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src/HOL/Library/Code_Target_Nat.thy

author | haftmann |

Fri, 15 Feb 2013 08:31:31 +0100 | |

changeset 51143 | 0a2371e7ced3 |

parent 51114 | 3e913a575dc6 |

child 52435 | 6646bb548c6b |

permissions | -rw-r--r-- |

two target language numeral types: integer and natural, as replacement for code_numeral;
former theory HOL/Library/Code_Numeral_Types replaces HOL/Code_Numeral;
refined stack of theories implementing int and/or nat by target language numerals;
reduced number of target language numeral types to exactly one

(* Title: HOL/Library/Code_Target_Nat.thy Author: Florian Haftmann, TU Muenchen *) header {* Implementation of natural numbers by target-language integers *} theory Code_Target_Nat imports Code_Abstract_Nat begin subsection {* Implementation for @{typ nat} *} lift_definition Nat :: "integer \<Rightarrow> nat" is nat . lemma [code_post]: "Nat 0 = 0" "Nat 1 = 1" "Nat (numeral k) = numeral k" by (transfer, simp)+ lemma [code_abbrev]: "integer_of_nat = of_nat" by transfer rule lemma [code_unfold]: "Int.nat (int_of_integer k) = nat_of_integer k" by transfer rule lemma [code abstype]: "Code_Target_Nat.Nat (integer_of_nat n) = n" by transfer simp lemma [code abstract]: "integer_of_nat (nat_of_integer k) = max 0 k" by transfer auto lemma [code_abbrev]: "nat_of_integer (numeral k) = nat_of_num k" by transfer (simp add: nat_of_num_numeral) lemma [code abstract]: "integer_of_nat (nat_of_num n) = integer_of_num n" by transfer (simp add: nat_of_num_numeral) lemma [code abstract]: "integer_of_nat 0 = 0" by transfer simp lemma [code abstract]: "integer_of_nat 1 = 1" by transfer simp lemma [code]: "Suc n = n + 1" by simp lemma [code abstract]: "integer_of_nat (m + n) = of_nat m + of_nat n" by transfer simp lemma [code abstract]: "integer_of_nat (m - n) = max 0 (of_nat m - of_nat n)" by transfer simp lemma [code abstract]: "integer_of_nat (m * n) = of_nat m * of_nat n" by transfer (simp add: of_nat_mult) lemma [code abstract]: "integer_of_nat (m div n) = of_nat m div of_nat n" by transfer (simp add: zdiv_int) lemma [code abstract]: "integer_of_nat (m mod n) = of_nat m mod of_nat n" by transfer (simp add: zmod_int) lemma [code]: "Divides.divmod_nat m n = (m div n, m mod n)" by (simp add: prod_eq_iff) lemma [code]: "HOL.equal m n = HOL.equal (of_nat m :: integer) (of_nat n)" by transfer (simp add: equal) lemma [code]: "m \<le> n \<longleftrightarrow> (of_nat m :: integer) \<le> of_nat n" by simp lemma [code]: "m < n \<longleftrightarrow> (of_nat m :: integer) < of_nat n" by simp lemma num_of_nat_code [code]: "num_of_nat = num_of_integer \<circ> of_nat" by transfer (simp add: fun_eq_iff) lemma (in semiring_1) of_nat_code: "of_nat n = (if n = 0 then 0 else let (m, q) = divmod_nat n 2; m' = 2 * of_nat m in if q = 0 then m' else m' + 1)" proof - from mod_div_equality have *: "of_nat n = of_nat (n div 2 * 2 + n mod 2)" by simp show ?thesis by (simp add: Let_def divmod_nat_div_mod mod_2_not_eq_zero_eq_one_nat of_nat_add [symmetric]) (simp add: * mult_commute of_nat_mult add_commute) qed declare of_nat_code [code] definition int_of_nat :: "nat \<Rightarrow> int" where [code_abbrev]: "int_of_nat = of_nat" lemma [code]: "int_of_nat n = int_of_integer (of_nat n)" by (simp add: int_of_nat_def) lemma [code abstract]: "integer_of_nat (nat k) = max 0 (integer_of_int k)" by transfer auto code_modulename SML Code_Target_Nat Arith code_modulename OCaml Code_Target_Nat Arith code_modulename Haskell Code_Target_Nat Arith end