author | haftmann |
Fri, 30 Oct 2009 18:32:40 +0100 | |
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parent 33343 | 2eb0b672ab40 |
child 34893 | ecdc526af73a |
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_post]: |
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"0 = (Numeral0 :: nat)" |
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by simp |
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lemma one_nat_code [code, code_unfold_post]: |
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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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text {* Specialized @{term "op div \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"} |
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and @{term "op mod \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"} operations. *} |
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definition divmod_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where |
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[code del]: "divmod_aux = divmod_nat" |
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lemma [code]: |
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"divmod_nat n m = (if m = 0 then (0, n) else divmod_aux n m)" |
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unfolding divmod_aux_def divmod_nat_div_mod by simp |
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lemma divmod_aux_code [code]: |
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"divmod_aux n m = (nat (of_nat n div of_nat m), nat (of_nat n mod of_nat m))" |
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unfolding divmod_aux_def divmod_nat_div_mod zdiv_int [symmetric] zmod_int [symmetric] by simp |
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lemma eq_nat_code [code]: |
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"eq_class.eq n m \<longleftrightarrow> eq_class.eq (of_nat n \<Colon> int) (of_nat m)" |
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by (simp add: eq) |
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lemma eq_nat_refl [code nbe]: |
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"eq_class.eq (n::nat) n \<longleftrightarrow> True" |
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by (rule HOL.eq_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: expand_fun_eq 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 = Name.variant (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.capply ct ct2)) (find_vars ct1) @ |
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map (apfst (Thm.capply 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.cabs cv ct), |
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SOME (Thm.cabs 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'])) (Variable.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_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_preproc1 = Code_Preproc.simple_functrans eqn_suc_preproc; |
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fun eqn_suc_preproc2 thy thms = eqn_suc_preproc thy thms |
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|> the_default thms; |
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fun remove_suc_clause thy thms = |
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let |
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val vname = Name.variant (map fst |
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(fold (Term.add_var_names o Thm.full_prop_of) thms [])) "x"; |
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fun find_var (t as Const (@{const_name Suc}, _) $ (v as Var _)) = SOME (t, v) |
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| find_var (t $ u) = (case find_var t of NONE => find_var u | x => x) |
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| find_var _ = NONE; |
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fun find_thm th = |
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let val th' = Conv.fconv_rule ObjectLogic.atomize th |
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in Option.map (pair (th, th')) (find_var (prop_of th')) end |
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in |
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case get_first find_thm thms of |
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NONE => thms |
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| SOME ((th, th'), (Sucv, v)) => |
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let |
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val cert = cterm_of (Thm.theory_of_thm th); |
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val th'' = ObjectLogic.rulify (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 (cert (lambda v (Abs ("x", HOLogic.natT, |
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abstract_over (Sucv, |
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HOLogic.dest_Trueprop (prop_of th')))))), |
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SOME (cert v)] @{thm Suc_clause})) |
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(Thm.forall_intr (cert v) th')) |
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in |
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remove_suc_clause thy (map (fn th''' => |
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if (op = o pairself prop_of) (th''', th) then th'' else th''') thms) |
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end |
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end; |
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fun clause_suc_preproc thy ths = |
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let |
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val dest = fst o HOLogic.dest_mem o HOLogic.dest_Trueprop |
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in |
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if forall (can (dest o concl_of)) ths andalso |
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exists (fn th => exists (exists_Const (fn (c, _) => c = @{const_name Suc})) |
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(map_filter (try dest) (concl_of th :: prems_of th))) ths |
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then remove_suc_clause thy ths else ths |
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end; |
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in |
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Codegen.add_preprocessor eqn_suc_preproc2 |
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#> Codegen.add_preprocessor clause_suc_preproc |
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#> Code_Preproc.add_functrans ("eqn_Suc", eqn_suc_preproc1) |
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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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assert 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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types_code |
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nat ("int") |
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attach (term_of) {* |
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val term_of_nat = HOLogic.mk_number HOLogic.natT; |
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*} |
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attach (test) {* |
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fun gen_nat i = |
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let val n = random_range 0 i |
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in (n, fn () => term_of_nat n) end; |
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*} |
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text {* |
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For Haskell we define our own @{typ nat} type. The reason |
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is that we have to distinguish type class instances |
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for @{typ nat} and @{typ int}. |
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*} |
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code_include Haskell "Nat" {* |
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newtype Nat = Nat Integer deriving (Show, Eq); |
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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) = (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_type nat |
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(Haskell "Nat.Nat") |
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code_instance nat :: eq |
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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_unfold_post]: |
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"nat (number_of i) = number_nat_inst.number_of_nat 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 true) ["SML", "OCaml", "Haskell"] |
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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 |
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returns its input value, provided that it is non-negative, and otherwise |
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returns @{text "0"}. |
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*} |
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definition int :: "nat \<Rightarrow> int" where |
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[code del]: "int = of_nat" |
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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)" |
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unfolding int_nat_number_of [folded int_def] .. |
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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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lemma of_nat_int [code_unfold_post]: |
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"of_nat = int" by (simp add: int_def) |
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lemma of_nat_aux_int [code_unfold]: |
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"of_nat_aux (\<lambda>i. i + 1) k 0 = int k" |
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by (simp add: int_def Nat.of_nat_code) |
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340 |
||
25931 | 341 |
code_const int |
342 |
(SML "_") |
|
343 |
(OCaml "_") |
|
344 |
||
345 |
consts_code |
|
346 |
int ("(_)") |
|
347 |
nat ("\<module>nat") |
|
348 |
attach {* |
|
349 |
fun nat i = if i < 0 then 0 else i; |
|
350 |
*} |
|
351 |
||
25967 | 352 |
code_const nat |
353 |
(SML "IntInf.max/ (/0,/ _)") |
|
354 |
(OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int") |
|
355 |
||
356 |
text {* For Haskell, things are slightly different again. *} |
|
357 |
||
358 |
code_const int and nat |
|
359 |
(Haskell "toInteger" and "fromInteger") |
|
25931 | 360 |
|
361 |
text {* Conversion from and to indices. *} |
|
362 |
||
31205
98370b26c2ce
String.literal replaces message_string, code_numeral replaces (code_)index
haftmann
parents:
31203
diff
changeset
|
363 |
code_const Code_Numeral.of_nat |
25967 | 364 |
(SML "IntInf.toInt") |
31377 | 365 |
(OCaml "_") |
27673 | 366 |
(Haskell "fromEnum") |
25967 | 367 |
|
31205
98370b26c2ce
String.literal replaces message_string, code_numeral replaces (code_)index
haftmann
parents:
31203
diff
changeset
|
368 |
code_const Code_Numeral.nat_of |
25931 | 369 |
(SML "IntInf.fromInt") |
31377 | 370 |
(OCaml "_") |
27673 | 371 |
(Haskell "toEnum") |
25931 | 372 |
|
373 |
text {* Using target language arithmetic operations whenever appropriate *} |
|
374 |
||
375 |
code_const "op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat" |
|
376 |
(SML "IntInf.+ ((_), (_))") |
|
377 |
(OCaml "Big'_int.add'_big'_int") |
|
378 |
(Haskell infixl 6 "+") |
|
379 |
||
380 |
code_const "op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat" |
|
381 |
(SML "IntInf.* ((_), (_))") |
|
382 |
(OCaml "Big'_int.mult'_big'_int") |
|
383 |
(Haskell infixl 7 "*") |
|
384 |
||
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26009
diff
changeset
|
385 |
code_const divmod_aux |
26009 | 386 |
(SML "IntInf.divMod/ ((_),/ (_))") |
387 |
(OCaml "Big'_int.quomod'_big'_int") |
|
388 |
(Haskell "divMod") |
|
25931 | 389 |
|
28346
b8390cd56b8f
discontinued special treatment of op = vs. eq_class.eq
haftmann
parents:
28228
diff
changeset
|
390 |
code_const "eq_class.eq \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool" |
25931 | 391 |
(SML "!((_ : IntInf.int) = _)") |
392 |
(OCaml "Big'_int.eq'_big'_int") |
|
393 |
(Haskell infixl 4 "==") |
|
394 |
||
395 |
code_const "op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool" |
|
396 |
(SML "IntInf.<= ((_), (_))") |
|
397 |
(OCaml "Big'_int.le'_big'_int") |
|
398 |
(Haskell infix 4 "<=") |
|
399 |
||
400 |
code_const "op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool" |
|
401 |
(SML "IntInf.< ((_), (_))") |
|
402 |
(OCaml "Big'_int.lt'_big'_int") |
|
403 |
(Haskell infix 4 "<") |
|
404 |
||
405 |
consts_code |
|
28522 | 406 |
"0::nat" ("0") |
407 |
"1::nat" ("1") |
|
25931 | 408 |
Suc ("(_ +/ 1)") |
409 |
"op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat" ("(_ +/ _)") |
|
410 |
"op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat" ("(_ */ _)") |
|
411 |
"op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool" ("(_ <=/ _)") |
|
412 |
"op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool" ("(_ </ _)") |
|
413 |
||
414 |
||
28228 | 415 |
text {* Evaluation *} |
416 |
||
28562 | 417 |
lemma [code, code del]: |
32657 | 418 |
"(Code_Evaluation.term_of \<Colon> nat \<Rightarrow> term) = Code_Evaluation.term_of" .. |
28228 | 419 |
|
32657 | 420 |
code_const "Code_Evaluation.term_of \<Colon> nat \<Rightarrow> term" |
28228 | 421 |
(SML "HOLogic.mk'_number/ HOLogic.natT") |
422 |
||
423 |
||
25931 | 424 |
text {* Module names *} |
23854 | 425 |
|
426 |
code_modulename SML |
|
33364 | 427 |
Efficient_Nat Arith |
23854 | 428 |
|
429 |
code_modulename OCaml |
|
33364 | 430 |
Efficient_Nat Arith |
23854 | 431 |
|
432 |
code_modulename Haskell |
|
33364 | 433 |
Efficient_Nat Arith |
23854 | 434 |
|
25931 | 435 |
hide const int |
23854 | 436 |
|
437 |
end |