src/HOL/Code_Numeral.thy
author kuncar
Tue, 25 Feb 2014 19:07:14 +0100
changeset 55736 f1ed1e9cd080
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child 55945 e96383acecf9
permissions -rw-r--r--
unregister lifting setup following the best practice of Lifting
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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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header {* Numeric types for code generation onto target language numerals only *}
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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 {* Type of target language integers *}
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typedef integer = "UNIV \<Colon> int set"
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  morphisms int_of_integer integer_of_int ..
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setup_lifting (no_code) 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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  "fun_rel HOL.eq pcr_integer (of_nat :: nat \<Rightarrow> int) (of_nat :: nat \<Rightarrow> integer)"
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  by (unfold of_nat_def [abs_def]) transfer_prover
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lemma [transfer_rule]:
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  "fun_rel HOL.eq pcr_integer (\<lambda>k :: int. k :: int) (of_int :: int \<Rightarrow> integer)"
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proof -
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  have "fun_rel HOL.eq pcr_integer (of_int :: int \<Rightarrow> int) (of_int :: int \<Rightarrow> integer)"
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    by (unfold of_int_of_nat [abs_def]) 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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  "fun_rel HOL.eq pcr_integer (numeral :: num \<Rightarrow> int) (numeral :: num \<Rightarrow> integer)"
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proof -
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  have "fun_rel HOL.eq pcr_integer (numeral :: num \<Rightarrow> int) (\<lambda>n. of_int (numeral n))"
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    by transfer_prover
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  then show ?thesis by simp
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qed
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lemma [transfer_rule]:
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  "fun_rel HOL.eq (fun_rel 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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instantiation integer :: "{ring_div, equal, linordered_idom}"
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begin
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lift_definition div_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
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  is "Divides.div :: int \<Rightarrow> int \<Rightarrow> int"
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  .
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declare div_integer.rep_eq [simp]
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lift_definition mod_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
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  is "Divides.mod :: int \<Rightarrow> int \<Rightarrow> int"
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  .
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declare mod_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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  "fun_rel pcr_integer (fun_rel 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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  "fun_rel pcr_integer (fun_rel 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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instance integer :: semiring_numeral_div
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  by intro_classes (transfer,
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    fact semiring_numeral_div_class.diff_invert_add1
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    semiring_numeral_div_class.le_add_diff_inverse2
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    semiring_numeral_div_class.mult_div_cancel
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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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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 {* Code theorems for target language integers *}
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text {* Constructors *}
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definition Pos :: "num \<Rightarrow> integer"
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where
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  [simp, code_abbrev]: "Pos = numeral"
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lemma [transfer_rule]:
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  "fun_rel HOL.eq pcr_integer numeral Pos"
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  by simp transfer_prover
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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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  "fun_rel 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 {* Auxiliary operations *}
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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)+
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   277
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   278
lift_definition sub :: "num \<Rightarrow> num \<Rightarrow> integer"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   279
  is "\<lambda>m n. numeral m - numeral n :: int"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   280
  .
26140
e45375135052 Zero/Suc recursion combinator for type index
haftmann
parents: 26086
diff changeset
   281
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   282
lemma sub_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   283
  "sub Num.One Num.One = 0"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   284
  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   285
  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   286
  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   287
  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   288
  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   289
  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   290
  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   291
  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   292
  by (transfer, simp add: dbl_def dbl_inc_def dbl_dec_def)+
28351
abfc66969d1f non left-linear equations for nbe
haftmann
parents: 28346
diff changeset
   293
24999
haftmann
parents:
diff changeset
   294
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   295
text {* Implementations *}
24999
haftmann
parents:
diff changeset
   296
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   297
lemma one_integer_code [code, code_unfold]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   298
  "1 = Pos Num.One"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   299
  by simp
24999
haftmann
parents:
diff changeset
   300
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   301
lemma plus_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   302
  "k + 0 = (k::integer)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   303
  "0 + l = (l::integer)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   304
  "Pos m + Pos n = Pos (m + n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   305
  "Pos m + Neg n = sub m n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   306
  "Neg m + Pos n = sub n m"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   307
  "Neg m + Neg n = Neg (m + n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   308
  by (transfer, simp)+
24999
haftmann
parents:
diff changeset
   309
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   310
lemma uminus_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   311
  "uminus 0 = (0::integer)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   312
  "uminus (Pos m) = Neg m"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   313
  "uminus (Neg m) = Pos m"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   314
  by simp_all
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   315
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   316
lemma minus_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   317
  "k - 0 = (k::integer)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   318
  "0 - l = uminus (l::integer)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   319
  "Pos m - Pos n = sub m n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   320
  "Pos m - Neg n = Pos (m + n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   321
  "Neg m - Pos n = Neg (m + n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   322
  "Neg m - Neg n = sub n m"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   323
  by (transfer, simp)+
46028
9f113cdf3d66 attribute code_abbrev superseedes code_unfold_post
haftmann
parents: 45694
diff changeset
   324
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   325
lemma abs_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   326
  "\<bar>k\<bar> = (if (k::integer) < 0 then - k else k)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   327
  by simp
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46664
diff changeset
   328
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   329
lemma sgn_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   330
  "sgn k = (if k = 0 then 0 else if (k::integer) < 0 then - 1 else 1)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46664
diff changeset
   331
  by simp
46028
9f113cdf3d66 attribute code_abbrev superseedes code_unfold_post
haftmann
parents: 45694
diff changeset
   332
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   333
lemma times_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   334
  "k * 0 = (0::integer)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   335
  "0 * l = (0::integer)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   336
  "Pos m * Pos n = Pos (m * n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   337
  "Pos m * Neg n = Neg (m * n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   338
  "Neg m * Pos n = Neg (m * n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   339
  "Neg m * Neg n = Pos (m * n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   340
  by simp_all
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   341
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   342
definition divmod_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer \<times> integer"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   343
where
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   344
  "divmod_integer k l = (k div l, k mod l)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   345
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   346
lemma fst_divmod [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   347
  "fst (divmod_integer k l) = k div l"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   348
  by (simp add: divmod_integer_def)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   349
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   350
lemma snd_divmod [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   351
  "snd (divmod_integer k l) = k mod l"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   352
  by (simp add: divmod_integer_def)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   353
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   354
definition divmod_abs :: "integer \<Rightarrow> integer \<Rightarrow> integer \<times> integer"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   355
where
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   356
  "divmod_abs k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   357
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   358
lemma fst_divmod_abs [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   359
  "fst (divmod_abs k l) = \<bar>k\<bar> div \<bar>l\<bar>"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   360
  by (simp add: divmod_abs_def)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   361
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   362
lemma snd_divmod_abs [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   363
  "snd (divmod_abs k l) = \<bar>k\<bar> mod \<bar>l\<bar>"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   364
  by (simp add: divmod_abs_def)
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   365
53069
d165213e3924 execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents: 52435
diff changeset
   366
lemma divmod_abs_code [code]:
d165213e3924 execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents: 52435
diff changeset
   367
  "divmod_abs (Pos k) (Pos l) = divmod k l"
d165213e3924 execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents: 52435
diff changeset
   368
  "divmod_abs (Neg k) (Neg l) = divmod k l"
d165213e3924 execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents: 52435
diff changeset
   369
  "divmod_abs (Neg k) (Pos l) = divmod k l"
d165213e3924 execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents: 52435
diff changeset
   370
  "divmod_abs (Pos k) (Neg l) = divmod k l"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   371
  "divmod_abs j 0 = (0, \<bar>j\<bar>)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   372
  "divmod_abs 0 j = (0, 0)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   373
  by (simp_all add: prod_eq_iff)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   374
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   375
lemma divmod_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   376
  "divmod_integer k l =
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   377
    (if k = 0 then (0, 0) else if l = 0 then (0, k) else
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   378
    (apsnd \<circ> times \<circ> sgn) l (if sgn k = sgn l
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   379
      then divmod_abs k l
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   380
      else (let (r, s) = divmod_abs k l in
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   381
        if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   382
proof -
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   383
  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"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   384
    by (auto simp add: sgn_if)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   385
  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
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   386
  show ?thesis
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 54489
diff changeset
   387
    by (simp add: prod_eq_iff integer_eq_iff case_prod_beta aux1)
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   388
      (auto simp add: zdiv_zminus1_eq_if zmod_zminus1_eq_if div_minus_right mod_minus_right aux2)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   389
qed
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   390
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   391
lemma div_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   392
  "k div l = fst (divmod_integer k l)"
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   393
  by simp
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   394
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   395
lemma mod_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   396
  "k mod l = snd (divmod_integer k l)"
25767
852bce03412a index now a copy of nat rather than int
haftmann
parents: 25691
diff changeset
   397
  by simp
24999
haftmann
parents:
diff changeset
   398
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   399
lemma equal_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   400
  "HOL.equal 0 (0::integer) \<longleftrightarrow> True"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   401
  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   402
  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   403
  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   404
  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   405
  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   406
  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   407
  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   408
  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   409
  by (simp_all add: equal)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   410
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   411
lemma equal_integer_refl [code nbe]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   412
  "HOL.equal (k::integer) k \<longleftrightarrow> True"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   413
  by (fact equal_refl)
31266
55e70b6d812e added lemma about 0 - 1
haftmann
parents: 31205
diff changeset
   414
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   415
lemma less_eq_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   416
  "0 \<le> (0::integer) \<longleftrightarrow> True"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   417
  "0 \<le> Pos l \<longleftrightarrow> True"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   418
  "0 \<le> Neg l \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   419
  "Pos k \<le> 0 \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   420
  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   421
  "Pos k \<le> Neg l \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   422
  "Neg k \<le> 0 \<longleftrightarrow> True"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   423
  "Neg k \<le> Pos l \<longleftrightarrow> True"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   424
  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   425
  by simp_all
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   426
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   427
lemma less_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   428
  "0 < (0::integer) \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   429
  "0 < Pos l \<longleftrightarrow> True"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   430
  "0 < Neg l \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   431
  "Pos k < 0 \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   432
  "Pos k < Pos l \<longleftrightarrow> k < l"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   433
  "Pos k < Neg l \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   434
  "Neg k < 0 \<longleftrightarrow> True"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   435
  "Neg k < Pos l \<longleftrightarrow> True"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   436
  "Neg k < Neg l \<longleftrightarrow> l < k"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   437
  by simp_all
26140
e45375135052 Zero/Suc recursion combinator for type index
haftmann
parents: 26086
diff changeset
   438
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   439
lift_definition integer_of_num :: "num \<Rightarrow> integer"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   440
  is "numeral :: num \<Rightarrow> int"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   441
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   442
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   443
lemma integer_of_num [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   444
  "integer_of_num num.One = 1"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   445
  "integer_of_num (num.Bit0 n) = (let k = integer_of_num n in k + k)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   446
  "integer_of_num (num.Bit1 n) = (let k = integer_of_num n in k + k + 1)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   447
  by (transfer, simp only: numeral.simps Let_def)+
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   448
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   449
lift_definition num_of_integer :: "integer \<Rightarrow> num"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   450
  is "num_of_nat \<circ> nat"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   451
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   452
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   453
lemma num_of_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   454
  "num_of_integer k = (if k \<le> 1 then Num.One
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   455
     else let
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   456
       (l, j) = divmod_integer k 2;
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   457
       l' = num_of_integer l;
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   458
       l'' = l' + l'
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   459
     in if j = 0 then l'' else l'' + Num.One)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   460
proof -
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   461
  {
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   462
    assume "int_of_integer k mod 2 = 1"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   463
    then have "nat (int_of_integer k mod 2) = nat 1" by simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   464
    moreover assume *: "1 < int_of_integer k"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   465
    ultimately have **: "nat (int_of_integer k) mod 2 = 1" by (simp add: nat_mod_distrib)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   466
    have "num_of_nat (nat (int_of_integer k)) =
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   467
      num_of_nat (2 * (nat (int_of_integer k) div 2) + nat (int_of_integer k) mod 2)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   468
      by simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   469
    then have "num_of_nat (nat (int_of_integer k)) =
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   470
      num_of_nat (nat (int_of_integer k) div 2 + nat (int_of_integer k) div 2 + nat (int_of_integer k) mod 2)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   471
      by (simp add: mult_2)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   472
    with ** have "num_of_nat (nat (int_of_integer k)) =
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   473
      num_of_nat (nat (int_of_integer k) div 2 + nat (int_of_integer k) div 2 + 1)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   474
      by simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   475
  }
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   476
  note aux = this
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   477
  show ?thesis
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 54489
diff changeset
   478
    by (auto simp add: num_of_integer_def nat_of_integer_def Let_def case_prod_beta
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   479
      not_le integer_eq_iff less_eq_integer_def
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   480
      nat_mult_distrib nat_div_distrib num_of_nat_One num_of_nat_plus_distrib
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   481
       mult_2 [where 'a=nat] aux add_One)
25918
haftmann
parents: 25767
diff changeset
   482
qed
haftmann
parents: 25767
diff changeset
   483
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   484
lemma nat_of_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   485
  "nat_of_integer k = (if k \<le> 0 then 0
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   486
     else let
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   487
       (l, j) = divmod_integer k 2;
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   488
       l' = nat_of_integer l;
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   489
       l'' = l' + l'
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   490
     in if j = 0 then l'' else l'' + 1)"
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   491
proof -
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   492
  obtain j where "k = integer_of_int j"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   493
  proof
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   494
    show "k = integer_of_int (int_of_integer k)" by simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   495
  qed
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   496
  moreover have "2 * (j div 2) = j - j mod 2"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   497
    by (simp add: zmult_div_cancel mult_commute)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   498
  ultimately show ?thesis
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   499
    by (auto simp add: split_def Let_def mod_integer_def nat_of_integer_def not_le
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   500
      nat_add_distrib [symmetric] Suc_nat_eq_nat_zadd1)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   501
      (auto simp add: mult_2 [symmetric])
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   502
qed
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   503
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   504
lemma int_of_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   505
  "int_of_integer k = (if k < 0 then - (int_of_integer (- k))
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   506
     else if k = 0 then 0
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   507
     else let
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   508
       (l, j) = divmod_integer k 2;
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   509
       l' = 2 * int_of_integer l
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   510
     in if j = 0 then l' else l' + 1)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   511
  by (auto simp add: split_def Let_def integer_eq_iff zmult_div_cancel)
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   512
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   513
lemma integer_of_int_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   514
  "integer_of_int k = (if k < 0 then - (integer_of_int (- k))
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   515
     else if k = 0 then 0
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   516
     else let
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   517
       (l, j) = divmod_int k 2;
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   518
       l' = 2 * integer_of_int l
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   519
     in if j = 0 then l' else l' + 1)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   520
  by (auto simp add: split_def Let_def integer_eq_iff zmult_div_cancel)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   521
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   522
hide_const (open) Pos Neg sub dup divmod_abs
46547
d1dcb91a512e use qualified constant names instead of suffixes (from Florian Haftmann)
huffman
parents: 46028
diff changeset
   523
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   524
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   525
subsection {* Serializer setup for target language integers *}
24999
haftmann
parents:
diff changeset
   526
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   527
code_reserved Eval int Integer abs
25767
852bce03412a index now a copy of nat rather than int
haftmann
parents: 25691
diff changeset
   528
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   529
code_printing
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   530
  type_constructor integer \<rightharpoonup>
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   531
    (SML) "IntInf.int"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   532
    and (OCaml) "Big'_int.big'_int"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   533
    and (Haskell) "Integer"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   534
    and (Scala) "BigInt"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   535
    and (Eval) "int"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   536
| class_instance integer :: equal \<rightharpoonup>
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   537
    (Haskell) -
24999
haftmann
parents:
diff changeset
   538
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   539
code_printing
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   540
  constant "0::integer" \<rightharpoonup>
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   541
    (SML) "0"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   542
    and (OCaml) "Big'_int.zero'_big'_int"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   543
    and (Haskell) "0"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   544
    and (Scala) "BigInt(0)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46664
diff changeset
   545
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   546
setup {*
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   547
  fold (Numeral.add_code @{const_name Code_Numeral.Pos}
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   548
    false Code_Printer.literal_numeral) ["SML", "OCaml", "Haskell", "Scala"]
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   549
*}
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   550
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   551
setup {*
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   552
  fold (Numeral.add_code @{const_name Code_Numeral.Neg}
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   553
    true Code_Printer.literal_numeral) ["SML", "OCaml", "Haskell", "Scala"]
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   554
*}
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   555
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   556
code_printing
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   557
  constant "plus :: integer \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   558
    (SML) "IntInf.+ ((_), (_))"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   559
    and (OCaml) "Big'_int.add'_big'_int"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   560
    and (Haskell) infixl 6 "+"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   561
    and (Scala) infixl 7 "+"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   562
    and (Eval) infixl 8 "+"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   563
| constant "uminus :: integer \<Rightarrow> _" \<rightharpoonup>
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   564
    (SML) "IntInf.~"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   565
    and (OCaml) "Big'_int.minus'_big'_int"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   566
    and (Haskell) "negate"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   567
    and (Scala) "!(- _)"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   568
    and (Eval) "~/ _"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   569
| constant "minus :: integer \<Rightarrow> _" \<rightharpoonup>
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   570
    (SML) "IntInf.- ((_), (_))"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   571
    and (OCaml) "Big'_int.sub'_big'_int"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   572
    and (Haskell) infixl 6 "-"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   573
    and (Scala) infixl 7 "-"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   574
    and (Eval) infixl 8 "-"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   575
| constant Code_Numeral.dup \<rightharpoonup>
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   576
    (SML) "IntInf.*/ (2,/ (_))"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   577
    and (OCaml) "Big'_int.mult'_big'_int/ (Big'_int.big'_int'_of'_int/ 2)"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   578
    and (Haskell) "!(2 * _)"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   579
    and (Scala) "!(2 * _)"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   580
    and (Eval) "!(2 * _)"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   581
| constant Code_Numeral.sub \<rightharpoonup>
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   582
    (SML) "!(raise/ Fail/ \"sub\")"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   583
    and (OCaml) "failwith/ \"sub\""
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   584
    and (Haskell) "error/ \"sub\""
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   585
    and (Scala) "!sys.error(\"sub\")"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   586
| constant "times :: integer \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   587
    (SML) "IntInf.* ((_), (_))"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   588
    and (OCaml) "Big'_int.mult'_big'_int"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   589
    and (Haskell) infixl 7 "*"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   590
    and (Scala) infixl 8 "*"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   591
    and (Eval) infixl 9 "*"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   592
| constant Code_Numeral.divmod_abs \<rightharpoonup>
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   593
    (SML) "IntInf.divMod/ (IntInf.abs _,/ IntInf.abs _)"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   594
    and (OCaml) "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   595
    and (Haskell) "divMod/ (abs _)/ (abs _)"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   596
    and (Scala) "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   597
    and (Eval) "Integer.div'_mod/ (abs _)/ (abs _)"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   598
| constant "HOL.equal :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   599
    (SML) "!((_ : IntInf.int) = _)"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   600
    and (OCaml) "Big'_int.eq'_big'_int"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   601
    and (Haskell) infix 4 "=="
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   602
    and (Scala) infixl 5 "=="
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   603
    and (Eval) infixl 6 "="
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   604
| constant "less_eq :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   605
    (SML) "IntInf.<= ((_), (_))"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   606
    and (OCaml) "Big'_int.le'_big'_int"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   607
    and (Haskell) infix 4 "<="
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   608
    and (Scala) infixl 4 "<="
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   609
    and (Eval) infixl 6 "<="
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   610
| constant "less :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   611
    (SML) "IntInf.< ((_), (_))"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   612
    and (OCaml) "Big'_int.lt'_big'_int"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   613
    and (Haskell) infix 4 "<"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   614
    and (Scala) infixl 4 "<"
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   615
    and (Eval) infixl 6 "<"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   616
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   617
code_identifier
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   618
  code_module Code_Numeral \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
46547
d1dcb91a512e use qualified constant names instead of suffixes (from Florian Haftmann)
huffman
parents: 46028
diff changeset
   619
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   620
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   621
subsection {* Type of target language naturals *}
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   622
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   623
typedef natural = "UNIV \<Colon> nat set"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   624
  morphisms nat_of_natural natural_of_nat ..
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   625
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   626
setup_lifting (no_code) type_definition_natural
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   627
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   628
lemma natural_eq_iff [termination_simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   629
  "m = n \<longleftrightarrow> nat_of_natural m = nat_of_natural n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   630
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   631
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   632
lemma natural_eqI:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   633
  "nat_of_natural m = nat_of_natural n \<Longrightarrow> m = n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   634
  using natural_eq_iff [of m n] by simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   635
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   636
lemma nat_of_natural_of_nat_inverse [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   637
  "nat_of_natural (natural_of_nat n) = n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   638
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   639
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   640
lemma natural_of_nat_of_natural_inverse [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   641
  "natural_of_nat (nat_of_natural n) = n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   642
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   643
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   644
instantiation natural :: "{comm_monoid_diff, semiring_1}"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   645
begin
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   646
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   647
lift_definition zero_natural :: natural
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   648
  is "0 :: nat"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   649
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   650
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   651
declare zero_natural.rep_eq [simp]
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   652
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   653
lift_definition one_natural :: natural
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   654
  is "1 :: nat"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   655
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   656
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   657
declare one_natural.rep_eq [simp]
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   658
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   659
lift_definition plus_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   660
  is "plus :: nat \<Rightarrow> nat \<Rightarrow> nat"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   661
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   662
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   663
declare plus_natural.rep_eq [simp]
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   664
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   665
lift_definition minus_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   666
  is "minus :: nat \<Rightarrow> nat \<Rightarrow> nat"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   667
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   668
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   669
declare minus_natural.rep_eq [simp]
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   670
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   671
lift_definition times_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   672
  is "times :: nat \<Rightarrow> nat \<Rightarrow> nat"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   673
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   674
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   675
declare times_natural.rep_eq [simp]
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   676
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   677
instance proof
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   678
qed (transfer, simp add: algebra_simps)+
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   679
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   680
end
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   681
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   682
lemma [transfer_rule]:
51375
d9e62d9c98de patch Isabelle ditribution to conform to changes regarding the parametricity
kuncar
parents: 51143
diff changeset
   683
  "fun_rel HOL.eq pcr_natural (\<lambda>n::nat. n) (of_nat :: nat \<Rightarrow> natural)"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   684
proof -
51375
d9e62d9c98de patch Isabelle ditribution to conform to changes regarding the parametricity
kuncar
parents: 51143
diff changeset
   685
  have "fun_rel HOL.eq pcr_natural (of_nat :: nat \<Rightarrow> nat) (of_nat :: nat \<Rightarrow> natural)"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   686
    by (unfold of_nat_def [abs_def]) transfer_prover
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   687
  then show ?thesis by (simp add: id_def)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   688
qed
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   689
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   690
lemma [transfer_rule]:
51375
d9e62d9c98de patch Isabelle ditribution to conform to changes regarding the parametricity
kuncar
parents: 51143
diff changeset
   691
  "fun_rel HOL.eq pcr_natural (numeral :: num \<Rightarrow> nat) (numeral :: num \<Rightarrow> natural)"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   692
proof -
51375
d9e62d9c98de patch Isabelle ditribution to conform to changes regarding the parametricity
kuncar
parents: 51143
diff changeset
   693
  have "fun_rel HOL.eq pcr_natural (numeral :: num \<Rightarrow> nat) (\<lambda>n. of_nat (numeral n))"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   694
    by transfer_prover
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   695
  then show ?thesis by simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   696
qed
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   697
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   698
lemma nat_of_natural_of_nat [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   699
  "nat_of_natural (of_nat n) = n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   700
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   701
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   702
lemma natural_of_nat_of_nat [simp, code_abbrev]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   703
  "natural_of_nat = of_nat"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   704
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   705
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   706
lemma of_nat_of_natural [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   707
  "of_nat (nat_of_natural n) = n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   708
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   709
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   710
lemma nat_of_natural_numeral [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   711
  "nat_of_natural (numeral k) = numeral k"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   712
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   713
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   714
instantiation natural :: "{semiring_div, equal, linordered_semiring}"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   715
begin
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   716
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   717
lift_definition div_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   718
  is "Divides.div :: nat \<Rightarrow> nat \<Rightarrow> nat"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   719
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   720
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   721
declare div_natural.rep_eq [simp]
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   722
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   723
lift_definition mod_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   724
  is "Divides.mod :: nat \<Rightarrow> nat \<Rightarrow> nat"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   725
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   726
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   727
declare mod_natural.rep_eq [simp]
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   728
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   729
lift_definition less_eq_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   730
  is "less_eq :: nat \<Rightarrow> nat \<Rightarrow> bool"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   731
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   732
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   733
declare less_eq_natural.rep_eq [termination_simp]
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   734
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   735
lift_definition less_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   736
  is "less :: nat \<Rightarrow> nat \<Rightarrow> bool"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   737
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   738
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   739
declare less_natural.rep_eq [termination_simp]
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   740
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   741
lift_definition equal_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   742
  is "HOL.equal :: nat \<Rightarrow> nat \<Rightarrow> bool"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   743
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   744
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   745
instance proof
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   746
qed (transfer, simp add: algebra_simps equal less_le_not_le [symmetric] linear)+
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   747
24999
haftmann
parents:
diff changeset
   748
end
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   749
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   750
lemma [transfer_rule]:
51375
d9e62d9c98de patch Isabelle ditribution to conform to changes regarding the parametricity
kuncar
parents: 51143
diff changeset
   751
  "fun_rel pcr_natural (fun_rel pcr_natural pcr_natural) (min :: _ \<Rightarrow> _ \<Rightarrow> nat) (min :: _ \<Rightarrow> _ \<Rightarrow> natural)"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   752
  by (unfold min_def [abs_def]) transfer_prover
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   753
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   754
lemma [transfer_rule]:
51375
d9e62d9c98de patch Isabelle ditribution to conform to changes regarding the parametricity
kuncar
parents: 51143
diff changeset
   755
  "fun_rel pcr_natural (fun_rel pcr_natural pcr_natural) (max :: _ \<Rightarrow> _ \<Rightarrow> nat) (max :: _ \<Rightarrow> _ \<Rightarrow> natural)"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   756
  by (unfold max_def [abs_def]) transfer_prover
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   757
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   758
lemma nat_of_natural_min [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   759
  "nat_of_natural (min k l) = min (nat_of_natural k) (nat_of_natural l)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   760
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   761
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   762
lemma nat_of_natural_max [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   763
  "nat_of_natural (max k l) = max (nat_of_natural k) (nat_of_natural l)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   764
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   765
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   766
lift_definition natural_of_integer :: "integer \<Rightarrow> natural"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   767
  is "nat :: int \<Rightarrow> nat"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   768
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   769
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   770
lift_definition integer_of_natural :: "natural \<Rightarrow> integer"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   771
  is "of_nat :: nat \<Rightarrow> int"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   772
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   773
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   774
lemma natural_of_integer_of_natural [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   775
  "natural_of_integer (integer_of_natural n) = n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   776
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   777
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   778
lemma integer_of_natural_of_integer [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   779
  "integer_of_natural (natural_of_integer k) = max 0 k"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   780
  by transfer auto
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   781
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   782
lemma int_of_integer_of_natural [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   783
  "int_of_integer (integer_of_natural n) = of_nat (nat_of_natural n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   784
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   785
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   786
lemma integer_of_natural_of_nat [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   787
  "integer_of_natural (of_nat n) = of_nat n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   788
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   789
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   790
lemma [measure_function]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   791
  "is_measure nat_of_natural"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   792
  by (rule is_measure_trivial)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   793
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   794
55416
dd7992d4a61a adapted theories to 'xxx_case' to 'case_xxx'
blanchet
parents: 55414
diff changeset
   795
subsection {* Inductive representation of target language naturals *}
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   796
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   797
lift_definition Suc :: "natural \<Rightarrow> natural"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   798
  is Nat.Suc
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   799
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   800
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   801
declare Suc.rep_eq [simp]
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   802
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   803
rep_datatype "0::natural" Suc
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   804
  by (transfer, fact nat.induct nat.inject nat.distinct)+
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   805
55416
dd7992d4a61a adapted theories to 'xxx_case' to 'case_xxx'
blanchet
parents: 55414
diff changeset
   806
lemma natural_cases [case_names nat, cases type: natural]:
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   807
  fixes m :: natural
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   808
  assumes "\<And>n. m = of_nat n \<Longrightarrow> P"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   809
  shows P
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   810
  using assms by transfer blast
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   811
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   812
lemma [simp, code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   813
  "natural_size = nat_of_natural"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   814
proof (rule ext)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   815
  fix n
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   816
  show "natural_size n = nat_of_natural n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   817
    by (induct n) simp_all
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   818
qed
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   819
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   820
lemma [simp, code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   821
  "size = nat_of_natural"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   822
proof (rule ext)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   823
  fix n
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   824
  show "size n = nat_of_natural n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   825
    by (induct n) simp_all
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   826
qed
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   827
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   828
lemma natural_decr [termination_simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   829
  "n \<noteq> 0 \<Longrightarrow> nat_of_natural n - Nat.Suc 0 < nat_of_natural n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   830
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   831
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   832
lemma natural_zero_minus_one:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   833
  "(0::natural) - 1 = 0"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   834
  by simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   835
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   836
lemma Suc_natural_minus_one:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   837
  "Suc n - 1 = n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   838
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   839
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   840
hide_const (open) Suc
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   841
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   842
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   843
subsection {* Code refinement for target language naturals *}
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   844
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   845
lift_definition Nat :: "integer \<Rightarrow> natural"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   846
  is nat
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   847
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   848
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   849
lemma [code_post]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   850
  "Nat 0 = 0"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   851
  "Nat 1 = 1"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   852
  "Nat (numeral k) = numeral k"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   853
  by (transfer, simp)+
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   854
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   855
lemma [code abstype]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   856
  "Nat (integer_of_natural n) = n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   857
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   858
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   859
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   860
  "integer_of_natural (natural_of_nat n) = of_nat n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   861
  by simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   862
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   863
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   864
  "integer_of_natural (natural_of_integer k) = max 0 k"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   865
  by simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   866
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   867
lemma [code_abbrev]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   868
  "natural_of_integer (Code_Numeral.Pos k) = numeral k"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   869
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   870
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   871
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   872
  "integer_of_natural 0 = 0"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   873
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   874
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   875
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   876
  "integer_of_natural 1 = 1"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   877
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   878
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   879
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   880
  "integer_of_natural (Code_Numeral.Suc n) = integer_of_natural n + 1"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   881
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   882
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   883
lemma [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   884
  "nat_of_natural = nat_of_integer \<circ> integer_of_natural"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   885
  by transfer (simp add: fun_eq_iff)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   886
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   887
lemma [code, code_unfold]:
55416
dd7992d4a61a adapted theories to 'xxx_case' to 'case_xxx'
blanchet
parents: 55414
diff changeset
   888
  "case_natural f g n = (if n = 0 then f else g (n - 1))"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   889
  by (cases n rule: natural.exhaust) (simp_all, simp add: Suc_def)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   890
55642
63beb38e9258 adapted to renaming of datatype 'cases' and 'recs' to 'case' and 'rec'
blanchet
parents: 55428
diff changeset
   891
declare natural.rec [code del]
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   892
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   893
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   894
  "integer_of_natural (m + n) = integer_of_natural m + integer_of_natural n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   895
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   896
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   897
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   898
  "integer_of_natural (m - n) = max 0 (integer_of_natural m - integer_of_natural n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   899
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   900
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   901
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   902
  "integer_of_natural (m * n) = integer_of_natural m * integer_of_natural n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   903
  by transfer (simp add: of_nat_mult)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   904
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   905
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   906
  "integer_of_natural (m div n) = integer_of_natural m div integer_of_natural n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   907
  by transfer (simp add: zdiv_int)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   908
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   909
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   910
  "integer_of_natural (m mod n) = integer_of_natural m mod integer_of_natural n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   911
  by transfer (simp add: zmod_int)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   912
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   913
lemma [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   914
  "HOL.equal m n \<longleftrightarrow> HOL.equal (integer_of_natural m) (integer_of_natural n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   915
  by transfer (simp add: equal)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   916
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   917
lemma [code nbe]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   918
  "HOL.equal n (n::natural) \<longleftrightarrow> True"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   919
  by (simp add: equal)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   920
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   921
lemma [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   922
  "m \<le> n \<longleftrightarrow> integer_of_natural m \<le> integer_of_natural n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   923
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   924
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   925
lemma [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   926
  "m < n \<longleftrightarrow> integer_of_natural m < integer_of_natural n"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   927
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   928
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   929
hide_const (open) Nat
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   930
55736
f1ed1e9cd080 unregister lifting setup following the best practice of Lifting
kuncar
parents: 55642
diff changeset
   931
lifting_update integer.lifting
f1ed1e9cd080 unregister lifting setup following the best practice of Lifting
kuncar
parents: 55642
diff changeset
   932
lifting_forget integer.lifting
f1ed1e9cd080 unregister lifting setup following the best practice of Lifting
kuncar
parents: 55642
diff changeset
   933
f1ed1e9cd080 unregister lifting setup following the best practice of Lifting
kuncar
parents: 55642
diff changeset
   934
lifting_update natural.lifting
f1ed1e9cd080 unregister lifting setup following the best practice of Lifting
kuncar
parents: 55642
diff changeset
   935
lifting_forget natural.lifting
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   936
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   937
code_reflect Code_Numeral
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   938
  datatypes natural = _
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   939
  functions integer_of_natural natural_of_integer
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   940
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   941
end
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   942