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