author | wenzelm |
Mon, 26 Feb 2007 23:18:24 +0100 | |
changeset 22360 | 26ead7ed4f4b |
parent 22320 | d5260836d662 |
child 22395 | b573f1f566e1 |
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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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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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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subsection {* Logical rewrites *} |
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Code generator plug-in for implementing natural numbers by integers.
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text {* |
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A int-to-nat conversion with domain |
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restricted to non-negative ints (in contrast to @{const nat}). |
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Note that this restriction has no logical relevance and |
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is just a kind of proof hint -- nothing prevents you from |
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writing nonsense like @{term "nat_of_int (-4)"} |
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*} |
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definition |
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nat_of_int :: "int \<Rightarrow> nat" where |
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"k \<ge> 0 \<Longrightarrow> nat_of_int k = nat k" |
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lemma nat_of_int_int: |
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"nat_of_int (int n) = n" |
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using zero_zle_int nat_of_int_def by simp |
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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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Code generator plug-in for implementing natural numbers by integers.
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*} |
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lemma [code unfold, code noinline]: "nat_case \<equiv> (\<lambda>f g n. if n = 0 then f else g (n - 1))" |
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proof - |
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have rewrite: "\<And>f g n. nat_case f g n = (if n = 0 then f else g (n - 1))" |
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proof - |
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fix f g n |
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show "nat_case f g n = (if n = 0 then f else g (n - 1))" |
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by (cases n) simp_all |
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qed |
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show "nat_case \<equiv> (\<lambda>f g n. if n = 0 then f else g (n - 1))" |
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by (rule eq_reflection ext rewrite)+ |
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qed |
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lemma [code inline]: |
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"nat_case f g n = (if n = 0 then f else g (nat_of_int (int n - 1)))" |
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by (cases n) (simp_all add: nat_of_int_int) |
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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 func]: "0 = nat_of_int 0" |
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by (simp add: nat_of_int_def) |
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lemma [code func, code inline]: "1 = nat_of_int 1" |
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by (simp add: nat_of_int_def) |
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lemma [code func]: "Suc n = n + 1" |
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by simp |
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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 func, code inline]: "m + n = nat_of_int (int m + int n)" |
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by (simp add: nat_of_int_def) |
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lemma [code, code inline]: "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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unfolding zmult_int by simp |
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lemma [code func, code inline]: "m * n = nat_of_int (int m * int n)" |
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proof - |
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have "int (m * n) = int m * int n" |
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by (induct m) (simp_all add: zadd_zmult_distrib) |
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then have "m * n = nat (int m * int n)" by auto |
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also have "\<dots> = nat_of_int (int m * int n)" |
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proof - |
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have "int m \<ge> 0" and "int n \<ge> 0" by auto |
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have "int m * int n \<ge> 0" by (rule split_mult_pos_le) auto |
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with nat_of_int_def show ?thesis by auto |
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qed |
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finally show ?thesis . |
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qed |
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lemma [code]: "m div n = nat (int m div int n)" |
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unfolding zdiv_int [symmetric] by simp |
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lemma [code func]: "m div n = fst (Divides.divmod m n)" |
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unfolding divmod_def by simp |
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lemma [code]: "m mod n = nat (int m mod int n)" |
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unfolding zmod_int [symmetric] by simp |
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lemma [code func]: "m mod n = snd (Divides.divmod m n)" |
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unfolding divmod_def by simp |
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lemma [code, code inline]: "(m < n) \<longleftrightarrow> (int m < int n)" |
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by simp |
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lemma [code func, code inline]: "(m \<le> n) \<longleftrightarrow> (int m \<le> int n)" |
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by simp |
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lemma [code func, code inline]: "m = n \<longleftrightarrow> int m = int n" |
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by simp |
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lemma [code func]: "nat k = (if k < 0 then 0 else nat_of_int k)" |
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proof (cases "k < 0") |
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case True then show ?thesis by simp |
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next |
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case False then show ?thesis by (simp add: nat_of_int_def) |
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qed |
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lemma [code func]: |
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"int_aux i n = (if int n = 0 then i else int_aux (i + 1) (nat_of_int (int n - 1)))" |
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proof - |
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have "0 < n \<Longrightarrow> int n = 1 + int (nat_of_int (int n - 1))" |
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proof - |
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assume prem: "n > 0" |
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then have "int n - 1 \<ge> 0" by auto |
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then have "nat_of_int (int n - 1) = nat (int n - 1)" by (simp add: nat_of_int_def) |
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with prem show "int n = 1 + int (nat_of_int (int n - 1))" by simp |
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qed |
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then show ?thesis unfolding int_aux_def by simp |
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qed |
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subsection {* Code generator setup for basic functions *} |
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text {* |
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@{typ nat} is no longer a datatype but embedded into the integers. |
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*} |
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code_const "0::nat" |
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(SML "!(0 : IntInf.int)") |
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(OCaml "Big'_int.big'_int'_of'_int/ 0") |
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(Haskell "0") |
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code_const "Suc" |
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(SML "IntInf.+ ((_), 1)") |
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(OCaml "Big_int.succ'_big'_int") |
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(Haskell "!((_) + 1)") |
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setup {* |
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CodegenData.del_datatype "nat" |
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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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val term_of_nat = HOLogic.mk_number HOLogic.natT o IntInf.fromInt; |
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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_type nat |
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(SML "IntInf.int") |
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(OCaml "Big'_int.big'_int") |
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(Haskell "Integer") |
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consts_code |
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"HOL.zero" :: nat ("0") |
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Suc ("(_ + 1)") |
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text {* |
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Since natural numbers are implemented |
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using integers, the coercion function @{const "int"} of type |
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@{typ "nat \<Rightarrow> int"} is simply implemented by the identity function, |
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likewise @{const nat_of_int} of type @{typ "int \<Rightarrow> nat"}. |
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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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consts_code |
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int ("(_)") |
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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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code_const int |
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(SML "_") |
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(OCaml "_") |
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(Haskell "_") |
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code_const nat_of_int |
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(SML "_") |
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(OCaml "_") |
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(Haskell "_") |
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hide const nat_of_int |
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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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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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Code generator plug-in for implementing natural numbers by integers.
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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 = Name.variant (map fst |
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(fold (Term.add_varnames o Thm.full_prop_of) thms [])) "x"; |
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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 (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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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 = 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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247 |
fun mk_thms (th, (ct, cv')) = |
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Code generator plug-in for implementing natural numbers by integers.
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let |
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val th' = |
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250 |
Thm.implies_elim |
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251 |
(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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252 |
(Drule.instantiate' |
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[SOME (ctyp_of_term ct)] [SOME (Thm.cabs cv ct), |
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254 |
SOME (Thm.cabs cv' (rhs_of th)), NONE, SOME cv'] |
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255 |
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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256 |
in |
21287 | 257 |
case map_filter (fn th'' => |
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|
258 |
SOME (th'', singleton |
21287 | 259 |
(Variable.trade (K (fn [th'''] => [th''' RS th'])) (Variable.thm_context th'')) th'') |
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260 |
handle THM _ => NONE) thms of |
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[] => NONE |
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262 |
| thps => |
19791 | 263 |
let val (ths1, ths2) = split_list thps |
22360
26ead7ed4f4b
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264 |
in SOME (subtract Thm.eq_thm (th :: ths1) thms @ ths2) end |
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Code generator plug-in for implementing natural numbers by integers.
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265 |
end |
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Code generator plug-in for implementing natural numbers by integers.
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266 |
in |
19791 | 267 |
case get_first mk_thms eqs of |
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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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270 |
end; |
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Code generator plug-in for implementing natural numbers by integers.
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271 |
|
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fun eqn_suc_preproc thy ths = |
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let |
274 |
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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275 |
in |
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Code generator plug-in for implementing natural numbers by integers.
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if forall (can dest) ths andalso |
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exists (contains_suc o dest) ths |
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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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279 |
end; |
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280 |
|
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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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283 |
val Suc_clause' = Thm.transfer thy Suc_clause; |
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replaced Term.variant(list) by Name.variant(_list);
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|
284 |
val vname = Name.variant (map fst |
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renamed add_term_varnames to Term.add_varnames (cf. Term.add_vars etc.);
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285 |
(fold (Term.add_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) |
287 |
| find_var (t $ u) = (case find_var t of NONE => find_var u | x => x) |
|
288 |
| find_var _ = NONE; |
|
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Code generator plug-in for implementing natural numbers by integers.
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289 |
fun find_thm th = |
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Code generator plug-in for implementing natural numbers by integers.
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290 |
let val th' = ObjectLogic.atomize_thm th |
15570 | 291 |
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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292 |
in |
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Code generator plug-in for implementing natural numbers by integers.
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293 |
case get_first find_thm thms of |
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NONE => thms |
295 |
| SOME ((th, th'), (Sucv, v)) => |
|
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Code generator plug-in for implementing natural numbers by integers.
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296 |
let |
16861 | 297 |
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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298 |
val th'' = ObjectLogic.rulify (Thm.implies_elim |
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299 |
(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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|
300 |
(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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|
302 |
abstract_over (Sucv, |
19828 | 303 |
HOLogic.dest_Trueprop (prop_of th')))))), |
15531 | 304 |
SOME (cert v)] Suc_clause')) |
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Code generator plug-in for implementing natural numbers by integers.
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|
305 |
(Thm.forall_intr (cert v) th')) |
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Code generator plug-in for implementing natural numbers by integers.
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|
306 |
in |
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|
307 |
remove_suc_clause thy (map (fn th''' => |
19617 | 308 |
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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|
309 |
end |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
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|
310 |
end; |
6c10fe1c0e17
Code generator plug-in for implementing natural numbers by integers.
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|
311 |
|
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Code generator plug-in for implementing natural numbers by integers.
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312 |
fun clause_suc_preproc thy ths = |
19791 | 313 |
let |
19828 | 314 |
val dest = fst o HOLogic.dest_mem o HOLogic.dest_Trueprop |
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Code generator plug-in for implementing natural numbers by integers.
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|
315 |
in |
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Code generator plug-in for implementing natural numbers by integers.
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|
316 |
if forall (can (dest o concl_of)) ths andalso |
19791 | 317 |
exists (fn th => member (op =) (foldr add_term_consts |
21287 | 318 |
[] (map_filter (try dest) (concl_of th :: prems_of th))) "Suc") ths |
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|
319 |
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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|
320 |
end; |
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Code generator plug-in for implementing natural numbers by integers.
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|
321 |
|
19791 | 322 |
end; (*local*) |
323 |
||
21546 | 324 |
local |
325 |
val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq"; |
|
326 |
val eq_reflection = thm "eq_reflection"; |
|
327 |
in fun lift_obj_eq f thy = |
|
19791 | 328 |
map (fn thm => thm RS meta_eq_to_obj_eq) |
329 |
#> f thy |
|
21546 | 330 |
#> map (fn thm => thm RS eq_reflection) |
331 |
end |
|
19791 | 332 |
*} |
333 |
||
334 |
setup {* |
|
19603 | 335 |
Codegen.add_preprocessor eqn_suc_preproc |
336 |
#> Codegen.add_preprocessor clause_suc_preproc |
|
22046 | 337 |
#> CodegenData.del_inline_proc "elim_number_of_nat" |
338 |
#> CodegenData.add_preproc ("eqn_Suc", lift_obj_eq eqn_suc_preproc) |
|
339 |
#> CodegenData.add_preproc ("clause_Suc", lift_obj_eq clause_suc_preproc) |
|
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Code generator plug-in for implementing natural numbers by integers.
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340 |
*} |
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Code generator plug-in for implementing natural numbers by integers.
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341 |
(*>*) |
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Code generator plug-in for implementing natural numbers by integers.
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|
342 |
|
21191 | 343 |
subsection {* Module names *} |
344 |
||
345 |
code_modulename SML |
|
346 |
Nat Integer |
|
347 |
EfficientNat Integer |
|
348 |
||
21911
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349 |
code_modulename OCaml |
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|
350 |
Nat Integer |
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|
351 |
EfficientNat Integer |
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|
352 |
|
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Code generator plug-in for implementing natural numbers by integers.
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|
353 |
end |