--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Code_Index.thy Fri Oct 12 10:26:18 2007 +0200
@@ -0,0 +1,255 @@
+(* ID: $Id$
+ Author: Florian Haftmann, TU Muenchen
+*)
+
+header {* Type of indices *}
+
+theory Code_Index
+imports PreList
+begin
+
+text {*
+ Indices are isomorphic to HOL @{typ int} but
+ mapped to target-language builtin integers
+*}
+
+subsection {* Datatype of indices *}
+
+datatype index = index_of_int int
+
+lemmas [code func del] = index.recs index.cases
+
+fun
+ int_of_index :: "index \<Rightarrow> int"
+where
+ "int_of_index (index_of_int k) = k"
+lemmas [code func del] = int_of_index.simps
+
+lemma index_id [simp]:
+ "index_of_int (int_of_index k) = k"
+ by (cases k) simp_all
+
+lemma index:
+ "(\<And>k\<Colon>index. PROP P k) \<equiv> (\<And>k\<Colon>int. PROP P (index_of_int k))"
+proof
+ fix k :: int
+ assume "\<And>k\<Colon>index. PROP P k"
+ then show "PROP P (index_of_int k)" .
+next
+ fix k :: index
+ assume "\<And>k\<Colon>int. PROP P (index_of_int k)"
+ then have "PROP P (index_of_int (int_of_index k))" .
+ then show "PROP P k" by simp
+qed
+
+lemma [code func]: "size (k\<Colon>index) = 0"
+ by (cases k) simp_all
+
+
+subsection {* Built-in integers as datatype on numerals *}
+
+instance index :: number
+ "number_of \<equiv> index_of_int" ..
+
+code_datatype "number_of \<Colon> int \<Rightarrow> index"
+
+lemma number_of_index_id [simp]:
+ "number_of (int_of_index k) = k"
+ unfolding number_of_index_def by simp
+
+lemma number_of_index_shift:
+ "number_of k = index_of_int (number_of k)"
+ by (simp add: number_of_is_id number_of_index_def)
+
+
+subsection {* Basic arithmetic *}
+
+instance index :: zero
+ [simp]: "0 \<equiv> index_of_int 0" ..
+lemmas [code func del] = zero_index_def
+
+instance index :: one
+ [simp]: "1 \<equiv> index_of_int 1" ..
+lemmas [code func del] = one_index_def
+
+instance index :: plus
+ [simp]: "k + l \<equiv> index_of_int (int_of_index k + int_of_index l)" ..
+lemmas [code func del] = plus_index_def
+lemma plus_index_code [code func]:
+ "index_of_int k + index_of_int l = index_of_int (k + l)"
+ unfolding plus_index_def by simp
+
+instance index :: minus
+ [simp]: "- k \<equiv> index_of_int (- int_of_index k)"
+ [simp]: "k - l \<equiv> index_of_int (int_of_index k - int_of_index l)" ..
+lemmas [code func del] = uminus_index_def minus_index_def
+lemma uminus_index_code [code func]:
+ "- index_of_int k \<equiv> index_of_int (- k)"
+ unfolding uminus_index_def by simp
+lemma minus_index_code [code func]:
+ "index_of_int k - index_of_int l = index_of_int (k - l)"
+ unfolding minus_index_def by simp
+
+instance index :: times
+ [simp]: "k * l \<equiv> index_of_int (int_of_index k * int_of_index l)" ..
+lemmas [code func del] = times_index_def
+lemma times_index_code [code func]:
+ "index_of_int k * index_of_int l = index_of_int (k * l)"
+ unfolding times_index_def by simp
+
+instance index :: ord
+ [simp]: "k \<le> l \<equiv> int_of_index k \<le> int_of_index l"
+ [simp]: "k < l \<equiv> int_of_index k < int_of_index l" ..
+lemmas [code func del] = less_eq_index_def less_index_def
+lemma less_eq_index_code [code func]:
+ "index_of_int k \<le> index_of_int l \<longleftrightarrow> k \<le> l"
+ unfolding less_eq_index_def by simp
+lemma less_index_code [code func]:
+ "index_of_int k < index_of_int l \<longleftrightarrow> k < l"
+ unfolding less_index_def by simp
+
+instance index :: ring_1
+ by default (auto simp add: left_distrib right_distrib)
+
+lemma of_nat_index: "of_nat n = index_of_int (of_nat n)"
+proof (induct n)
+ case 0 show ?case by simp
+next
+ case (Suc n)
+ then have "int_of_index (index_of_int (int n))
+ = int_of_index (of_nat n)" by simp
+ then have "int n = int_of_index (of_nat n)" by simp
+ then show ?case by simp
+qed
+
+instance index :: number_ring
+ by default
+ (simp_all add: left_distrib number_of_index_def of_int_of_nat of_nat_index)
+
+lemma zero_index_code [code inline, code func]:
+ "(0\<Colon>index) = Numeral0"
+ by simp
+
+lemma one_index_code [code inline, code func]:
+ "(1\<Colon>index) = Numeral1"
+ by simp
+
+instance index :: abs
+ "\<bar>k\<bar> \<equiv> if k < 0 then -k else k" ..
+
+lemma index_of_int [code func]:
+ "index_of_int k = (if k = 0 then 0
+ else if k = -1 then -1
+ else let (l, m) = divAlg (k, 2) in 2 * index_of_int l +
+ (if m = 0 then 0 else 1))"
+ by (simp add: number_of_index_shift Let_def split_def divAlg_mod_div) arith
+
+
+subsection {* Conversion to and from @{typ nat} *}
+
+definition
+ nat_of_index :: "index \<Rightarrow> nat"
+where
+ [code func del]: "nat_of_index = nat o int_of_index"
+
+definition
+ nat_of_index_aux :: "index \<Rightarrow> nat \<Rightarrow> nat" where
+ [code func del]: "nat_of_index_aux i n = nat_of_index i + n"
+
+lemma nat_of_index_aux_code [code]:
+ "nat_of_index_aux i n = (if i \<le> 0 then n else nat_of_index_aux (i - 1) (Suc n))"
+ by (auto simp add: nat_of_index_aux_def nat_of_index_def)
+
+lemma nat_of_index_code [code]:
+ "nat_of_index i = nat_of_index_aux i 0"
+ by (simp add: nat_of_index_aux_def)
+
+definition
+ index_of_nat :: "nat \<Rightarrow> index"
+where
+ [code func del]: "index_of_nat = index_of_int o of_nat"
+
+lemma index_of_nat [code func]:
+ "index_of_nat 0 = 0"
+ "index_of_nat (Suc n) = index_of_nat n + 1"
+ unfolding index_of_nat_def by simp_all
+
+lemma index_nat_id [simp]:
+ "nat_of_index (index_of_nat n) = n"
+ "index_of_nat (nat_of_index i) = (if i \<le> 0 then 0 else i)"
+ unfolding index_of_nat_def nat_of_index_def by simp_all
+
+
+subsection {* ML interface *}
+
+ML {*
+structure Index =
+struct
+
+fun mk k = @{term index_of_int} $ HOLogic.mk_number @{typ index} k;
+
+end;
+*}
+
+
+subsection {* Code serialization *}
+
+code_type index
+ (SML "int")
+ (OCaml "int")
+ (Haskell "Integer")
+
+code_instance index :: eq
+ (Haskell -)
+
+setup {*
+ fold (fn target => CodeTarget.add_pretty_numeral target true
+ @{const_name number_index_inst.number_of_index}
+ @{const_name Numeral.B0} @{const_name Numeral.B1}
+ @{const_name Numeral.Pls} @{const_name Numeral.Min}
+ @{const_name Numeral.Bit}
+ ) ["SML", "OCaml", "Haskell"]
+*}
+
+code_reserved SML int
+code_reserved OCaml int
+
+code_const "op + \<Colon> index \<Rightarrow> index \<Rightarrow> index"
+ (SML "Int.+ ((_), (_))")
+ (OCaml "Pervasives.+")
+ (Haskell infixl 6 "+")
+
+code_const "uminus \<Colon> index \<Rightarrow> index"
+ (SML "Int.~")
+ (OCaml "Pervasives.~-")
+ (Haskell "negate")
+
+code_const "op - \<Colon> index \<Rightarrow> index \<Rightarrow> index"
+ (SML "Int.- ((_), (_))")
+ (OCaml "Pervasives.-")
+ (Haskell infixl 6 "-")
+
+code_const "op * \<Colon> index \<Rightarrow> index \<Rightarrow> index"
+ (SML "Int.* ((_), (_))")
+ (OCaml "Pervasives.*")
+ (Haskell infixl 7 "*")
+
+code_const "op = \<Colon> index \<Rightarrow> index \<Rightarrow> bool"
+ (SML "!((_ : Int.int) = _)")
+ (OCaml "!((_ : Pervasives.int) = _)")
+ (Haskell infixl 4 "==")
+
+code_const "op \<le> \<Colon> index \<Rightarrow> index \<Rightarrow> bool"
+ (SML "Int.<= ((_), (_))")
+ (OCaml "!((_ : Pervasives.int) <= _)")
+ (Haskell infix 4 "<=")
+
+code_const "op < \<Colon> index \<Rightarrow> index \<Rightarrow> bool"
+ (SML "Int.< ((_), (_))")
+ (OCaml "!((_ : Pervasives.int) < _)")
+ (Haskell infix 4 "<")
+
+code_reserved SML Int
+code_reserved OCaml Pervasives
+
+end