--- a/src/HOL/Library/Numeral_Type.thy Wed Mar 04 10:43:39 2009 +0100
+++ b/src/HOL/Library/Numeral_Type.thy Wed Mar 04 10:45:52 2009 +0100
@@ -42,36 +42,87 @@
end
*}
-lemma card_unit: "CARD(unit) = 1"
+lemma card_unit [simp]: "CARD(unit) = 1"
unfolding UNIV_unit by simp
-lemma card_bool: "CARD(bool) = 2"
+lemma card_bool [simp]: "CARD(bool) = 2"
unfolding UNIV_bool by simp
-lemma card_prod: "CARD('a::finite \<times> 'b::finite) = CARD('a) * CARD('b)"
+lemma card_prod [simp]: "CARD('a \<times> 'b) = CARD('a::finite) * CARD('b::finite)"
unfolding UNIV_Times_UNIV [symmetric] by (simp only: card_cartesian_product)
-lemma card_sum: "CARD('a::finite + 'b::finite) = CARD('a) + CARD('b)"
+lemma card_sum [simp]: "CARD('a + 'b) = CARD('a::finite) + CARD('b::finite)"
unfolding UNIV_Plus_UNIV [symmetric] by (simp only: finite card_Plus)
-lemma card_option: "CARD('a::finite option) = Suc CARD('a)"
+lemma card_option [simp]: "CARD('a option) = Suc CARD('a::finite)"
unfolding insert_None_conv_UNIV [symmetric]
apply (subgoal_tac "(None::'a option) \<notin> range Some")
- apply (simp add: finite card_image)
+ apply (simp add: card_image)
apply fast
done
-lemma card_set: "CARD('a::finite set) = 2 ^ CARD('a)"
+lemma card_set [simp]: "CARD('a set) = 2 ^ CARD('a::finite)"
unfolding Pow_UNIV [symmetric]
by (simp only: card_Pow finite numeral_2_eq_2)
+lemma card_nat [simp]: "CARD(nat) = 0"
+ by (simp add: infinite_UNIV_nat card_eq_0_iff)
+
+
+subsection {* Classes with at least 1 and 2 *}
+
+text {* Class finite already captures "at least 1" *}
+
+lemma zero_less_card_finite [simp]: "0 < CARD('a::finite)"
+ unfolding neq0_conv [symmetric] by simp
+
+lemma one_le_card_finite [simp]: "Suc 0 \<le> CARD('a::finite)"
+ by (simp add: less_Suc_eq_le [symmetric])
+
+text {* Class for cardinality "at least 2" *}
+
+class card2 = finite +
+ assumes two_le_card: "2 \<le> CARD('a)"
+
+lemma one_less_card: "Suc 0 < CARD('a::card2)"
+ using two_le_card [where 'a='a] by simp
+
+lemma one_less_int_card: "1 < int CARD('a::card2)"
+ using one_less_card [where 'a='a] by simp
+
subsection {* Numeral Types *}
typedef (open) num0 = "UNIV :: nat set" ..
typedef (open) num1 = "UNIV :: unit set" ..
-typedef (open) 'a bit0 = "UNIV :: (bool * 'a) set" ..
-typedef (open) 'a bit1 = "UNIV :: (bool * 'a) option set" ..
+
+typedef (open) 'a bit0 = "{0 ..< 2 * int CARD('a::finite)}"
+proof
+ show "0 \<in> {0 ..< 2 * int CARD('a)}"
+ by simp
+qed
+
+typedef (open) 'a bit1 = "{0 ..< 1 + 2 * int CARD('a::finite)}"
+proof
+ show "0 \<in> {0 ..< 1 + 2 * int CARD('a)}"
+ by simp
+qed
+
+lemma card_num0 [simp]: "CARD (num0) = 0"
+ unfolding type_definition.card [OF type_definition_num0]
+ by simp
+
+lemma card_num1 [simp]: "CARD(num1) = 1"
+ unfolding type_definition.card [OF type_definition_num1]
+ by (simp only: card_unit)
+
+lemma card_bit0 [simp]: "CARD('a bit0) = 2 * CARD('a::finite)"
+ unfolding type_definition.card [OF type_definition_bit0]
+ by simp
+
+lemma card_bit1 [simp]: "CARD('a bit1) = Suc (2 * CARD('a::finite))"
+ unfolding type_definition.card [OF type_definition_bit1]
+ by simp
instance num1 :: finite
proof
@@ -80,46 +131,263 @@
using finite by (rule finite_imageI)
qed
-instance bit0 :: (finite) finite
+instance bit0 :: (finite) card2
proof
show "finite (UNIV::'a bit0 set)"
unfolding type_definition.univ [OF type_definition_bit0]
- using finite by (rule finite_imageI)
+ by simp
+ show "2 \<le> CARD('a bit0)"
+ by simp
qed
-instance bit1 :: (finite) finite
+instance bit1 :: (finite) card2
proof
show "finite (UNIV::'a bit1 set)"
unfolding type_definition.univ [OF type_definition_bit1]
- using finite by (rule finite_imageI)
+ by simp
+ show "2 \<le> CARD('a bit1)"
+ by simp
qed
-lemma card_num1: "CARD(num1) = 1"
- unfolding type_definition.card [OF type_definition_num1]
- by (simp only: card_unit)
+
+subsection {* Locale for modular arithmetic subtypes *}
+
+locale mod_type =
+ fixes n :: int
+ and Rep :: "'a::{zero,one,plus,times,uminus,minus,power} \<Rightarrow> int"
+ and Abs :: "int \<Rightarrow> 'a::{zero,one,plus,times,uminus,minus,power}"
+ assumes type: "type_definition Rep Abs {0..<n}"
+ and size1: "1 < n"
+ and zero_def: "0 = Abs 0"
+ and one_def: "1 = Abs 1"
+ and add_def: "x + y = Abs ((Rep x + Rep y) mod n)"
+ and mult_def: "x * y = Abs ((Rep x * Rep y) mod n)"
+ and diff_def: "x - y = Abs ((Rep x - Rep y) mod n)"
+ and minus_def: "- x = Abs ((- Rep x) mod n)"
+ and power_def: "x ^ k = Abs (Rep x ^ k mod n)"
+begin
+
+lemma size0: "0 < n"
+by (cut_tac size1, simp)
+
+lemmas definitions =
+ zero_def one_def add_def mult_def minus_def diff_def power_def
+
+lemma Rep_less_n: "Rep x < n"
+by (rule type_definition.Rep [OF type, simplified, THEN conjunct2])
+
+lemma Rep_le_n: "Rep x \<le> n"
+by (rule Rep_less_n [THEN order_less_imp_le])
+
+lemma Rep_inject_sym: "x = y \<longleftrightarrow> Rep x = Rep y"
+by (rule type_definition.Rep_inject [OF type, symmetric])
+
+lemma Rep_inverse: "Abs (Rep x) = x"
+by (rule type_definition.Rep_inverse [OF type])
+
+lemma Abs_inverse: "m \<in> {0..<n} \<Longrightarrow> Rep (Abs m) = m"
+by (rule type_definition.Abs_inverse [OF type])
+
+lemma Rep_Abs_mod: "Rep (Abs (m mod n)) = m mod n"
+by (simp add: Abs_inverse IntDiv.pos_mod_conj [OF size0])
+
+lemma Rep_Abs_0: "Rep (Abs 0) = 0"
+by (simp add: Abs_inverse size0)
+
+lemma Rep_0: "Rep 0 = 0"
+by (simp add: zero_def Rep_Abs_0)
+
+lemma Rep_Abs_1: "Rep (Abs 1) = 1"
+by (simp add: Abs_inverse size1)
+
+lemma Rep_1: "Rep 1 = 1"
+by (simp add: one_def Rep_Abs_1)
-lemma card_bit0: "CARD('a::finite bit0) = 2 * CARD('a)"
- unfolding type_definition.card [OF type_definition_bit0]
- by (simp only: card_prod card_bool)
+lemma Rep_mod: "Rep x mod n = Rep x"
+apply (rule_tac x=x in type_definition.Abs_cases [OF type])
+apply (simp add: type_definition.Abs_inverse [OF type])
+apply (simp add: mod_pos_pos_trivial)
+done
+
+lemmas Rep_simps =
+ Rep_inject_sym Rep_inverse Rep_Abs_mod Rep_mod Rep_Abs_0 Rep_Abs_1
+
+lemma comm_ring_1: "OFCLASS('a, comm_ring_1_class)"
+apply (intro_classes, unfold definitions)
+apply (simp_all add: Rep_simps zmod_simps ring_simps)
+done
+
+lemma recpower: "OFCLASS('a, recpower_class)"
+apply (intro_classes, unfold definitions)
+apply (simp_all add: Rep_simps zmod_simps add_ac mult_assoc
+ mod_pos_pos_trivial size1)
+done
+
+end
+
+locale mod_ring = mod_type +
+ constrains n :: int
+ and Rep :: "'a::{number_ring,power} \<Rightarrow> int"
+ and Abs :: "int \<Rightarrow> 'a::{number_ring,power}"
+begin
-lemma card_bit1: "CARD('a::finite bit1) = Suc (2 * CARD('a))"
- unfolding type_definition.card [OF type_definition_bit1]
- by (simp only: card_prod card_option card_bool)
+lemma of_nat_eq: "of_nat k = Abs (int k mod n)"
+apply (induct k)
+apply (simp add: zero_def)
+apply (simp add: Rep_simps add_def one_def zmod_simps add_ac)
+done
+
+lemma of_int_eq: "of_int z = Abs (z mod n)"
+apply (cases z rule: int_diff_cases)
+apply (simp add: Rep_simps of_nat_eq diff_def zmod_simps)
+done
+
+lemma Rep_number_of:
+ "Rep (number_of w) = number_of w mod n"
+by (simp add: number_of_eq of_int_eq Rep_Abs_mod)
+
+lemma iszero_number_of:
+ "iszero (number_of w::'a) \<longleftrightarrow> number_of w mod n = 0"
+by (simp add: Rep_simps number_of_eq of_int_eq iszero_def zero_def)
+
+lemma cases:
+ assumes 1: "\<And>z. \<lbrakk>(x::'a) = of_int z; 0 \<le> z; z < n\<rbrakk> \<Longrightarrow> P"
+ shows "P"
+apply (cases x rule: type_definition.Abs_cases [OF type])
+apply (rule_tac z="y" in 1)
+apply (simp_all add: of_int_eq mod_pos_pos_trivial)
+done
+
+lemma induct:
+ "(\<And>z. \<lbrakk>0 \<le> z; z < n\<rbrakk> \<Longrightarrow> P (of_int z)) \<Longrightarrow> P (x::'a)"
+by (cases x rule: cases) simp
+
+end
+
+
+subsection {* Number ring instances *}
-lemma card_num0: "CARD (num0) = 0"
- by (simp add: infinite_UNIV_nat card_eq_0_iff type_definition.card [OF type_definition_num0])
+text {*
+ Unfortunately a number ring instance is not possible for
+ @{typ num1}, since 0 and 1 are not distinct.
+*}
+
+instantiation num1 :: "{comm_ring,comm_monoid_mult,number,recpower}"
+begin
+
+lemma num1_eq_iff: "(x::num1) = (y::num1) \<longleftrightarrow> True"
+ by (induct x, induct y) simp
+
+instance proof
+qed (simp_all add: num1_eq_iff)
+
+end
+
+instantiation
+ bit0 and bit1 :: (finite) "{zero,one,plus,times,uminus,minus,power}"
+begin
+
+definition Abs_bit0' :: "int \<Rightarrow> 'a bit0" where
+ "Abs_bit0' x = Abs_bit0 (x mod int CARD('a bit0))"
+
+definition Abs_bit1' :: "int \<Rightarrow> 'a bit1" where
+ "Abs_bit1' x = Abs_bit1 (x mod int CARD('a bit1))"
+
+definition "0 = Abs_bit0 0"
+definition "1 = Abs_bit0 1"
+definition "x + y = Abs_bit0' (Rep_bit0 x + Rep_bit0 y)"
+definition "x * y = Abs_bit0' (Rep_bit0 x * Rep_bit0 y)"
+definition "x - y = Abs_bit0' (Rep_bit0 x - Rep_bit0 y)"
+definition "- x = Abs_bit0' (- Rep_bit0 x)"
+definition "x ^ k = Abs_bit0' (Rep_bit0 x ^ k)"
+
+definition "0 = Abs_bit1 0"
+definition "1 = Abs_bit1 1"
+definition "x + y = Abs_bit1' (Rep_bit1 x + Rep_bit1 y)"
+definition "x * y = Abs_bit1' (Rep_bit1 x * Rep_bit1 y)"
+definition "x - y = Abs_bit1' (Rep_bit1 x - Rep_bit1 y)"
+definition "- x = Abs_bit1' (- Rep_bit1 x)"
+definition "x ^ k = Abs_bit1' (Rep_bit1 x ^ k)"
+
+instance ..
+
+end
-lemmas card_univ_simps [simp] =
- card_unit
- card_bool
- card_prod
- card_sum
- card_option
- card_set
- card_num1
- card_bit0
- card_bit1
- card_num0
+interpretation bit0!:
+ mod_type "int CARD('a::finite bit0)"
+ "Rep_bit0 :: 'a::finite bit0 \<Rightarrow> int"
+ "Abs_bit0 :: int \<Rightarrow> 'a::finite bit0"
+apply (rule mod_type.intro)
+apply (simp add: int_mult type_definition_bit0)
+apply (rule one_less_int_card)
+apply (rule zero_bit0_def)
+apply (rule one_bit0_def)
+apply (rule plus_bit0_def [unfolded Abs_bit0'_def])
+apply (rule times_bit0_def [unfolded Abs_bit0'_def])
+apply (rule minus_bit0_def [unfolded Abs_bit0'_def])
+apply (rule uminus_bit0_def [unfolded Abs_bit0'_def])
+apply (rule power_bit0_def [unfolded Abs_bit0'_def])
+done
+
+interpretation bit1!:
+ mod_type "int CARD('a::finite bit1)"
+ "Rep_bit1 :: 'a::finite bit1 \<Rightarrow> int"
+ "Abs_bit1 :: int \<Rightarrow> 'a::finite bit1"
+apply (rule mod_type.intro)
+apply (simp add: int_mult type_definition_bit1)
+apply (rule one_less_int_card)
+apply (rule zero_bit1_def)
+apply (rule one_bit1_def)
+apply (rule plus_bit1_def [unfolded Abs_bit1'_def])
+apply (rule times_bit1_def [unfolded Abs_bit1'_def])
+apply (rule minus_bit1_def [unfolded Abs_bit1'_def])
+apply (rule uminus_bit1_def [unfolded Abs_bit1'_def])
+apply (rule power_bit1_def [unfolded Abs_bit1'_def])
+done
+
+instance bit0 :: (finite) "{comm_ring_1,recpower}"
+ by (rule bit0.comm_ring_1 bit0.recpower)+
+
+instance bit1 :: (finite) "{comm_ring_1,recpower}"
+ by (rule bit1.comm_ring_1 bit1.recpower)+
+
+instantiation bit0 and bit1 :: (finite) number_ring
+begin
+
+definition "(number_of w :: _ bit0) = of_int w"
+
+definition "(number_of w :: _ bit1) = of_int w"
+
+instance proof
+qed (rule number_of_bit0_def number_of_bit1_def)+
+
+end
+
+interpretation bit0!:
+ mod_ring "int CARD('a::finite bit0)"
+ "Rep_bit0 :: 'a::finite bit0 \<Rightarrow> int"
+ "Abs_bit0 :: int \<Rightarrow> 'a::finite bit0"
+ ..
+
+interpretation bit1!:
+ mod_ring "int CARD('a::finite bit1)"
+ "Rep_bit1 :: 'a::finite bit1 \<Rightarrow> int"
+ "Abs_bit1 :: int \<Rightarrow> 'a::finite bit1"
+ ..
+
+text {* Set up cases, induction, and arithmetic *}
+
+lemmas bit0_cases [case_names of_int, cases type: bit0] = bit0.cases
+lemmas bit1_cases [case_names of_int, cases type: bit1] = bit1.cases
+
+lemmas bit0_induct [case_names of_int, induct type: bit0] = bit0.induct
+lemmas bit1_induct [case_names of_int, induct type: bit1] = bit1.induct
+
+lemmas bit0_iszero_number_of [simp] = bit0.iszero_number_of
+lemmas bit1_iszero_number_of [simp] = bit1.iszero_number_of
+
+declare power_Suc [where ?'a="'a::finite bit0", standard, simp]
+declare power_Suc [where ?'a="'a::finite bit1", standard, simp]
subsection {* Syntax *}
@@ -184,42 +452,10 @@
in [("bit0", bit_tr' 0), ("bit1", bit_tr' 1)] end;
*}
-
-subsection {* Classes with at least 1 and 2 *}
-
-text {* Class finite already captures "at least 1" *}
-
-lemma zero_less_card_finite [simp]:
- "0 < CARD('a::finite)"
-proof (cases "CARD('a::finite) = 0")
- case False thus ?thesis by (simp del: card_0_eq)
-next
- case True
- thus ?thesis by (simp add: finite)
-qed
-
-lemma one_le_card_finite [simp]:
- "Suc 0 <= CARD('a::finite)"
- by (simp add: less_Suc_eq_le [symmetric] zero_less_card_finite)
-
-
-text {* Class for cardinality "at least 2" *}
-
-class card2 = finite +
- assumes two_le_card: "2 <= CARD('a)"
-
-lemma one_less_card: "Suc 0 < CARD('a::card2)"
- using two_le_card [where 'a='a] by simp
-
-instance bit0 :: (finite) card2
- by intro_classes (simp add: one_le_card_finite)
-
-instance bit1 :: (finite) card2
- by intro_classes (simp add: one_le_card_finite)
-
subsection {* Examples *}
lemma "CARD(0) = 0" by simp
lemma "CARD(17) = 17" by simp
+lemma "8 * 11 ^ 3 - 6 = (2::5)" by simp
end