author | wenzelm |
Wed, 15 Feb 2012 23:19:30 +0100 | |
changeset 46497 | 89ccf66aa73d |
parent 46028 | 9f113cdf3d66 |
child 47108 | 2a1953f0d20d |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Efficient_Nat.thy |
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Author: Stefan Berghofer, Florian Haftmann, TU Muenchen |
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*) |
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header {* Implementation of natural numbers by target-language integers *} |
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theory Efficient_Nat |
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imports Code_Integer Main |
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begin |
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text {* |
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When generating code for functions on natural numbers, the |
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canonical representation using @{term "0::nat"} and |
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@{term Suc} is unsuitable for computations involving large |
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numbers. The efficiency of the generated code can be improved |
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drastically by implementing natural numbers by target-language |
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integers. To do this, just include this theory. |
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*} |
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subsection {* Basic arithmetic *} |
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text {* |
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Most standard arithmetic functions on natural numbers are implemented |
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using their counterparts on the integers: |
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*} |
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code_datatype number_nat_inst.number_of_nat |
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lemma zero_nat_code [code, code_unfold]: |
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"0 = (Numeral0 :: nat)" |
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by simp |
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lemma one_nat_code [code, code_unfold]: |
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"1 = (Numeral1 :: nat)" |
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by simp |
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lemma Suc_code [code]: |
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"Suc n = n + 1" |
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by simp |
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lemma plus_nat_code [code]: |
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"n + m = nat (of_nat n + of_nat m)" |
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by simp |
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lemma minus_nat_code [code]: |
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"n - m = nat (of_nat n - of_nat m)" |
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by simp |
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lemma times_nat_code [code]: |
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"n * m = nat (of_nat n * of_nat m)" |
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unfolding of_nat_mult [symmetric] by simp |
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lemma divmod_nat_code [code]: |
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"divmod_nat n m = map_pair nat nat (pdivmod (of_nat n) (of_nat m))" |
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by (simp add: map_pair_def split_def pdivmod_def nat_div_distrib nat_mod_distrib divmod_nat_div_mod) |
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lemma eq_nat_code [code]: |
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"HOL.equal n m \<longleftrightarrow> HOL.equal (of_nat n \<Colon> int) (of_nat m)" |
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by (simp add: equal) |
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lemma eq_nat_refl [code nbe]: |
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"HOL.equal (n::nat) n \<longleftrightarrow> True" |
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by (rule equal_refl) |
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lemma less_eq_nat_code [code]: |
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"n \<le> m \<longleftrightarrow> (of_nat n \<Colon> int) \<le> of_nat m" |
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by simp |
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lemma less_nat_code [code]: |
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"n < m \<longleftrightarrow> (of_nat n \<Colon> int) < of_nat m" |
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by simp |
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subsection {* Case analysis *} |
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text {* |
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Case analysis on natural numbers is rephrased using a conditional |
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expression: |
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*} |
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lemma [code, code_unfold]: |
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"nat_case = (\<lambda>f g n. if n = 0 then f else g (n - 1))" |
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by (auto simp add: fun_eq_iff dest!: gr0_implies_Suc) |
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subsection {* Preprocessors *} |
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text {* |
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In contrast to @{term "Suc n"}, the term @{term "n + (1::nat)"} is no longer |
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a constructor term. Therefore, all occurrences of this term in a position |
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where a pattern is expected (i.e.\ on the left-hand side of a recursion |
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equation or in the arguments of an inductive relation in an introduction |
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rule) must be eliminated. |
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This can be accomplished by applying the following transformation rules: |
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*} |
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lemma Suc_if_eq: "(\<And>n. f (Suc n) \<equiv> h n) \<Longrightarrow> f 0 \<equiv> g \<Longrightarrow> |
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f n \<equiv> if n = 0 then g else h (n - 1)" |
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by (rule eq_reflection) (cases n, simp_all) |
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lemma Suc_clause: "(\<And>n. P n (Suc n)) \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> P (n - 1) n" |
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by (cases n) simp_all |
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text {* |
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The rules above are built into a preprocessor that is plugged into |
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the code generator. Since the preprocessor for introduction rules |
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does not know anything about modes, some of the modes that worked |
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for the canonical representation of natural numbers may no longer work. |
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*} |
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(*<*) |
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setup {* |
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let |
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fun remove_suc thy thms = |
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let |
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val vname = singleton (Name.variant_list (map fst |
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(fold (Term.add_var_names o Thm.full_prop_of) thms []))) "n"; |
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val cv = cterm_of thy (Var ((vname, 0), HOLogic.natT)); |
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fun lhs_of th = snd (Thm.dest_comb |
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(fst (Thm.dest_comb (cprop_of th)))); |
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fun rhs_of th = snd (Thm.dest_comb (cprop_of th)); |
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fun find_vars ct = (case term_of ct of |
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(Const (@{const_name Suc}, _) $ Var _) => [(cv, snd (Thm.dest_comb ct))] |
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| _ $ _ => |
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let val (ct1, ct2) = Thm.dest_comb ct |
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in |
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map (apfst (fn ct => Thm.apply ct ct2)) (find_vars ct1) @ |
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map (apfst (Thm.apply ct1)) (find_vars ct2) |
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end |
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| _ => []); |
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val eqs = maps |
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(fn th => map (pair th) (find_vars (lhs_of th))) thms; |
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fun mk_thms (th, (ct, cv')) = |
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let |
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val th' = |
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Thm.implies_elim |
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(Conv.fconv_rule (Thm.beta_conversion true) |
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(Drule.instantiate' |
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[SOME (ctyp_of_term ct)] [SOME (Thm.lambda cv ct), |
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SOME (Thm.lambda cv' (rhs_of th)), NONE, SOME cv'] |
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@{thm Suc_if_eq})) (Thm.forall_intr cv' th) |
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in |
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case map_filter (fn th'' => |
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SOME (th'', singleton |
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(Variable.trade (K (fn [th'''] => [th''' RS th'])) |
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(Variable.global_thm_context th'')) th'') |
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handle THM _ => NONE) thms of |
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[] => NONE |
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| thps => |
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let val (ths1, ths2) = split_list thps |
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in SOME (subtract Thm.eq_thm (th :: ths1) thms @ ths2) end |
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end |
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in get_first mk_thms eqs end; |
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fun eqn_suc_base_preproc thy thms = |
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let |
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val dest = fst o Logic.dest_equals o prop_of; |
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val contains_suc = exists_Const (fn (c, _) => c = @{const_name Suc}); |
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in |
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if forall (can dest) thms andalso exists (contains_suc o dest) thms |
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then thms |> perhaps_loop (remove_suc thy) |> (Option.map o map) Drule.zero_var_indexes |
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else NONE |
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end; |
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val eqn_suc_preproc = Code_Preproc.simple_functrans eqn_suc_base_preproc; |
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in |
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Code_Preproc.add_functrans ("eqn_Suc", eqn_suc_preproc) |
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end; |
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*} |
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(*>*) |
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subsection {* Target language setup *} |
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text {* |
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For ML, we map @{typ nat} to target language integers, where we |
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ensure that values are always non-negative. |
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*} |
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code_type nat |
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(SML "IntInf.int") |
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(OCaml "Big'_int.big'_int") |
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(Eval "int") |
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text {* |
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For Haskell and Scala we define our own @{typ nat} type. The reason |
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is that we have to distinguish type class instances for @{typ nat} |
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and @{typ int}. |
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*} |
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code_include Haskell "Nat" |
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{*newtype Nat = Nat Integer deriving (Eq, Show, Read); |
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instance Num Nat where { |
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fromInteger k = Nat (if k >= 0 then k else 0); |
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Nat n + Nat m = Nat (n + m); |
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Nat n - Nat m = fromInteger (n - m); |
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Nat n * Nat m = Nat (n * m); |
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abs n = n; |
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signum _ = 1; |
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negate n = error "negate Nat"; |
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}; |
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instance Ord Nat where { |
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Nat n <= Nat m = n <= m; |
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Nat n < Nat m = n < m; |
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}; |
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instance Real Nat where { |
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toRational (Nat n) = toRational n; |
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}; |
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instance Enum Nat where { |
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toEnum k = fromInteger (toEnum k); |
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fromEnum (Nat n) = fromEnum n; |
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}; |
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instance Integral Nat where { |
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toInteger (Nat n) = n; |
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divMod n m = quotRem n m; |
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quotRem (Nat n) (Nat m) |
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| (m == 0) = (0, Nat n) |
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| otherwise = (Nat k, Nat l) where (k, l) = quotRem n m; |
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}; |
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*} |
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code_reserved Haskell Nat |
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code_include Scala "Nat" |
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{*object Nat { |
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def apply(numeral: BigInt): Nat = new Nat(numeral max 0) |
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def apply(numeral: Int): Nat = Nat(BigInt(numeral)) |
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def apply(numeral: String): Nat = Nat(BigInt(numeral)) |
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} |
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class Nat private(private val value: BigInt) { |
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override def hashCode(): Int = this.value.hashCode() |
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override def equals(that: Any): Boolean = that match { |
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case that: Nat => this equals that |
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case _ => false |
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} |
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override def toString(): String = this.value.toString |
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def equals(that: Nat): Boolean = this.value == that.value |
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def as_BigInt: BigInt = this.value |
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def as_Int: Int = if (this.value >= scala.Int.MinValue && this.value <= scala.Int.MaxValue) |
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this.value.intValue |
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else error("Int value out of range: " + this.value.toString) |
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def +(that: Nat): Nat = new Nat(this.value + that.value) |
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def -(that: Nat): Nat = Nat(this.value - that.value) |
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def *(that: Nat): Nat = new Nat(this.value * that.value) |
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def /%(that: Nat): (Nat, Nat) = if (that.value == 0) (new Nat(0), this) |
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else { |
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val (k, l) = this.value /% that.value |
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(new Nat(k), new Nat(l)) |
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} |
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def <=(that: Nat): Boolean = this.value <= that.value |
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def <(that: Nat): Boolean = this.value < that.value |
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} |
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*} |
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code_reserved Scala Nat |
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code_type nat |
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(Haskell "Nat.Nat") |
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(Scala "Nat") |
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code_instance nat :: equal |
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(Haskell -) |
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text {* |
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Natural numerals. |
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*} |
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lemma [code_abbrev]: |
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"number_nat_inst.number_of_nat i = nat (number_of i)" |
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-- {* this interacts as desired with @{thm nat_number_of_def} *} |
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by (simp add: number_nat_inst.number_of_nat) |
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setup {* |
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fold (Numeral.add_code @{const_name number_nat_inst.number_of_nat} |
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false Code_Printer.literal_positive_numeral) ["SML", "OCaml", "Haskell", "Scala"] |
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*} |
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text {* |
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Since natural numbers are implemented |
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using integers in ML, the coercion function @{const "of_nat"} of type |
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@{typ "nat \<Rightarrow> int"} is simply implemented by the identity function. |
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For the @{const nat} function for converting an integer to a natural |
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number, we give a specific implementation using an ML function that |
305 |
returns its input value, provided that it is non-negative, and otherwise |
|
306 |
returns @{text "0"}. |
|
307 |
*} |
|
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||
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definition int :: "nat \<Rightarrow> int" where |
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[code del, code_abbrev]: "int = of_nat" |
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|
28562 | 312 |
lemma int_code' [code]: |
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"int (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)" |
314 |
unfolding int_nat_number_of [folded int_def] .. |
|
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||
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lemma nat_code' [code]: |
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"nat (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)" |
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unfolding nat_number_of_def number_of_is_id neg_def by simp |
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|
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lemma of_nat_int: (* FIXME delete candidate *) |
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"of_nat = int" by (simp add: int_def) |
322 |
||
32073 | 323 |
lemma of_nat_aux_int [code_unfold]: |
324 |
"of_nat_aux (\<lambda>i. i + 1) k 0 = int k" |
|
325 |
by (simp add: int_def Nat.of_nat_code) |
|
326 |
||
25931 | 327 |
code_const int |
328 |
(SML "_") |
|
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(OCaml "_") |
|
330 |
||
25967 | 331 |
code_const nat |
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(SML "IntInf.max/ (0,/ _)") |
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(OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int") |
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(Eval "Integer.max/ _/ 0") |
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|
35689 | 336 |
text {* For Haskell and Scala, things are slightly different again. *} |
25967 | 337 |
|
338 |
code_const int and nat |
|
339 |
(Haskell "toInteger" and "fromInteger") |
|
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(Scala "!_.as'_BigInt" and "Nat") |
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|
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text {* Conversion from and to code numerals. *} |
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|
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code_const Code_Numeral.of_nat |
25967 | 345 |
(SML "IntInf.toInt") |
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(OCaml "_") |
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(Haskell "!(fromInteger/ ./ toInteger)") |
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(Scala "!Natural(_.as'_BigInt)") |
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(Eval "_") |
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|
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code_const Code_Numeral.nat_of |
25931 | 352 |
(SML "IntInf.fromInt") |
31377 | 353 |
(OCaml "_") |
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(Haskell "!(fromInteger/ ./ toInteger)") |
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(Scala "!Nat(_.as'_BigInt)") |
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(Eval "_") |
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|
358 |
text {* Using target language arithmetic operations whenever appropriate *} |
|
359 |
||
360 |
code_const "op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat" |
|
361 |
(SML "IntInf.+ ((_), (_))") |
|
362 |
(OCaml "Big'_int.add'_big'_int") |
|
363 |
(Haskell infixl 6 "+") |
|
34899 | 364 |
(Scala infixl 7 "+") |
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|
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(Eval infixl 8 "+") |
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|
367 |
code_const "op - \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat" |
|
368 |
(Haskell infixl 6 "-") |
|
369 |
(Scala infixl 7 "-") |
|
25931 | 370 |
|
371 |
code_const "op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat" |
|
372 |
(SML "IntInf.* ((_), (_))") |
|
373 |
(OCaml "Big'_int.mult'_big'_int") |
|
374 |
(Haskell infixl 7 "*") |
|
34899 | 375 |
(Scala infixl 8 "*") |
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|
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(Eval infixl 9 "*") |
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|
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code_const divmod_nat |
26009 | 379 |
(SML "IntInf.divMod/ ((_),/ (_))") |
380 |
(OCaml "Big'_int.quomod'_big'_int") |
|
381 |
(Haskell "divMod") |
|
34899 | 382 |
(Scala infixl 8 "/%") |
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|
383 |
(Eval "Integer.div'_mod") |
25931 | 384 |
|
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|
385 |
code_const "HOL.equal \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool" |
25931 | 386 |
(SML "!((_ : IntInf.int) = _)") |
387 |
(OCaml "Big'_int.eq'_big'_int") |
|
39272 | 388 |
(Haskell infix 4 "==") |
34899 | 389 |
(Scala infixl 5 "==") |
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|
390 |
(Eval infixl 6 "=") |
25931 | 391 |
|
392 |
code_const "op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool" |
|
393 |
(SML "IntInf.<= ((_), (_))") |
|
394 |
(OCaml "Big'_int.le'_big'_int") |
|
395 |
(Haskell infix 4 "<=") |
|
34899 | 396 |
(Scala infixl 4 "<=") |
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|
397 |
(Eval infixl 6 "<=") |
25931 | 398 |
|
399 |
code_const "op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool" |
|
400 |
(SML "IntInf.< ((_), (_))") |
|
401 |
(OCaml "Big'_int.lt'_big'_int") |
|
402 |
(Haskell infix 4 "<") |
|
34899 | 403 |
(Scala infixl 4 "<") |
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|
404 |
(Eval infixl 6 "<") |
25931 | 405 |
|
406 |
||
28228 | 407 |
text {* Evaluation *} |
408 |
||
28562 | 409 |
lemma [code, code del]: |
32657 | 410 |
"(Code_Evaluation.term_of \<Colon> nat \<Rightarrow> term) = Code_Evaluation.term_of" .. |
28228 | 411 |
|
32657 | 412 |
code_const "Code_Evaluation.term_of \<Colon> nat \<Rightarrow> term" |
28228 | 413 |
(SML "HOLogic.mk'_number/ HOLogic.natT") |
414 |
||
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|
415 |
text {* Evaluation with @{text "Quickcheck_Narrowing"} does not work, as |
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changeset
|
416 |
@{text "code_module"} is very aggressive leading to bad Haskell code. |
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|
417 |
Therefore, we simply deactivate the narrowing-based quickcheck from here on. |
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|
418 |
*} |
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|
419 |
|
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changeset
|
420 |
declare [[quickcheck_narrowing_active = false]] |
28228 | 421 |
|
25931 | 422 |
text {* Module names *} |
23854 | 423 |
|
424 |
code_modulename SML |
|
33364 | 425 |
Efficient_Nat Arith |
23854 | 426 |
|
427 |
code_modulename OCaml |
|
33364 | 428 |
Efficient_Nat Arith |
23854 | 429 |
|
430 |
code_modulename Haskell |
|
33364 | 431 |
Efficient_Nat Arith |
23854 | 432 |
|
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|
433 |
hide_const int |
23854 | 434 |
|
435 |
end |