src/HOL/Code_Numeral.thy
author haftmann
Sun Oct 08 22:28:21 2017 +0200 (20 months ago)
changeset 66806 a4e82b58d833
parent 66801 f3fda9777f9a
child 66815 93c6632ddf44
permissions -rw-r--r--
abolished (semi)ring_div in favour of euclidean_(semi)ring_cancel
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(*  Title:      HOL/Code_Numeral.thy
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    Author:     Florian Haftmann, TU Muenchen
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*)
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section \<open>Numeric types for code generation onto target language numerals only\<close>
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theory Code_Numeral
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imports Nat_Transfer Divides Lifting
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begin
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subsection \<open>Type of target language integers\<close>
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typedef integer = "UNIV :: int set"
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  morphisms int_of_integer integer_of_int ..
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setup_lifting type_definition_integer
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lemma integer_eq_iff:
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  "k = l \<longleftrightarrow> int_of_integer k = int_of_integer l"
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  by transfer rule
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lemma integer_eqI:
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  "int_of_integer k = int_of_integer l \<Longrightarrow> k = l"
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  using integer_eq_iff [of k l] by simp
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lemma int_of_integer_integer_of_int [simp]:
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  "int_of_integer (integer_of_int k) = k"
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  by transfer rule
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lemma integer_of_int_int_of_integer [simp]:
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  "integer_of_int (int_of_integer k) = k"
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  by transfer rule
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instantiation integer :: ring_1
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begin
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lift_definition zero_integer :: integer
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  is "0 :: int"
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  .
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declare zero_integer.rep_eq [simp]
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lift_definition one_integer :: integer
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  is "1 :: int"
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  .
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declare one_integer.rep_eq [simp]
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lift_definition plus_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
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  is "plus :: int \<Rightarrow> int \<Rightarrow> int"
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  .
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declare plus_integer.rep_eq [simp]
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lift_definition uminus_integer :: "integer \<Rightarrow> integer"
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  is "uminus :: int \<Rightarrow> int"
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  .
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declare uminus_integer.rep_eq [simp]
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lift_definition minus_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
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  is "minus :: int \<Rightarrow> int \<Rightarrow> int"
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  .
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declare minus_integer.rep_eq [simp]
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lift_definition times_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
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  is "times :: int \<Rightarrow> int \<Rightarrow> int"
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  .
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declare times_integer.rep_eq [simp]
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instance proof
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qed (transfer, simp add: algebra_simps)+
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end
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instance integer :: Rings.dvd ..
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lemma [transfer_rule]:
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  "rel_fun pcr_integer (rel_fun pcr_integer HOL.iff) Rings.dvd Rings.dvd"
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  unfolding dvd_def by transfer_prover
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lemma [transfer_rule]:
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  "rel_fun HOL.eq pcr_integer (of_nat :: nat \<Rightarrow> int) (of_nat :: nat \<Rightarrow> integer)"
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  by (rule transfer_rule_of_nat) transfer_prover+
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lemma [transfer_rule]:
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  "rel_fun HOL.eq pcr_integer (\<lambda>k :: int. k :: int) (of_int :: int \<Rightarrow> integer)"
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proof -
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  have "rel_fun HOL.eq pcr_integer (of_int :: int \<Rightarrow> int) (of_int :: int \<Rightarrow> integer)"
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    by (rule transfer_rule_of_int) transfer_prover+
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  then show ?thesis by (simp add: id_def)
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qed
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lemma [transfer_rule]:
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  "rel_fun HOL.eq pcr_integer (numeral :: num \<Rightarrow> int) (numeral :: num \<Rightarrow> integer)"
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  by (rule transfer_rule_numeral) transfer_prover+
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lemma [transfer_rule]:
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  "rel_fun HOL.eq (rel_fun HOL.eq pcr_integer) (Num.sub :: _ \<Rightarrow> _ \<Rightarrow> int) (Num.sub :: _ \<Rightarrow> _ \<Rightarrow> integer)"
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  by (unfold Num.sub_def [abs_def]) transfer_prover
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lemma int_of_integer_of_nat [simp]:
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  "int_of_integer (of_nat n) = of_nat n"
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  by transfer rule
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lift_definition integer_of_nat :: "nat \<Rightarrow> integer"
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  is "of_nat :: nat \<Rightarrow> int"
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  .
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lemma integer_of_nat_eq_of_nat [code]:
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  "integer_of_nat = of_nat"
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  by transfer rule
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lemma int_of_integer_integer_of_nat [simp]:
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  "int_of_integer (integer_of_nat n) = of_nat n"
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  by transfer rule
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lift_definition nat_of_integer :: "integer \<Rightarrow> nat"
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  is Int.nat
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  .
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lemma nat_of_integer_of_nat [simp]:
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  "nat_of_integer (of_nat n) = n"
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  by transfer simp
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lemma int_of_integer_of_int [simp]:
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  "int_of_integer (of_int k) = k"
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  by transfer simp
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lemma nat_of_integer_integer_of_nat [simp]:
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  "nat_of_integer (integer_of_nat n) = n"
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  by transfer simp
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lemma integer_of_int_eq_of_int [simp, code_abbrev]:
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  "integer_of_int = of_int"
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  by transfer (simp add: fun_eq_iff)
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lemma of_int_integer_of [simp]:
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  "of_int (int_of_integer k) = (k :: integer)"
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  by transfer rule
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lemma int_of_integer_numeral [simp]:
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  "int_of_integer (numeral k) = numeral k"
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  by transfer rule
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lemma int_of_integer_sub [simp]:
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  "int_of_integer (Num.sub k l) = Num.sub k l"
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  by transfer rule
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definition integer_of_num :: "num \<Rightarrow> integer"
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  where [simp]: "integer_of_num = numeral"
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lemma integer_of_num [code]:
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  "integer_of_num Num.One = 1"
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  "integer_of_num (Num.Bit0 n) = (let k = integer_of_num n in k + k)"
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  "integer_of_num (Num.Bit1 n) = (let k = integer_of_num n in k + k + 1)"
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  by (simp_all only: integer_of_num_def numeral.simps Let_def)
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lemma integer_of_num_triv:
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  "integer_of_num Num.One = 1"
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  "integer_of_num (Num.Bit0 Num.One) = 2"
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  by simp_all
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instantiation integer :: "{linordered_idom, equal}"
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begin
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lift_definition abs_integer :: "integer \<Rightarrow> integer"
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  is "abs :: int \<Rightarrow> int"
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  .
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declare abs_integer.rep_eq [simp]
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lift_definition sgn_integer :: "integer \<Rightarrow> integer"
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  is "sgn :: int \<Rightarrow> int"
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  .
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declare sgn_integer.rep_eq [simp]
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lift_definition less_eq_integer :: "integer \<Rightarrow> integer \<Rightarrow> bool"
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  is "less_eq :: int \<Rightarrow> int \<Rightarrow> bool"
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  .
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lift_definition less_integer :: "integer \<Rightarrow> integer \<Rightarrow> bool"
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  is "less :: int \<Rightarrow> int \<Rightarrow> bool"
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  .
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lift_definition equal_integer :: "integer \<Rightarrow> integer \<Rightarrow> bool"
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  is "HOL.equal :: int \<Rightarrow> int \<Rightarrow> bool"
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  .
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instance
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  by standard (transfer, simp add: algebra_simps equal less_le_not_le [symmetric] mult_strict_right_mono linear)+
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end
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lemma [transfer_rule]:
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  "rel_fun pcr_integer (rel_fun pcr_integer pcr_integer) (min :: _ \<Rightarrow> _ \<Rightarrow> int) (min :: _ \<Rightarrow> _ \<Rightarrow> integer)"
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  by (unfold min_def [abs_def]) transfer_prover
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lemma [transfer_rule]:
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  "rel_fun pcr_integer (rel_fun pcr_integer pcr_integer) (max :: _ \<Rightarrow> _ \<Rightarrow> int) (max :: _ \<Rightarrow> _ \<Rightarrow> integer)"
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  by (unfold max_def [abs_def]) transfer_prover
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lemma int_of_integer_min [simp]:
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  "int_of_integer (min k l) = min (int_of_integer k) (int_of_integer l)"
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  by transfer rule
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lemma int_of_integer_max [simp]:
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  "int_of_integer (max k l) = max (int_of_integer k) (int_of_integer l)"
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  by transfer rule
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lemma nat_of_integer_non_positive [simp]:
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  "k \<le> 0 \<Longrightarrow> nat_of_integer k = 0"
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  by transfer simp
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lemma of_nat_of_integer [simp]:
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  "of_nat (nat_of_integer k) = max 0 k"
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  by transfer auto
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instantiation integer :: normalization_semidom
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begin
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lift_definition normalize_integer :: "integer \<Rightarrow> integer"
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  is "normalize :: int \<Rightarrow> int"
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  .
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declare normalize_integer.rep_eq [simp]
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lift_definition unit_factor_integer :: "integer \<Rightarrow> integer"
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  is "unit_factor :: int \<Rightarrow> int"
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  .
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declare unit_factor_integer.rep_eq [simp]
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lift_definition divide_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
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  is "divide :: int \<Rightarrow> int \<Rightarrow> int"
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  .
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declare divide_integer.rep_eq [simp]
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instance
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  by (standard; transfer)
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    (auto simp add: mult_sgn_abs sgn_mult abs_eq_iff')
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end
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instantiation integer :: unique_euclidean_ring
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begin
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lift_definition modulo_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
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  is "modulo :: int \<Rightarrow> int \<Rightarrow> int"
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  .
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declare modulo_integer.rep_eq [simp]
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lift_definition euclidean_size_integer :: "integer \<Rightarrow> nat"
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  is "euclidean_size :: int \<Rightarrow> nat"
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  .
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declare euclidean_size_integer.rep_eq [simp]
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lift_definition uniqueness_constraint_integer :: "integer \<Rightarrow> integer \<Rightarrow> bool"
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  is "uniqueness_constraint :: int \<Rightarrow> int \<Rightarrow> bool"
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  .
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declare uniqueness_constraint_integer.rep_eq [simp]
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instance
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  by (standard; transfer)
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    (use mult_le_mono2 [of 1] in \<open>auto simp add: sgn_mult_abs abs_mult sgn_mult abs_mod_less sgn_mod nat_mult_distrib\<close>, rule div_eqI, simp_all)
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end
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lemma [code]:
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  "euclidean_size = nat_of_integer \<circ> abs"
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  by (simp add: fun_eq_iff nat_of_integer.rep_eq)
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lemma [code]:
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  "uniqueness_constraint (k :: integer) l \<longleftrightarrow> unit_factor k = unit_factor l"
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  by (simp add: integer_eq_iff)
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instantiation integer :: unique_euclidean_semiring_numeral
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begin
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definition divmod_integer :: "num \<Rightarrow> num \<Rightarrow> integer \<times> integer"
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where
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  divmod_integer'_def: "divmod_integer m n = (numeral m div numeral n, numeral m mod numeral n)"
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definition divmod_step_integer :: "num \<Rightarrow> integer \<times> integer \<Rightarrow> integer \<times> integer"
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where
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  "divmod_step_integer l qr = (let (q, r) = qr
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    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
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    else (2 * q, r))"
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instance proof
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  show "divmod m n = (numeral m div numeral n :: integer, numeral m mod numeral n)"
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    for m n by (fact divmod_integer'_def)
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  show "divmod_step l qr = (let (q, r) = qr
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    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
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    else (2 * q, r))" for l and qr :: "integer \<times> integer"
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    by (fact divmod_step_integer_def)
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qed (transfer,
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  fact le_add_diff_inverse2
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  unique_euclidean_semiring_numeral_class.div_less
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  unique_euclidean_semiring_numeral_class.mod_less
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  unique_euclidean_semiring_numeral_class.div_positive
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  unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
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  unique_euclidean_semiring_numeral_class.pos_mod_bound
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  unique_euclidean_semiring_numeral_class.pos_mod_sign
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  unique_euclidean_semiring_numeral_class.mod_mult2_eq
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  unique_euclidean_semiring_numeral_class.div_mult2_eq
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  unique_euclidean_semiring_numeral_class.discrete)+
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end
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declare divmod_algorithm_code [where ?'a = integer,
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  folded integer_of_num_def, unfolded integer_of_num_triv, 
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  code]
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lemma integer_of_nat_0: "integer_of_nat 0 = 0"
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by transfer simp
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lemma integer_of_nat_1: "integer_of_nat 1 = 1"
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by transfer simp
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lemma integer_of_nat_numeral:
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  "integer_of_nat (numeral n) = numeral n"
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by transfer simp
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subsection \<open>Code theorems for target language integers\<close>
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text \<open>Constructors\<close>
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definition Pos :: "num \<Rightarrow> integer"
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where
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  [simp, code_post]: "Pos = numeral"
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lemma [transfer_rule]:
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  "rel_fun HOL.eq pcr_integer numeral Pos"
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  by simp transfer_prover
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lemma Pos_fold [code_unfold]:
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  "numeral Num.One = Pos Num.One"
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  "numeral (Num.Bit0 k) = Pos (Num.Bit0 k)"
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  "numeral (Num.Bit1 k) = Pos (Num.Bit1 k)"
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  by simp_all
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definition Neg :: "num \<Rightarrow> integer"
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where
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  [simp, code_abbrev]: "Neg n = - Pos n"
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   354
haftmann@51143
   355
lemma [transfer_rule]:
blanchet@55945
   356
  "rel_fun HOL.eq pcr_integer (\<lambda>n. - numeral n) Neg"
haftmann@54489
   357
  by (simp add: Neg_def [abs_def]) transfer_prover
haftmann@51143
   358
haftmann@51143
   359
code_datatype "0::integer" Pos Neg
haftmann@51143
   360
haftmann@64994
   361
  
haftmann@64994
   362
text \<open>A further pair of constructors for generated computations\<close>
haftmann@64994
   363
haftmann@64994
   364
context
haftmann@64994
   365
begin  
haftmann@64994
   366
haftmann@64994
   367
qualified definition positive :: "num \<Rightarrow> integer"
haftmann@64994
   368
  where [simp]: "positive = numeral"
haftmann@64994
   369
haftmann@64994
   370
qualified definition negative :: "num \<Rightarrow> integer"
haftmann@64994
   371
  where [simp]: "negative = uminus \<circ> numeral"
haftmann@64994
   372
haftmann@64994
   373
lemma [code_computation_unfold]:
haftmann@64994
   374
  "numeral = positive"
haftmann@64994
   375
  "Pos = positive"
haftmann@64994
   376
  "Neg = negative"
haftmann@64994
   377
  by (simp_all add: fun_eq_iff)
haftmann@64994
   378
haftmann@64994
   379
end
haftmann@64994
   380
haftmann@51143
   381
wenzelm@60758
   382
text \<open>Auxiliary operations\<close>
haftmann@51143
   383
haftmann@51143
   384
lift_definition dup :: "integer \<Rightarrow> integer"
haftmann@51143
   385
  is "\<lambda>k::int. k + k"
haftmann@51143
   386
  .
haftmann@26140
   387
haftmann@51143
   388
lemma dup_code [code]:
haftmann@51143
   389
  "dup 0 = 0"
haftmann@51143
   390
  "dup (Pos n) = Pos (Num.Bit0 n)"
haftmann@51143
   391
  "dup (Neg n) = Neg (Num.Bit0 n)"
haftmann@54489
   392
  by (transfer, simp only: numeral_Bit0 minus_add_distrib)+
haftmann@51143
   393
haftmann@51143
   394
lift_definition sub :: "num \<Rightarrow> num \<Rightarrow> integer"
haftmann@51143
   395
  is "\<lambda>m n. numeral m - numeral n :: int"
haftmann@51143
   396
  .
haftmann@26140
   397
haftmann@51143
   398
lemma sub_code [code]:
haftmann@51143
   399
  "sub Num.One Num.One = 0"
haftmann@51143
   400
  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
haftmann@51143
   401
  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
haftmann@51143
   402
  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
haftmann@51143
   403
  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
haftmann@51143
   404
  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
haftmann@51143
   405
  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
haftmann@51143
   406
  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
haftmann@51143
   407
  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
haftmann@51143
   408
  by (transfer, simp add: dbl_def dbl_inc_def dbl_dec_def)+
haftmann@28351
   409
haftmann@24999
   410
wenzelm@60758
   411
text \<open>Implementations\<close>
haftmann@24999
   412
haftmann@51143
   413
lemma one_integer_code [code, code_unfold]:
haftmann@51143
   414
  "1 = Pos Num.One"
haftmann@51143
   415
  by simp
haftmann@24999
   416
haftmann@51143
   417
lemma plus_integer_code [code]:
haftmann@51143
   418
  "k + 0 = (k::integer)"
haftmann@51143
   419
  "0 + l = (l::integer)"
haftmann@51143
   420
  "Pos m + Pos n = Pos (m + n)"
haftmann@51143
   421
  "Pos m + Neg n = sub m n"
haftmann@51143
   422
  "Neg m + Pos n = sub n m"
haftmann@51143
   423
  "Neg m + Neg n = Neg (m + n)"
haftmann@51143
   424
  by (transfer, simp)+
haftmann@24999
   425
haftmann@51143
   426
lemma uminus_integer_code [code]:
haftmann@51143
   427
  "uminus 0 = (0::integer)"
haftmann@51143
   428
  "uminus (Pos m) = Neg m"
haftmann@51143
   429
  "uminus (Neg m) = Pos m"
haftmann@51143
   430
  by simp_all
haftmann@28708
   431
haftmann@51143
   432
lemma minus_integer_code [code]:
haftmann@51143
   433
  "k - 0 = (k::integer)"
haftmann@51143
   434
  "0 - l = uminus (l::integer)"
haftmann@51143
   435
  "Pos m - Pos n = sub m n"
haftmann@51143
   436
  "Pos m - Neg n = Pos (m + n)"
haftmann@51143
   437
  "Neg m - Pos n = Neg (m + n)"
haftmann@51143
   438
  "Neg m - Neg n = sub n m"
haftmann@51143
   439
  by (transfer, simp)+
haftmann@46028
   440
haftmann@51143
   441
lemma abs_integer_code [code]:
haftmann@51143
   442
  "\<bar>k\<bar> = (if (k::integer) < 0 then - k else k)"
haftmann@51143
   443
  by simp
huffman@47108
   444
haftmann@51143
   445
lemma sgn_integer_code [code]:
haftmann@51143
   446
  "sgn k = (if k = 0 then 0 else if (k::integer) < 0 then - 1 else 1)"
huffman@47108
   447
  by simp
haftmann@46028
   448
haftmann@51143
   449
lemma times_integer_code [code]:
haftmann@51143
   450
  "k * 0 = (0::integer)"
haftmann@51143
   451
  "0 * l = (0::integer)"
haftmann@51143
   452
  "Pos m * Pos n = Pos (m * n)"
haftmann@51143
   453
  "Pos m * Neg n = Neg (m * n)"
haftmann@51143
   454
  "Neg m * Pos n = Neg (m * n)"
haftmann@51143
   455
  "Neg m * Neg n = Pos (m * n)"
haftmann@51143
   456
  by simp_all
haftmann@51143
   457
haftmann@64592
   458
lemma normalize_integer_code [code]:
haftmann@64592
   459
  "normalize = (abs :: integer \<Rightarrow> integer)"
haftmann@64592
   460
  by transfer simp
haftmann@64592
   461
haftmann@64592
   462
lemma unit_factor_integer_code [code]:
haftmann@64592
   463
  "unit_factor = (sgn :: integer \<Rightarrow> integer)"
haftmann@64592
   464
  by transfer simp
haftmann@64592
   465
haftmann@51143
   466
definition divmod_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer \<times> integer"
haftmann@51143
   467
where
haftmann@51143
   468
  "divmod_integer k l = (k div l, k mod l)"
haftmann@51143
   469
haftmann@66801
   470
lemma fst_divmod_integer [simp]:
haftmann@51143
   471
  "fst (divmod_integer k l) = k div l"
haftmann@51143
   472
  by (simp add: divmod_integer_def)
haftmann@51143
   473
haftmann@66801
   474
lemma snd_divmod_integer [simp]:
haftmann@51143
   475
  "snd (divmod_integer k l) = k mod l"
haftmann@51143
   476
  by (simp add: divmod_integer_def)
haftmann@51143
   477
haftmann@51143
   478
definition divmod_abs :: "integer \<Rightarrow> integer \<Rightarrow> integer \<times> integer"
haftmann@51143
   479
where
haftmann@51143
   480
  "divmod_abs k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
haftmann@51143
   481
haftmann@51143
   482
lemma fst_divmod_abs [simp]:
haftmann@51143
   483
  "fst (divmod_abs k l) = \<bar>k\<bar> div \<bar>l\<bar>"
haftmann@51143
   484
  by (simp add: divmod_abs_def)
haftmann@51143
   485
haftmann@51143
   486
lemma snd_divmod_abs [simp]:
haftmann@51143
   487
  "snd (divmod_abs k l) = \<bar>k\<bar> mod \<bar>l\<bar>"
haftmann@51143
   488
  by (simp add: divmod_abs_def)
haftmann@28708
   489
haftmann@53069
   490
lemma divmod_abs_code [code]:
haftmann@53069
   491
  "divmod_abs (Pos k) (Pos l) = divmod k l"
haftmann@53069
   492
  "divmod_abs (Neg k) (Neg l) = divmod k l"
haftmann@53069
   493
  "divmod_abs (Neg k) (Pos l) = divmod k l"
haftmann@53069
   494
  "divmod_abs (Pos k) (Neg l) = divmod k l"
haftmann@51143
   495
  "divmod_abs j 0 = (0, \<bar>j\<bar>)"
haftmann@51143
   496
  "divmod_abs 0 j = (0, 0)"
haftmann@51143
   497
  by (simp_all add: prod_eq_iff)
haftmann@51143
   498
haftmann@51143
   499
lemma divmod_integer_code [code]:
haftmann@51143
   500
  "divmod_integer k l =
haftmann@51143
   501
    (if k = 0 then (0, 0) else if l = 0 then (0, k) else
haftmann@51143
   502
    (apsnd \<circ> times \<circ> sgn) l (if sgn k = sgn l
haftmann@51143
   503
      then divmod_abs k l
haftmann@51143
   504
      else (let (r, s) = divmod_abs k l in
haftmann@51143
   505
        if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
haftmann@51143
   506
proof -
haftmann@51143
   507
  have aux1: "\<And>k l::int. sgn k = sgn l \<longleftrightarrow> k = 0 \<and> l = 0 \<or> 0 < l \<and> 0 < k \<or> l < 0 \<and> k < 0"
haftmann@51143
   508
    by (auto simp add: sgn_if)
haftmann@51143
   509
  have aux2: "\<And>q::int. - int_of_integer k = int_of_integer l * q \<longleftrightarrow> int_of_integer k = int_of_integer l * - q" by auto
haftmann@51143
   510
  show ?thesis
blanchet@55414
   511
    by (simp add: prod_eq_iff integer_eq_iff case_prod_beta aux1)
haftmann@51143
   512
      (auto simp add: zdiv_zminus1_eq_if zmod_zminus1_eq_if div_minus_right mod_minus_right aux2)
haftmann@51143
   513
qed
haftmann@51143
   514
haftmann@51143
   515
lemma div_integer_code [code]:
haftmann@51143
   516
  "k div l = fst (divmod_integer k l)"
haftmann@28708
   517
  by simp
haftmann@28708
   518
haftmann@51143
   519
lemma mod_integer_code [code]:
haftmann@51143
   520
  "k mod l = snd (divmod_integer k l)"
haftmann@25767
   521
  by simp
haftmann@24999
   522
haftmann@51143
   523
lemma equal_integer_code [code]:
haftmann@51143
   524
  "HOL.equal 0 (0::integer) \<longleftrightarrow> True"
haftmann@51143
   525
  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
haftmann@51143
   526
  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
haftmann@51143
   527
  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
haftmann@51143
   528
  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
haftmann@51143
   529
  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
haftmann@51143
   530
  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
haftmann@51143
   531
  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
haftmann@51143
   532
  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
haftmann@51143
   533
  by (simp_all add: equal)
haftmann@51143
   534
haftmann@51143
   535
lemma equal_integer_refl [code nbe]:
haftmann@51143
   536
  "HOL.equal (k::integer) k \<longleftrightarrow> True"
haftmann@51143
   537
  by (fact equal_refl)
haftmann@31266
   538
haftmann@51143
   539
lemma less_eq_integer_code [code]:
haftmann@51143
   540
  "0 \<le> (0::integer) \<longleftrightarrow> True"
haftmann@51143
   541
  "0 \<le> Pos l \<longleftrightarrow> True"
haftmann@51143
   542
  "0 \<le> Neg l \<longleftrightarrow> False"
haftmann@51143
   543
  "Pos k \<le> 0 \<longleftrightarrow> False"
haftmann@51143
   544
  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
haftmann@51143
   545
  "Pos k \<le> Neg l \<longleftrightarrow> False"
haftmann@51143
   546
  "Neg k \<le> 0 \<longleftrightarrow> True"
haftmann@51143
   547
  "Neg k \<le> Pos l \<longleftrightarrow> True"
haftmann@51143
   548
  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
haftmann@51143
   549
  by simp_all
haftmann@51143
   550
haftmann@51143
   551
lemma less_integer_code [code]:
haftmann@51143
   552
  "0 < (0::integer) \<longleftrightarrow> False"
haftmann@51143
   553
  "0 < Pos l \<longleftrightarrow> True"
haftmann@51143
   554
  "0 < Neg l \<longleftrightarrow> False"
haftmann@51143
   555
  "Pos k < 0 \<longleftrightarrow> False"
haftmann@51143
   556
  "Pos k < Pos l \<longleftrightarrow> k < l"
haftmann@51143
   557
  "Pos k < Neg l \<longleftrightarrow> False"
haftmann@51143
   558
  "Neg k < 0 \<longleftrightarrow> True"
haftmann@51143
   559
  "Neg k < Pos l \<longleftrightarrow> True"
haftmann@51143
   560
  "Neg k < Neg l \<longleftrightarrow> l < k"
haftmann@51143
   561
  by simp_all
haftmann@26140
   562
haftmann@51143
   563
lift_definition num_of_integer :: "integer \<Rightarrow> num"
haftmann@51143
   564
  is "num_of_nat \<circ> nat"
haftmann@51143
   565
  .
haftmann@51143
   566
haftmann@51143
   567
lemma num_of_integer_code [code]:
haftmann@51143
   568
  "num_of_integer k = (if k \<le> 1 then Num.One
haftmann@51143
   569
     else let
haftmann@51143
   570
       (l, j) = divmod_integer k 2;
haftmann@51143
   571
       l' = num_of_integer l;
haftmann@51143
   572
       l'' = l' + l'
haftmann@51143
   573
     in if j = 0 then l'' else l'' + Num.One)"
haftmann@51143
   574
proof -
haftmann@51143
   575
  {
haftmann@51143
   576
    assume "int_of_integer k mod 2 = 1"
haftmann@51143
   577
    then have "nat (int_of_integer k mod 2) = nat 1" by simp
haftmann@51143
   578
    moreover assume *: "1 < int_of_integer k"
haftmann@51143
   579
    ultimately have **: "nat (int_of_integer k) mod 2 = 1" by (simp add: nat_mod_distrib)
haftmann@51143
   580
    have "num_of_nat (nat (int_of_integer k)) =
haftmann@51143
   581
      num_of_nat (2 * (nat (int_of_integer k) div 2) + nat (int_of_integer k) mod 2)"
haftmann@51143
   582
      by simp
haftmann@51143
   583
    then have "num_of_nat (nat (int_of_integer k)) =
haftmann@51143
   584
      num_of_nat (nat (int_of_integer k) div 2 + nat (int_of_integer k) div 2 + nat (int_of_integer k) mod 2)"
haftmann@51143
   585
      by (simp add: mult_2)
haftmann@51143
   586
    with ** have "num_of_nat (nat (int_of_integer k)) =
haftmann@51143
   587
      num_of_nat (nat (int_of_integer k) div 2 + nat (int_of_integer k) div 2 + 1)"
haftmann@51143
   588
      by simp
haftmann@51143
   589
  }
haftmann@51143
   590
  note aux = this
haftmann@51143
   591
  show ?thesis
blanchet@55414
   592
    by (auto simp add: num_of_integer_def nat_of_integer_def Let_def case_prod_beta
haftmann@51143
   593
      not_le integer_eq_iff less_eq_integer_def
haftmann@51143
   594
      nat_mult_distrib nat_div_distrib num_of_nat_One num_of_nat_plus_distrib
haftmann@51143
   595
       mult_2 [where 'a=nat] aux add_One)
haftmann@25918
   596
qed
haftmann@25918
   597
haftmann@51143
   598
lemma nat_of_integer_code [code]:
haftmann@51143
   599
  "nat_of_integer k = (if k \<le> 0 then 0
haftmann@51143
   600
     else let
haftmann@51143
   601
       (l, j) = divmod_integer k 2;
haftmann@51143
   602
       l' = nat_of_integer l;
haftmann@51143
   603
       l'' = l' + l'
haftmann@51143
   604
     in if j = 0 then l'' else l'' + 1)"
haftmann@33340
   605
proof -
haftmann@51143
   606
  obtain j where "k = integer_of_int j"
haftmann@51143
   607
  proof
haftmann@51143
   608
    show "k = integer_of_int (int_of_integer k)" by simp
haftmann@51143
   609
  qed
haftmann@51143
   610
  moreover have "2 * (j div 2) = j - j mod 2"
haftmann@64246
   611
    by (simp add: minus_mod_eq_mult_div [symmetric] mult.commute)
haftmann@51143
   612
  ultimately show ?thesis
haftmann@63950
   613
    by (auto simp add: split_def Let_def modulo_integer_def nat_of_integer_def not_le
haftmann@51143
   614
      nat_add_distrib [symmetric] Suc_nat_eq_nat_zadd1)
haftmann@51143
   615
      (auto simp add: mult_2 [symmetric])
haftmann@33340
   616
qed
haftmann@28708
   617
haftmann@51143
   618
lemma int_of_integer_code [code]:
haftmann@51143
   619
  "int_of_integer k = (if k < 0 then - (int_of_integer (- k))
haftmann@51143
   620
     else if k = 0 then 0
haftmann@51143
   621
     else let
haftmann@51143
   622
       (l, j) = divmod_integer k 2;
haftmann@51143
   623
       l' = 2 * int_of_integer l
haftmann@51143
   624
     in if j = 0 then l' else l' + 1)"
haftmann@64246
   625
  by (auto simp add: split_def Let_def integer_eq_iff minus_mod_eq_mult_div [symmetric])
haftmann@28708
   626
haftmann@51143
   627
lemma integer_of_int_code [code]:
haftmann@51143
   628
  "integer_of_int k = (if k < 0 then - (integer_of_int (- k))
haftmann@51143
   629
     else if k = 0 then 0
haftmann@51143
   630
     else let
haftmann@60868
   631
       l = 2 * integer_of_int (k div 2);
haftmann@60868
   632
       j = k mod 2
haftmann@60868
   633
     in if j = 0 then l else l + 1)"
haftmann@64246
   634
  by (auto simp add: split_def Let_def integer_eq_iff minus_mod_eq_mult_div [symmetric])
haftmann@51143
   635
haftmann@51143
   636
hide_const (open) Pos Neg sub dup divmod_abs
huffman@46547
   637
haftmann@28708
   638
wenzelm@60758
   639
subsection \<open>Serializer setup for target language integers\<close>
haftmann@24999
   640
haftmann@51143
   641
code_reserved Eval int Integer abs
haftmann@25767
   642
haftmann@52435
   643
code_printing
haftmann@52435
   644
  type_constructor integer \<rightharpoonup>
haftmann@52435
   645
    (SML) "IntInf.int"
haftmann@52435
   646
    and (OCaml) "Big'_int.big'_int"
haftmann@52435
   647
    and (Haskell) "Integer"
haftmann@52435
   648
    and (Scala) "BigInt"
haftmann@52435
   649
    and (Eval) "int"
haftmann@52435
   650
| class_instance integer :: equal \<rightharpoonup>
haftmann@52435
   651
    (Haskell) -
haftmann@24999
   652
haftmann@52435
   653
code_printing
haftmann@52435
   654
  constant "0::integer" \<rightharpoonup>
haftmann@58400
   655
    (SML) "!(0/ :/ IntInf.int)"
haftmann@52435
   656
    and (OCaml) "Big'_int.zero'_big'_int"
haftmann@58400
   657
    and (Haskell) "!(0/ ::/ Integer)"
haftmann@52435
   658
    and (Scala) "BigInt(0)"
huffman@47108
   659
wenzelm@60758
   660
setup \<open>
haftmann@58399
   661
  fold (fn target =>
haftmann@58399
   662
    Numeral.add_code @{const_name Code_Numeral.Pos} I Code_Printer.literal_numeral target
haftmann@58399
   663
    #> Numeral.add_code @{const_name Code_Numeral.Neg} (op ~) Code_Printer.literal_numeral target)
haftmann@58399
   664
    ["SML", "OCaml", "Haskell", "Scala"]
wenzelm@60758
   665
\<close>
haftmann@51143
   666
haftmann@52435
   667
code_printing
haftmann@52435
   668
  constant "plus :: integer \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
haftmann@52435
   669
    (SML) "IntInf.+ ((_), (_))"
haftmann@52435
   670
    and (OCaml) "Big'_int.add'_big'_int"
haftmann@52435
   671
    and (Haskell) infixl 6 "+"
haftmann@52435
   672
    and (Scala) infixl 7 "+"
haftmann@52435
   673
    and (Eval) infixl 8 "+"
haftmann@52435
   674
| constant "uminus :: integer \<Rightarrow> _" \<rightharpoonup>
haftmann@52435
   675
    (SML) "IntInf.~"
haftmann@52435
   676
    and (OCaml) "Big'_int.minus'_big'_int"
haftmann@52435
   677
    and (Haskell) "negate"
haftmann@52435
   678
    and (Scala) "!(- _)"
haftmann@52435
   679
    and (Eval) "~/ _"
haftmann@52435
   680
| constant "minus :: integer \<Rightarrow> _" \<rightharpoonup>
haftmann@52435
   681
    (SML) "IntInf.- ((_), (_))"
haftmann@52435
   682
    and (OCaml) "Big'_int.sub'_big'_int"
haftmann@52435
   683
    and (Haskell) infixl 6 "-"
haftmann@52435
   684
    and (Scala) infixl 7 "-"
haftmann@52435
   685
    and (Eval) infixl 8 "-"
haftmann@52435
   686
| constant Code_Numeral.dup \<rightharpoonup>
haftmann@52435
   687
    (SML) "IntInf.*/ (2,/ (_))"
haftmann@52435
   688
    and (OCaml) "Big'_int.mult'_big'_int/ (Big'_int.big'_int'_of'_int/ 2)"
haftmann@52435
   689
    and (Haskell) "!(2 * _)"
haftmann@52435
   690
    and (Scala) "!(2 * _)"
haftmann@52435
   691
    and (Eval) "!(2 * _)"
haftmann@52435
   692
| constant Code_Numeral.sub \<rightharpoonup>
haftmann@52435
   693
    (SML) "!(raise/ Fail/ \"sub\")"
haftmann@52435
   694
    and (OCaml) "failwith/ \"sub\""
haftmann@52435
   695
    and (Haskell) "error/ \"sub\""
haftmann@52435
   696
    and (Scala) "!sys.error(\"sub\")"
haftmann@52435
   697
| constant "times :: integer \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
haftmann@52435
   698
    (SML) "IntInf.* ((_), (_))"
haftmann@52435
   699
    and (OCaml) "Big'_int.mult'_big'_int"
haftmann@52435
   700
    and (Haskell) infixl 7 "*"
haftmann@52435
   701
    and (Scala) infixl 8 "*"
haftmann@52435
   702
    and (Eval) infixl 9 "*"
haftmann@52435
   703
| constant Code_Numeral.divmod_abs \<rightharpoonup>
haftmann@52435
   704
    (SML) "IntInf.divMod/ (IntInf.abs _,/ IntInf.abs _)"
haftmann@52435
   705
    and (OCaml) "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)"
haftmann@52435
   706
    and (Haskell) "divMod/ (abs _)/ (abs _)"
haftmann@52435
   707
    and (Scala) "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))"
haftmann@52435
   708
    and (Eval) "Integer.div'_mod/ (abs _)/ (abs _)"
haftmann@52435
   709
| constant "HOL.equal :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
haftmann@52435
   710
    (SML) "!((_ : IntInf.int) = _)"
haftmann@52435
   711
    and (OCaml) "Big'_int.eq'_big'_int"
haftmann@52435
   712
    and (Haskell) infix 4 "=="
haftmann@52435
   713
    and (Scala) infixl 5 "=="
haftmann@52435
   714
    and (Eval) infixl 6 "="
haftmann@52435
   715
| constant "less_eq :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
haftmann@52435
   716
    (SML) "IntInf.<= ((_), (_))"
haftmann@52435
   717
    and (OCaml) "Big'_int.le'_big'_int"
haftmann@52435
   718
    and (Haskell) infix 4 "<="
haftmann@52435
   719
    and (Scala) infixl 4 "<="
haftmann@52435
   720
    and (Eval) infixl 6 "<="
haftmann@52435
   721
| constant "less :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
haftmann@52435
   722
    (SML) "IntInf.< ((_), (_))"
haftmann@52435
   723
    and (OCaml) "Big'_int.lt'_big'_int"
haftmann@52435
   724
    and (Haskell) infix 4 "<"
haftmann@52435
   725
    and (Scala) infixl 4 "<"
haftmann@52435
   726
    and (Eval) infixl 6 "<"
Andreas@61857
   727
| constant "abs :: integer \<Rightarrow> _" \<rightharpoonup>
Andreas@61857
   728
    (SML) "IntInf.abs"
Andreas@61857
   729
    and (OCaml) "Big'_int.abs'_big'_int"
Andreas@61857
   730
    and (Haskell) "Prelude.abs"
Andreas@61857
   731
    and (Scala) "_.abs"
Andreas@61857
   732
    and (Eval) "abs"
haftmann@51143
   733
haftmann@52435
   734
code_identifier
haftmann@52435
   735
  code_module Code_Numeral \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
huffman@46547
   736
haftmann@51143
   737
wenzelm@60758
   738
subsection \<open>Type of target language naturals\<close>
haftmann@51143
   739
wenzelm@61076
   740
typedef natural = "UNIV :: nat set"
haftmann@51143
   741
  morphisms nat_of_natural natural_of_nat ..
haftmann@51143
   742
haftmann@59487
   743
setup_lifting type_definition_natural
haftmann@51143
   744
haftmann@51143
   745
lemma natural_eq_iff [termination_simp]:
haftmann@51143
   746
  "m = n \<longleftrightarrow> nat_of_natural m = nat_of_natural n"
haftmann@51143
   747
  by transfer rule
haftmann@51143
   748
haftmann@51143
   749
lemma natural_eqI:
haftmann@51143
   750
  "nat_of_natural m = nat_of_natural n \<Longrightarrow> m = n"
haftmann@51143
   751
  using natural_eq_iff [of m n] by simp
haftmann@51143
   752
haftmann@51143
   753
lemma nat_of_natural_of_nat_inverse [simp]:
haftmann@51143
   754
  "nat_of_natural (natural_of_nat n) = n"
haftmann@51143
   755
  by transfer rule
haftmann@51143
   756
haftmann@51143
   757
lemma natural_of_nat_of_natural_inverse [simp]:
haftmann@51143
   758
  "natural_of_nat (nat_of_natural n) = n"
haftmann@51143
   759
  by transfer rule
haftmann@51143
   760
haftmann@51143
   761
instantiation natural :: "{comm_monoid_diff, semiring_1}"
haftmann@51143
   762
begin
haftmann@51143
   763
haftmann@51143
   764
lift_definition zero_natural :: natural
haftmann@51143
   765
  is "0 :: nat"
haftmann@51143
   766
  .
haftmann@51143
   767
haftmann@51143
   768
declare zero_natural.rep_eq [simp]
haftmann@51143
   769
haftmann@51143
   770
lift_definition one_natural :: natural
haftmann@51143
   771
  is "1 :: nat"
haftmann@51143
   772
  .
haftmann@51143
   773
haftmann@51143
   774
declare one_natural.rep_eq [simp]
haftmann@51143
   775
haftmann@51143
   776
lift_definition plus_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
haftmann@51143
   777
  is "plus :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@51143
   778
  .
haftmann@51143
   779
haftmann@51143
   780
declare plus_natural.rep_eq [simp]
haftmann@51143
   781
haftmann@51143
   782
lift_definition minus_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
haftmann@51143
   783
  is "minus :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@51143
   784
  .
haftmann@51143
   785
haftmann@51143
   786
declare minus_natural.rep_eq [simp]
haftmann@51143
   787
haftmann@51143
   788
lift_definition times_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
haftmann@51143
   789
  is "times :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@51143
   790
  .
haftmann@51143
   791
haftmann@51143
   792
declare times_natural.rep_eq [simp]
haftmann@51143
   793
haftmann@51143
   794
instance proof
haftmann@51143
   795
qed (transfer, simp add: algebra_simps)+
haftmann@51143
   796
haftmann@51143
   797
end
haftmann@51143
   798
haftmann@64241
   799
instance natural :: Rings.dvd ..
haftmann@64241
   800
haftmann@64241
   801
lemma [transfer_rule]:
haftmann@64241
   802
  "rel_fun pcr_natural (rel_fun pcr_natural HOL.iff) Rings.dvd Rings.dvd"
haftmann@64241
   803
  unfolding dvd_def by transfer_prover
haftmann@64241
   804
haftmann@51143
   805
lemma [transfer_rule]:
blanchet@55945
   806
  "rel_fun HOL.eq pcr_natural (\<lambda>n::nat. n) (of_nat :: nat \<Rightarrow> natural)"
haftmann@51143
   807
proof -
blanchet@55945
   808
  have "rel_fun HOL.eq pcr_natural (of_nat :: nat \<Rightarrow> nat) (of_nat :: nat \<Rightarrow> natural)"
haftmann@51143
   809
    by (unfold of_nat_def [abs_def]) transfer_prover
haftmann@51143
   810
  then show ?thesis by (simp add: id_def)
haftmann@51143
   811
qed
haftmann@51143
   812
haftmann@51143
   813
lemma [transfer_rule]:
blanchet@55945
   814
  "rel_fun HOL.eq pcr_natural (numeral :: num \<Rightarrow> nat) (numeral :: num \<Rightarrow> natural)"
haftmann@51143
   815
proof -
blanchet@55945
   816
  have "rel_fun HOL.eq pcr_natural (numeral :: num \<Rightarrow> nat) (\<lambda>n. of_nat (numeral n))"
haftmann@51143
   817
    by transfer_prover
haftmann@51143
   818
  then show ?thesis by simp
haftmann@51143
   819
qed
haftmann@51143
   820
haftmann@51143
   821
lemma nat_of_natural_of_nat [simp]:
haftmann@51143
   822
  "nat_of_natural (of_nat n) = n"
haftmann@51143
   823
  by transfer rule
haftmann@51143
   824
haftmann@51143
   825
lemma natural_of_nat_of_nat [simp, code_abbrev]:
haftmann@51143
   826
  "natural_of_nat = of_nat"
haftmann@51143
   827
  by transfer rule
haftmann@51143
   828
haftmann@51143
   829
lemma of_nat_of_natural [simp]:
haftmann@51143
   830
  "of_nat (nat_of_natural n) = n"
haftmann@51143
   831
  by transfer rule
haftmann@51143
   832
haftmann@51143
   833
lemma nat_of_natural_numeral [simp]:
haftmann@51143
   834
  "nat_of_natural (numeral k) = numeral k"
haftmann@51143
   835
  by transfer rule
haftmann@51143
   836
haftmann@64592
   837
instantiation natural :: "{linordered_semiring, equal}"
haftmann@51143
   838
begin
haftmann@51143
   839
haftmann@51143
   840
lift_definition less_eq_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
haftmann@51143
   841
  is "less_eq :: nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@51143
   842
  .
haftmann@51143
   843
haftmann@51143
   844
declare less_eq_natural.rep_eq [termination_simp]
haftmann@51143
   845
haftmann@51143
   846
lift_definition less_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
haftmann@51143
   847
  is "less :: nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@51143
   848
  .
haftmann@51143
   849
haftmann@51143
   850
declare less_natural.rep_eq [termination_simp]
haftmann@51143
   851
haftmann@51143
   852
lift_definition equal_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
haftmann@51143
   853
  is "HOL.equal :: nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@51143
   854
  .
haftmann@51143
   855
haftmann@51143
   856
instance proof
haftmann@51143
   857
qed (transfer, simp add: algebra_simps equal less_le_not_le [symmetric] linear)+
haftmann@51143
   858
haftmann@24999
   859
end
haftmann@46664
   860
haftmann@51143
   861
lemma [transfer_rule]:
blanchet@55945
   862
  "rel_fun pcr_natural (rel_fun pcr_natural pcr_natural) (min :: _ \<Rightarrow> _ \<Rightarrow> nat) (min :: _ \<Rightarrow> _ \<Rightarrow> natural)"
haftmann@51143
   863
  by (unfold min_def [abs_def]) transfer_prover
haftmann@51143
   864
haftmann@51143
   865
lemma [transfer_rule]:
blanchet@55945
   866
  "rel_fun pcr_natural (rel_fun pcr_natural pcr_natural) (max :: _ \<Rightarrow> _ \<Rightarrow> nat) (max :: _ \<Rightarrow> _ \<Rightarrow> natural)"
haftmann@51143
   867
  by (unfold max_def [abs_def]) transfer_prover
haftmann@51143
   868
haftmann@51143
   869
lemma nat_of_natural_min [simp]:
haftmann@51143
   870
  "nat_of_natural (min k l) = min (nat_of_natural k) (nat_of_natural l)"
haftmann@51143
   871
  by transfer rule
haftmann@51143
   872
haftmann@51143
   873
lemma nat_of_natural_max [simp]:
haftmann@51143
   874
  "nat_of_natural (max k l) = max (nat_of_natural k) (nat_of_natural l)"
haftmann@51143
   875
  by transfer rule
haftmann@51143
   876
haftmann@66806
   877
instantiation natural :: unique_euclidean_semiring
haftmann@64592
   878
begin
haftmann@64592
   879
haftmann@64592
   880
lift_definition normalize_natural :: "natural \<Rightarrow> natural"
haftmann@64592
   881
  is "normalize :: nat \<Rightarrow> nat"
haftmann@64592
   882
  .
haftmann@64592
   883
haftmann@64592
   884
declare normalize_natural.rep_eq [simp]
haftmann@64592
   885
haftmann@64592
   886
lift_definition unit_factor_natural :: "natural \<Rightarrow> natural"
haftmann@64592
   887
  is "unit_factor :: nat \<Rightarrow> nat"
haftmann@64592
   888
  .
haftmann@64592
   889
haftmann@64592
   890
declare unit_factor_natural.rep_eq [simp]
haftmann@64592
   891
haftmann@64592
   892
lift_definition divide_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
haftmann@64592
   893
  is "divide :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@64592
   894
  .
haftmann@64592
   895
haftmann@64592
   896
declare divide_natural.rep_eq [simp]
haftmann@64592
   897
haftmann@64592
   898
lift_definition modulo_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
haftmann@64592
   899
  is "modulo :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@64592
   900
  .
haftmann@64592
   901
haftmann@64592
   902
declare modulo_natural.rep_eq [simp]
haftmann@64592
   903
haftmann@66806
   904
lift_definition euclidean_size_natural :: "natural \<Rightarrow> nat"
haftmann@66806
   905
  is "euclidean_size :: nat \<Rightarrow> nat"
haftmann@66806
   906
  .
haftmann@66806
   907
haftmann@66806
   908
declare euclidean_size_natural.rep_eq [simp]
haftmann@66806
   909
haftmann@66806
   910
lift_definition uniqueness_constraint_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
haftmann@66806
   911
  is "uniqueness_constraint :: nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@66806
   912
  .
haftmann@66806
   913
haftmann@66806
   914
declare uniqueness_constraint_natural.rep_eq [simp]
haftmann@66806
   915
haftmann@64592
   916
instance
haftmann@66806
   917
  by (standard; transfer)
haftmann@66806
   918
    (auto simp add: algebra_simps unit_factor_nat_def gr0_conv_Suc)
haftmann@64592
   919
haftmann@64592
   920
end
haftmann@64592
   921
haftmann@66806
   922
lemma [code]:
haftmann@66806
   923
  "euclidean_size = nat_of_natural"
haftmann@66806
   924
  by (simp add: fun_eq_iff)
haftmann@66806
   925
haftmann@66806
   926
lemma [code]:
haftmann@66806
   927
  "uniqueness_constraint = (\<top> :: natural \<Rightarrow> natural \<Rightarrow> bool)"
haftmann@66806
   928
  by (simp add: fun_eq_iff)
haftmann@66806
   929
haftmann@51143
   930
lift_definition natural_of_integer :: "integer \<Rightarrow> natural"
haftmann@51143
   931
  is "nat :: int \<Rightarrow> nat"
haftmann@51143
   932
  .
haftmann@51143
   933
haftmann@51143
   934
lift_definition integer_of_natural :: "natural \<Rightarrow> integer"
haftmann@51143
   935
  is "of_nat :: nat \<Rightarrow> int"
haftmann@51143
   936
  .
haftmann@51143
   937
haftmann@51143
   938
lemma natural_of_integer_of_natural [simp]:
haftmann@51143
   939
  "natural_of_integer (integer_of_natural n) = n"
haftmann@51143
   940
  by transfer simp
haftmann@51143
   941
haftmann@51143
   942
lemma integer_of_natural_of_integer [simp]:
haftmann@51143
   943
  "integer_of_natural (natural_of_integer k) = max 0 k"
haftmann@51143
   944
  by transfer auto
haftmann@51143
   945
haftmann@51143
   946
lemma int_of_integer_of_natural [simp]:
haftmann@51143
   947
  "int_of_integer (integer_of_natural n) = of_nat (nat_of_natural n)"
haftmann@51143
   948
  by transfer rule
haftmann@51143
   949
haftmann@51143
   950
lemma integer_of_natural_of_nat [simp]:
haftmann@51143
   951
  "integer_of_natural (of_nat n) = of_nat n"
haftmann@51143
   952
  by transfer rule
haftmann@51143
   953
haftmann@51143
   954
lemma [measure_function]:
haftmann@51143
   955
  "is_measure nat_of_natural"
haftmann@51143
   956
  by (rule is_measure_trivial)
haftmann@51143
   957
haftmann@51143
   958
wenzelm@60758
   959
subsection \<open>Inductive representation of target language naturals\<close>
haftmann@51143
   960
haftmann@51143
   961
lift_definition Suc :: "natural \<Rightarrow> natural"
haftmann@51143
   962
  is Nat.Suc
haftmann@51143
   963
  .
haftmann@51143
   964
haftmann@51143
   965
declare Suc.rep_eq [simp]
haftmann@51143
   966
blanchet@58306
   967
old_rep_datatype "0::natural" Suc
haftmann@51143
   968
  by (transfer, fact nat.induct nat.inject nat.distinct)+
haftmann@51143
   969
blanchet@55416
   970
lemma natural_cases [case_names nat, cases type: natural]:
haftmann@51143
   971
  fixes m :: natural
haftmann@51143
   972
  assumes "\<And>n. m = of_nat n \<Longrightarrow> P"
haftmann@51143
   973
  shows P
haftmann@51143
   974
  using assms by transfer blast
haftmann@51143
   975
blanchet@58390
   976
lemma [simp, code]: "size_natural = nat_of_natural"
blanchet@58390
   977
proof (rule ext)
blanchet@58390
   978
  fix n
blanchet@58390
   979
  show "size_natural n = nat_of_natural n"
blanchet@58390
   980
    by (induct n) simp_all
blanchet@58390
   981
qed
blanchet@58379
   982
blanchet@58390
   983
lemma [simp, code]: "size = nat_of_natural"
blanchet@58390
   984
proof (rule ext)
blanchet@58390
   985
  fix n
blanchet@58390
   986
  show "size n = nat_of_natural n"
blanchet@58390
   987
    by (induct n) simp_all
blanchet@58390
   988
qed
blanchet@58379
   989
haftmann@51143
   990
lemma natural_decr [termination_simp]:
haftmann@51143
   991
  "n \<noteq> 0 \<Longrightarrow> nat_of_natural n - Nat.Suc 0 < nat_of_natural n"
haftmann@51143
   992
  by transfer simp
haftmann@51143
   993
blanchet@58379
   994
lemma natural_zero_minus_one: "(0::natural) - 1 = 0"
blanchet@58379
   995
  by (rule zero_diff)
haftmann@51143
   996
blanchet@58379
   997
lemma Suc_natural_minus_one: "Suc n - 1 = n"
haftmann@51143
   998
  by transfer simp
haftmann@51143
   999
haftmann@51143
  1000
hide_const (open) Suc
haftmann@51143
  1001
haftmann@51143
  1002
wenzelm@60758
  1003
subsection \<open>Code refinement for target language naturals\<close>
haftmann@51143
  1004
haftmann@51143
  1005
lift_definition Nat :: "integer \<Rightarrow> natural"
haftmann@51143
  1006
  is nat
haftmann@51143
  1007
  .
haftmann@51143
  1008
haftmann@51143
  1009
lemma [code_post]:
haftmann@51143
  1010
  "Nat 0 = 0"
haftmann@51143
  1011
  "Nat 1 = 1"
haftmann@51143
  1012
  "Nat (numeral k) = numeral k"
haftmann@51143
  1013
  by (transfer, simp)+
haftmann@51143
  1014
haftmann@51143
  1015
lemma [code abstype]:
haftmann@51143
  1016
  "Nat (integer_of_natural n) = n"
haftmann@51143
  1017
  by transfer simp
haftmann@51143
  1018
haftmann@63174
  1019
lemma [code]:
haftmann@63174
  1020
  "natural_of_nat n = natural_of_integer (integer_of_nat n)"
haftmann@63174
  1021
  by transfer simp
haftmann@51143
  1022
haftmann@51143
  1023
lemma [code abstract]:
haftmann@51143
  1024
  "integer_of_natural (natural_of_integer k) = max 0 k"
haftmann@51143
  1025
  by simp
haftmann@51143
  1026
haftmann@51143
  1027
lemma [code_abbrev]:
haftmann@51143
  1028
  "natural_of_integer (Code_Numeral.Pos k) = numeral k"
haftmann@51143
  1029
  by transfer simp
haftmann@51143
  1030
haftmann@51143
  1031
lemma [code abstract]:
haftmann@51143
  1032
  "integer_of_natural 0 = 0"
haftmann@51143
  1033
  by transfer simp
haftmann@51143
  1034
haftmann@51143
  1035
lemma [code abstract]:
haftmann@51143
  1036
  "integer_of_natural 1 = 1"
haftmann@51143
  1037
  by transfer simp
haftmann@51143
  1038
haftmann@51143
  1039
lemma [code abstract]:
haftmann@51143
  1040
  "integer_of_natural (Code_Numeral.Suc n) = integer_of_natural n + 1"
haftmann@51143
  1041
  by transfer simp
haftmann@51143
  1042
haftmann@51143
  1043
lemma [code]:
haftmann@51143
  1044
  "nat_of_natural = nat_of_integer \<circ> integer_of_natural"
haftmann@51143
  1045
  by transfer (simp add: fun_eq_iff)
haftmann@51143
  1046
haftmann@51143
  1047
lemma [code, code_unfold]:
blanchet@55416
  1048
  "case_natural f g n = (if n = 0 then f else g (n - 1))"
haftmann@51143
  1049
  by (cases n rule: natural.exhaust) (simp_all, simp add: Suc_def)
haftmann@51143
  1050
blanchet@55642
  1051
declare natural.rec [code del]
haftmann@51143
  1052
haftmann@51143
  1053
lemma [code abstract]:
haftmann@51143
  1054
  "integer_of_natural (m + n) = integer_of_natural m + integer_of_natural n"
haftmann@51143
  1055
  by transfer simp
haftmann@51143
  1056
haftmann@51143
  1057
lemma [code abstract]:
haftmann@51143
  1058
  "integer_of_natural (m - n) = max 0 (integer_of_natural m - integer_of_natural n)"
haftmann@51143
  1059
  by transfer simp
haftmann@51143
  1060
haftmann@51143
  1061
lemma [code abstract]:
haftmann@51143
  1062
  "integer_of_natural (m * n) = integer_of_natural m * integer_of_natural n"
haftmann@64592
  1063
  by transfer simp
haftmann@64592
  1064
haftmann@64592
  1065
lemma [code]:
haftmann@64592
  1066
  "normalize n = n" for n :: natural
haftmann@64592
  1067
  by transfer simp
haftmann@64592
  1068
haftmann@64592
  1069
lemma [code]:
haftmann@64592
  1070
  "unit_factor n = of_bool (n \<noteq> 0)" for n :: natural
haftmann@64592
  1071
proof (cases "n = 0")
haftmann@64592
  1072
  case True
haftmann@64592
  1073
  then show ?thesis
haftmann@64592
  1074
    by simp
haftmann@64592
  1075
next
haftmann@64592
  1076
  case False
haftmann@64592
  1077
  then have "unit_factor n = 1"
haftmann@64592
  1078
  proof transfer
haftmann@64592
  1079
    fix n :: nat
haftmann@64592
  1080
    assume "n \<noteq> 0"
haftmann@64592
  1081
    then obtain m where "n = Suc m"
haftmann@64592
  1082
      by (cases n) auto
haftmann@64592
  1083
    then show "unit_factor n = 1"
haftmann@64592
  1084
      by simp
haftmann@64592
  1085
  qed
haftmann@64592
  1086
  with False show ?thesis
haftmann@64592
  1087
    by simp
haftmann@64592
  1088
qed
haftmann@51143
  1089
haftmann@51143
  1090
lemma [code abstract]:
haftmann@51143
  1091
  "integer_of_natural (m div n) = integer_of_natural m div integer_of_natural n"
haftmann@51143
  1092
  by transfer (simp add: zdiv_int)
haftmann@51143
  1093
haftmann@51143
  1094
lemma [code abstract]:
haftmann@51143
  1095
  "integer_of_natural (m mod n) = integer_of_natural m mod integer_of_natural n"
haftmann@51143
  1096
  by transfer (simp add: zmod_int)
haftmann@51143
  1097
haftmann@51143
  1098
lemma [code]:
haftmann@51143
  1099
  "HOL.equal m n \<longleftrightarrow> HOL.equal (integer_of_natural m) (integer_of_natural n)"
haftmann@51143
  1100
  by transfer (simp add: equal)
haftmann@51143
  1101
blanchet@58379
  1102
lemma [code nbe]: "HOL.equal n (n::natural) \<longleftrightarrow> True"
blanchet@58379
  1103
  by (rule equal_class.equal_refl)
haftmann@51143
  1104
blanchet@58379
  1105
lemma [code]: "m \<le> n \<longleftrightarrow> integer_of_natural m \<le> integer_of_natural n"
haftmann@51143
  1106
  by transfer simp
haftmann@51143
  1107
blanchet@58379
  1108
lemma [code]: "m < n \<longleftrightarrow> integer_of_natural m < integer_of_natural n"
haftmann@51143
  1109
  by transfer simp
haftmann@51143
  1110
haftmann@51143
  1111
hide_const (open) Nat
haftmann@51143
  1112
kuncar@55736
  1113
lifting_update integer.lifting
kuncar@55736
  1114
lifting_forget integer.lifting
kuncar@55736
  1115
kuncar@55736
  1116
lifting_update natural.lifting
kuncar@55736
  1117
lifting_forget natural.lifting
haftmann@51143
  1118
haftmann@51143
  1119
code_reflect Code_Numeral
haftmann@63174
  1120
  datatypes natural
haftmann@63174
  1121
  functions "Code_Numeral.Suc" "0 :: natural" "1 :: natural"
haftmann@63174
  1122
    "plus :: natural \<Rightarrow> _" "minus :: natural \<Rightarrow> _"
haftmann@63174
  1123
    "times :: natural \<Rightarrow> _" "divide :: natural \<Rightarrow> _"
haftmann@63950
  1124
    "modulo :: natural \<Rightarrow> _"
haftmann@63174
  1125
    integer_of_natural natural_of_integer
haftmann@51143
  1126
haftmann@51143
  1127
end