author | haftmann |
Tue, 06 Jun 2006 15:02:55 +0200 | |
changeset 19791 | ab326de16ad5 |
parent 19617 | 7cb4b67d4b97 |
child 19828 | f1dccc547595 |
permissions | -rw-r--r-- |
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Code generator plug-in for implementing natural numbers by integers.
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(* Title: HOL/Library/EfficientNat.thy |
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Code generator plug-in for implementing natural numbers by integers.
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ID: $Id$ |
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Code generator plug-in for implementing natural numbers by integers.
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Author: Stefan Berghofer, TU Muenchen |
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Code generator plug-in for implementing natural numbers by integers.
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*) |
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Code generator plug-in for implementing natural numbers by integers.
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Code generator plug-in for implementing natural numbers by integers.
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header {* Implementation of natural numbers by integers *} |
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Code generator plug-in for implementing natural numbers by integers.
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|
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Code generator plug-in for implementing natural numbers by integers.
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theory EfficientNat |
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Code generator plug-in for implementing natural numbers by integers.
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imports Main |
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Code generator plug-in for implementing natural numbers by integers.
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begin |
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Code generator plug-in for implementing natural numbers by integers.
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Code generator plug-in for implementing natural numbers by integers.
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text {* |
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Code generator plug-in for implementing natural numbers by integers.
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When generating code for functions on natural numbers, the canonical |
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Code generator plug-in for implementing natural numbers by integers.
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representation using @{term "0::nat"} and @{term "Suc"} is unsuitable for |
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Code generator plug-in for implementing natural numbers by integers.
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computations involving large numbers. The efficiency of the generated |
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Code generator plug-in for implementing natural numbers by integers.
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code can be improved drastically by implementing natural numbers by |
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integers. To do this, just include this theory. |
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Code generator plug-in for implementing natural numbers by integers.
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*} |
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Code generator plug-in for implementing natural numbers by integers.
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Code generator plug-in for implementing natural numbers by integers.
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subsection {* Basic functions *} |
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Code generator plug-in for implementing natural numbers by integers.
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|
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Code generator plug-in for implementing natural numbers by integers.
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text {* |
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The implementation of @{term "0::nat"} and @{term "Suc"} using the |
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ML integers is straightforward. Since natural numbers are implemented |
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Code generator plug-in for implementing natural numbers by integers.
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using integers, the coercion function @{term "int"} of type |
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@{typ "nat \<Rightarrow> int"} is simply implemented by the identity function. |
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Code generator plug-in for implementing natural numbers by integers.
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For the @{term "nat"} function for converting an integer to a natural |
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Code generator plug-in for implementing natural numbers by integers.
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number, we give a specific implementation using an ML function that |
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Code generator plug-in for implementing natural numbers by integers.
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returns its input value, provided that it is non-negative, and otherwise |
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Code generator plug-in for implementing natural numbers by integers.
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returns @{text "0"}. |
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Code generator plug-in for implementing natural numbers by integers.
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*} |
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Code generator plug-in for implementing natural numbers by integers.
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types_code |
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nat ("int") |
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attach (term_of) {* |
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fun term_of_nat 0 = Const ("0", HOLogic.natT) |
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| term_of_nat 1 = Const ("1", HOLogic.natT) |
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| term_of_nat i = HOLogic.number_of_const HOLogic.natT $ |
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HOLogic.mk_binum (IntInf.fromInt i); |
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*} |
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attach (test) {* |
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fun gen_nat i = random_range 0 i; |
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*} |
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Code generator plug-in for implementing natural numbers by integers.
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consts_code |
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Code generator plug-in for implementing natural numbers by integers.
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0 :: nat ("0") |
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Code generator plug-in for implementing natural numbers by integers.
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Suc ("(_ + 1)") |
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nat ("\<module>nat") |
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attach {* |
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fun nat i = if i < 0 then 0 else i; |
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Code generator plug-in for implementing natural numbers by integers.
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*} |
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int ("(_)") |
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code_syntax_tyco nat |
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ml (target_atom "{* int *}") |
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haskell (target_atom "{* int *}") |
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code_syntax_const |
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"0 :: nat" |
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ml (target_atom "{* 0 :: int *}") |
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haskell (target_atom "{* 0 :: int *}") |
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Suc |
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ml (target_atom "(__ + 1)") |
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haskell (target_atom "(__ + 1)") |
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nat |
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ml ("{* abs :: int \<Rightarrow> int *}") |
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haskell ("{* abs :: int \<Rightarrow> int *}") |
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int |
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ml ("_") |
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haskell ("_") |
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Code generator plug-in for implementing natural numbers by integers.
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text {* |
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Case analysis on natural numbers is rephrased using a conditional |
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Code generator plug-in for implementing natural numbers by integers.
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expression: |
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Code generator plug-in for implementing natural numbers by integers.
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*} |
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Code generator plug-in for implementing natural numbers by integers.
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Code generator plug-in for implementing natural numbers by integers.
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lemma [code unfold]: "nat_case \<equiv> (\<lambda>f g n. if n = 0 then f else g (n - 1))" |
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Code generator plug-in for implementing natural numbers by integers.
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apply (rule eq_reflection ext)+ |
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Code generator plug-in for implementing natural numbers by integers.
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apply (case_tac n) |
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apply simp_all |
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Code generator plug-in for implementing natural numbers by integers.
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done |
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Code generator plug-in for implementing natural numbers by integers.
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Code generator plug-in for implementing natural numbers by integers.
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text {* |
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Most standard arithmetic functions on natural numbers are implemented |
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Code generator plug-in for implementing natural numbers by integers.
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using their counterparts on the integers: |
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Code generator plug-in for implementing natural numbers by integers.
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*} |
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declare |
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Nat.plus.simps [code del] |
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Nat.minus.simps [code del] |
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diff_0_eq_0 [code del] |
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diff_Suc_Suc [code del] |
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Nat.times.simps [code del] |
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lemma [code]: "m + n = nat (int m + int n)" |
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by arith |
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lemma [code]: "m - n = nat (int m - int n)" |
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by arith |
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lemma [code]: "m * n = nat (int m * int n)" |
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by (simp add: zmult_int) |
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lemma [code]: "m div n = nat (int m div int n)" |
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Code generator plug-in for implementing natural numbers by integers.
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by (simp add: zdiv_int [symmetric]) |
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Code generator plug-in for implementing natural numbers by integers.
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lemma [code]: "m mod n = nat (int m mod int n)" |
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by (simp add: zmod_int [symmetric]) |
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lemma [code]: "(m < n) = (int m < int n)" |
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by simp |
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lemma [code fun]: |
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"((0::nat) = 0) = True" |
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"(m = n) = (int m = int n)" |
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by simp_all |
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Code generator plug-in for implementing natural numbers by integers.
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Code generator plug-in for implementing natural numbers by integers.
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subsection {* Preprocessors *} |
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Code generator plug-in for implementing natural numbers by integers.
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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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Code generator plug-in for implementing natural numbers by integers.
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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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Code generator plug-in for implementing natural numbers by integers.
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rule) must be eliminated. |
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Code generator plug-in for implementing natural numbers by integers.
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This can be accomplished by applying the following transformation rules: |
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Code generator plug-in for implementing natural numbers by integers.
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*} |
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theorem Suc_if_eq: "(\<And>n. f (Suc n) = h n) \<Longrightarrow> f 0 = g \<Longrightarrow> |
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f n = (if n = 0 then g else h (n - 1))" |
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by (case_tac n) simp_all |
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theorem Suc_clause: "(\<And>n. P n (Suc n)) \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> P (n - 1) n" |
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by (case_tac 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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Code generator plug-in for implementing natural numbers by integers.
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*} |
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(*<*) |
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ML {* |
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local |
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val Suc_if_eq = thm "Suc_if_eq"; |
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val Suc_clause = thm "Suc_clause"; |
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fun contains_suc t = member (op =) (term_consts t) "Suc"; |
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in |
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fun remove_suc thy thms = |
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let |
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val Suc_if_eq' = Thm.transfer thy Suc_if_eq; |
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val vname = variant (map fst |
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(fold (add_term_varnames o Thm.full_prop_of) thms [])) "x"; |
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val cv = cterm_of Main.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 (snd (Thm.dest_comb (cprop_of th)))))); |
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fun rhs_of th = snd (Thm.dest_comb (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 ("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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160 |
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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Code generator plug-in for implementing natural numbers by integers.
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162 |
end |
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Code generator plug-in for implementing natural numbers by integers.
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| _ => []); |
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164 |
val eqs = List.concat (map |
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Code generator plug-in for implementing natural numbers by integers.
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(fn th => map (pair th) (find_vars (lhs_of th))) thms); |
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166 |
fun mk_thms (th, (ct, cv')) = |
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Code generator plug-in for implementing natural numbers by integers.
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167 |
let |
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val th' = |
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169 |
Thm.implies_elim |
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(Drule.fconv_rule (Thm.beta_conversion true) |
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Code generator plug-in for implementing natural numbers by integers.
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(Drule.instantiate' |
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[SOME (ctyp_of_term ct)] [SOME (Thm.cabs cv ct), |
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173 |
SOME (Thm.cabs cv' (rhs_of th)), NONE, SOME cv'] |
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174 |
Suc_if_eq')) (Thm.forall_intr cv' th) |
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Code generator plug-in for implementing natural numbers by integers.
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175 |
in |
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176 |
case List.mapPartial (fn th'' => |
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177 |
SOME (th'', standard (Drule.freeze_all th'' RS th')) |
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178 |
handle THM _ => NONE) thms of |
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179 |
[] => NONE |
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180 |
| thps => |
19791 | 181 |
let val (ths1, ths2) = split_list thps |
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182 |
in SOME (gen_rems eq_thm (thms, th :: ths1) @ ths2) end |
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Code generator plug-in for implementing natural numbers by integers.
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183 |
end |
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Code generator plug-in for implementing natural numbers by integers.
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184 |
in |
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case get_first mk_thms eqs of |
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186 |
NONE => thms |
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| SOME x => remove_suc thy x |
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Code generator plug-in for implementing natural numbers by integers.
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188 |
end; |
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Code generator plug-in for implementing natural numbers by integers.
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189 |
|
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Code generator plug-in for implementing natural numbers by integers.
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190 |
fun eqn_suc_preproc thy ths = |
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let |
192 |
val dest = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of |
|
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Code generator plug-in for implementing natural numbers by integers.
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193 |
in |
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Code generator plug-in for implementing natural numbers by integers.
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194 |
if forall (can dest) ths andalso |
19791 | 195 |
exists (contains_suc o dest) ths |
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196 |
then remove_suc thy ths else ths |
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Code generator plug-in for implementing natural numbers by integers.
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197 |
end; |
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Code generator plug-in for implementing natural numbers by integers.
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198 |
|
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199 |
fun remove_suc_clause thy thms = |
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Code generator plug-in for implementing natural numbers by integers.
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let |
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201 |
val Suc_clause' = Thm.transfer thy Suc_clause; |
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Code generator plug-in for implementing natural numbers by integers.
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202 |
val vname = variant (map fst |
16861 | 203 |
(fold (add_term_varnames o Thm.full_prop_of) thms [])) "x"; |
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fun find_var (t as Const ("Suc", _) $ (v as Var _)) = SOME (t, v) |
205 |
| find_var (t $ u) = (case find_var t of NONE => find_var u | x => x) |
|
206 |
| find_var _ = NONE; |
|
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Code generator plug-in for implementing natural numbers by integers.
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207 |
fun find_thm th = |
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Code generator plug-in for implementing natural numbers by integers.
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208 |
let val th' = ObjectLogic.atomize_thm th |
15570 | 209 |
in Option.map (pair (th, th')) (find_var (prop_of th')) end |
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Code generator plug-in for implementing natural numbers by integers.
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210 |
in |
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Code generator plug-in for implementing natural numbers by integers.
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|
211 |
case get_first find_thm thms of |
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NONE => thms |
213 |
| SOME ((th, th'), (Sucv, v)) => |
|
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Code generator plug-in for implementing natural numbers by integers.
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214 |
let |
16861 | 215 |
val cert = cterm_of (Thm.theory_of_thm th); |
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Code generator plug-in for implementing natural numbers by integers.
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216 |
val th'' = ObjectLogic.rulify (Thm.implies_elim |
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217 |
(Drule.fconv_rule (Thm.beta_conversion true) |
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Code generator plug-in for implementing natural numbers by integers.
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218 |
(Drule.instantiate' [] |
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[SOME (cert (lambda v (Abs ("x", HOLogic.natT, |
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Code generator plug-in for implementing natural numbers by integers.
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|
220 |
abstract_over (Sucv, |
19791 | 221 |
(HOLogic.dest_Trueprop o prop_of) th'))))), |
15531 | 222 |
SOME (cert v)] Suc_clause')) |
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Code generator plug-in for implementing natural numbers by integers.
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|
223 |
(Thm.forall_intr (cert v) th')) |
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Code generator plug-in for implementing natural numbers by integers.
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|
224 |
in |
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|
225 |
remove_suc_clause thy (map (fn th''' => |
19617 | 226 |
if (op = o pairself prop_of) (th''', th) then th'' else th''') thms) |
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Code generator plug-in for implementing natural numbers by integers.
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227 |
end |
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Code generator plug-in for implementing natural numbers by integers.
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228 |
end; |
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Code generator plug-in for implementing natural numbers by integers.
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|
229 |
|
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Code generator plug-in for implementing natural numbers by integers.
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230 |
fun clause_suc_preproc thy ths = |
19791 | 231 |
let |
232 |
val dest = fst o HOLogic.dest_mem o ObjectLogic.drop_judgment thy |
|
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Code generator plug-in for implementing natural numbers by integers.
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|
233 |
in |
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Code generator plug-in for implementing natural numbers by integers.
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|
234 |
if forall (can (dest o concl_of)) ths andalso |
19791 | 235 |
exists (fn th => member (op =) (foldr add_term_consts |
236 |
[] (List.mapPartial (try dest) (concl_of th :: prems_of th))) "Suc") ths |
|
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237 |
then remove_suc_clause thy ths else ths |
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Code generator plug-in for implementing natural numbers by integers.
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|
238 |
end; |
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Code generator plug-in for implementing natural numbers by integers.
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|
239 |
|
19791 | 240 |
end; (*local*) |
241 |
||
242 |
fun lift_obj_eq f thy = |
|
243 |
map (fn thm => thm RS meta_eq_to_obj_eq) |
|
244 |
#> f thy |
|
245 |
#> map (fn thm => thm RS HOL.eq_reflection) |
|
246 |
*} |
|
247 |
||
248 |
setup {* |
|
19603 | 249 |
Codegen.add_preprocessor eqn_suc_preproc |
250 |
#> Codegen.add_preprocessor clause_suc_preproc |
|
19791 | 251 |
#> CodegenTheorems.add_preproc (lift_obj_eq eqn_suc_preproc) |
252 |
#> CodegenTheorems.add_preproc (lift_obj_eq clause_suc_preproc) |
|
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Code generator plug-in for implementing natural numbers by integers.
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253 |
*} |
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Code generator plug-in for implementing natural numbers by integers.
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|
254 |
(*>*) |
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Code generator plug-in for implementing natural numbers by integers.
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255 |
|
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Code generator plug-in for implementing natural numbers by integers.
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|
256 |
end |