author | haftmann |
Fri, 01 Sep 2006 08:36:51 +0200 | |
changeset 20453 | 855f07fabd76 |
parent 20356 | 21e7e9093940 |
child 20523 | 36a59e5d0039 |
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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|
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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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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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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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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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*} |
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int ("(_)") |
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definition |
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"nat_plus m n = nat (int m + int n)" |
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"nat_minus m n = nat (int m - int n)" |
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"nat_mult m n = nat (int m * int n)" |
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"nat_div m n = nat (fst (divAlg (int m, int n)))" |
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"nat_mod m n = nat (snd (divAlg (int m, int n)))" |
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"nat_less m n = (int m < int n)" |
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"nat_less_equal m n = (int m <= int n)" |
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"nat_eq m n = (int m = int n)" |
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code_type nat |
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(SML target_atom "IntInf.int") |
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(Haskell target_atom "Integer") |
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code_const "0::nat" and Suc (*all constructors of nat must be hidden*) |
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(SML target_atom "(0 :: IntInf.int)" and "IntInf.+ (_, 1)") |
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(Haskell target_atom "0" and "(+) 1 _") |
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code_const nat and int |
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(SML target_atom "(fn n : IntInf.int => if n < 0 then 0 else n)" and "_") |
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(Haskell target_atom "(\\n :: Int -> if n < 0 then 0 else n)" and "_") |
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text {* |
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Eliminate @{const "0::nat"} and @{const "1::nat"} |
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by unfolding in place. |
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*} |
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||
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lemma [code inline]: |
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"0 = nat 0" |
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"1 = nat 1" |
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by (simp_all add: expand_fun_eq) |
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||
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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 unfold]: "nat_case \<equiv> (\<lambda>f g n. if n = 0 then f else g (n - 1))" |
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apply (rule eq_reflection ext)+ |
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apply (case_tac n) |
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apply simp_all |
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done |
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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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using their counterparts on the integers: |
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*} |
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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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by (simp add: zdiv_int [symmetric]) |
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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, code inline]: "m + n = nat_plus m n" |
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unfolding nat_plus_def by arith |
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lemma [code fun, code inline]: "m - n = nat_minus m n" |
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unfolding nat_minus_def by arith |
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lemma [code fun, code inline]: "m * n = nat_mult m n" |
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unfolding nat_mult_def by (simp add: zmult_int) |
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lemma [code fun, code inline]: "m div n = nat_div m n" |
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unfolding nat_div_def div_def [symmetric] zdiv_int [symmetric] by simp |
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lemma [code fun, code inline]: "m mod n = nat_mod m n" |
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unfolding nat_mod_def mod_def [symmetric] zmod_int [symmetric] by simp |
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lemma [code fun, code inline]: "(m < n) = nat_less m n" |
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unfolding nat_less_def by simp |
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lemma [code fun, code inline]: "(m <= n) = nat_less_equal m n" |
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unfolding nat_less_equal_def by simp |
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lemma [code fun, code inline]: "(m = n) = nat_eq m n" |
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unfolding nat_eq_def by simp |
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lemma [code fun]: |
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"int_aux i n = (if nat_eq n 0 then i else int_aux (i + 1) (nat_minus n 1))" |
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unfolding nat_eq_def nat_minus_def int_aux_def by simp |
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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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Code generator plug-in for implementing natural numbers by integers.
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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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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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*} |
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(*<*) |
19791 | 162 |
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ML {* |
19791 | 164 |
local |
165 |
val Suc_if_eq = thm "Suc_if_eq"; |
|
166 |
val Suc_clause = thm "Suc_clause"; |
|
167 |
fun contains_suc t = member (op =) (term_consts t) "Suc"; |
|
168 |
in |
|
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6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
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parents:
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changeset
|
169 |
|
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Preprocessors now transfer theorems to current theory in order to
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|
170 |
fun remove_suc thy thms = |
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6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
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changeset
|
171 |
let |
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Preprocessors now transfer theorems to current theory in order to
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|
172 |
val Suc_if_eq' = Thm.transfer thy Suc_if_eq; |
20071
8f3e1ddb50e6
replaced Term.variant(list) by Name.variant(_list);
wenzelm
parents:
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diff
changeset
|
173 |
val vname = Name.variant (map fst |
20196
9a19e4de6e2e
renamed add_term_varnames to Term.add_varnames (cf. Term.add_vars etc.);
wenzelm
parents:
20105
diff
changeset
|
174 |
(fold (Term.add_varnames o Thm.full_prop_of) thms [])) "x"; |
16861 | 175 |
val cv = cterm_of Main.thy (Var ((vname, 0), HOLogic.natT)); |
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Code generator plug-in for implementing natural numbers by integers.
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parents:
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|
176 |
fun lhs_of th = snd (Thm.dest_comb |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
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parents:
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changeset
|
177 |
(fst (Thm.dest_comb (snd (Thm.dest_comb (cprop_of th)))))); |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
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parents:
diff
changeset
|
178 |
fun rhs_of th = snd (Thm.dest_comb (snd (Thm.dest_comb (cprop_of th)))); |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
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parents:
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changeset
|
179 |
fun find_vars ct = (case term_of ct of |
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Rewrote function remove_suc, since it failed on some equations
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|
180 |
(Const ("Suc", _) $ Var _) => [(cv, snd (Thm.dest_comb ct))] |
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Code generator plug-in for implementing natural numbers by integers.
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|
181 |
| _ $ _ => |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
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parents:
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changeset
|
182 |
let val (ct1, ct2) = Thm.dest_comb ct |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
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parents:
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changeset
|
183 |
in |
16900
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Rewrote function remove_suc, since it failed on some equations
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parents:
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diff
changeset
|
184 |
map (apfst (fn ct => Thm.capply ct ct2)) (find_vars ct1) @ |
e294033d1c0f
Rewrote function remove_suc, since it failed on some equations
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parents:
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diff
changeset
|
185 |
map (apfst (Thm.capply ct1)) (find_vars ct2) |
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6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
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parents:
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changeset
|
186 |
end |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
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parents:
diff
changeset
|
187 |
| _ => []); |
16900
e294033d1c0f
Rewrote function remove_suc, since it failed on some equations
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parents:
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changeset
|
188 |
val eqs = List.concat (map |
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
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parents:
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changeset
|
189 |
(fn th => map (pair th) (find_vars (lhs_of th))) thms); |
16900
e294033d1c0f
Rewrote function remove_suc, since it failed on some equations
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parents:
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changeset
|
190 |
fun mk_thms (th, (ct, cv')) = |
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6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
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parents:
diff
changeset
|
191 |
let |
16900
e294033d1c0f
Rewrote function remove_suc, since it failed on some equations
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parents:
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diff
changeset
|
192 |
val th' = |
e294033d1c0f
Rewrote function remove_suc, since it failed on some equations
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parents:
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diff
changeset
|
193 |
Thm.implies_elim |
15396
0a36ccb79481
Preprocessors now transfer theorems to current theory in order to
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parents:
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diff
changeset
|
194 |
(Drule.fconv_rule (Thm.beta_conversion true) |
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6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
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parents:
diff
changeset
|
195 |
(Drule.instantiate' |
15531 | 196 |
[SOME (ctyp_of_term ct)] [SOME (Thm.cabs cv ct), |
16900
e294033d1c0f
Rewrote function remove_suc, since it failed on some equations
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parents:
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changeset
|
197 |
SOME (Thm.cabs cv' (rhs_of th)), NONE, SOME cv'] |
e294033d1c0f
Rewrote function remove_suc, since it failed on some equations
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parents:
16861
diff
changeset
|
198 |
Suc_if_eq')) (Thm.forall_intr cv' th) |
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6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
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parents:
diff
changeset
|
199 |
in |
16900
e294033d1c0f
Rewrote function remove_suc, since it failed on some equations
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parents:
16861
diff
changeset
|
200 |
case List.mapPartial (fn th'' => |
20287
8cbcb46c3c09
replaced obsolete standard/freeze_all by Variable.trade;
wenzelm
parents:
20196
diff
changeset
|
201 |
SOME (th'', singleton |
8cbcb46c3c09
replaced obsolete standard/freeze_all by Variable.trade;
wenzelm
parents:
20196
diff
changeset
|
202 |
(Variable.trade (Variable.thm_context th'') (fn [th'''] => [th''' RS th'])) th'') |
16900
e294033d1c0f
Rewrote function remove_suc, since it failed on some equations
berghofe
parents:
16861
diff
changeset
|
203 |
handle THM _ => NONE) thms of |
e294033d1c0f
Rewrote function remove_suc, since it failed on some equations
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parents:
16861
diff
changeset
|
204 |
[] => NONE |
e294033d1c0f
Rewrote function remove_suc, since it failed on some equations
berghofe
parents:
16861
diff
changeset
|
205 |
| thps => |
19791 | 206 |
let val (ths1, ths2) = split_list thps |
16900
e294033d1c0f
Rewrote function remove_suc, since it failed on some equations
berghofe
parents:
16861
diff
changeset
|
207 |
in SOME (gen_rems eq_thm (thms, th :: ths1) @ ths2) end |
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
208 |
end |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
209 |
in |
19791 | 210 |
case get_first mk_thms eqs of |
16900
e294033d1c0f
Rewrote function remove_suc, since it failed on some equations
berghofe
parents:
16861
diff
changeset
|
211 |
NONE => thms |
e294033d1c0f
Rewrote function remove_suc, since it failed on some equations
berghofe
parents:
16861
diff
changeset
|
212 |
| SOME x => remove_suc thy x |
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
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changeset
|
213 |
end; |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
214 |
|
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
215 |
fun eqn_suc_preproc thy ths = |
19791 | 216 |
let |
217 |
val dest = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of |
|
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
218 |
in |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
219 |
if forall (can dest) ths andalso |
19791 | 220 |
exists (contains_suc o dest) ths |
15396
0a36ccb79481
Preprocessors now transfer theorems to current theory in order to
berghofe
parents:
15323
diff
changeset
|
221 |
then remove_suc thy ths else ths |
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
222 |
end; |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
223 |
|
15396
0a36ccb79481
Preprocessors now transfer theorems to current theory in order to
berghofe
parents:
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diff
changeset
|
224 |
fun remove_suc_clause thy thms = |
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
225 |
let |
15396
0a36ccb79481
Preprocessors now transfer theorems to current theory in order to
berghofe
parents:
15323
diff
changeset
|
226 |
val Suc_clause' = Thm.transfer thy Suc_clause; |
20071
8f3e1ddb50e6
replaced Term.variant(list) by Name.variant(_list);
wenzelm
parents:
19889
diff
changeset
|
227 |
val vname = Name.variant (map fst |
20196
9a19e4de6e2e
renamed add_term_varnames to Term.add_varnames (cf. Term.add_vars etc.);
wenzelm
parents:
20105
diff
changeset
|
228 |
(fold (Term.add_varnames o Thm.full_prop_of) thms [])) "x"; |
15531 | 229 |
fun find_var (t as Const ("Suc", _) $ (v as Var _)) = SOME (t, v) |
230 |
| find_var (t $ u) = (case find_var t of NONE => find_var u | x => x) |
|
231 |
| find_var _ = NONE; |
|
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
232 |
fun find_thm th = |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
233 |
let val th' = ObjectLogic.atomize_thm th |
15570 | 234 |
in Option.map (pair (th, th')) (find_var (prop_of th')) end |
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
235 |
in |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
236 |
case get_first find_thm thms of |
15531 | 237 |
NONE => thms |
238 |
| SOME ((th, th'), (Sucv, v)) => |
|
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
239 |
let |
16861 | 240 |
val cert = cterm_of (Thm.theory_of_thm th); |
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
241 |
val th'' = ObjectLogic.rulify (Thm.implies_elim |
15396
0a36ccb79481
Preprocessors now transfer theorems to current theory in order to
berghofe
parents:
15323
diff
changeset
|
242 |
(Drule.fconv_rule (Thm.beta_conversion true) |
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
243 |
(Drule.instantiate' [] |
15531 | 244 |
[SOME (cert (lambda v (Abs ("x", HOLogic.natT, |
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
245 |
abstract_over (Sucv, |
19828 | 246 |
HOLogic.dest_Trueprop (prop_of th')))))), |
15531 | 247 |
SOME (cert v)] Suc_clause')) |
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
248 |
(Thm.forall_intr (cert v) th')) |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
249 |
in |
15396
0a36ccb79481
Preprocessors now transfer theorems to current theory in order to
berghofe
parents:
15323
diff
changeset
|
250 |
remove_suc_clause thy (map (fn th''' => |
19617 | 251 |
if (op = o pairself prop_of) (th''', th) then th'' else th''') thms) |
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
252 |
end |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
253 |
end; |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
254 |
|
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
255 |
fun clause_suc_preproc thy ths = |
19791 | 256 |
let |
19828 | 257 |
val dest = fst o HOLogic.dest_mem o HOLogic.dest_Trueprop |
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
258 |
in |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
259 |
if forall (can (dest o concl_of)) ths andalso |
19791 | 260 |
exists (fn th => member (op =) (foldr add_term_consts |
261 |
[] (List.mapPartial (try dest) (concl_of th :: prems_of th))) "Suc") ths |
|
15396
0a36ccb79481
Preprocessors now transfer theorems to current theory in order to
berghofe
parents:
15323
diff
changeset
|
262 |
then remove_suc_clause thy ths else ths |
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
263 |
end; |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
264 |
|
19791 | 265 |
end; (*local*) |
266 |
||
267 |
fun lift_obj_eq f thy = |
|
268 |
map (fn thm => thm RS meta_eq_to_obj_eq) |
|
269 |
#> f thy |
|
270 |
#> map (fn thm => thm RS HOL.eq_reflection) |
|
271 |
*} |
|
272 |
||
273 |
setup {* |
|
19603 | 274 |
Codegen.add_preprocessor eqn_suc_preproc |
275 |
#> Codegen.add_preprocessor clause_suc_preproc |
|
19791 | 276 |
#> CodegenTheorems.add_preproc (lift_obj_eq eqn_suc_preproc) |
277 |
#> CodegenTheorems.add_preproc (lift_obj_eq clause_suc_preproc) |
|
15323
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
278 |
*} |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
279 |
(*>*) |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
280 |
|
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
berghofe
parents:
diff
changeset
|
281 |
end |