(*
Author: Jeremy Dawson and Gerwin Klein, NICTA
Consequences of type definition theorems, and of extended type definition.
*)
section \<open>Type Definition Theorems\<close>
theory Misc_Typedef
imports Main Word Bit_Comprehension Bits_Int
begin
subsection "More lemmas about normal type definitions"
lemma tdD1: "type_definition Rep Abs A \<Longrightarrow> \<forall>x. Rep x \<in> A"
and tdD2: "type_definition Rep Abs A \<Longrightarrow> \<forall>x. Abs (Rep x) = x"
and tdD3: "type_definition Rep Abs A \<Longrightarrow> \<forall>y. y \<in> A \<longrightarrow> Rep (Abs y) = y"
by (auto simp: type_definition_def)
lemma td_nat_int: "type_definition int nat (Collect ((\<le>) 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 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x = y"
by (simp add: Abs_inject)
lemma Abs_inverse': "r \<in> A \<Longrightarrow> Abs r = a \<Longrightarrow> Rep a = r"
by (safe elim!: Abs_inverse)
lemma Rep_comp_inverse: "Rep \<circ> f = g \<Longrightarrow> Abs \<circ> g = f"
using Rep_inverse by auto
lemma Rep_eqD [elim!]: "Rep x = Rep y \<Longrightarrow> x = y"
by simp
lemma Rep_inverse': "Rep a = r \<Longrightarrow> Abs r = a"
by (safe intro!: Rep_inverse)
lemma comp_Abs_inverse: "f \<circ> Abs = g \<Longrightarrow> g \<circ> Rep = f"
using Rep_inverse by auto
lemma set_Rep: "A = range Rep"
proof (rule set_eqI)
show "x \<in> A \<longleftrightarrow> x \<in> range Rep" for x
by (auto dest: Abs_inverse [of x, symmetric])
qed
lemma set_Rep_Abs: "A = range (Rep \<circ> Abs)"
proof (rule set_eqI)
show "x \<in> A \<longleftrightarrow> x \<in> range (Rep \<circ> 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 \<circ> fa = fr \<circ> Rep \<or> fa \<circ> Abs = Abs \<circ> fr \<Longrightarrow> Abs \<circ> fr \<circ> 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 \<circ> fa \<circ> Abs = fr \<Longrightarrow> Rep \<circ> fa = fr \<circ> Rep \<and> fa \<circ> Abs = Abs \<circ> fr"
by auto
end
interpretation nat_int: type_definition int nat "Collect ((\<le>) 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 \<circ> norm = norm \<Longrightarrow>
fr \<circ> norm = norm \<circ> fr \<longleftrightarrow> norm \<circ> fr \<circ> norm = fr \<circ> norm \<and> norm \<circ> fr \<circ> norm = norm \<circ> fr"
apply safe
apply (simp_all add: comp_assoc)
apply (simp_all add: o_assoc)
done
lemma fn_comm_power: "fa \<circ> tr = tr \<circ> fr \<Longrightarrow> fa ^^ n \<circ> tr = tr \<circ> 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: "\<And>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 \<circ> Abs = f \<Longrightarrow> f (Rep x) = g x"
by (drule comp_Abs_inverse [symmetric]) simp
lemma eq_norm': "Rep \<circ> 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 \<in> A \<longleftrightarrow> w = norm w"
by (auto simp: set_Rep_Abs eq_norm' eq_norm [symmetric])
lemma inverse_norm: "Abs n = w \<longleftrightarrow> Rep w = norm n"
apply (rule iffI)
apply (clarsimp simp add: eq_norm)
apply (simp add: eq_norm' [symmetric])
done
lemma norm_eq_iff: "norm x = norm y \<longleftrightarrow> Abs x = Abs y"
by (simp add: eq_norm' [symmetric])
lemma norm_comps:
"Abs \<circ> norm = Abs"
"norm \<circ> Rep = Rep"
"norm \<circ> norm = norm"
by (auto simp: eq_norm' [symmetric] o_def)
lemmas norm_norm [simp] = norm_comps
lemma fns5: "Rep \<circ> fa \<circ> Abs = fr \<Longrightarrow> fr \<circ> norm = fr \<and> norm \<circ> fr = fr"
by (fold eq_norm') auto
text \<open>
following give conditions for converses to \<open>td_fns1\<close>
\<^item> the condition \<open>norm \<circ> fr \<circ> norm = fr \<circ> norm\<close> says that
\<open>fr\<close> takes normalised arguments to normalised results
\<^item> \<open>norm \<circ> fr \<circ> norm = norm \<circ> fr\<close> says that \<open>fr\<close>
takes norm-equivalent arguments to norm-equivalent results
\<^item> \<open>fr \<circ> norm = fr\<close> says that \<open>fr\<close>
takes norm-equivalent arguments to the same result
\<^item> \<open>norm \<circ> fr = fr\<close> says that \<open>fr\<close> takes any argument to a normalised result
\<close>
lemma fns2: "Abs \<circ> fr \<circ> Rep = fa \<Longrightarrow> norm \<circ> fr \<circ> norm = fr \<circ> norm \<longleftrightarrow> Rep \<circ> fa = fr \<circ> Rep"
apply (fold eq_norm')
apply safe
prefer 2
apply (simp add: o_assoc)
apply (rule ext)
apply (drule_tac x="Rep x" in fun_cong)
apply auto
done
lemma fns3: "Abs \<circ> fr \<circ> Rep = fa \<Longrightarrow> norm \<circ> fr \<circ> norm = norm \<circ> fr \<longleftrightarrow> fa \<circ> Abs = Abs \<circ> fr"
apply (fold eq_norm')
apply safe
prefer 2
apply (simp add: comp_assoc)
apply (rule ext)
apply (drule_tac f="a \<circ> b" for a b in fun_cong)
apply simp
done
lemma fns: "fr \<circ> norm = norm \<circ> fr \<Longrightarrow> fa \<circ> Abs = Abs \<circ> fr \<longleftrightarrow> Rep \<circ> fa = fr \<circ> Rep"
apply safe
apply (frule fns1b)
prefer 2
apply (frule fns1a)
apply (rule fns3 [THEN iffD1])
prefer 3
apply (rule fns2 [THEN iffD1])
apply (simp_all add: comp_assoc)
apply (simp_all add: o_assoc)
done
lemma range_norm: "range (Rep \<circ> Abs) = A"
by (simp add: set_Rep_Abs)
end
lemmas td_ext_def' =
td_ext_def [unfolded type_definition_def td_ext_axioms_def]
subsection \<open>Type-definition locale instantiations\<close>
definition uints :: "nat \<Rightarrow> int set"
\<comment> \<open>the sets of integers representing the words\<close>
where "uints n = range (take_bit n)"
definition sints :: "nat \<Rightarrow> int set"
where "sints n = range (signed_take_bit (n - 1))"
lemma uints_num: "uints n = {i. 0 \<le> i \<and> i < 2 ^ n}"
by (simp add: uints_def range_bintrunc)
lemma sints_num: "sints n = {i. - (2 ^ (n - 1)) \<le> i \<and> i < 2 ^ (n - 1)}"
by (simp add: sints_def range_sbintrunc)
definition unats :: "nat \<Rightarrow> nat set"
where "unats n = {i. i < 2 ^ n}"
\<comment> \<open>naturals\<close>
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 \<Rightarrow> int) word_of_int (uints (LENGTH('a::len)))
(\<lambda>w::int. w mod 2 ^ LENGTH('a))"
apply (unfold td_ext_def')
apply transfer
apply (simp add: uints_num take_bit_eq_mod)
done
interpretation word_uint:
td_ext
"uint::'a::len word \<Rightarrow> int"
word_of_int
"uints (LENGTH('a::len))"
"\<lambda>w. 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 \<Rightarrow> 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 \<Rightarrow> 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) \<Longrightarrow>
td_ext (unat :: 'a word \<Rightarrow> nat) of_nat (unats n) (\<lambda>i. 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 \<Rightarrow> nat"
of_nat
"unats (LENGTH('a::len))"
"\<lambda>i. i mod 2 ^ LENGTH('a::len)"
by (rule td_ext_unat)
lemmas td_unat = word_unat.td_thm
lemmas unat_lt2p [iff] = word_unat.Rep [unfolded unats_def mem_Collect_eq]
lemma unat_le: "y \<le> unat z \<Longrightarrow> y \<in> unats (LENGTH('a))"
for z :: "'a::len word"
apply (unfold unats_def)
apply clarsimp
apply (rule xtrans, rule unat_lt2p, assumption)
done
lemma td_ext_sbin:
"td_ext (sint :: 'a word \<Rightarrow> 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 \<Rightarrow> int) word_of_int (sints (LENGTH('a::len)))
(\<lambda>w. (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 \<open>
We do \<open>sint\<close> before \<open>sbin\<close>, before \<open>sint\<close> is the user version
and interpretations do not produce thm duplicates. I.e.
we get the name \<open>word_sint.Rep_eqD\<close>, but not \<open>word_sbin.Req_eqD\<close>,
because the latter is the same thm as the former.
\<close>
interpretation word_sint:
td_ext
"sint ::'a::len word \<Rightarrow> int"
word_of_int
"sints (LENGTH('a::len))"
"\<lambda>w. (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 \<Rightarrow> 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 (\<lambda>w. w mod 2 ^ n)"
by (fact uints_def [unfolded no_bintr_alt1])
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq]
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq]
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
"(!!) :: 'a::len word \<Rightarrow> nat \<Rightarrow> bool"
set_bits
"{f. \<forall>i. f i \<longrightarrow> i < LENGTH('a::len)}"
"(\<lambda>h i. h i \<and> i < LENGTH('a::len))"
by standard (auto simp add: test_bit_word_eq bit_imp_le_length bit_set_bits_word_iff set_bits_bit_eq)
lemmas td_nth = test_bit.td_thm
end