src/HOL/Code_Numeral.thy
author haftmann
Wed Oct 12 21:48:53 2016 +0200 (2016-10-12)
changeset 64178 12e6c3bbb488
parent 63950 cdc1e59aa513
child 64241 430d74089d4d
permissions -rw-r--r--
transfer lifting rule for numeral
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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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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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lift_definition integer_of_num :: "num \<Rightarrow> integer"
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  is "numeral :: num \<Rightarrow> int"
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  .
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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 (transfer, simp only: numeral.simps Let_def)+
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lemma numeral_unfold_integer_of_num:
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  "numeral = integer_of_num"
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  by (simp add: integer_of_num_def map_fun_def fun_eq_iff)
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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 (transfer, simp)+
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instantiation integer :: "{ring_div, equal, linordered_idom}"
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begin
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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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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 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 proof
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qed (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 :: semiring_numeral_div
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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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  semiring_numeral_div_class.div_less
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  semiring_numeral_div_class.mod_less
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  semiring_numeral_div_class.div_positive
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  semiring_numeral_div_class.mod_less_eq_dividend
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  semiring_numeral_div_class.pos_mod_bound
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  semiring_numeral_div_class.pos_mod_sign
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  semiring_numeral_div_class.mod_mult2_eq
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  semiring_numeral_div_class.div_mult2_eq
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  semiring_numeral_div_class.discrete)+
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end
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declare divmod_algorithm_code [where ?'a = integer,
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  unfolded numeral_unfold_integer_of_num, 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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lemma [transfer_rule]:
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  "rel_fun HOL.eq pcr_integer (\<lambda>n. - numeral n) Neg"
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  by (simp add: Neg_def [abs_def]) transfer_prover
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code_datatype "0::integer" Pos Neg
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text \<open>Auxiliary operations\<close>
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lift_definition dup :: "integer \<Rightarrow> integer"
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  is "\<lambda>k::int. k + k"
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  .
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lemma dup_code [code]:
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  "dup 0 = 0"
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  "dup (Pos n) = Pos (Num.Bit0 n)"
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  "dup (Neg n) = Neg (Num.Bit0 n)"
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  by (transfer, simp only: numeral_Bit0 minus_add_distrib)+
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lift_definition sub :: "num \<Rightarrow> num \<Rightarrow> integer"
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  is "\<lambda>m n. numeral m - numeral n :: int"
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  .
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lemma sub_code [code]:
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  "sub Num.One Num.One = 0"
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  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
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  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
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  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
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  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
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  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
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  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
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  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
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  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
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  by (transfer, simp add: dbl_def dbl_inc_def dbl_dec_def)+
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text \<open>Implementations\<close>
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lemma one_integer_code [code, code_unfold]:
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  "1 = Pos Num.One"
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  by simp
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lemma plus_integer_code [code]:
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  "k + 0 = (k::integer)"
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  "0 + l = (l::integer)"
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  "Pos m + Pos n = Pos (m + n)"
haftmann@51143
   349
  "Pos m + Neg n = sub m n"
haftmann@51143
   350
  "Neg m + Pos n = sub n m"
haftmann@51143
   351
  "Neg m + Neg n = Neg (m + n)"
haftmann@51143
   352
  by (transfer, simp)+
haftmann@24999
   353
haftmann@51143
   354
lemma uminus_integer_code [code]:
haftmann@51143
   355
  "uminus 0 = (0::integer)"
haftmann@51143
   356
  "uminus (Pos m) = Neg m"
haftmann@51143
   357
  "uminus (Neg m) = Pos m"
haftmann@51143
   358
  by simp_all
haftmann@28708
   359
haftmann@51143
   360
lemma minus_integer_code [code]:
haftmann@51143
   361
  "k - 0 = (k::integer)"
haftmann@51143
   362
  "0 - l = uminus (l::integer)"
haftmann@51143
   363
  "Pos m - Pos n = sub m n"
haftmann@51143
   364
  "Pos m - Neg n = Pos (m + n)"
haftmann@51143
   365
  "Neg m - Pos n = Neg (m + n)"
haftmann@51143
   366
  "Neg m - Neg n = sub n m"
haftmann@51143
   367
  by (transfer, simp)+
haftmann@46028
   368
haftmann@51143
   369
lemma abs_integer_code [code]:
haftmann@51143
   370
  "\<bar>k\<bar> = (if (k::integer) < 0 then - k else k)"
haftmann@51143
   371
  by simp
huffman@47108
   372
haftmann@51143
   373
lemma sgn_integer_code [code]:
haftmann@51143
   374
  "sgn k = (if k = 0 then 0 else if (k::integer) < 0 then - 1 else 1)"
huffman@47108
   375
  by simp
haftmann@46028
   376
haftmann@51143
   377
lemma times_integer_code [code]:
haftmann@51143
   378
  "k * 0 = (0::integer)"
haftmann@51143
   379
  "0 * l = (0::integer)"
haftmann@51143
   380
  "Pos m * Pos n = Pos (m * n)"
haftmann@51143
   381
  "Pos m * Neg n = Neg (m * n)"
haftmann@51143
   382
  "Neg m * Pos n = Neg (m * n)"
haftmann@51143
   383
  "Neg m * Neg n = Pos (m * n)"
haftmann@51143
   384
  by simp_all
haftmann@51143
   385
haftmann@51143
   386
definition divmod_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer \<times> integer"
haftmann@51143
   387
where
haftmann@51143
   388
  "divmod_integer k l = (k div l, k mod l)"
haftmann@51143
   389
haftmann@51143
   390
lemma fst_divmod [simp]:
haftmann@51143
   391
  "fst (divmod_integer k l) = k div l"
haftmann@51143
   392
  by (simp add: divmod_integer_def)
haftmann@51143
   393
haftmann@51143
   394
lemma snd_divmod [simp]:
haftmann@51143
   395
  "snd (divmod_integer k l) = k mod l"
haftmann@51143
   396
  by (simp add: divmod_integer_def)
haftmann@51143
   397
haftmann@51143
   398
definition divmod_abs :: "integer \<Rightarrow> integer \<Rightarrow> integer \<times> integer"
haftmann@51143
   399
where
haftmann@51143
   400
  "divmod_abs k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
haftmann@51143
   401
haftmann@51143
   402
lemma fst_divmod_abs [simp]:
haftmann@51143
   403
  "fst (divmod_abs k l) = \<bar>k\<bar> div \<bar>l\<bar>"
haftmann@51143
   404
  by (simp add: divmod_abs_def)
haftmann@51143
   405
haftmann@51143
   406
lemma snd_divmod_abs [simp]:
haftmann@51143
   407
  "snd (divmod_abs k l) = \<bar>k\<bar> mod \<bar>l\<bar>"
haftmann@51143
   408
  by (simp add: divmod_abs_def)
haftmann@28708
   409
haftmann@53069
   410
lemma divmod_abs_code [code]:
haftmann@53069
   411
  "divmod_abs (Pos k) (Pos l) = divmod k l"
haftmann@53069
   412
  "divmod_abs (Neg k) (Neg l) = divmod k l"
haftmann@53069
   413
  "divmod_abs (Neg k) (Pos l) = divmod k l"
haftmann@53069
   414
  "divmod_abs (Pos k) (Neg l) = divmod k l"
haftmann@51143
   415
  "divmod_abs j 0 = (0, \<bar>j\<bar>)"
haftmann@51143
   416
  "divmod_abs 0 j = (0, 0)"
haftmann@51143
   417
  by (simp_all add: prod_eq_iff)
haftmann@51143
   418
haftmann@51143
   419
lemma divmod_integer_code [code]:
haftmann@51143
   420
  "divmod_integer k l =
haftmann@51143
   421
    (if k = 0 then (0, 0) else if l = 0 then (0, k) else
haftmann@51143
   422
    (apsnd \<circ> times \<circ> sgn) l (if sgn k = sgn l
haftmann@51143
   423
      then divmod_abs k l
haftmann@51143
   424
      else (let (r, s) = divmod_abs k l in
haftmann@51143
   425
        if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
haftmann@51143
   426
proof -
haftmann@51143
   427
  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
   428
    by (auto simp add: sgn_if)
haftmann@51143
   429
  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
   430
  show ?thesis
blanchet@55414
   431
    by (simp add: prod_eq_iff integer_eq_iff case_prod_beta aux1)
haftmann@51143
   432
      (auto simp add: zdiv_zminus1_eq_if zmod_zminus1_eq_if div_minus_right mod_minus_right aux2)
haftmann@51143
   433
qed
haftmann@51143
   434
haftmann@51143
   435
lemma div_integer_code [code]:
haftmann@51143
   436
  "k div l = fst (divmod_integer k l)"
haftmann@28708
   437
  by simp
haftmann@28708
   438
haftmann@51143
   439
lemma mod_integer_code [code]:
haftmann@51143
   440
  "k mod l = snd (divmod_integer k l)"
haftmann@25767
   441
  by simp
haftmann@24999
   442
haftmann@51143
   443
lemma equal_integer_code [code]:
haftmann@51143
   444
  "HOL.equal 0 (0::integer) \<longleftrightarrow> True"
haftmann@51143
   445
  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
haftmann@51143
   446
  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
haftmann@51143
   447
  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
haftmann@51143
   448
  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
haftmann@51143
   449
  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
haftmann@51143
   450
  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
haftmann@51143
   451
  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
haftmann@51143
   452
  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
haftmann@51143
   453
  by (simp_all add: equal)
haftmann@51143
   454
haftmann@51143
   455
lemma equal_integer_refl [code nbe]:
haftmann@51143
   456
  "HOL.equal (k::integer) k \<longleftrightarrow> True"
haftmann@51143
   457
  by (fact equal_refl)
haftmann@31266
   458
haftmann@51143
   459
lemma less_eq_integer_code [code]:
haftmann@51143
   460
  "0 \<le> (0::integer) \<longleftrightarrow> True"
haftmann@51143
   461
  "0 \<le> Pos l \<longleftrightarrow> True"
haftmann@51143
   462
  "0 \<le> Neg l \<longleftrightarrow> False"
haftmann@51143
   463
  "Pos k \<le> 0 \<longleftrightarrow> False"
haftmann@51143
   464
  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
haftmann@51143
   465
  "Pos k \<le> Neg l \<longleftrightarrow> False"
haftmann@51143
   466
  "Neg k \<le> 0 \<longleftrightarrow> True"
haftmann@51143
   467
  "Neg k \<le> Pos l \<longleftrightarrow> True"
haftmann@51143
   468
  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
haftmann@51143
   469
  by simp_all
haftmann@51143
   470
haftmann@51143
   471
lemma less_integer_code [code]:
haftmann@51143
   472
  "0 < (0::integer) \<longleftrightarrow> False"
haftmann@51143
   473
  "0 < Pos l \<longleftrightarrow> True"
haftmann@51143
   474
  "0 < Neg l \<longleftrightarrow> False"
haftmann@51143
   475
  "Pos k < 0 \<longleftrightarrow> False"
haftmann@51143
   476
  "Pos k < Pos l \<longleftrightarrow> k < l"
haftmann@51143
   477
  "Pos k < Neg l \<longleftrightarrow> False"
haftmann@51143
   478
  "Neg k < 0 \<longleftrightarrow> True"
haftmann@51143
   479
  "Neg k < Pos l \<longleftrightarrow> True"
haftmann@51143
   480
  "Neg k < Neg l \<longleftrightarrow> l < k"
haftmann@51143
   481
  by simp_all
haftmann@26140
   482
haftmann@51143
   483
lift_definition num_of_integer :: "integer \<Rightarrow> num"
haftmann@51143
   484
  is "num_of_nat \<circ> nat"
haftmann@51143
   485
  .
haftmann@51143
   486
haftmann@51143
   487
lemma num_of_integer_code [code]:
haftmann@51143
   488
  "num_of_integer k = (if k \<le> 1 then Num.One
haftmann@51143
   489
     else let
haftmann@51143
   490
       (l, j) = divmod_integer k 2;
haftmann@51143
   491
       l' = num_of_integer l;
haftmann@51143
   492
       l'' = l' + l'
haftmann@51143
   493
     in if j = 0 then l'' else l'' + Num.One)"
haftmann@51143
   494
proof -
haftmann@51143
   495
  {
haftmann@51143
   496
    assume "int_of_integer k mod 2 = 1"
haftmann@51143
   497
    then have "nat (int_of_integer k mod 2) = nat 1" by simp
haftmann@51143
   498
    moreover assume *: "1 < int_of_integer k"
haftmann@51143
   499
    ultimately have **: "nat (int_of_integer k) mod 2 = 1" by (simp add: nat_mod_distrib)
haftmann@51143
   500
    have "num_of_nat (nat (int_of_integer k)) =
haftmann@51143
   501
      num_of_nat (2 * (nat (int_of_integer k) div 2) + nat (int_of_integer k) mod 2)"
haftmann@51143
   502
      by simp
haftmann@51143
   503
    then have "num_of_nat (nat (int_of_integer k)) =
haftmann@51143
   504
      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
   505
      by (simp add: mult_2)
haftmann@51143
   506
    with ** have "num_of_nat (nat (int_of_integer k)) =
haftmann@51143
   507
      num_of_nat (nat (int_of_integer k) div 2 + nat (int_of_integer k) div 2 + 1)"
haftmann@51143
   508
      by simp
haftmann@51143
   509
  }
haftmann@51143
   510
  note aux = this
haftmann@51143
   511
  show ?thesis
blanchet@55414
   512
    by (auto simp add: num_of_integer_def nat_of_integer_def Let_def case_prod_beta
haftmann@51143
   513
      not_le integer_eq_iff less_eq_integer_def
haftmann@51143
   514
      nat_mult_distrib nat_div_distrib num_of_nat_One num_of_nat_plus_distrib
haftmann@51143
   515
       mult_2 [where 'a=nat] aux add_One)
haftmann@25918
   516
qed
haftmann@25918
   517
haftmann@51143
   518
lemma nat_of_integer_code [code]:
haftmann@51143
   519
  "nat_of_integer k = (if k \<le> 0 then 0
haftmann@51143
   520
     else let
haftmann@51143
   521
       (l, j) = divmod_integer k 2;
haftmann@51143
   522
       l' = nat_of_integer l;
haftmann@51143
   523
       l'' = l' + l'
haftmann@51143
   524
     in if j = 0 then l'' else l'' + 1)"
haftmann@33340
   525
proof -
haftmann@51143
   526
  obtain j where "k = integer_of_int j"
haftmann@51143
   527
  proof
haftmann@51143
   528
    show "k = integer_of_int (int_of_integer k)" by simp
haftmann@51143
   529
  qed
haftmann@51143
   530
  moreover have "2 * (j div 2) = j - j mod 2"
haftmann@57512
   531
    by (simp add: zmult_div_cancel mult.commute)
haftmann@51143
   532
  ultimately show ?thesis
haftmann@63950
   533
    by (auto simp add: split_def Let_def modulo_integer_def nat_of_integer_def not_le
haftmann@51143
   534
      nat_add_distrib [symmetric] Suc_nat_eq_nat_zadd1)
haftmann@51143
   535
      (auto simp add: mult_2 [symmetric])
haftmann@33340
   536
qed
haftmann@28708
   537
haftmann@51143
   538
lemma int_of_integer_code [code]:
haftmann@51143
   539
  "int_of_integer k = (if k < 0 then - (int_of_integer (- k))
haftmann@51143
   540
     else if k = 0 then 0
haftmann@51143
   541
     else let
haftmann@51143
   542
       (l, j) = divmod_integer k 2;
haftmann@51143
   543
       l' = 2 * int_of_integer l
haftmann@51143
   544
     in if j = 0 then l' else l' + 1)"
haftmann@51143
   545
  by (auto simp add: split_def Let_def integer_eq_iff zmult_div_cancel)
haftmann@28708
   546
haftmann@51143
   547
lemma integer_of_int_code [code]:
haftmann@51143
   548
  "integer_of_int k = (if k < 0 then - (integer_of_int (- k))
haftmann@51143
   549
     else if k = 0 then 0
haftmann@51143
   550
     else let
haftmann@60868
   551
       l = 2 * integer_of_int (k div 2);
haftmann@60868
   552
       j = k mod 2
haftmann@60868
   553
     in if j = 0 then l else l + 1)"
haftmann@51143
   554
  by (auto simp add: split_def Let_def integer_eq_iff zmult_div_cancel)
haftmann@51143
   555
haftmann@51143
   556
hide_const (open) Pos Neg sub dup divmod_abs
huffman@46547
   557
haftmann@28708
   558
wenzelm@60758
   559
subsection \<open>Serializer setup for target language integers\<close>
haftmann@24999
   560
haftmann@51143
   561
code_reserved Eval int Integer abs
haftmann@25767
   562
haftmann@52435
   563
code_printing
haftmann@52435
   564
  type_constructor integer \<rightharpoonup>
haftmann@52435
   565
    (SML) "IntInf.int"
haftmann@52435
   566
    and (OCaml) "Big'_int.big'_int"
haftmann@52435
   567
    and (Haskell) "Integer"
haftmann@52435
   568
    and (Scala) "BigInt"
haftmann@52435
   569
    and (Eval) "int"
haftmann@52435
   570
| class_instance integer :: equal \<rightharpoonup>
haftmann@52435
   571
    (Haskell) -
haftmann@24999
   572
haftmann@52435
   573
code_printing
haftmann@52435
   574
  constant "0::integer" \<rightharpoonup>
haftmann@58400
   575
    (SML) "!(0/ :/ IntInf.int)"
haftmann@52435
   576
    and (OCaml) "Big'_int.zero'_big'_int"
haftmann@58400
   577
    and (Haskell) "!(0/ ::/ Integer)"
haftmann@52435
   578
    and (Scala) "BigInt(0)"
huffman@47108
   579
wenzelm@60758
   580
setup \<open>
haftmann@58399
   581
  fold (fn target =>
haftmann@58399
   582
    Numeral.add_code @{const_name Code_Numeral.Pos} I Code_Printer.literal_numeral target
haftmann@58399
   583
    #> Numeral.add_code @{const_name Code_Numeral.Neg} (op ~) Code_Printer.literal_numeral target)
haftmann@58399
   584
    ["SML", "OCaml", "Haskell", "Scala"]
wenzelm@60758
   585
\<close>
haftmann@51143
   586
haftmann@52435
   587
code_printing
haftmann@52435
   588
  constant "plus :: integer \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
haftmann@52435
   589
    (SML) "IntInf.+ ((_), (_))"
haftmann@52435
   590
    and (OCaml) "Big'_int.add'_big'_int"
haftmann@52435
   591
    and (Haskell) infixl 6 "+"
haftmann@52435
   592
    and (Scala) infixl 7 "+"
haftmann@52435
   593
    and (Eval) infixl 8 "+"
haftmann@52435
   594
| constant "uminus :: integer \<Rightarrow> _" \<rightharpoonup>
haftmann@52435
   595
    (SML) "IntInf.~"
haftmann@52435
   596
    and (OCaml) "Big'_int.minus'_big'_int"
haftmann@52435
   597
    and (Haskell) "negate"
haftmann@52435
   598
    and (Scala) "!(- _)"
haftmann@52435
   599
    and (Eval) "~/ _"
haftmann@52435
   600
| constant "minus :: integer \<Rightarrow> _" \<rightharpoonup>
haftmann@52435
   601
    (SML) "IntInf.- ((_), (_))"
haftmann@52435
   602
    and (OCaml) "Big'_int.sub'_big'_int"
haftmann@52435
   603
    and (Haskell) infixl 6 "-"
haftmann@52435
   604
    and (Scala) infixl 7 "-"
haftmann@52435
   605
    and (Eval) infixl 8 "-"
haftmann@52435
   606
| constant Code_Numeral.dup \<rightharpoonup>
haftmann@52435
   607
    (SML) "IntInf.*/ (2,/ (_))"
haftmann@52435
   608
    and (OCaml) "Big'_int.mult'_big'_int/ (Big'_int.big'_int'_of'_int/ 2)"
haftmann@52435
   609
    and (Haskell) "!(2 * _)"
haftmann@52435
   610
    and (Scala) "!(2 * _)"
haftmann@52435
   611
    and (Eval) "!(2 * _)"
haftmann@52435
   612
| constant Code_Numeral.sub \<rightharpoonup>
haftmann@52435
   613
    (SML) "!(raise/ Fail/ \"sub\")"
haftmann@52435
   614
    and (OCaml) "failwith/ \"sub\""
haftmann@52435
   615
    and (Haskell) "error/ \"sub\""
haftmann@52435
   616
    and (Scala) "!sys.error(\"sub\")"
haftmann@52435
   617
| constant "times :: integer \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
haftmann@52435
   618
    (SML) "IntInf.* ((_), (_))"
haftmann@52435
   619
    and (OCaml) "Big'_int.mult'_big'_int"
haftmann@52435
   620
    and (Haskell) infixl 7 "*"
haftmann@52435
   621
    and (Scala) infixl 8 "*"
haftmann@52435
   622
    and (Eval) infixl 9 "*"
haftmann@52435
   623
| constant Code_Numeral.divmod_abs \<rightharpoonup>
haftmann@52435
   624
    (SML) "IntInf.divMod/ (IntInf.abs _,/ IntInf.abs _)"
haftmann@52435
   625
    and (OCaml) "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)"
haftmann@52435
   626
    and (Haskell) "divMod/ (abs _)/ (abs _)"
haftmann@52435
   627
    and (Scala) "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))"
haftmann@52435
   628
    and (Eval) "Integer.div'_mod/ (abs _)/ (abs _)"
haftmann@52435
   629
| constant "HOL.equal :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
haftmann@52435
   630
    (SML) "!((_ : IntInf.int) = _)"
haftmann@52435
   631
    and (OCaml) "Big'_int.eq'_big'_int"
haftmann@52435
   632
    and (Haskell) infix 4 "=="
haftmann@52435
   633
    and (Scala) infixl 5 "=="
haftmann@52435
   634
    and (Eval) infixl 6 "="
haftmann@52435
   635
| constant "less_eq :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
haftmann@52435
   636
    (SML) "IntInf.<= ((_), (_))"
haftmann@52435
   637
    and (OCaml) "Big'_int.le'_big'_int"
haftmann@52435
   638
    and (Haskell) infix 4 "<="
haftmann@52435
   639
    and (Scala) infixl 4 "<="
haftmann@52435
   640
    and (Eval) infixl 6 "<="
haftmann@52435
   641
| constant "less :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
haftmann@52435
   642
    (SML) "IntInf.< ((_), (_))"
haftmann@52435
   643
    and (OCaml) "Big'_int.lt'_big'_int"
haftmann@52435
   644
    and (Haskell) infix 4 "<"
haftmann@52435
   645
    and (Scala) infixl 4 "<"
haftmann@52435
   646
    and (Eval) infixl 6 "<"
Andreas@61857
   647
| constant "abs :: integer \<Rightarrow> _" \<rightharpoonup>
Andreas@61857
   648
    (SML) "IntInf.abs"
Andreas@61857
   649
    and (OCaml) "Big'_int.abs'_big'_int"
Andreas@61857
   650
    and (Haskell) "Prelude.abs"
Andreas@61857
   651
    and (Scala) "_.abs"
Andreas@61857
   652
    and (Eval) "abs"
haftmann@51143
   653
haftmann@52435
   654
code_identifier
haftmann@52435
   655
  code_module Code_Numeral \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
huffman@46547
   656
haftmann@51143
   657
wenzelm@60758
   658
subsection \<open>Type of target language naturals\<close>
haftmann@51143
   659
wenzelm@61076
   660
typedef natural = "UNIV :: nat set"
haftmann@51143
   661
  morphisms nat_of_natural natural_of_nat ..
haftmann@51143
   662
haftmann@59487
   663
setup_lifting type_definition_natural
haftmann@51143
   664
haftmann@51143
   665
lemma natural_eq_iff [termination_simp]:
haftmann@51143
   666
  "m = n \<longleftrightarrow> nat_of_natural m = nat_of_natural n"
haftmann@51143
   667
  by transfer rule
haftmann@51143
   668
haftmann@51143
   669
lemma natural_eqI:
haftmann@51143
   670
  "nat_of_natural m = nat_of_natural n \<Longrightarrow> m = n"
haftmann@51143
   671
  using natural_eq_iff [of m n] by simp
haftmann@51143
   672
haftmann@51143
   673
lemma nat_of_natural_of_nat_inverse [simp]:
haftmann@51143
   674
  "nat_of_natural (natural_of_nat n) = n"
haftmann@51143
   675
  by transfer rule
haftmann@51143
   676
haftmann@51143
   677
lemma natural_of_nat_of_natural_inverse [simp]:
haftmann@51143
   678
  "natural_of_nat (nat_of_natural n) = n"
haftmann@51143
   679
  by transfer rule
haftmann@51143
   680
haftmann@51143
   681
instantiation natural :: "{comm_monoid_diff, semiring_1}"
haftmann@51143
   682
begin
haftmann@51143
   683
haftmann@51143
   684
lift_definition zero_natural :: natural
haftmann@51143
   685
  is "0 :: nat"
haftmann@51143
   686
  .
haftmann@51143
   687
haftmann@51143
   688
declare zero_natural.rep_eq [simp]
haftmann@51143
   689
haftmann@51143
   690
lift_definition one_natural :: natural
haftmann@51143
   691
  is "1 :: nat"
haftmann@51143
   692
  .
haftmann@51143
   693
haftmann@51143
   694
declare one_natural.rep_eq [simp]
haftmann@51143
   695
haftmann@51143
   696
lift_definition plus_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
haftmann@51143
   697
  is "plus :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@51143
   698
  .
haftmann@51143
   699
haftmann@51143
   700
declare plus_natural.rep_eq [simp]
haftmann@51143
   701
haftmann@51143
   702
lift_definition minus_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
haftmann@51143
   703
  is "minus :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@51143
   704
  .
haftmann@51143
   705
haftmann@51143
   706
declare minus_natural.rep_eq [simp]
haftmann@51143
   707
haftmann@51143
   708
lift_definition times_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
haftmann@51143
   709
  is "times :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@51143
   710
  .
haftmann@51143
   711
haftmann@51143
   712
declare times_natural.rep_eq [simp]
haftmann@51143
   713
haftmann@51143
   714
instance proof
haftmann@51143
   715
qed (transfer, simp add: algebra_simps)+
haftmann@51143
   716
haftmann@51143
   717
end
haftmann@51143
   718
haftmann@51143
   719
lemma [transfer_rule]:
blanchet@55945
   720
  "rel_fun HOL.eq pcr_natural (\<lambda>n::nat. n) (of_nat :: nat \<Rightarrow> natural)"
haftmann@51143
   721
proof -
blanchet@55945
   722
  have "rel_fun HOL.eq pcr_natural (of_nat :: nat \<Rightarrow> nat) (of_nat :: nat \<Rightarrow> natural)"
haftmann@51143
   723
    by (unfold of_nat_def [abs_def]) transfer_prover
haftmann@51143
   724
  then show ?thesis by (simp add: id_def)
haftmann@51143
   725
qed
haftmann@51143
   726
haftmann@51143
   727
lemma [transfer_rule]:
blanchet@55945
   728
  "rel_fun HOL.eq pcr_natural (numeral :: num \<Rightarrow> nat) (numeral :: num \<Rightarrow> natural)"
haftmann@51143
   729
proof -
blanchet@55945
   730
  have "rel_fun HOL.eq pcr_natural (numeral :: num \<Rightarrow> nat) (\<lambda>n. of_nat (numeral n))"
haftmann@51143
   731
    by transfer_prover
haftmann@51143
   732
  then show ?thesis by simp
haftmann@51143
   733
qed
haftmann@51143
   734
haftmann@51143
   735
lemma nat_of_natural_of_nat [simp]:
haftmann@51143
   736
  "nat_of_natural (of_nat n) = n"
haftmann@51143
   737
  by transfer rule
haftmann@51143
   738
haftmann@51143
   739
lemma natural_of_nat_of_nat [simp, code_abbrev]:
haftmann@51143
   740
  "natural_of_nat = of_nat"
haftmann@51143
   741
  by transfer rule
haftmann@51143
   742
haftmann@51143
   743
lemma of_nat_of_natural [simp]:
haftmann@51143
   744
  "of_nat (nat_of_natural n) = n"
haftmann@51143
   745
  by transfer rule
haftmann@51143
   746
haftmann@51143
   747
lemma nat_of_natural_numeral [simp]:
haftmann@51143
   748
  "nat_of_natural (numeral k) = numeral k"
haftmann@51143
   749
  by transfer rule
haftmann@51143
   750
haftmann@51143
   751
instantiation natural :: "{semiring_div, equal, linordered_semiring}"
haftmann@51143
   752
begin
haftmann@51143
   753
haftmann@60352
   754
lift_definition divide_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
haftmann@60352
   755
  is "divide :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@51143
   756
  .
haftmann@51143
   757
haftmann@60352
   758
declare divide_natural.rep_eq [simp]
haftmann@51143
   759
haftmann@63950
   760
lift_definition modulo_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
haftmann@63950
   761
  is "modulo :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@51143
   762
  .
haftmann@51143
   763
haftmann@63950
   764
declare modulo_natural.rep_eq [simp]
haftmann@51143
   765
haftmann@51143
   766
lift_definition less_eq_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
haftmann@51143
   767
  is "less_eq :: nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@51143
   768
  .
haftmann@51143
   769
haftmann@51143
   770
declare less_eq_natural.rep_eq [termination_simp]
haftmann@51143
   771
haftmann@51143
   772
lift_definition less_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
haftmann@51143
   773
  is "less :: nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@51143
   774
  .
haftmann@51143
   775
haftmann@51143
   776
declare less_natural.rep_eq [termination_simp]
haftmann@51143
   777
haftmann@51143
   778
lift_definition equal_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
haftmann@51143
   779
  is "HOL.equal :: nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@51143
   780
  .
haftmann@51143
   781
haftmann@51143
   782
instance proof
haftmann@51143
   783
qed (transfer, simp add: algebra_simps equal less_le_not_le [symmetric] linear)+
haftmann@51143
   784
haftmann@24999
   785
end
haftmann@46664
   786
haftmann@51143
   787
lemma [transfer_rule]:
blanchet@55945
   788
  "rel_fun pcr_natural (rel_fun pcr_natural pcr_natural) (min :: _ \<Rightarrow> _ \<Rightarrow> nat) (min :: _ \<Rightarrow> _ \<Rightarrow> natural)"
haftmann@51143
   789
  by (unfold min_def [abs_def]) transfer_prover
haftmann@51143
   790
haftmann@51143
   791
lemma [transfer_rule]:
blanchet@55945
   792
  "rel_fun pcr_natural (rel_fun pcr_natural pcr_natural) (max :: _ \<Rightarrow> _ \<Rightarrow> nat) (max :: _ \<Rightarrow> _ \<Rightarrow> natural)"
haftmann@51143
   793
  by (unfold max_def [abs_def]) transfer_prover
haftmann@51143
   794
haftmann@51143
   795
lemma nat_of_natural_min [simp]:
haftmann@51143
   796
  "nat_of_natural (min k l) = min (nat_of_natural k) (nat_of_natural l)"
haftmann@51143
   797
  by transfer rule
haftmann@51143
   798
haftmann@51143
   799
lemma nat_of_natural_max [simp]:
haftmann@51143
   800
  "nat_of_natural (max k l) = max (nat_of_natural k) (nat_of_natural l)"
haftmann@51143
   801
  by transfer rule
haftmann@51143
   802
haftmann@51143
   803
lift_definition natural_of_integer :: "integer \<Rightarrow> natural"
haftmann@51143
   804
  is "nat :: int \<Rightarrow> nat"
haftmann@51143
   805
  .
haftmann@51143
   806
haftmann@51143
   807
lift_definition integer_of_natural :: "natural \<Rightarrow> integer"
haftmann@51143
   808
  is "of_nat :: nat \<Rightarrow> int"
haftmann@51143
   809
  .
haftmann@51143
   810
haftmann@51143
   811
lemma natural_of_integer_of_natural [simp]:
haftmann@51143
   812
  "natural_of_integer (integer_of_natural n) = n"
haftmann@51143
   813
  by transfer simp
haftmann@51143
   814
haftmann@51143
   815
lemma integer_of_natural_of_integer [simp]:
haftmann@51143
   816
  "integer_of_natural (natural_of_integer k) = max 0 k"
haftmann@51143
   817
  by transfer auto
haftmann@51143
   818
haftmann@51143
   819
lemma int_of_integer_of_natural [simp]:
haftmann@51143
   820
  "int_of_integer (integer_of_natural n) = of_nat (nat_of_natural n)"
haftmann@51143
   821
  by transfer rule
haftmann@51143
   822
haftmann@51143
   823
lemma integer_of_natural_of_nat [simp]:
haftmann@51143
   824
  "integer_of_natural (of_nat n) = of_nat n"
haftmann@51143
   825
  by transfer rule
haftmann@51143
   826
haftmann@51143
   827
lemma [measure_function]:
haftmann@51143
   828
  "is_measure nat_of_natural"
haftmann@51143
   829
  by (rule is_measure_trivial)
haftmann@51143
   830
haftmann@51143
   831
wenzelm@60758
   832
subsection \<open>Inductive representation of target language naturals\<close>
haftmann@51143
   833
haftmann@51143
   834
lift_definition Suc :: "natural \<Rightarrow> natural"
haftmann@51143
   835
  is Nat.Suc
haftmann@51143
   836
  .
haftmann@51143
   837
haftmann@51143
   838
declare Suc.rep_eq [simp]
haftmann@51143
   839
blanchet@58306
   840
old_rep_datatype "0::natural" Suc
haftmann@51143
   841
  by (transfer, fact nat.induct nat.inject nat.distinct)+
haftmann@51143
   842
blanchet@55416
   843
lemma natural_cases [case_names nat, cases type: natural]:
haftmann@51143
   844
  fixes m :: natural
haftmann@51143
   845
  assumes "\<And>n. m = of_nat n \<Longrightarrow> P"
haftmann@51143
   846
  shows P
haftmann@51143
   847
  using assms by transfer blast
haftmann@51143
   848
blanchet@58390
   849
lemma [simp, code]: "size_natural = nat_of_natural"
blanchet@58390
   850
proof (rule ext)
blanchet@58390
   851
  fix n
blanchet@58390
   852
  show "size_natural n = nat_of_natural n"
blanchet@58390
   853
    by (induct n) simp_all
blanchet@58390
   854
qed
blanchet@58379
   855
blanchet@58390
   856
lemma [simp, code]: "size = nat_of_natural"
blanchet@58390
   857
proof (rule ext)
blanchet@58390
   858
  fix n
blanchet@58390
   859
  show "size n = nat_of_natural n"
blanchet@58390
   860
    by (induct n) simp_all
blanchet@58390
   861
qed
blanchet@58379
   862
haftmann@51143
   863
lemma natural_decr [termination_simp]:
haftmann@51143
   864
  "n \<noteq> 0 \<Longrightarrow> nat_of_natural n - Nat.Suc 0 < nat_of_natural n"
haftmann@51143
   865
  by transfer simp
haftmann@51143
   866
blanchet@58379
   867
lemma natural_zero_minus_one: "(0::natural) - 1 = 0"
blanchet@58379
   868
  by (rule zero_diff)
haftmann@51143
   869
blanchet@58379
   870
lemma Suc_natural_minus_one: "Suc n - 1 = n"
haftmann@51143
   871
  by transfer simp
haftmann@51143
   872
haftmann@51143
   873
hide_const (open) Suc
haftmann@51143
   874
haftmann@51143
   875
wenzelm@60758
   876
subsection \<open>Code refinement for target language naturals\<close>
haftmann@51143
   877
haftmann@51143
   878
lift_definition Nat :: "integer \<Rightarrow> natural"
haftmann@51143
   879
  is nat
haftmann@51143
   880
  .
haftmann@51143
   881
haftmann@51143
   882
lemma [code_post]:
haftmann@51143
   883
  "Nat 0 = 0"
haftmann@51143
   884
  "Nat 1 = 1"
haftmann@51143
   885
  "Nat (numeral k) = numeral k"
haftmann@51143
   886
  by (transfer, simp)+
haftmann@51143
   887
haftmann@51143
   888
lemma [code abstype]:
haftmann@51143
   889
  "Nat (integer_of_natural n) = n"
haftmann@51143
   890
  by transfer simp
haftmann@51143
   891
haftmann@63174
   892
lemma [code]:
haftmann@63174
   893
  "natural_of_nat n = natural_of_integer (integer_of_nat n)"
haftmann@63174
   894
  by transfer simp
haftmann@51143
   895
haftmann@51143
   896
lemma [code abstract]:
haftmann@51143
   897
  "integer_of_natural (natural_of_integer k) = max 0 k"
haftmann@51143
   898
  by simp
haftmann@51143
   899
haftmann@51143
   900
lemma [code_abbrev]:
haftmann@51143
   901
  "natural_of_integer (Code_Numeral.Pos k) = numeral k"
haftmann@51143
   902
  by transfer simp
haftmann@51143
   903
haftmann@51143
   904
lemma [code abstract]:
haftmann@51143
   905
  "integer_of_natural 0 = 0"
haftmann@51143
   906
  by transfer simp
haftmann@51143
   907
haftmann@51143
   908
lemma [code abstract]:
haftmann@51143
   909
  "integer_of_natural 1 = 1"
haftmann@51143
   910
  by transfer simp
haftmann@51143
   911
haftmann@51143
   912
lemma [code abstract]:
haftmann@51143
   913
  "integer_of_natural (Code_Numeral.Suc n) = integer_of_natural n + 1"
haftmann@51143
   914
  by transfer simp
haftmann@51143
   915
haftmann@51143
   916
lemma [code]:
haftmann@51143
   917
  "nat_of_natural = nat_of_integer \<circ> integer_of_natural"
haftmann@51143
   918
  by transfer (simp add: fun_eq_iff)
haftmann@51143
   919
haftmann@51143
   920
lemma [code, code_unfold]:
blanchet@55416
   921
  "case_natural f g n = (if n = 0 then f else g (n - 1))"
haftmann@51143
   922
  by (cases n rule: natural.exhaust) (simp_all, simp add: Suc_def)
haftmann@51143
   923
blanchet@55642
   924
declare natural.rec [code del]
haftmann@51143
   925
haftmann@51143
   926
lemma [code abstract]:
haftmann@51143
   927
  "integer_of_natural (m + n) = integer_of_natural m + integer_of_natural n"
haftmann@51143
   928
  by transfer simp
haftmann@51143
   929
haftmann@51143
   930
lemma [code abstract]:
haftmann@51143
   931
  "integer_of_natural (m - n) = max 0 (integer_of_natural m - integer_of_natural n)"
haftmann@51143
   932
  by transfer simp
haftmann@51143
   933
haftmann@51143
   934
lemma [code abstract]:
haftmann@51143
   935
  "integer_of_natural (m * n) = integer_of_natural m * integer_of_natural n"
haftmann@51143
   936
  by transfer (simp add: of_nat_mult)
haftmann@51143
   937
haftmann@51143
   938
lemma [code abstract]:
haftmann@51143
   939
  "integer_of_natural (m div n) = integer_of_natural m div integer_of_natural n"
haftmann@51143
   940
  by transfer (simp add: zdiv_int)
haftmann@51143
   941
haftmann@51143
   942
lemma [code abstract]:
haftmann@51143
   943
  "integer_of_natural (m mod n) = integer_of_natural m mod integer_of_natural n"
haftmann@51143
   944
  by transfer (simp add: zmod_int)
haftmann@51143
   945
haftmann@51143
   946
lemma [code]:
haftmann@51143
   947
  "HOL.equal m n \<longleftrightarrow> HOL.equal (integer_of_natural m) (integer_of_natural n)"
haftmann@51143
   948
  by transfer (simp add: equal)
haftmann@51143
   949
blanchet@58379
   950
lemma [code nbe]: "HOL.equal n (n::natural) \<longleftrightarrow> True"
blanchet@58379
   951
  by (rule equal_class.equal_refl)
haftmann@51143
   952
blanchet@58379
   953
lemma [code]: "m \<le> n \<longleftrightarrow> integer_of_natural m \<le> integer_of_natural n"
haftmann@51143
   954
  by transfer simp
haftmann@51143
   955
blanchet@58379
   956
lemma [code]: "m < n \<longleftrightarrow> integer_of_natural m < integer_of_natural n"
haftmann@51143
   957
  by transfer simp
haftmann@51143
   958
haftmann@51143
   959
hide_const (open) Nat
haftmann@51143
   960
kuncar@55736
   961
lifting_update integer.lifting
kuncar@55736
   962
lifting_forget integer.lifting
kuncar@55736
   963
kuncar@55736
   964
lifting_update natural.lifting
kuncar@55736
   965
lifting_forget natural.lifting
haftmann@51143
   966
haftmann@51143
   967
code_reflect Code_Numeral
haftmann@63174
   968
  datatypes natural
haftmann@63174
   969
  functions "Code_Numeral.Suc" "0 :: natural" "1 :: natural"
haftmann@63174
   970
    "plus :: natural \<Rightarrow> _" "minus :: natural \<Rightarrow> _"
haftmann@63174
   971
    "times :: natural \<Rightarrow> _" "divide :: natural \<Rightarrow> _"
haftmann@63950
   972
    "modulo :: natural \<Rightarrow> _"
haftmann@63174
   973
    integer_of_natural natural_of_integer
haftmann@51143
   974
haftmann@51143
   975
end