(* Title: HOL/Library/Efficient_Nat.thy
Author: Stefan Berghofer, Florian Haftmann, TU Muenchen
*)
header {* Implementation of natural numbers by target-language integers *}
theory Efficient_Nat
imports Code_Index Code_Integer
begin
text {*
When generating code for functions on natural numbers, the
canonical representation using @{term "0::nat"} and
@{term "Suc"} is unsuitable for computations involving large
numbers. The efficiency of the generated code can be improved
drastically by implementing natural numbers by target-language
integers. To do this, just include this theory.
*}
subsection {* Basic arithmetic *}
text {*
Most standard arithmetic functions on natural numbers are implemented
using their counterparts on the integers:
*}
code_datatype number_nat_inst.number_of_nat
lemma zero_nat_code [code, code inline]:
"0 = (Numeral0 :: nat)"
by simp
lemmas [code post] = zero_nat_code [symmetric]
lemma one_nat_code [code, code inline]:
"1 = (Numeral1 :: nat)"
by simp
lemmas [code post] = one_nat_code [symmetric]
lemma Suc_code [code]:
"Suc n = n + 1"
by simp
lemma plus_nat_code [code]:
"n + m = nat (of_nat n + of_nat m)"
by simp
lemma minus_nat_code [code]:
"n - m = nat (of_nat n - of_nat m)"
by simp
lemma times_nat_code [code]:
"n * m = nat (of_nat n * of_nat m)"
unfolding of_nat_mult [symmetric] by simp
text {* Specialized @{term "op div \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"}
and @{term "op mod \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"} operations. *}
definition divmod_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
[code del]: "divmod_aux = Divides.divmod"
lemma [code]:
"Divides.divmod n m = (if m = 0 then (0, n) else divmod_aux n m)"
unfolding divmod_aux_def divmod_div_mod by simp
lemma divmod_aux_code [code]:
"divmod_aux n m = (nat (of_nat n div of_nat m), nat (of_nat n mod of_nat m))"
unfolding divmod_aux_def divmod_div_mod zdiv_int [symmetric] zmod_int [symmetric] by simp
lemma eq_nat_code [code]:
"eq_class.eq n m \<longleftrightarrow> eq_class.eq (of_nat n \<Colon> int) (of_nat m)"
by (simp add: eq)
lemma eq_nat_refl [code nbe]:
"eq_class.eq (n::nat) n \<longleftrightarrow> True"
by (rule HOL.eq_refl)
lemma less_eq_nat_code [code]:
"n \<le> m \<longleftrightarrow> (of_nat n \<Colon> int) \<le> of_nat m"
by simp
lemma less_nat_code [code]:
"n < m \<longleftrightarrow> (of_nat n \<Colon> int) < of_nat m"
by simp
subsection {* Case analysis *}
text {*
Case analysis on natural numbers is rephrased using a conditional
expression:
*}
lemma [code, code unfold]:
"nat_case = (\<lambda>f g n. if n = 0 then f else g (n - 1))"
by (auto simp add: expand_fun_eq dest!: gr0_implies_Suc)
subsection {* Preprocessors *}
text {*
In contrast to @{term "Suc n"}, the term @{term "n + (1::nat)"} is no longer
a constructor term. Therefore, all occurrences of this term in a position
where a pattern is expected (i.e.\ on the left-hand side of a recursion
equation or in the arguments of an inductive relation in an introduction
rule) must be eliminated.
This can be accomplished by applying the following transformation rules:
*}
lemma Suc_if_eq: "(\<And>n. f (Suc n) = h n) \<Longrightarrow> f 0 = g \<Longrightarrow>
f n = (if n = 0 then g else h (n - 1))"
by (case_tac n) simp_all
lemma Suc_clause: "(\<And>n. P n (Suc n)) \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> P (n - 1) n"
by (case_tac n) simp_all
text {*
The rules above are built into a preprocessor that is plugged into
the code generator. Since the preprocessor for introduction rules
does not know anything about modes, some of the modes that worked
for the canonical representation of natural numbers may no longer work.
*}
(*<*)
setup {*
let
fun remove_suc thy thms =
let
val vname = Name.variant (map fst
(fold (Term.add_var_names o Thm.full_prop_of) thms [])) "x";
val cv = cterm_of thy (Var ((vname, 0), HOLogic.natT));
fun lhs_of th = snd (Thm.dest_comb
(fst (Thm.dest_comb (snd (Thm.dest_comb (cprop_of th))))));
fun rhs_of th = snd (Thm.dest_comb (snd (Thm.dest_comb (cprop_of th))));
fun find_vars ct = (case term_of ct of
(Const ("Suc", _) $ Var _) => [(cv, snd (Thm.dest_comb ct))]
| _ $ _ =>
let val (ct1, ct2) = Thm.dest_comb ct
in
map (apfst (fn ct => Thm.capply ct ct2)) (find_vars ct1) @
map (apfst (Thm.capply ct1)) (find_vars ct2)
end
| _ => []);
val eqs = maps
(fn th => map (pair th) (find_vars (lhs_of th))) thms;
fun mk_thms (th, (ct, cv')) =
let
val th' =
Thm.implies_elim
(Conv.fconv_rule (Thm.beta_conversion true)
(Drule.instantiate'
[SOME (ctyp_of_term ct)] [SOME (Thm.cabs cv ct),
SOME (Thm.cabs cv' (rhs_of th)), NONE, SOME cv']
@{thm Suc_if_eq})) (Thm.forall_intr cv' th)
in
case map_filter (fn th'' =>
SOME (th'', singleton
(Variable.trade (K (fn [th'''] => [th''' RS th'])) (Variable.thm_context th'')) th'')
handle THM _ => NONE) thms of
[] => NONE
| thps =>
let val (ths1, ths2) = split_list thps
in SOME (subtract Thm.eq_thm (th :: ths1) thms @ ths2) end
end
in case get_first mk_thms eqs of
NONE => thms
| SOME x => remove_suc thy x
end;
fun eqn_suc_preproc thy ths =
let
val dest = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of;
val contains_suc = exists_Const (fn (c, _) => c = @{const_name Suc});
in
if forall (can dest) ths andalso
exists (contains_suc o dest) ths
then remove_suc thy ths else ths
end;
fun remove_suc_clause thy thms =
let
val vname = Name.variant (map fst
(fold (Term.add_var_names o Thm.full_prop_of) thms [])) "x";
fun find_var (t as Const (@{const_name Suc}, _) $ (v as Var _)) = SOME (t, v)
| find_var (t $ u) = (case find_var t of NONE => find_var u | x => x)
| find_var _ = NONE;
fun find_thm th =
let val th' = Conv.fconv_rule ObjectLogic.atomize th
in Option.map (pair (th, th')) (find_var (prop_of th')) end
in
case get_first find_thm thms of
NONE => thms
| SOME ((th, th'), (Sucv, v)) =>
let
val cert = cterm_of (Thm.theory_of_thm th);
val th'' = ObjectLogic.rulify (Thm.implies_elim
(Conv.fconv_rule (Thm.beta_conversion true)
(Drule.instantiate' []
[SOME (cert (lambda v (Abs ("x", HOLogic.natT,
abstract_over (Sucv,
HOLogic.dest_Trueprop (prop_of th')))))),
SOME (cert v)] @{thm Suc_clause}))
(Thm.forall_intr (cert v) th'))
in
remove_suc_clause thy (map (fn th''' =>
if (op = o pairself prop_of) (th''', th) then th'' else th''') thms)
end
end;
fun clause_suc_preproc thy ths =
let
val dest = fst o HOLogic.dest_mem o HOLogic.dest_Trueprop
in
if forall (can (dest o concl_of)) ths andalso
exists (fn th => exists (exists_Const (fn (c, _) => c = @{const_name Suc}))
(map_filter (try dest) (concl_of th :: prems_of th))) ths
then remove_suc_clause thy ths else ths
end;
fun lift f thy eqns1 =
let
val eqns2 = burrow_fst Drule.zero_var_indexes_list eqns1;
val thms3 = try (map fst
#> map (fn thm => thm RS @{thm meta_eq_to_obj_eq})
#> f thy
#> map (fn thm => thm RS @{thm eq_reflection})
#> map (Conv.fconv_rule Drule.beta_eta_conversion)) eqns2;
val thms4 = Option.map Drule.zero_var_indexes_list thms3;
in case thms4
of NONE => NONE
| SOME thms4 => if Thm.eq_thms (map fst eqns2, thms4)
then NONE else SOME (map (apfst (AxClass.overload thy) o Code_Unit.mk_eqn thy) thms4)
end
in
Codegen.add_preprocessor eqn_suc_preproc
#> Codegen.add_preprocessor clause_suc_preproc
#> Code.add_functrans ("eqn_Suc", lift eqn_suc_preproc)
#> Code.add_functrans ("clause_Suc", lift clause_suc_preproc)
end;
*}
(*>*)
subsection {* Target language setup *}
text {*
For ML, we map @{typ nat} to target language integers, where we
assert that values are always non-negative.
*}
code_type nat
(SML "IntInf.int")
(OCaml "Big'_int.big'_int")
types_code
nat ("int")
attach (term_of) {*
val term_of_nat = HOLogic.mk_number HOLogic.natT;
*}
attach (test) {*
fun gen_nat i =
let val n = random_range 0 i
in (n, fn () => term_of_nat n) end;
*}
text {*
For Haskell we define our own @{typ nat} type. The reason
is that we have to distinguish type class instances
for @{typ nat} and @{typ int}.
*}
code_include Haskell "Nat" {*
newtype Nat = Nat Integer deriving (Show, Eq);
instance Num Nat where {
fromInteger k = Nat (if k >= 0 then k else 0);
Nat n + Nat m = Nat (n + m);
Nat n - Nat m = fromInteger (n - m);
Nat n * Nat m = Nat (n * m);
abs n = n;
signum _ = 1;
negate n = error "negate Nat";
};
instance Ord Nat where {
Nat n <= Nat m = n <= m;
Nat n < Nat m = n < m;
};
instance Real Nat where {
toRational (Nat n) = toRational n;
};
instance Enum Nat where {
toEnum k = fromInteger (toEnum k);
fromEnum (Nat n) = fromEnum n;
};
instance Integral Nat where {
toInteger (Nat n) = n;
divMod n m = quotRem n m;
quotRem (Nat n) (Nat m) = (Nat k, Nat l) where (k, l) = quotRem n m;
};
*}
code_reserved Haskell Nat
code_type nat
(Haskell "Nat")
code_instance nat :: eq
(Haskell -)
text {*
Natural numerals.
*}
lemma [code inline, symmetric, code post]:
"nat (number_of i) = number_nat_inst.number_of_nat i"
-- {* this interacts as desired with @{thm nat_number_of_def} *}
by (simp add: number_nat_inst.number_of_nat)
setup {*
fold (Numeral.add_code @{const_name number_nat_inst.number_of_nat}
true false) ["SML", "OCaml", "Haskell"]
*}
text {*
Since natural numbers are implemented
using integers in ML, the coercion function @{const "of_nat"} of type
@{typ "nat \<Rightarrow> int"} is simply implemented by the identity function.
For the @{const "nat"} function for converting an integer to a natural
number, we give a specific implementation using an ML function that
returns its input value, provided that it is non-negative, and otherwise
returns @{text "0"}.
*}
definition
int :: "nat \<Rightarrow> int"
where
[code del]: "int = of_nat"
lemma int_code' [code]:
"int (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
unfolding int_nat_number_of [folded int_def] ..
lemma nat_code' [code]:
"nat (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
unfolding nat_number_of_def number_of_is_id neg_def by simp
lemma of_nat_int [code unfold]:
"of_nat = int" by (simp add: int_def)
declare of_nat_int [symmetric, code post]
code_const int
(SML "_")
(OCaml "_")
consts_code
int ("(_)")
nat ("\<module>nat")
attach {*
fun nat i = if i < 0 then 0 else i;
*}
code_const nat
(SML "IntInf.max/ (/0,/ _)")
(OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int")
text {* For Haskell, things are slightly different again. *}
code_const int and nat
(Haskell "toInteger" and "fromInteger")
text {* Conversion from and to indices. *}
code_const index_of_nat
(SML "IntInf.toInt")
(OCaml "Big'_int.int'_of'_big'_int")
(Haskell "fromEnum")
code_const nat_of_index
(SML "IntInf.fromInt")
(OCaml "Big'_int.big'_int'_of'_int")
(Haskell "toEnum")
text {* Using target language arithmetic operations whenever appropriate *}
code_const "op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
(SML "IntInf.+ ((_), (_))")
(OCaml "Big'_int.add'_big'_int")
(Haskell infixl 6 "+")
code_const "op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
(SML "IntInf.* ((_), (_))")
(OCaml "Big'_int.mult'_big'_int")
(Haskell infixl 7 "*")
code_const divmod_aux
(SML "IntInf.divMod/ ((_),/ (_))")
(OCaml "Big'_int.quomod'_big'_int")
(Haskell "divMod")
code_const "eq_class.eq \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
(SML "!((_ : IntInf.int) = _)")
(OCaml "Big'_int.eq'_big'_int")
(Haskell infixl 4 "==")
code_const "op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
(SML "IntInf.<= ((_), (_))")
(OCaml "Big'_int.le'_big'_int")
(Haskell infix 4 "<=")
code_const "op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
(SML "IntInf.< ((_), (_))")
(OCaml "Big'_int.lt'_big'_int")
(Haskell infix 4 "<")
consts_code
"0::nat" ("0")
"1::nat" ("1")
Suc ("(_ +/ 1)")
"op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat" ("(_ +/ _)")
"op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat" ("(_ */ _)")
"op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool" ("(_ <=/ _)")
"op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool" ("(_ </ _)")
text {* Evaluation *}
lemma [code, code del]:
"(Code_Eval.term_of \<Colon> nat \<Rightarrow> term) = Code_Eval.term_of" ..
code_const "Code_Eval.term_of \<Colon> nat \<Rightarrow> term"
(SML "HOLogic.mk'_number/ HOLogic.natT")
text {* Module names *}
code_modulename SML
Nat Integer
Divides Integer
Ring_and_Field Integer
Efficient_Nat Integer
code_modulename OCaml
Nat Integer
Divides Integer
Ring_and_Field Integer
Efficient_Nat Integer
code_modulename Haskell
Nat Integer
Divides Integer
Ring_and_Field Integer
Efficient_Nat Integer
hide const int
end