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 Cardinality
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begin
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subsection {* Numeral Types *}
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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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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 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 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_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 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 {* Ring class instances *}
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text {*
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  Unfortunately @{text ring_1} 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,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 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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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_numeral [simp] = bit0.iszero_numeral
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lemmas bit1_iszero_numeral [simp] = bit1.iszero_numeral
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declare eq_numeral_iff_iszero [where 'a="('a::finite) bit0", standard, simp]
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declare eq_numeral_iff_iszero [where 'a="('a::finite) bit1", standard, simp]
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subsection {* Order instances *}
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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: "op <\<^sup>+\<^sup>+ = op <"
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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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subsection {* Code setup and type classes for code generation *}
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text {* Code setup for @{typ num0} and @{typ num1} *}
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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
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definition equal_num0 :: "num0 \<Rightarrow> num0 \<Rightarrow> bool"
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  where "equal_num0 = op ="
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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 = op ="
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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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text {* Code setup for @{typ "'a bit0"} and @{typ "'a bit1"} *}
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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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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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  (metis bit0.Rep_Abs_mod bit0.Rep_less_n card_bit0 of_nat_numeral zmult_int)
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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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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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lemma Abs_bit1'_code [code abstract]:
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  "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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  (metis of_nat_0_less_iff of_nat_Suc of_nat_mult of_nat_numeral pos_mod_conj zero_less_Suc)
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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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instantiation bit0 and bit1 :: (finite) equal begin
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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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instance
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  by intro_classes (simp_all add: equal_bit0_def equal_bit1_def Rep_bit0_inject Rep_bit1_inject)
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end
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   408
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instantiation bit0 :: (finite) enum begin
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definition "(enum_class.enum :: 'a bit0 list) = map (Abs_bit0' \<circ> int) (upt 0 (CARD('a bit0)))"
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definition "enum_class.enum_all P = (\<forall>b :: 'a bit0 \<in> set enum_class.enum. P b)"
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definition "enum_class.enum_ex P = (\<exists>b :: 'a bit0 \<in> set enum_class.enum. P b)"
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instance
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proof(intro_classes)
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  show "distinct (enum_class.enum :: 'a bit0 list)"
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    by(auto intro!: inj_onI simp add: Abs_bit0'_def Abs_bit0_inject zmod_int[symmetric] enum_bit0_def distinct_map)
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   418
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  show univ_eq: "(UNIV :: 'a bit0 set) = set enum_class.enum"
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    unfolding enum_bit0_def type_definition.Abs_image[OF type_definition_bit0, symmetric]
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    by(simp add: image_comp inj_on_Abs_bit0 card_image image_int_atLeastLessThan)
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      (auto intro!: image_cong[OF refl] simp add: Abs_bit0'_def mod_pos_pos_trivial)
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   423
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  fix P :: "'a bit0 \<Rightarrow> bool"
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  show "enum_class.enum_all P = Ball UNIV P"
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    and "enum_class.enum_ex P = Bex UNIV P"
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    by(simp_all add: enum_all_bit0_def enum_ex_bit0_def univ_eq)
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qed
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end
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instantiation bit1 :: (finite) enum begin
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definition "(enum_class.enum :: 'a bit1 list) = map (Abs_bit1' \<circ> int) (upt 0 (CARD('a bit1)))"
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definition "enum_class.enum_all P = (\<forall>b :: 'a bit1 \<in> set enum_class.enum. P b)"
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diff changeset
   435
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
   436
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   437
instance
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   438
proof(intro_classes)
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   439
  show "distinct (enum_class.enum :: 'a bit1 list)"
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   440
    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)
Andreas Lochbihler
parents: 49834
diff changeset
   441
      (clarsimp simp add: Abs_bit1_inject)
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   442
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   443
  show univ_eq: "(UNIV :: 'a bit1 set) = set enum_class.enum"
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   444
    unfolding enum_bit1_def type_definition.Abs_image[OF type_definition_bit1, symmetric]
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   445
    by(simp add: image_comp inj_on_Abs_bit1 card_image image_int_atLeastLessThan)
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   446
      (auto intro!: image_cong[OF refl] simp add: Abs_bit1'_def mod_pos_pos_trivial)
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   447
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   448
  fix P :: "'a bit1 \<Rightarrow> bool"
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   449
  show "enum_class.enum_all P = Ball UNIV P"
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   450
    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
   451
    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
   452
qed
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   453
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   454
end
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   455
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   456
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
   457
definition "finite_UNIV = Phantom('a bit0) True"
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   458
definition "finite_UNIV = Phantom('a bit1) True"
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   459
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
   460
end
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   461
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   462
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
   463
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
   464
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
   465
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
   466
end
b14ee572cc7b more type class instances for Numeral_Type (contributed by Jesus Aransay)
Andreas Lochbihler
parents: 49834
diff changeset
   467
24332
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   468
subsection {* Syntax *}
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   469
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   470
syntax
46236
ae79f2978a67 position constraints for numerals enable PIDE markup;
wenzelm
parents: 37653
diff changeset
   471
  "_NumeralType" :: "num_token => type"  ("_")
24332
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   472
  "_NumeralType0" :: type ("0")
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   473
  "_NumeralType1" :: type ("1")
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   474
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   475
translations
35362
828a42fb7445 explicit @{type_syntax} markup;
wenzelm
parents: 35115
diff changeset
   476
  (type) "1" == (type) "num1"
828a42fb7445 explicit @{type_syntax} markup;
wenzelm
parents: 35115
diff changeset
   477
  (type) "0" == (type) "num0"
24332
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   478
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   479
parse_translation {*
52143
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   480
  let
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   481
    fun mk_bintype n =
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   482
      let
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   483
        fun mk_bit 0 = Syntax.const @{type_syntax bit0}
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   484
          | mk_bit 1 = Syntax.const @{type_syntax bit1};
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   485
        fun bin_of n =
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   486
          if n = 1 then Syntax.const @{type_syntax num1}
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   487
          else if n = 0 then Syntax.const @{type_syntax num0}
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   488
          else if n = ~1 then raise TERM ("negative type numeral", [])
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   489
          else
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   490
            let val (q, r) = Integer.div_mod n 2;
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   491
            in mk_bit r $ bin_of q end;
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   492
      in bin_of n end;
24332
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   493
52143
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   494
    fun numeral_tr [Free (str, _)] = mk_bintype (the (Int.fromString str))
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   495
      | 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
   496
52143
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   497
  in [(@{syntax_const "_NumeralType"}, K numeral_tr)] end;
24332
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   498
*}
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   499
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   500
print_translation {*
52143
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   501
  let
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   502
    fun int_of [] = 0
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   503
      | 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
   504
52143
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   505
    fun bin_of (Const (@{type_syntax num0}, _)) = []
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   506
      | bin_of (Const (@{type_syntax num1}, _)) = [1]
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   507
      | bin_of (Const (@{type_syntax bit0}, _) $ bs) = 0 :: bin_of bs
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   508
      | bin_of (Const (@{type_syntax bit1}, _) $ bs) = 1 :: bin_of bs
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   509
      | 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
   510
52143
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   511
    fun bit_tr' b [t] =
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   512
          let
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   513
            val rev_digs = b :: bin_of t handle TERM _ => raise Match
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   514
            val i = int_of rev_digs;
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   515
            val num = string_of_int (abs i);
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   516
          in
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   517
            Syntax.const @{syntax_const "_NumeralType"} $ Syntax.free num
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   518
          end
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   519
      | bit_tr' b _ = raise Match;
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   520
  in
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51288
diff changeset
   521
   [(@{type_syntax bit0}, K (bit_tr' 0)),
52147
wenzelm
parents: 52143
diff changeset
   522
    (@{type_syntax bit1}, K (bit_tr' 1))]
wenzelm
parents: 52143
diff changeset
   523
  end;
24332
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   524
*}
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   525
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   526
subsection {* Examples *}
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   527
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   528
lemma "CARD(0) = 0" by simp
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   529
lemma "CARD(17) = 17" by simp
29997
f6756c097c2d number_ring instances for numeral types
huffman
parents: 29629
diff changeset
   530
lemma "8 * 11 ^ 3 - 6 = (2::5)" by simp
28920
4ed4b8b1988d fix typed print translation for card UNIV
huffman
parents: 27487
diff changeset
   531
24332
e3a2b75b1cf9 boolean algebras as locales and numbers as types by Brian Huffman
kleing
parents:
diff changeset
   532
end