# Theory Word_Lib.Enumeration

```(*
* Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
*
*)

section "Enumeration extensions and alternative definition"

theory Enumeration
imports Main
begin

abbreviation
"enum ≡ enum_class.enum"
abbreviation
"enum_all ≡ enum_class.enum_all"
abbreviation
"enum_ex ≡ enum_class.enum_ex"

primrec (nonexhaustive)
the_index :: "'a list ⇒ 'a ⇒ nat"
where
"the_index (x # xs) y = (if x = y then 0 else Suc (the_index xs y))"

lemma the_index_bounded:
"x ∈ set xs ⟹ the_index xs x < length xs"
by (induct xs, clarsimp+)

lemma nth_the_index:
"x ∈ set xs ⟹ xs ! the_index xs x = x"
by (induct xs, clarsimp+)

lemma distinct_the_index_is_index[simp]:
"⟦ distinct xs ; n < length xs ⟧ ⟹ the_index xs (xs ! n) = n"
by (meson nth_eq_iff_index_eq nth_mem nth_the_index the_index_bounded)

lemma the_index_last_distinct:
"distinct xs ∧ xs ≠ [] ⟹ the_index xs (last xs) = length xs - 1"
apply safe
apply (subgoal_tac "xs ! (length xs - 1) = last xs")
apply (subgoal_tac "xs ! the_index xs (last xs) = last xs")
apply (subst nth_eq_iff_index_eq[symmetric])
apply assumption
apply (rule the_index_bounded)
apply simp_all
apply (rule nth_the_index)
apply simp
apply (induct xs, auto)
done

context enum begin

(* These two are added for historical reasons. *)
lemmas enum_surj[simp] = enum_UNIV
declare enum_distinct[simp]

lemma enum_nonempty[simp]: "(enum :: 'a list) ≠ []"
using enum_surj by fastforce

definition
maxBound :: 'a where
"maxBound ≡ last enum"

definition
minBound :: 'a where
"minBound ≡ hd enum"

definition
toEnum :: "nat ⇒ 'a" where
"toEnum n ≡ if n < length (enum :: 'a list) then enum ! n else the None"

definition
fromEnum :: "'a ⇒ nat" where
"fromEnum x ≡ the_index enum x"

lemma maxBound_is_length:
"fromEnum maxBound = length (enum :: 'a list) - 1"

lemma maxBound_less_length:
"(x ≤ fromEnum maxBound) = (x < length (enum :: 'a list))"
unfolding maxBound_is_length by (cases "length enum") auto

lemma maxBound_is_bound [simp]:
unfolding maxBound_less_length
by (fastforce simp: fromEnum_def intro: the_index_bounded)

lemma to_from_enum [simp]:
fixes x :: 'a
shows "toEnum (fromEnum x) = x"
proof -
have "x ∈ set enum" by simp
qed

lemma from_to_enum [simp]:

lemma map_enum:
fixes x :: 'a
shows "map f enum ! fromEnum x = f x"
proof -
by (rule maxBound_is_bound)
then have "fromEnum x < length (enum::'a list)"
then have "map f enum ! fromEnum x = f (enum ! fromEnum x)" by simp
also
have "x ∈ set enum" by simp
then have "enum ! fromEnum x = x"
finally
show ?thesis .
qed

definition
assocs :: "('a ⇒ 'b) ⇒ ('a × 'b) list" where
"assocs f ≡ map (λx. (x, f x)) enum"

end

(* For historical naming reasons. *)
lemmas enum_bool = enum_bool_def

class enum_alt =
fixes enum_alt :: "nat ⇒ 'a option"

class enumeration_alt = enum_alt +
assumes enum_alt_one_bound:
"enum_alt x = (None :: 'a option) ⟹ enum_alt (Suc x) = (None :: 'a option)"
assumes enum_alt_surj:
"range enum_alt ∪ {None} = UNIV"
assumes enum_alt_inj:
"(enum_alt x :: 'a option) = enum_alt y ⟹ (x = y) ∨ (enum_alt x = (None :: 'a option))"
begin

lemma enum_alt_inj_2:
assumes "enum_alt x = (enum_alt y :: 'a option)"
"enum_alt x ≠ (None :: 'a option)"
shows "x = y"
proof -
from assms
have "(x = y) ∨ (enum_alt x = (None :: 'a option))" by (fastforce intro!: enum_alt_inj)
with assms show ?thesis by clarsimp
qed

lemma enum_alt_surj_2:
"∃x. enum_alt x = Some y"
proof -
have "Some y ∈ range enum_alt ∪ {None}" by (subst enum_alt_surj) simp
then have "Some y ∈ range enum_alt" by simp
then show ?thesis by auto
qed

end

definition
alt_from_ord :: "'a list ⇒ nat ⇒ 'a option"
where
"alt_from_ord L ≡ λn. if (n < length L) then Some (L ! n) else None"

lemma handy_if_lemma: "((if P then Some A else None) = Some B) = (P ∧ (A = B))"
by simp

class enumeration_both = enum_alt + enum +
assumes enum_alt_rel: "enum_alt = alt_from_ord enum"

instance enumeration_both < enumeration_alt
apply (intro_classes; simp add: enum_alt_rel alt_from_ord_def)
apply auto[1]
apply (safe; simp)[1]
apply (rule rev_image_eqI; simp)
apply (rule the_index_bounded; simp)
apply (subst nth_the_index; simp)
apply (clarsimp simp: handy_if_lemma)
apply (subst nth_eq_iff_index_eq[symmetric]; simp)
done

instantiation bool :: enumeration_both
begin
definition enum_alt_bool: "enum_alt ≡ alt_from_ord [False, True]"
instance by (intro_classes, simp add: enum_bool_def enum_alt_bool)
end

definition
toEnumAlt :: "nat ⇒ ('a :: enum_alt)" where
"toEnumAlt n ≡ the (enum_alt n)"

definition
fromEnumAlt :: "('a :: enum_alt) ⇒ nat" where
"fromEnumAlt x ≡ THE n. enum_alt n = Some x"

definition
upto_enum :: "('a :: enumeration_alt) ⇒ 'a ⇒ 'a list" ("(1[_ .e. _])") where
"upto_enum n m ≡ map toEnumAlt [fromEnumAlt n ..< Suc (fromEnumAlt m)]"

apply (rule ext)
apply (rule theI2)
apply (rule conjI)
apply (clarify, rule nth_the_index, simp)
apply (rule the_index_bounded, simp)
apply auto
done

lemma toEnum_alt_red[simp]:
"toEnumAlt = (toEnum :: nat ⇒ 'a :: enumeration_both)"
by (rule ext) (simp add: enum_alt_rel alt_from_ord_def toEnum_def toEnumAlt_def)

lemma upto_enum_red:
"[(n :: ('a :: enumeration_both)) .e. m] = map toEnum [fromEnum n ..< Suc (fromEnum m)]"
unfolding upto_enum_def by simp

instantiation nat :: enumeration_alt
begin
definition enum_alt_nat: "enum_alt ≡ Some"
instance by (intro_classes; simp add: enum_alt_nat UNIV_option_conv)
end

lemma toEnumAlt_nat[simp]: "toEnumAlt = id"
by (rule ext) (simp add: toEnumAlt_def enum_alt_nat)

lemma upto_enum_nat[simp]: "[n .e. m] = [n ..< Suc m]"
by (subst upto_enum_def) simp

definition
zipE1 :: "'a :: enum_alt ⇒ 'b list ⇒ ('a × 'b) list"
where
"zipE1 x L ≡ zip (map toEnumAlt [fromEnumAlt x ..< fromEnumAlt x + length L]) L"

definition
zipE2 :: "'a :: enum_alt ⇒ 'a ⇒ 'b list ⇒ ('a × 'b) list"
where
"zipE2 x xn L ≡ zip (map (λn. toEnumAlt (fromEnumAlt x + (fromEnumAlt xn - fromEnumAlt x) * n))
[0 ..< length L]) L"

definition
zipE3 :: "'a list ⇒ 'b :: enum_alt ⇒ ('a × 'b) list"
where
"zipE3 L x ≡ zip L (map toEnumAlt [fromEnumAlt x ..< fromEnumAlt x + length L])"

definition
zipE4 :: "'a list ⇒ 'b :: enum_alt ⇒ 'b ⇒ ('a × 'b) list"
where
"zipE4 L x xn ≡ zip L (map (λn. toEnumAlt (fromEnumAlt x + (fromEnumAlt xn - fromEnumAlt x) * n))
[0 ..< length L])"

lemma to_from_enum_alt[simp]:
"toEnumAlt (fromEnumAlt x) = (x :: 'a :: enumeration_alt)"
proof -
have rl: "⋀a b. a = Some b ⟹ the a = b" by simp
show ?thesis
by (rule rl, rule theI') (metis enum_alt_inj enum_alt_surj_2 not_None_eq)
qed

lemma upto_enum_triv [simp]: "[x .e. x] = [x]"
unfolding upto_enum_def by simp

fixes v :: "'a :: enum"
shows "n ≤ fromEnum (maxBound :: 'a) ⟹ (toEnum n = v) = (n = fromEnum v)"
by auto

lemma le_imp_diff_le:
"(j::nat) ≤ k ⟹ j - n ≤ k"
by simp

fixes start :: "'a :: enumeration_both"
assumes "n < length [start .e. end]"
shows "fromEnum ([start .e. end] ! n) = fromEnum start + n"
proof -
have less_sub: "⋀m k m' n. ⟦ (n::nat) < m - k ; m ≤ m' ⟧ ⟹ n < m' - k" by fastforce
note upt_Suc[simp del]
show ?thesis using assms
by (fastforce simp: upto_enum_red
dest: less_sub[where m'="Suc (fromEnum maxBound)"] intro: maxBound_is_bound)
qed

lemma length_upto_enum_le_maxBound:
fixes start :: "'a :: enumeration_both"
shows "length [start .e. end] ≤ Suc (fromEnum (maxBound :: 'a))"
apply (clarsimp simp add: upto_enum_red split: if_splits)
apply (rule le_imp_diff_le[OF maxBound_is_bound[of "end"]])
done

lemma less_length_upto_enum_maxBoundD:
fixes start :: "'a :: enumeration_both"
assumes "n < length [start .e. end]"
shows "n ≤ fromEnum (maxBound :: 'a)"
using assms
le_trans[OF _ le_imp_diff_le[OF maxBound_is_bound[of "end"]]]
split: if_splits)