈ S; f x = f y ⟧ ⟹ x = y ⟧ ⟹ card (f ` S) = card S"
  by (induct rule: finite_induct) (auto simp: card_insert_if)

lemma has_size_image_inj:
  "⟦ has_size S n; ⋀x y. x ∈ S ∧ y ∈ S ∧ f x = f y ⟹ x = y ⟧ ⟹ has_size (f ` S) n"
  by (simp add: has_size_def card_image_inj)

lemma has_size_0[simp]:
  "has_size S 0 = (S = {})"
  by (auto simp: has_size_def)

lemma has_size_suc:
  "has_size S (Suc n) = (S ≠ {} ∧ (∀a. a ∈ S ⟶ has_size (S - {a}) n))"
  unfolding has_size_def
  by (metis Diff_empty Suc_not_Zero bot_least card_Suc_Diff1 card_gt_0_iff finite_Diff_insert
            nat.inject neq0_conv subsetI subset_antisym)

lemma has_index:
  "⟦ finite S; card S = n ⟧ ⟹
   (∃f. (∀m. m < n ⟶ f m ∈ S) ∧ (∀x. x∈S ⟶ (∃!m. m < n ∧ f m = x)))"
proof (induct n arbitrary: S)
  case 0 thus ?case by (auto simp: card_eq_0_iff)
next
  case (Suc n)
  then obtain b B where
    S: "S = insert b B ∧ b ∉ B ∧ card B = n ∧ (n = 0 ⟶ B = {})"
    by (auto simp: card_Suc_eq)
  with ‹finite S› Suc.hyps [of B]
  obtain f where IH: "(∀m<n. f m ∈ B) ∧ (∀x. x ∈ B ⟶ (∃!m. m < n ∧ f m = x))" by auto
  define f' where "f' ≡ λm. if m = card B then b else f m"
  from Suc.prems S IH
  have "(∀m<Suc n. f' m ∈ S) ∧ (∀x. x ∈ S ⟶ (∃!m. m < Suc n ∧ f' m = x))"
    unfolding f'_def
    apply (clarsimp)
    apply (rule conjI, metis less_SucE)
    apply (metis less_SucE less_SucI nat_neq_iff)
    done
  thus ?case by blast
qed

lemma finite_index_works:
  "∃!n. n < CARD('n::finite) ∧ finite_index n = (i::'n)"
proof -
  have "∃f::nat ⇒ 'n. ∀i. ∃!n. n < CARD('n) ∧ f n = i"
    using has_index[where S = "UNIV :: 'n set"] by simp
  hence "∀i. ∃!n. n < CARD('n::finite) ∧ finite_index n = (i::'n)"
    unfolding finite_index_def by (rule someI_ex)
  thus ?thesis ..
qed

lemma finite_index_inj:
  "⟦ i < CARD('a::finite); j < CARD('a) ⟧ ⟹
   ((finite_index i :: 'a) = finite_index j) = (i = j)"
  using finite_index_works[where i = "finite_index j"] by blast

lemma forall_finite_index:
  "(∀k::'a::finite. P k) = (∀i. i < CARD('a) ⟶ P (finite_index i))"
  by (metis finite_index_works)

section ‹Finite Cartesian Products›

typedef ('a,'n::finite) array
    (‹(‹open_block notation=‹mixfix array››_[_])› [30,0] 31) = "UNIV :: ('n => 'a) set"
  by simp
setup_lifting type_definition_array

lift_definition index :: "('a,'n::finite) array ⇒ nat ⇒ 'a"
    (‹(‹open_block notation=‹mixfix array index››_.[_])› [900,0] 901) is
  "λf i. f (finite_index i)".
 
lemma index_legacy_def: "index x i = Rep_array x (finite_index i)"
  by (transfer) simp

theorem array_index_eq:
  "((x::'a['n::finite]) = y) = (∀i. i < CARD('n) ⟶ x.[i] = y.[i])"
  by (auto dest!: forall_finite_index [THEN iffD2]
           simp: index_def Rep_array_inject [symmetric])

(* fixme: legacy name *)
lemmas cart_eq = array_index_eq

lemma array_ext:
  fixes x :: "'a['n::finite]"
  shows "(⋀i. i < CARD('n) ⟹ x.[i] = y.[i]) ⟹ x = y"
  by (simp add: array_index_eq)

definition FCP :: "(nat ⇒ 'a) ⇒ 'a['b::finite]" (binder ‹ARRAY › 10) where
  "FCP ≡ λg. SOME a. ∀i. i < CARD('b) ⟶ a.[i] = g i"

definition update :: "'a['n::finite] ⇒ nat ⇒ 'a ⇒ 'a['n]" where
  "update f i x ≡ FCP ((index f)(i := x))"

definition fupdate :: "nat ⇒ ('a ⇒ 'a) ⇒ 'a['b::finite] ⇒ 'a['b]" where
  "fupdate i f x ≡ update x i (f (index x i))"

lemma fcp_beta[simp]: "i < CARD('n :: finite) ⟹ (FCP g :: 'a['n]).[i] = g i" 
proof -
  have "∀i < CARD('n). (FCP g :: 'a['n]).[i] = g i"
    unfolding FCP_def
  proof (rule someI_ex)
    let ?g' = "λk::'n. g (SOME i. i < CARD('n) ∧ finite_index i = k)"
    have "∀i<CARD('n). (Abs_array ?g').[i] = g i"
      by (clarsimp simp: index_def Abs_array_inverse)
        (metis (mono_tags, lifting) finite_index_inj someI_ex)
    thus "∃x::'a['n]. ∀i<CARD('n). x.[i] = g i" ..
  qed
  thus "i < CARD('n :: finite) ⟹ (FCP g :: 'a['n]).[i] = g i" by auto
qed

lemma fcp_unique:
  "(∀i. i < CARD('b::finite) ⟶ index f i = g i) =
   (FCP g = (f :: ('a,'b) array))"
  by (fastforce simp: cart_eq)

lemma fcp_eta[simp]:
  "(ARRAY i. g.[i]) = g"
  by (simp add: cart_eq)

lemma index_update[simp]:
  "n < CARD('b::finite) ⟹ index (update (f::'a['b]) n x) n = x"
  by (simp add: update_def)

lemma index_update2[simp]:
  "⟦ k < CARD('b::finite); n ≠ k ⟧ ⟹ index (update (f::'a['b]) n x) k = index f k"
  by (simp add: update_def)

lemma update_update[simp]:
  "update (update f n x) n y = update f n y"
  by (simp add: cart_eq update_def)

lemma update_comm[simp]:
  "n ≠ m ⟹ update (update f m v) n v' = update (update f n v') m v"
  by (simp add: cart_eq update_def)

lemma update_index_same [simp]:
  "update v n (index v n) = v"
  by (simp add: cart_eq update_def)

function foldli0 :: "(nat ⇒ 'acc ⇒ 'a ⇒ 'acc) ⇒ 'acc ⇒ nat ⇒ 'a['index::finite] ⇒ 'acc" where
  "foldli0 f a i arr = (if CARD('index) ≤ i then a else foldli0 f (f i a (index arr i)) (i+1) arr)"
  by pat_completeness auto

termination
  by (relation "measure (λ(f,a,i,(arr::'b['c::finite])). CARD('c) - i)") auto

definition foldli :: "(nat ⇒ 'acc ⇒ 'a ⇒ 'acc) ⇒ 'acc ⇒ ('a,'i::finite) array ⇒ 'acc" where
  "foldli f a arr = foldli0 f a 0 arr"

(* for a traditional word presentation, with MSB on the left, you'd
   want a fold that numbered in the reverse direction (with element 0
   on the right rather than the left) *)

definition map_array :: "('a ⇒ 'b) ⇒ 'a['n::finite] ⇒ 'b['n]" where
  "map_array f a ≡ ARRAY i. f (a.[i])"

definition map_array2 :: "('a ⇒ 'b ⇒ 'c) ⇒ 'a['n::finite] ⇒ 'b['n] ⇒ 'c['n]" where
  "map_array2 f a b ≡ ARRAY i. f (a.[i]) (b.[i])"

definition zip_array :: "'a['b::finite] ⇒ 'c['b] ⇒ ('a × 'c)['b]" where
  "zip_array ≡ map_array2 Pair"

definition list_array :: "('a,'n::finite) array ⇒ 'a list" where
  "list_array a = map (index a) [0..<CARD('n)]"

lift_definition set_array :: "'a['n::finite] ⇒ 'a set" is range .

lemma set_array_list:
  "set_array a = set (list_array a)"
  by (simp add: list_array_def index_def set_array.rep_eq image_def)
     (metis atLeast0LessThan finite_index_works lessThan_iff)

definition rel_array :: "('a ⇒ 'b ⇒ bool) ⇒ 'a['n::finite] ⇒ 'b['n] ⇒ bool" where
  "rel_array f a b ≡ ∀i < CARD('n). f (a.[i]) (b.[i])"

lemma map_array_index:
  fixes a :: "'a['n::finite]"
  shows "n < CARD('n) ⟹ (map_array f a).[n] = f (a.[n])"
  by (metis fcp_beta map_array_def)

lemma map_array2_index:
  fixes a :: "'a['n::finite]"
  shows "n < CARD('n) ⟹ (map_array2 f a b).[n] = f (a.[n]) (b.[n])"
  by (metis fcp_beta map_array2_def)

lemma zip_array_index:
  fixes a :: "'a['n::finite]"
  shows "n < CARD('n) ⟹ (zip_array a b).[n] = (a.[n],b.[n])"
  by (simp add: zip_array_def map_array2_index)

lemma map_array_id[simp]:
  "map_array id = id"
  by (auto simp: map_array_index array_ext)

lemma map_array_comp:
  "map_array (g ∘ f) = map_array g ∘ map_array f"
  by (auto simp: map_array_def intro!: array_ext)

lemma list_array_nth:
  fixes a :: "'a['n::finite]"
  shows "n < CARD('n) ⟹ list_array a ! n = index a n"
  by (simp add: list_array_def)

lemma list_array_length [simp]:
  "length (list_array (a :: 'a['n::finite])) = CARD('n)"
  by (simp add: list_array_def)

lemma in_set_array_index_conv:
  "(z ∈ set_array (a :: 'a['n::finite])) = (∃n < CARD('n). z = a.[n])"
  by (metis in_set_conv_nth list_array_length list_array_nth nth_mem set_array_list)

lemma in_set_arrayE [elim!]:
  "⟦ z ∈ set_array (a :: 'a['n::finite]); ⋀n . ⟦n < CARD('n); z = a.[n]⟧ ⟹ P ⟧ ⟹ P"
  by (metis in_set_array_index_conv)

lemma map_array_setI:
  "(⋀z. z ∈ set_array x ⟹ f z = g z) ⟹ map_array f x = map_array g x"
  by (rule array_ext) (auto simp: map_array_index in_set_array_index_conv)

lemma list_array_map_array:
  "list_array (map_array f a) = map f (list_array a)"
  by (simp add: list_array_def map_array_index)

lemma list_array_FCP [simp]:
  "list_array (FCP f :: 'a['n]) = map f [0..<CARD('n::finite)]"
  by (simp add: list_array_def)

lemma set_array_FCP [simp]:
  "set_array (FCP f :: 'a['n]) = f ` {0..< CARD('n::finite)}"
  by (auto simp: set_array_list)

lemma map_array_set_img:
  "set_array ∘ map_array f = (`) f ∘ set_array"
  by (rule ext) (auto simp: map_array_def in_set_array_index_conv intro!: imageI)

lemma fcp_cong [cong]:
  "(⋀i. i < CARD('n::finite) ⟹ f i = g i) ⟹ FCP f = (FCP g :: 'a['n])"
  by (rule array_ext) simp

bnf "('a,'n::finite) array"
  map: map_array
  sets: set_array
  bd: "card_suc (BNF_Cardinal_Arithmetic.csum natLeq (card_of (UNIV :: 'n set)))"
  rel: rel_array
proof -
  show "map_array id = id" by simp
next
  fix f :: "'a ⇒ 'b" and g :: "'b ⇒ 'c"
  show "map_array (g ∘ f) = map_array g ∘ map_array f"
    by (rule map_array_comp)
next
  fix x :: "'a['n::finite]" and f :: "'a ⇒ 'b" and g
  assume "⋀z. z ∈ set_array x ⟹ f z = g z"
  thus "map_array f x = map_array g x"
    by (rule map_array_setI)
next
  fix f :: "'a ⇒ 'b"
  show "set_array ∘ map_array f = (`) f ∘ set_array"
    by (rule map_array_set_img)
next
  show "card_order (card_suc (BNF_Cardinal_Arithmetic.csum natLeq (card_of UNIV)))"
    by (simp add: card_of_card_order_on card_order_card_suc card_order_csum natLeq_card_order)
next
  show " BNF_Cardinal_Arithmetic.cinfinite (card_suc (BNF_Cardinal_Arithmetic.csum natLeq (card_of UNIV)))"
    by (simp add: Card_order_csum Cinfinite_card_suc card_of_card_order_on card_order_csum 
        cinfinite_csum natLeq_Cinfinite natLeq_card_order)
next
  fix R :: "'a ⇒ 'b ⇒ bool" and S :: "'b ⇒ 'c ⇒ bool"
  show "rel_array R OO rel_array S ≤ rel_array (R OO S)"
    by (force simp: rel_array_def)
next
  fix R :: "'a ⇒ 'b ⇒ bool"
  { fix a :: "('a × 'b)['n::finite]" and i
    have "⟦set_array a ⊆ {(x, y). R x y}; i < CARD('n)⟧ ⟹ R (fst (a.[i])) (snd (a.[i]))"
      by (meson Collect_case_prodD in_set_array_index_conv subset_iff)
  } note conv = this
  show "rel_array R =
         (λx y. ∃z. set_array z ⊆ {(x, y). R x y} ∧ map_array fst z = x ∧ map_array snd z = y)"
    unfolding rel_array_def
    apply (intro ext iffI)
    subgoal for a b
     apply (rule exI [where x="zip_array a b"])
      by (auto intro!: array_ext simp: conv map_array_index zip_array_index)
    subgoal 
      by (auto intro!: array_ext simp: conv map_array_index zip_array_index)
    done
next
  show "regularCard (card_suc (BNF_Cardinal_Arithmetic.csum natLeq (card_of UNIV)))"
    by (simp add: Card_order_csum card_of_card_order_on card_order_csum cinfinite_csum 
        natLeq_Cinfinite natLeq_card_order regularCard_card_suc)
next
  fix x :: "'a['n::finite]"
  let ?U = "UNIV :: 'n set"
  have "ordLeq3 (card_of (set_array x)) (card_of ?U)" by transfer (rule card_of_image)
  also
  have "ordLeq3 (card_of ?U) (BNF_Cardinal_Arithmetic.csum natLeq (card_of ?U))"
   by (rule ordLeq_csum2) (rule card_of_Card_order)
  finally
  have "ordLeq3 (card_of (set_array x)) (BNF_Cardinal_Arithmetic.csum natLeq (card_of ?U))" .
  then show "ordLess2 (card_of (set_array x))
          (card_suc (BNF_Cardinal_Arithmetic.csum natLeq (card_of ?U)))"
    using ordLeq_ordLess_trans card_suc_greater card_order_csum natLeq_card_order
          card_of_card_order_on  
    by blast
qed

lemma arr_fupdate_same: "i < CARD('n) ⟹ 
  index (fupdate i f (x::'a['n::finite])) i = f (index x i)"
  by (simp add: fupdate_def)

lemma arr_fupdate_other: "j < CARD('n) ⟹ i ≠ j ⟹ 
  index (fupdate i f (x::'a['n::finite])) j = (index x j)"
  by (simp add: fupdate_def)

lemmas arr_fupdate_simps = arr_fupdate_same arr_fupdate_other

lemma surj_finite_index: "surj finite_index"
  by (metis finite_index_works surj_def)

lemma finite_index_inj_on: "inj_on (finite_index::nat⇒'a::finite) {i. i < CARD('a)}"
  by (simp add: finite_index_inj inj_on_def)

lemma finite_index_bij_betw: "bij_betw (finite_index::nat⇒'a::finite) {i. i < CARD('a)} UNIV"
  using finite_index_inj_on
  by (metis (mono_tags, lifting) UNIV_I bij_betw_def finite_index_works imageI mem_Collect_eq subsetI subset_antisym)

end