# Theory Word_Lib.Typedef_Morphisms

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

(*
Author: Jeremy Dawson and Gerwin Klein, NICTA

Consequences of type definition theorems, and of extended type definition.
*)

section ‹Type Definition Theorems›

theory Typedef_Morphisms
imports Main "HOL-Library.Word" More_Int Bit_Comprehension
begin

subsection "More lemmas about normal type definitions"

lemma tdD1: "type_definition Rep Abs A ⟹ ∀x. Rep x ∈ A"
and tdD2: "type_definition Rep Abs A ⟹ ∀x. Abs (Rep x) = x"
and tdD3: "type_definition Rep Abs A ⟹ ∀y. y ∈ A ⟶ Rep (Abs y) = y"
by (auto simp: type_definition_def)

lemma td_nat_int: "type_definition int nat (Collect ((≤) 0))"
unfolding type_definition_def by auto

context type_definition
begin

declare Rep [iff] Rep_inverse [simp] Rep_inject [simp]

lemma Abs_eqD: "Abs x = Abs y ⟹ x ∈ A ⟹ y ∈ A ⟹ x = y"

lemma Abs_inverse': "r ∈ A ⟹ Abs r = a ⟹ Rep a = r"
by (safe elim!: Abs_inverse)

lemma Rep_comp_inverse: "Rep ∘ f = g ⟹ Abs ∘ g = f"
using Rep_inverse by auto

lemma Rep_eqD [elim!]: "Rep x = Rep y ⟹ x = y"
by simp

lemma Rep_inverse': "Rep a = r ⟹ Abs r = a"
by (safe intro!: Rep_inverse)

lemma comp_Abs_inverse: "f ∘ Abs = g ⟹ g ∘ Rep = f"
using Rep_inverse by auto

lemma set_Rep: "A = range Rep"
proof (rule set_eqI)
show "x ∈ A ⟷ x ∈ range Rep" for x
by (auto dest: Abs_inverse [of x, symmetric])
qed

lemma set_Rep_Abs: "A = range (Rep ∘ Abs)"
proof (rule set_eqI)
show "x ∈ A ⟷ x ∈ range (Rep ∘ Abs)" for x
by (auto dest: Abs_inverse [of x, symmetric])
qed

lemma Abs_inj_on: "inj_on Abs A"
unfolding inj_on_def
by (auto dest: Abs_inject [THEN iffD1])

lemma image: "Abs ` A = UNIV"
by (fact Abs_image)

lemmas td_thm = type_definition_axioms

lemma fns1: "Rep ∘ fa = fr ∘ Rep ∨ fa ∘ Abs = Abs ∘ fr ⟹ Abs ∘ fr ∘ Rep = fa"
by (auto dest: Rep_comp_inverse elim: comp_Abs_inverse simp: o_assoc)

lemmas fns1a = disjI1 [THEN fns1]
lemmas fns1b = disjI2 [THEN fns1]

lemma fns4: "Rep ∘ fa ∘ Abs = fr ⟹ Rep ∘ fa = fr ∘ Rep ∧ fa ∘ Abs = Abs ∘ fr"
by auto

end

interpretation nat_int: type_definition int nat "Collect ((≤) 0)"
by (rule td_nat_int)

declare
nat_int.Rep_cases [cases del]
nat_int.Abs_cases [cases del]
nat_int.Rep_induct [induct del]
nat_int.Abs_induct [induct del]

subsection "Extended form of type definition predicate"

lemma td_conds:
"norm ∘ norm = norm ⟹
fr ∘ norm = norm ∘ fr ⟷ norm ∘ fr ∘ norm = fr ∘ norm ∧ norm ∘ fr ∘ norm = norm ∘ fr"
apply safe
done

lemma fn_comm_power: "fa ∘ tr = tr ∘ fr ⟹ fa ^^ n ∘ tr = tr ∘ fr ^^ n"
apply (rule ext)
apply (induct n)
apply (auto dest: fun_cong)
done

lemmas fn_comm_power' =
ext [THEN fn_comm_power, THEN fun_cong, unfolded o_def]

locale td_ext = type_definition +
fixes norm
assumes eq_norm: "⋀x. Rep (Abs x) = norm x"
begin

lemma Abs_norm [simp]: "Abs (norm x) = Abs x"
using eq_norm [of x] by (auto elim: Rep_inverse')

lemma td_th: "g ∘ Abs = f ⟹ f (Rep x) = g x"
by (drule comp_Abs_inverse [symmetric]) simp

lemma eq_norm': "Rep ∘ Abs = norm"
by (auto simp: eq_norm)

lemma norm_Rep [simp]: "norm (Rep x) = Rep x"
by (auto simp: eq_norm' intro: td_th)

lemmas td = td_thm

lemma set_iff_norm: "w ∈ A ⟷ w = norm w"
by (auto simp: set_Rep_Abs eq_norm' eq_norm [symmetric])

lemma inverse_norm: "Abs n = w ⟷ Rep w = norm n"
apply (rule iffI)
done

lemma norm_eq_iff: "norm x = norm y ⟷ Abs x = Abs y"

lemma norm_comps:
"Abs ∘ norm = Abs"
"norm ∘ Rep = Rep"
"norm ∘ norm = norm"
by (auto simp: eq_norm' [symmetric] o_def)

lemmas norm_norm [simp] = norm_comps

lemma fns5: "Rep ∘ fa ∘ Abs = fr ⟹ fr ∘ norm = fr ∧ norm ∘ fr = fr"
by (fold eq_norm') auto

text ‹
following give conditions for converses to ‹td_fns1›
▪ the condition ‹norm ∘ fr ∘ norm = fr ∘ norm› says that
‹fr› takes normalised arguments to normalised results
▪ ‹norm ∘ fr ∘ norm = norm ∘ fr› says that ‹fr›
takes norm-equivalent arguments to norm-equivalent results
▪ ‹fr ∘ norm = fr› says that ‹fr›
takes norm-equivalent arguments to the same result
▪ ‹norm ∘ fr = fr› says that ‹fr› takes any argument to a normalised result
›
lemma fns2: "Abs ∘ fr ∘ Rep = fa ⟹ norm ∘ fr ∘ norm = fr ∘ norm ⟷ Rep ∘ fa = fr ∘ Rep"
apply (fold eq_norm')
apply safe
prefer 2
apply (rule ext)
apply (drule_tac x="Rep x" in fun_cong)
apply auto
done

lemma fns3: "Abs ∘ fr ∘ Rep = fa ⟹ norm ∘ fr ∘ norm = norm ∘ fr ⟷ fa ∘ Abs = Abs ∘ fr"
apply (fold eq_norm')
apply safe
prefer 2
apply (rule ext)
apply (drule_tac f="a ∘ b" for a b in fun_cong)
apply simp
done

lemma fns: "fr ∘ norm = norm ∘ fr ⟹ fa ∘ Abs = Abs ∘ fr ⟷ Rep ∘ fa = fr ∘ Rep"
apply safe
apply (frule fns1b)
prefer 2
apply (frule fns1a)
apply (rule fns3 [THEN iffD1])
prefer 3
apply (rule fns2 [THEN iffD1])
done

lemma range_norm: "range (Rep ∘ Abs) = A"

end

lemmas td_ext_def' =
td_ext_def [unfolded type_definition_def td_ext_axioms_def]

subsection ‹Type-definition locale instantiations›

definition uints :: "nat ⇒ int set"
― ‹the sets of integers representing the words›
where "uints n = range (take_bit n)"

definition sints :: "nat ⇒ int set"
where "sints n = range (signed_take_bit (n - 1))"

lemma uints_num: "uints n = {i. 0 ≤ i ∧ i < 2 ^ n}"

lemma sints_num: "sints n = {i. - (2 ^ (n - 1)) ≤ i ∧ i < 2 ^ (n - 1)}"

definition unats :: "nat ⇒ nat set"
where "unats n = {i. i < 2 ^ n}"

― ‹naturals›
lemma uints_unats: "uints n = int ` unats n"
apply (unfold unats_def uints_num)
apply safe
apply (rule_tac image_eqI)
apply (erule_tac nat_0_le [symmetric])
by auto

lemma unats_uints: "unats n = nat ` uints n"
by (auto simp: uints_unats image_iff)

lemma td_ext_uint:
"td_ext (uint :: 'a word ⇒ int) word_of_int (uints (LENGTH('a::len)))
(λw::int. w mod 2 ^ LENGTH('a))"
apply (unfold td_ext_def')
apply transfer
done

interpretation word_uint:
td_ext
"uint::'a::len word ⇒ int"
word_of_int
"uints (LENGTH('a::len))"
"λw. w mod 2 ^ LENGTH('a::len)"
by (fact td_ext_uint)

lemmas td_uint = word_uint.td_thm
lemmas int_word_uint = word_uint.eq_norm

lemma td_ext_ubin:
"td_ext (uint :: 'a word ⇒ int) word_of_int (uints (LENGTH('a::len)))
(take_bit (LENGTH('a)))"
apply standard
apply transfer
apply simp
done

interpretation word_ubin:
td_ext
"uint::'a::len word ⇒ int"
word_of_int
"uints (LENGTH('a::len))"
"take_bit (LENGTH('a::len))"
by (fact td_ext_ubin)

lemma td_ext_unat [OF refl]:
"n = LENGTH('a::len) ⟹
td_ext (unat :: 'a word ⇒ nat) of_nat (unats n) (λi. i mod 2 ^ n)"
apply (standard; transfer)
apply (simp_all add: unats_def take_bit_of_nat take_bit_nat_eq_self_iff
flip: take_bit_eq_mod)
done

lemmas unat_of_nat = td_ext_unat [THEN td_ext.eq_norm]

interpretation word_unat:
td_ext
"unat::'a::len word ⇒ nat"
of_nat
"unats (LENGTH('a::len))"
"λi. i mod 2 ^ LENGTH('a::len)"
by (rule td_ext_unat)

lemmas td_unat = word_unat.td_thm

lemma unat_le: "y ≤ unat z ⟹ y ∈ unats (LENGTH('a))"
for z :: "'a::len word"
apply (unfold unats_def)
apply clarsimp
apply (metis le_unat_uoi unsigned_less)
done

lemma td_ext_sbin:
"td_ext (sint :: 'a word ⇒ int) word_of_int (sints (LENGTH('a::len)))
(signed_take_bit (LENGTH('a) - 1))"
by (standard; transfer) (auto simp add: sints_def)

lemma td_ext_sint:
"td_ext (sint :: 'a word ⇒ int) word_of_int (sints (LENGTH('a::len)))
(λw. (w + 2 ^ (LENGTH('a) - 1)) mod 2 ^ LENGTH('a) -
2 ^ (LENGTH('a) - 1))"
using td_ext_sbin [where ?'a = 'a] by (simp add: no_sbintr_alt2)

text ‹
We do ‹sint› before ‹sbin›, before ‹sint› is the user version
and interpretations do not produce thm duplicates. I.e.
we get the name ‹word_sint.Rep_eqD›, but not ‹word_sbin.Req_eqD›,
because the latter is the same thm as the former.
›
interpretation word_sint:
td_ext
"sint ::'a::len word ⇒ int"
word_of_int
"sints (LENGTH('a::len))"
"λw. (w + 2^(LENGTH('a::len) - 1)) mod 2^LENGTH('a::len) -
2 ^ (LENGTH('a::len) - 1)"
by (rule td_ext_sint)

interpretation word_sbin:
td_ext
"sint ::'a::len word ⇒ int"
word_of_int
"sints (LENGTH('a::len))"
"signed_take_bit (LENGTH('a::len) - 1)"
by (rule td_ext_sbin)

lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm]

lemmas td_sint = word_sint.td

lemma uints_mod: "uints n = range (λw. w mod 2 ^ n)"
by (fact uints_def [unfolded no_bintr_alt1])

lemmas bintr_num =
word_ubin.norm_eq_iff [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
lemmas sbintr_num =
word_sbin.norm_eq_iff [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b

lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]]
lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]]

interpretation test_bit:
td_ext
"bit :: 'a::len word ⇒ nat ⇒ bool"
set_bits
"{f. ∀i. f i ⟶ i < LENGTH('a::len)}"
"(λh i. h i ∧ i < LENGTH('a::len))"
by standard (auto simp add: bit_imp_le_length bit_set_bits_word_iff set_bits_bit_eq)

lemmas td_nth = test_bit.td_thm

lemma sints_subset:
"m ≤ n ⟹ sints m ⊆ sints n"
apply clarsimp
apply (rule conjI)
apply (erule order_trans[rotated])
apply simp
apply (erule order_less_le_trans)
apply simp
done

lemma uints_mono_iff: "uints l ⊆ uints m ⟷ l ≤ m"
using power_increasing_iff[of "2::int" l m]
apply (auto simp: uints_num subset_iff simp del: power_increasing_iff)
apply (meson less_irrefl not_le zero_le_numeral zero_le_power)
done

lemmas uints_monoI = uints_mono_iff[THEN iffD2]

lemma Bit_in_uints_Suc: "of_bool c + 2 * w ∈ uints (Suc m)" if "w ∈ uints m"
using that
by (auto simp: uints_num)

lemma Bit_in_uintsI: "of_bool c + 2 * w ∈ uints m" if "w ∈ uints (m - 1)" "m > 0"
using Bit_in_uints_Suc[OF that(1)] that(2)
by auto

lemma bin_cat_in_uintsI:
‹concat_bit n b a ∈ uints m› if ‹a ∈ uints l› ‹l + n ≤ m›
proof -
from ‹a ∈ uints l› have ‹0 ≤ a› ‹a < 2 ^ l›
by (auto simp add: uints_def range_bintrunc)
define q where ‹q = m - n›
with ‹l + n ≤ m› have ‹m = n + q›
by simp
from ‹q = m - n› ‹l + n ≤ m› have ‹l ≤ q›
by simp
then have ‹(2::int) ^ l ≤ 2 ^ q›
by simp
with ‹a < 2 ^ l› have ‹a < 2 ^ q›
by linarith
have ‹take_bit n b < 2 ^ n * 2 ^ q›
using take_bit_int_less_exp [of n b]
by (rule order.strict_trans2) simp
then have ‹take_bit n b < 2 ^ (n + q)›