src/HOL/Library/Numeral_Type.thy
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(*  Title:      HOL/Library/Numeral_Type.thy
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    Author:     Brian Huffman
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*)
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header {* Numeral Syntax for Types *}
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theory Numeral_Type
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imports Main
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begin
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subsection {* Preliminary lemmas *}
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(* These should be moved elsewhere *)
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lemma (in type_definition) univ:
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  "UNIV = Abs ` A"
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proof
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  show "Abs ` A \<subseteq> UNIV" by (rule subset_UNIV)
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  show "UNIV \<subseteq> Abs ` A"
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  proof
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    fix x :: 'b
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    have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
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    moreover have "Rep x \<in> A" by (rule Rep)
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    ultimately show "x \<in> Abs ` A" by (rule image_eqI)
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  qed
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qed
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lemma (in type_definition) card: "card (UNIV :: 'b set) = card A"
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  by (simp add: univ card_image inj_on_def Abs_inject)
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subsection {* Cardinalities of types *}
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syntax "_type_card" :: "type => nat" ("(1CARD/(1'(_')))")
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translations "CARD(t)" => "CONST card (CONST UNIV \<Colon> t set)"
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typed_print_translation {*
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let
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  fun card_univ_tr' show_sorts _ [Const (@{const_syntax UNIV}, Type(_, [T, _]))] =
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    Syntax.const @{syntax_const "_type_card"} $ Syntax.term_of_typ show_sorts T;
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in [(@{const_syntax card}, card_univ_tr')]
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end
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*}
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lemma card_unit [simp]: "CARD(unit) = 1"
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  unfolding UNIV_unit by simp
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lemma card_bool [simp]: "CARD(bool) = 2"
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  unfolding UNIV_bool by simp
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lemma card_prod [simp]: "CARD('a \<times> 'b) = CARD('a::finite) * CARD('b::finite)"
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  unfolding UNIV_Times_UNIV [symmetric] by (simp only: card_cartesian_product)
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lemma card_sum [simp]: "CARD('a + 'b) = CARD('a::finite) + CARD('b::finite)"
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  unfolding UNIV_Plus_UNIV [symmetric] by (simp only: finite card_Plus)
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lemma card_option [simp]: "CARD('a option) = Suc CARD('a::finite)"
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  unfolding UNIV_option_conv
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  apply (subgoal_tac "(None::'a option) \<notin> range Some")
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  apply (simp add: card_image)
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  apply fast
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  done
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lemma card_set [simp]: "CARD('a set) = 2 ^ CARD('a::finite)"
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  unfolding Pow_UNIV [symmetric]
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  by (simp only: card_Pow finite numeral_2_eq_2)
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lemma card_nat [simp]: "CARD(nat) = 0"
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  by (simp add: infinite_UNIV_nat card_eq_0_iff)
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subsection {* Classes with at least 1 and 2  *}
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text {* Class finite already captures "at least 1" *}
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lemma zero_less_card_finite [simp]: "0 < CARD('a::finite)"
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  unfolding neq0_conv [symmetric] by simp
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lemma one_le_card_finite [simp]: "Suc 0 \<le> CARD('a::finite)"
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  by (simp add: less_Suc_eq_le [symmetric])
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text {* Class for cardinality "at least 2" *}
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class card2 = finite + 
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  assumes two_le_card: "2 \<le> CARD('a)"
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lemma one_less_card: "Suc 0 < CARD('a::card2)"
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  using two_le_card [where 'a='a] by simp
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lemma one_less_int_card: "1 < int CARD('a::card2)"
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  using one_less_card [where 'a='a] by simp
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subsection {* Numeral Types *}
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typedef (open) num0 = "UNIV :: nat set" ..
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typedef (open) num1 = "UNIV :: unit set" ..
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typedef (open) 'a bit0 = "{0 ..< 2 * int CARD('a::finite)}"
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proof
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  show "0 \<in> {0 ..< 2 * int CARD('a)}"
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    by simp
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qed
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typedef (open) 'a bit1 = "{0 ..< 1 + 2 * int CARD('a::finite)}"
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proof
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  show "0 \<in> {0 ..< 1 + 2 * int CARD('a)}"
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    by simp
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qed
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lemma card_num0 [simp]: "CARD (num0) = 0"
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  unfolding type_definition.card [OF type_definition_num0]
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  by simp
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lemma card_num1 [simp]: "CARD(num1) = 1"
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  unfolding type_definition.card [OF type_definition_num1]
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  by (simp only: card_unit)
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lemma card_bit0 [simp]: "CARD('a bit0) = 2 * CARD('a::finite)"
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  unfolding type_definition.card [OF type_definition_bit0]
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  by simp
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lemma card_bit1 [simp]: "CARD('a bit1) = Suc (2 * CARD('a::finite))"
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  unfolding type_definition.card [OF type_definition_bit1]
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  by simp
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instance num1 :: finite
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proof
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  show "finite (UNIV::num1 set)"
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    unfolding type_definition.univ [OF type_definition_num1]
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    using finite by (rule finite_imageI)
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qed
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instance bit0 :: (finite) card2
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proof
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  show "finite (UNIV::'a bit0 set)"
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    unfolding type_definition.univ [OF type_definition_bit0]
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    by simp
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  show "2 \<le> CARD('a bit0)"
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    by simp
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qed
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instance bit1 :: (finite) card2
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proof
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  show "finite (UNIV::'a bit1 set)"
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    unfolding type_definition.univ [OF type_definition_bit1]
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    by simp
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  show "2 \<le> CARD('a bit1)"
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    by simp
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qed
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subsection {* Locale for modular arithmetic subtypes *}
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locale mod_type =
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  fixes n :: int
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  and Rep :: "'a::{zero,one,plus,times,uminus,minus} \<Rightarrow> int"
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  and Abs :: "int \<Rightarrow> 'a::{zero,one,plus,times,uminus,minus}"
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  assumes type: "type_definition Rep Abs {0..<n}"
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  and size1: "1 < n"
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  and zero_def: "0 = Abs 0"
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  and one_def:  "1 = Abs 1"
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  and add_def:  "x + y = Abs ((Rep x + Rep y) mod n)"
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  and mult_def: "x * y = Abs ((Rep x * Rep y) mod n)"
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  and diff_def: "x - y = Abs ((Rep x - Rep y) mod n)"
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  and minus_def: "- x = Abs ((- Rep x) mod n)"
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begin
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lemma size0: "0 < n"
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using size1 by simp
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lemmas definitions =
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  zero_def one_def add_def mult_def minus_def diff_def
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lemma Rep_less_n: "Rep x < n"
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by (rule type_definition.Rep [OF type, simplified, THEN conjunct2])
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lemma Rep_le_n: "Rep x \<le> n"
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by (rule Rep_less_n [THEN order_less_imp_le])
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lemma Rep_inject_sym: "x = y \<longleftrightarrow> Rep x = Rep y"
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by (rule type_definition.Rep_inject [OF type, symmetric])
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lemma Rep_inverse: "Abs (Rep x) = x"
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by (rule type_definition.Rep_inverse [OF type])
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lemma Abs_inverse: "m \<in> {0..<n} \<Longrightarrow> Rep (Abs m) = m"
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by (rule type_definition.Abs_inverse [OF type])
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lemma Rep_Abs_mod: "Rep (Abs (m mod n)) = m mod n"
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by (simp add: Abs_inverse pos_mod_conj [OF size0])
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lemma Rep_Abs_0: "Rep (Abs 0) = 0"
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by (simp add: Abs_inverse size0)
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lemma Rep_0: "Rep 0 = 0"
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by (simp add: zero_def Rep_Abs_0)
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lemma Rep_Abs_1: "Rep (Abs 1) = 1"
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by (simp add: Abs_inverse size1)
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lemma Rep_1: "Rep 1 = 1"
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by (simp add: one_def Rep_Abs_1)
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lemma Rep_mod: "Rep x mod n = Rep x"
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apply (rule_tac x=x in type_definition.Abs_cases [OF type])
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apply (simp add: type_definition.Abs_inverse [OF type])
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apply (simp add: mod_pos_pos_trivial)
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done
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lemmas Rep_simps =
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  Rep_inject_sym Rep_inverse Rep_Abs_mod Rep_mod Rep_Abs_0 Rep_Abs_1
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lemma comm_ring_1: "OFCLASS('a, comm_ring_1_class)"
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apply (intro_classes, unfold definitions)
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apply (simp_all add: Rep_simps zmod_simps ring_simps)
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done
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end
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locale mod_ring = mod_type +
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  constrains n :: int
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  and Rep :: "'a::{number_ring} \<Rightarrow> int"
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  and Abs :: "int \<Rightarrow> 'a::{number_ring}"
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begin
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lemma of_nat_eq: "of_nat k = Abs (int k mod n)"
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apply (induct k)
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apply (simp add: zero_def)
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apply (simp add: Rep_simps add_def one_def zmod_simps add_ac)
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done
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lemma of_int_eq: "of_int z = Abs (z mod n)"
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apply (cases z rule: int_diff_cases)
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apply (simp add: Rep_simps of_nat_eq diff_def zmod_simps)
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done
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lemma Rep_number_of:
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  "Rep (number_of w) = number_of w mod n"
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by (simp add: number_of_eq of_int_eq Rep_Abs_mod)
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lemma iszero_number_of:
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  "iszero (number_of w::'a) \<longleftrightarrow> number_of w mod n = 0"
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by (simp add: Rep_simps number_of_eq of_int_eq iszero_def zero_def)
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lemma cases:
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  assumes 1: "\<And>z. \<lbrakk>(x::'a) = of_int z; 0 \<le> z; z < n\<rbrakk> \<Longrightarrow> P"
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  shows "P"
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apply (cases x rule: type_definition.Abs_cases [OF type])
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apply (rule_tac z="y" in 1)
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apply (simp_all add: of_int_eq mod_pos_pos_trivial)
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done
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lemma induct:
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  "(\<And>z. \<lbrakk>0 \<le> z; z < n\<rbrakk> \<Longrightarrow> P (of_int z)) \<Longrightarrow> P (x::'a)"
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by (cases x rule: cases) simp
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end
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subsection {* Number ring instances *}
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text {*
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  Unfortunately a number ring instance is not possible for
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  @{typ num1}, since 0 and 1 are not distinct.
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*}
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instantiation num1 :: "{comm_ring,comm_monoid_mult,number}"
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begin
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lemma num1_eq_iff: "(x::num1) = (y::num1) \<longleftrightarrow> True"
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  by (induct x, induct y) simp
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instance proof
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qed (simp_all add: num1_eq_iff)
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end
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instantiation
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  bit0 and bit1 :: (finite) "{zero,one,plus,times,uminus,minus}"
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begin
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definition Abs_bit0' :: "int \<Rightarrow> 'a bit0" where
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  "Abs_bit0' x = Abs_bit0 (x mod int CARD('a bit0))"
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definition Abs_bit1' :: "int \<Rightarrow> 'a bit1" where
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  "Abs_bit1' x = Abs_bit1 (x mod int CARD('a bit1))"
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definition "0 = Abs_bit0 0"
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definition "1 = Abs_bit0 1"
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definition "x + y = Abs_bit0' (Rep_bit0 x + Rep_bit0 y)"
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definition "x * y = Abs_bit0' (Rep_bit0 x * Rep_bit0 y)"
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definition "x - y = Abs_bit0' (Rep_bit0 x - Rep_bit0 y)"
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definition "- x = Abs_bit0' (- Rep_bit0 x)"
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definition "0 = Abs_bit1 0"
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definition "1 = Abs_bit1 1"
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definition "x + y = Abs_bit1' (Rep_bit1 x + Rep_bit1 y)"
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definition "x * y = Abs_bit1' (Rep_bit1 x * Rep_bit1 y)"
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definition "x - y = Abs_bit1' (Rep_bit1 x - Rep_bit1 y)"
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definition "- x = Abs_bit1' (- Rep_bit1 x)"
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instance ..
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end
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interpretation bit0:
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  mod_type "int CARD('a::finite bit0)"
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           "Rep_bit0 :: 'a::finite bit0 \<Rightarrow> int"
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           "Abs_bit0 :: int \<Rightarrow> 'a::finite bit0"
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apply (rule mod_type.intro)
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apply (simp add: int_mult type_definition_bit0)
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apply (rule one_less_int_card)
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apply (rule zero_bit0_def)
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apply (rule one_bit0_def)
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apply (rule plus_bit0_def [unfolded Abs_bit0'_def])
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apply (rule times_bit0_def [unfolded Abs_bit0'_def])
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apply (rule minus_bit0_def [unfolded Abs_bit0'_def])
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apply (rule uminus_bit0_def [unfolded Abs_bit0'_def])
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done
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interpretation bit1:
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  mod_type "int CARD('a::finite bit1)"
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           "Rep_bit1 :: 'a::finite bit1 \<Rightarrow> int"
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           "Abs_bit1 :: int \<Rightarrow> 'a::finite bit1"
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apply (rule mod_type.intro)
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apply (simp add: int_mult type_definition_bit1)
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apply (rule one_less_int_card)
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apply (rule zero_bit1_def)
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apply (rule one_bit1_def)
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apply (rule plus_bit1_def [unfolded Abs_bit1'_def])
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apply (rule times_bit1_def [unfolded Abs_bit1'_def])
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apply (rule minus_bit1_def [unfolded Abs_bit1'_def])
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apply (rule uminus_bit1_def [unfolded Abs_bit1'_def])
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done
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instance bit0 :: (finite) comm_ring_1
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  by (rule bit0.comm_ring_1)+
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instance bit1 :: (finite) comm_ring_1
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  by (rule bit1.comm_ring_1)+
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instantiation bit0 and bit1 :: (finite) number_ring
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begin
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definition "(number_of w :: _ bit0) = of_int w"
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definition "(number_of w :: _ bit1) = of_int w"
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instance proof
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qed (rule number_of_bit0_def number_of_bit1_def)+
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end
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interpretation bit0:
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  mod_ring "int CARD('a::finite bit0)"
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           "Rep_bit0 :: 'a::finite bit0 \<Rightarrow> int"
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           "Abs_bit0 :: int \<Rightarrow> 'a::finite bit0"
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  ..
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interpretation bit1:
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  mod_ring "int CARD('a::finite bit1)"
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           "Rep_bit1 :: 'a::finite bit1 \<Rightarrow> int"
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           "Abs_bit1 :: int \<Rightarrow> 'a::finite bit1"
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  ..
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text {* Set up cases, induction, and arithmetic *}
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lemmas bit0_cases [case_names of_int, cases type: bit0] = bit0.cases
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lemmas bit1_cases [case_names of_int, cases type: bit1] = bit1.cases
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lemmas bit0_induct [case_names of_int, induct type: bit0] = bit0.induct
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lemmas bit1_induct [case_names of_int, induct type: bit1] = bit1.induct
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lemmas bit0_iszero_number_of [simp] = bit0.iszero_number_of
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lemmas bit1_iszero_number_of [simp] = bit1.iszero_number_of
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subsection {* Syntax *}
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syntax
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  "_NumeralType" :: "num_const => type"  ("_")
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  "_NumeralType0" :: type ("0")
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  "_NumeralType1" :: type ("1")
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translations
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  (type) "1" == (type) "num1"
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  (type) "0" == (type) "num0"
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parse_translation {*
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let
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fun mk_bintype n =
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  let
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    fun mk_bit 0 = Syntax.const @{type_syntax bit0}
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      | mk_bit 1 = Syntax.const @{type_syntax bit1};
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    fun bin_of n =
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      if n = 1 then Syntax.const @{type_syntax num1}
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      else if n = 0 then Syntax.const @{type_syntax num0}
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      else if n = ~1 then raise TERM ("negative type numeral", [])
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      else
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        let val (q, r) = Integer.div_mod n 2;
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        in mk_bit r $ bin_of q end;
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  in bin_of n end;
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fun numeral_tr (*"_NumeralType"*) [Const (str, _)] =
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      mk_bintype (the (Int.fromString str))
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  | numeral_tr (*"_NumeralType"*) ts = raise TERM ("numeral_tr", ts);
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in [(@{syntax_const "_NumeralType"}, numeral_tr)] end;
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*}
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print_translation {*
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let
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fun int_of [] = 0
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  | int_of (b :: bs) = b + 2 * int_of bs;
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fun bin_of (Const (@{type_syntax num0}, _)) = []
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  | bin_of (Const (@{type_syntax num1}, _)) = [1]
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  | bin_of (Const (@{type_syntax bit0}, _) $ bs) = 0 :: bin_of bs
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  | bin_of (Const (@{type_syntax bit1}, _) $ bs) = 1 :: bin_of bs
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  | bin_of t = raise TERM ("bin_of", [t]);
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fun bit_tr' b [t] =
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      let
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        val rev_digs = b :: bin_of t handle TERM _ => raise Match
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        val i = int_of rev_digs;
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        val num = string_of_int (abs i);
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      in
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        Syntax.const @{syntax_const "_NumeralType"} $ Syntax.free num
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      end
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  | bit_tr' b _ = raise Match;
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in [(@{type_syntax bit0}, bit_tr' 0), (@{type_syntax bit1}, bit_tr' 1)] end;
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*}
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subsection {* Examples *}
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lemma "CARD(0) = 0" by simp
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lemma "CARD(17) = 17" by simp
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lemma "8 * 11 ^ 3 - 6 = (2::5)" by simp
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end