author | nipkow |
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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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section \<open>Numeral Syntax for Types\<close> |
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theory Numeral_Type |
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imports Cardinality |
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begin |
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subsection \<open>Numeral Types\<close> |
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typedef num0 = "UNIV :: nat set" .. |
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typedef num1 = "UNIV :: unit set" .. |
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typedef '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 '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 infinite_num0: "\<not> finite (UNIV :: num0 set)" |
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using card_num0[unfolded card_eq_0_iff] |
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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_UNIV_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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instantiation num1 :: CARD_1 |
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begin |
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instance |
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proof |
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show "CARD(num1) = 1" by auto |
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qed |
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end |
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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 \<open>Locales for for modular arithmetic subtypes\<close> |
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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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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 mod_simps field_simps) |
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done |
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end |
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locale mod_ring = mod_type n Rep Abs |
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for n :: int |
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and Rep :: "'a::{comm_ring_1} \<Rightarrow> int" |
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and Abs :: "int \<Rightarrow> 'a::{comm_ring_1}" |
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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 mod_simps ac_simps) |
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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 mod_simps) |
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done |
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lemma Rep_numeral: |
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"Rep (numeral w) = numeral w mod n" |
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using of_int_eq [of "numeral w"] |
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by (simp add: Rep_inject_sym Rep_Abs_mod) |
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lemma iszero_numeral: |
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"iszero (numeral w::'a) \<longleftrightarrow> numeral w mod n = 0" |
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by (simp add: Rep_inject_sym Rep_numeral Rep_0 iszero_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) |
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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 \<open>Ring class instances\<close> |
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text \<open> |
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Unfortunately \<open>ring_1\<close> instance is not possible for |
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\<^typ>\<open>num1\<close>, since 0 and 1 are not distinct. |
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\<close> |
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instantiation num1 :: "{comm_ring,comm_monoid_mult,numeral}" |
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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 |
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by standard (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: 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: 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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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 \<open>Set up cases, induction, and arithmetic\<close> |
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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_numeral [simp] = bit0.iszero_numeral |
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lemmas bit1_iszero_numeral [simp] = bit1.iszero_numeral |
29997 | 295 |
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lemmas [simp] = eq_numeral_iff_iszero [where 'a="'a bit0"] for dummy :: "'a::finite" |
297 |
lemmas [simp] = eq_numeral_iff_iszero [where 'a="'a bit1"] for dummy :: "'a::finite" |
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subsection \<open>Order instances\<close> |
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instantiation bit0 and bit1 :: (finite) linorder begin |
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definition "a < b \<longleftrightarrow> Rep_bit0 a < Rep_bit0 b" |
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definition "a \<le> b \<longleftrightarrow> Rep_bit0 a \<le> Rep_bit0 b" |
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definition "a < b \<longleftrightarrow> Rep_bit1 a < Rep_bit1 b" |
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definition "a \<le> b \<longleftrightarrow> Rep_bit1 a \<le> Rep_bit1 b" |
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instance |
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by(intro_classes) |
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(auto simp add: less_eq_bit0_def less_bit0_def less_eq_bit1_def less_bit1_def Rep_bit0_inject Rep_bit1_inject) |
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end |
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lemma (in preorder) tranclp_less: "(<) \<^sup>+\<^sup>+ = (<)" |
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by(auto simp add: fun_eq_iff intro: less_trans elim: tranclp.induct) |
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instance bit0 and bit1 :: (finite) wellorder |
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proof - |
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have "wf {(x :: 'a bit0, y). x < y}" |
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by(auto simp add: trancl_def tranclp_less intro!: finite_acyclic_wf acyclicI) |
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thus "OFCLASS('a bit0, wellorder_class)" |
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by(rule wf_wellorderI) intro_classes |
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next |
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have "wf {(x :: 'a bit1, y). x < y}" |
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by(auto simp add: trancl_def tranclp_less intro!: finite_acyclic_wf acyclicI) |
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thus "OFCLASS('a bit1, wellorder_class)" |
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by(rule wf_wellorderI) intro_classes |
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qed |
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60500 | 328 |
subsection \<open>Code setup and type classes for code generation\<close> |
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text \<open>Code setup for \<^typ>\<open>num0\<close> and \<^typ>\<open>num1\<close>\<close> |
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definition Num0 :: num0 where "Num0 = Abs_num0 0" |
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code_datatype Num0 |
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instantiation num0 :: equal begin |
52143 | 336 |
definition equal_num0 :: "num0 \<Rightarrow> num0 \<Rightarrow> bool" |
67399 | 337 |
where "equal_num0 = (=)" |
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instance by intro_classes (simp add: equal_num0_def) |
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end |
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lemma equal_num0_code [code]: |
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"equal_class.equal Num0 Num0 = True" |
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by(rule equal_refl) |
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code_datatype "1 :: num1" |
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instantiation num1 :: equal begin |
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definition equal_num1 :: "num1 \<Rightarrow> num1 \<Rightarrow> bool" |
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where "equal_num1 = (=)" |
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instance by intro_classes (simp add: equal_num1_def) |
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end |
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lemma equal_num1_code [code]: |
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"equal_class.equal (1 :: num1) 1 = True" |
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by(rule equal_refl) |
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instantiation num1 :: enum begin |
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definition "enum_class.enum = [1 :: num1]" |
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definition "enum_class.enum_all P = P (1 :: num1)" |
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definition "enum_class.enum_ex P = P (1 :: num1)" |
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instance |
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by intro_classes |
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(auto simp add: enum_num1_def enum_all_num1_def enum_ex_num1_def num1_eq_iff Ball_def, |
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(metis (full_types) num1_eq_iff)+) |
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end |
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instantiation num0 and num1 :: card_UNIV begin |
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definition "finite_UNIV = Phantom(num0) False" |
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definition "card_UNIV = Phantom(num0) 0" |
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definition "finite_UNIV = Phantom(num1) True" |
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definition "card_UNIV = Phantom(num1) 1" |
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instance |
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by intro_classes |
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(simp_all add: finite_UNIV_num0_def card_UNIV_num0_def infinite_num0 finite_UNIV_num1_def card_UNIV_num1_def) |
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end |
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|
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text \<open>Code setup for \<^typ>\<open>'a bit0\<close> and \<^typ>\<open>'a bit1\<close>\<close> |
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|
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declare |
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bit0.Rep_inverse[code abstype] |
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bit0.Rep_0[code abstract] |
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bit0.Rep_1[code abstract] |
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|
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lemma Abs_bit0'_code [code abstract]: |
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"Rep_bit0 (Abs_bit0' x :: 'a :: finite bit0) = x mod int (CARD('a bit0))" |
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by(auto simp add: Abs_bit0'_def intro!: Abs_bit0_inverse) |
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|
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lemma inj_on_Abs_bit0: |
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"inj_on (Abs_bit0 :: int \<Rightarrow> 'a bit0) {0..<2 * int CARD('a :: finite)}" |
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by(auto intro: inj_onI simp add: Abs_bit0_inject) |
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393 |
declare |
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bit1.Rep_inverse[code abstype] |
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bit1.Rep_0[code abstract] |
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bit1.Rep_1[code abstract] |
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397 |
|
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398 |
lemma Abs_bit1'_code [code abstract]: |
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399 |
"Rep_bit1 (Abs_bit1' x :: 'a :: finite bit1) = x mod int (CARD('a bit1))" |
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by(auto simp add: Abs_bit1'_def intro!: Abs_bit1_inverse) |
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|
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lemma inj_on_Abs_bit1: |
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"inj_on (Abs_bit1 :: int \<Rightarrow> 'a bit1) {0..<1 + 2 * int CARD('a :: finite)}" |
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by(auto intro: inj_onI simp add: Abs_bit1_inject) |
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405 |
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instantiation bit0 and bit1 :: (finite) equal begin |
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407 |
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definition "equal_class.equal x y \<longleftrightarrow> Rep_bit0 x = Rep_bit0 y" |
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definition "equal_class.equal x y \<longleftrightarrow> Rep_bit1 x = Rep_bit1 y" |
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410 |
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instance |
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more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
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diff
changeset
|
412 |
by intro_classes (simp_all add: equal_bit0_def equal_bit1_def Rep_bit0_inject Rep_bit1_inject) |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
413 |
|
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
414 |
end |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
415 |
|
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
416 |
instantiation bit0 :: (finite) enum begin |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
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diff
changeset
|
417 |
definition "(enum_class.enum :: 'a bit0 list) = map (Abs_bit0' \<circ> int) (upt 0 (CARD('a bit0)))" |
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more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
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diff
changeset
|
418 |
definition "enum_class.enum_all P = (\<forall>b :: 'a bit0 \<in> set enum_class.enum. P b)" |
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more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
49834
diff
changeset
|
419 |
definition "enum_class.enum_ex P = (\<exists>b :: 'a bit0 \<in> set enum_class.enum. P b)" |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
49834
diff
changeset
|
420 |
|
69661 | 421 |
instance proof |
51153
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more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
422 |
show "distinct (enum_class.enum :: 'a bit0 list)" |
66936 | 423 |
by (simp add: enum_bit0_def distinct_map inj_on_def Abs_bit0'_def Abs_bit0_inject) |
51153
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more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
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diff
changeset
|
424 |
|
69661 | 425 |
let ?Abs = "Abs_bit0 :: _ \<Rightarrow> 'a bit0" |
426 |
interpret type_definition Rep_bit0 ?Abs "{0..<2 * int CARD('a)}" |
|
427 |
by (fact type_definition_bit0) |
|
428 |
have "UNIV = ?Abs ` {0..<2 * int CARD('a)}" |
|
429 |
by (simp add: Abs_image) |
|
430 |
also have "\<dots> = ?Abs ` (int ` {0..<2 * CARD('a)})" |
|
431 |
by (simp add: image_int_atLeastLessThan) |
|
432 |
also have "\<dots> = (?Abs \<circ> int) ` {0..<2 * CARD('a)}" |
|
433 |
by (simp add: image_image cong: image_cong) |
|
434 |
also have "\<dots> = set enum_class.enum" |
|
435 |
by (simp add: enum_bit0_def Abs_bit0'_def cong: image_cong_simp) |
|
436 |
finally show univ_eq: "(UNIV :: 'a bit0 set) = set enum_class.enum" . |
|
51153
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more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
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diff
changeset
|
437 |
|
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
438 |
fix P :: "'a bit0 \<Rightarrow> bool" |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
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diff
changeset
|
439 |
show "enum_class.enum_all P = Ball UNIV P" |
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more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
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diff
changeset
|
440 |
and "enum_class.enum_ex P = Bex UNIV P" |
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more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
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diff
changeset
|
441 |
by(simp_all add: enum_all_bit0_def enum_ex_bit0_def univ_eq) |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
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diff
changeset
|
442 |
qed |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
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diff
changeset
|
443 |
|
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
444 |
end |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
445 |
|
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
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diff
changeset
|
446 |
instantiation bit1 :: (finite) enum begin |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
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diff
changeset
|
447 |
definition "(enum_class.enum :: 'a bit1 list) = map (Abs_bit1' \<circ> int) (upt 0 (CARD('a bit1)))" |
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more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
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diff
changeset
|
448 |
definition "enum_class.enum_all P = (\<forall>b :: 'a bit1 \<in> set enum_class.enum. P b)" |
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more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
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diff
changeset
|
449 |
definition "enum_class.enum_ex P = (\<exists>b :: 'a bit1 \<in> set enum_class.enum. P b)" |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
450 |
|
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
49834
diff
changeset
|
451 |
instance |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
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diff
changeset
|
452 |
proof(intro_classes) |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
49834
diff
changeset
|
453 |
show "distinct (enum_class.enum :: 'a bit1 list)" |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
49834
diff
changeset
|
454 |
by(simp only: Abs_bit1'_def zmod_int[symmetric] enum_bit1_def distinct_map Suc_eq_plus1 card_bit1 o_apply inj_on_def) |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
49834
diff
changeset
|
455 |
(clarsimp simp add: Abs_bit1_inject) |
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more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
49834
diff
changeset
|
456 |
|
69661 | 457 |
let ?Abs = "Abs_bit1 :: _ \<Rightarrow> 'a bit1" |
458 |
interpret type_definition Rep_bit1 ?Abs "{0..<1 + 2 * int CARD('a)}" |
|
459 |
by (fact type_definition_bit1) |
|
460 |
have "UNIV = ?Abs ` {0..<1 + 2 * int CARD('a)}" |
|
461 |
by (simp add: Abs_image) |
|
462 |
also have "\<dots> = ?Abs ` (int ` {0..<1 + 2 * CARD('a)})" |
|
463 |
by (simp add: image_int_atLeastLessThan) |
|
464 |
also have "\<dots> = (?Abs \<circ> int) ` {0..<1 + 2 * CARD('a)}" |
|
465 |
by (simp add: image_image cong: image_cong) |
|
466 |
finally show univ_eq: "(UNIV :: 'a bit1 set) = set enum_class.enum" |
|
467 |
by (simp only: enum_bit1_def set_map set_upt) (simp add: Abs_bit1'_def cong: image_cong_simp) |
|
51153
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more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
49834
diff
changeset
|
468 |
|
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
469 |
fix P :: "'a bit1 \<Rightarrow> bool" |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
470 |
show "enum_class.enum_all P = Ball UNIV P" |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
49834
diff
changeset
|
471 |
and "enum_class.enum_ex P = Bex UNIV P" |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
472 |
by(simp_all add: enum_all_bit1_def enum_ex_bit1_def univ_eq) |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
473 |
qed |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
474 |
|
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
475 |
end |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
476 |
|
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
477 |
instantiation bit0 and bit1 :: (finite) finite_UNIV begin |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
478 |
definition "finite_UNIV = Phantom('a bit0) True" |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
49834
diff
changeset
|
479 |
definition "finite_UNIV = Phantom('a bit1) True" |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
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parents:
49834
diff
changeset
|
480 |
instance by intro_classes (simp_all add: finite_UNIV_bit0_def finite_UNIV_bit1_def) |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
481 |
end |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
482 |
|
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
483 |
instantiation bit0 and bit1 :: ("{finite,card_UNIV}") card_UNIV begin |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
484 |
definition "card_UNIV = Phantom('a bit0) (2 * of_phantom (card_UNIV :: 'a card_UNIV))" |
51175
9f472d5f112c
simplify definition as sort constraints ensure finiteness (thanks to Jesus Aransay)
Andreas Lochbihler
parents:
51153
diff
changeset
|
485 |
definition "card_UNIV = Phantom('a bit1) (1 + 2 * of_phantom (card_UNIV :: 'a card_UNIV))" |
51153
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
486 |
instance by intro_classes (simp_all add: card_UNIV_bit0_def card_UNIV_bit1_def card_UNIV) |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
487 |
end |
b14ee572cc7b
more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents:
49834
diff
changeset
|
488 |
|
60500 | 489 |
subsection \<open>Syntax\<close> |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
490 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
491 |
syntax |
46236
ae79f2978a67
position constraints for numerals enable PIDE markup;
wenzelm
parents:
37653
diff
changeset
|
492 |
"_NumeralType" :: "num_token => type" ("_") |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
493 |
"_NumeralType0" :: type ("0") |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
494 |
"_NumeralType1" :: type ("1") |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
495 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
496 |
translations |
35362 | 497 |
(type) "1" == (type) "num1" |
498 |
(type) "0" == (type) "num0" |
|
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
499 |
|
60500 | 500 |
parse_translation \<open> |
52143 | 501 |
let |
502 |
fun mk_bintype n = |
|
503 |
let |
|
69593 | 504 |
fun mk_bit 0 = Syntax.const \<^type_syntax>\<open>bit0\<close> |
505 |
| mk_bit 1 = Syntax.const \<^type_syntax>\<open>bit1\<close>; |
|
52143 | 506 |
fun bin_of n = |
69593 | 507 |
if n = 1 then Syntax.const \<^type_syntax>\<open>num1\<close> |
508 |
else if n = 0 then Syntax.const \<^type_syntax>\<open>num0\<close> |
|
52143 | 509 |
else if n = ~1 then raise TERM ("negative type numeral", []) |
510 |
else |
|
511 |
let val (q, r) = Integer.div_mod n 2; |
|
512 |
in mk_bit r $ bin_of q end; |
|
513 |
in bin_of n end; |
|
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
514 |
|
52143 | 515 |
fun numeral_tr [Free (str, _)] = mk_bintype (the (Int.fromString str)) |
516 |
| numeral_tr ts = raise TERM ("numeral_tr", ts); |
|
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
517 |
|
69593 | 518 |
in [(\<^syntax_const>\<open>_NumeralType\<close>, K numeral_tr)] end |
60500 | 519 |
\<close> |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
520 |
|
60500 | 521 |
print_translation \<open> |
52143 | 522 |
let |
523 |
fun int_of [] = 0 |
|
524 |
| int_of (b :: bs) = b + 2 * int_of bs; |
|
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
525 |
|
69593 | 526 |
fun bin_of (Const (\<^type_syntax>\<open>num0\<close>, _)) = [] |
527 |
| bin_of (Const (\<^type_syntax>\<open>num1\<close>, _)) = [1] |
|
528 |
| bin_of (Const (\<^type_syntax>\<open>bit0\<close>, _) $ bs) = 0 :: bin_of bs |
|
529 |
| bin_of (Const (\<^type_syntax>\<open>bit1\<close>, _) $ bs) = 1 :: bin_of bs |
|
52143 | 530 |
| bin_of t = raise TERM ("bin_of", [t]); |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
531 |
|
52143 | 532 |
fun bit_tr' b [t] = |
533 |
let |
|
534 |
val rev_digs = b :: bin_of t handle TERM _ => raise Match |
|
535 |
val i = int_of rev_digs; |
|
536 |
val num = string_of_int (abs i); |
|
537 |
in |
|
69593 | 538 |
Syntax.const \<^syntax_const>\<open>_NumeralType\<close> $ Syntax.free num |
52143 | 539 |
end |
540 |
| bit_tr' b _ = raise Match; |
|
541 |
in |
|
69593 | 542 |
[(\<^type_syntax>\<open>bit0\<close>, K (bit_tr' 0)), |
543 |
(\<^type_syntax>\<open>bit1\<close>, K (bit_tr' 1))] |
|
69216
1a52baa70aed
clarified ML_Context.expression: it is a closed expression, not a let-declaration -- thus source positions are more accurate (amending d8849cfad60f, 162a4c2e97bc);
wenzelm
parents:
67411
diff
changeset
|
544 |
end |
60500 | 545 |
\<close> |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
546 |
|
60500 | 547 |
subsection \<open>Examples\<close> |
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
548 |
|
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
549 |
lemma "CARD(0) = 0" by simp |
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
550 |
lemma "CARD(17) = 17" by simp |
29997 | 551 |
lemma "8 * 11 ^ 3 - 6 = (2::5)" by simp |
28920 | 552 |
|
24332
e3a2b75b1cf9
boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff
changeset
|
553 |
end |