author | haftmann |
Wed, 13 Jul 2011 23:49:56 +0200 | |
changeset 43816 | 05ab37be94ed |
parent 37653 | 847e95ca9b0a |
child 46236 | ae79f2978a67 |
permissions | -rw-r--r-- |
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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 Cardinality |
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begin |
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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 {* Locales for 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 field_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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351 |
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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 |