src/HOL/Code_Numeral.thy
author Manuel Eberl <eberlm@in.tum.de>
Tue, 19 Jan 2016 11:19:25 +0100
changeset 62201 eca7b38c8ee5
parent 61857 542f2c6da692
child 63174 57c0d60e491c
permissions -rw-r--r--
Added approximation of powr to NEWS/CONTRIBUTORS
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(*  Title:      HOL/Code_Numeral.thy
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    Author:     Florian Haftmann, TU Muenchen
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*)
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section \<open>Numeric types for code generation onto target language numerals only\<close>
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theory Code_Numeral
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imports Nat_Transfer Divides Lifting
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begin
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subsection \<open>Type of target language integers\<close>
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typedef integer = "UNIV :: int set"
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  morphisms int_of_integer integer_of_int ..
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setup_lifting type_definition_integer
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lemma integer_eq_iff:
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  "k = l \<longleftrightarrow> int_of_integer k = int_of_integer l"
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  by transfer rule
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lemma integer_eqI:
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  "int_of_integer k = int_of_integer l \<Longrightarrow> k = l"
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  using integer_eq_iff [of k l] by simp
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lemma int_of_integer_integer_of_int [simp]:
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  "int_of_integer (integer_of_int k) = k"
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  by transfer rule
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lemma integer_of_int_int_of_integer [simp]:
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  "integer_of_int (int_of_integer k) = k"
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  by transfer rule
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instantiation integer :: ring_1
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begin
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lift_definition zero_integer :: integer
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  is "0 :: int"
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  .
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declare zero_integer.rep_eq [simp]
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lift_definition one_integer :: integer
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  is "1 :: int"
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  .
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declare one_integer.rep_eq [simp]
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lift_definition plus_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
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  is "plus :: int \<Rightarrow> int \<Rightarrow> int"
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  .
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declare plus_integer.rep_eq [simp]
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lift_definition uminus_integer :: "integer \<Rightarrow> integer"
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  is "uminus :: int \<Rightarrow> int"
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  .
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declare uminus_integer.rep_eq [simp]
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lift_definition minus_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
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  is "minus :: int \<Rightarrow> int \<Rightarrow> int"
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  .
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declare minus_integer.rep_eq [simp]
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lift_definition times_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
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  is "times :: int \<Rightarrow> int \<Rightarrow> int"
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  .
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declare times_integer.rep_eq [simp]
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instance proof
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qed (transfer, simp add: algebra_simps)+
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end
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lemma [transfer_rule]:
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  "rel_fun HOL.eq pcr_integer (of_nat :: nat \<Rightarrow> int) (of_nat :: nat \<Rightarrow> integer)"
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  by (unfold of_nat_def [abs_def]) transfer_prover
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lemma [transfer_rule]:
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  "rel_fun HOL.eq pcr_integer (\<lambda>k :: int. k :: int) (of_int :: int \<Rightarrow> integer)"
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proof -
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  have "rel_fun HOL.eq pcr_integer (of_int :: int \<Rightarrow> int) (of_int :: int \<Rightarrow> integer)"
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    by (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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  "rel_fun HOL.eq pcr_integer (numeral :: num \<Rightarrow> int) (numeral :: num \<Rightarrow> integer)"
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proof -
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  have "rel_fun 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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  "rel_fun HOL.eq (rel_fun HOL.eq pcr_integer) (Num.sub :: _ \<Rightarrow> _ \<Rightarrow> int) (Num.sub :: _ \<Rightarrow> _ \<Rightarrow> integer)"
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  by (unfold Num.sub_def [abs_def]) transfer_prover
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lemma int_of_integer_of_nat [simp]:
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  "int_of_integer (of_nat n) = of_nat n"
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  by transfer rule
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lift_definition integer_of_nat :: "nat \<Rightarrow> integer"
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  is "of_nat :: nat \<Rightarrow> int"
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  .
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lemma integer_of_nat_eq_of_nat [code]:
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  "integer_of_nat = of_nat"
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  by transfer rule
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lemma int_of_integer_integer_of_nat [simp]:
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  "int_of_integer (integer_of_nat n) = of_nat n"
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  by transfer rule
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lift_definition nat_of_integer :: "integer \<Rightarrow> nat"
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  is Int.nat
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  .
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lemma nat_of_integer_of_nat [simp]:
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  "nat_of_integer (of_nat n) = n"
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  by transfer simp
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lemma int_of_integer_of_int [simp]:
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  "int_of_integer (of_int k) = k"
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  by transfer simp
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lemma nat_of_integer_integer_of_nat [simp]:
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  "nat_of_integer (integer_of_nat n) = n"
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  by transfer simp
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lemma integer_of_int_eq_of_int [simp, code_abbrev]:
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  "integer_of_int = of_int"
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  by transfer (simp add: fun_eq_iff)
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lemma of_int_integer_of [simp]:
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  "of_int (int_of_integer k) = (k :: integer)"
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  by transfer rule
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lemma int_of_integer_numeral [simp]:
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  "int_of_integer (numeral k) = numeral k"
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  by transfer rule
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lemma int_of_integer_sub [simp]:
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  "int_of_integer (Num.sub k l) = Num.sub k l"
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  by transfer rule
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lift_definition integer_of_num :: "num \<Rightarrow> integer"
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  is "numeral :: num \<Rightarrow> int"
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  .
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lemma integer_of_num [code]:
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  "integer_of_num num.One = 1"
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  "integer_of_num (num.Bit0 n) = (let k = integer_of_num n in k + k)"
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  "integer_of_num (num.Bit1 n) = (let k = integer_of_num n in k + k + 1)"
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  by (transfer, simp only: numeral.simps Let_def)+
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lemma numeral_unfold_integer_of_num:
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  "numeral = integer_of_num"
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  by (simp add: integer_of_num_def map_fun_def fun_eq_iff)
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lemma integer_of_num_triv:
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  "integer_of_num Num.One = 1"
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  "integer_of_num (Num.Bit0 Num.One) = 2"
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  by (transfer, simp)+
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instantiation integer :: "{ring_div, equal, linordered_idom}"
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begin
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lift_definition divide_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
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  is "divide :: int \<Rightarrow> int \<Rightarrow> int"
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  .
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declare divide_integer.rep_eq [simp]
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lift_definition 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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  "rel_fun pcr_integer (rel_fun pcr_integer pcr_integer) (min :: _ \<Rightarrow> _ \<Rightarrow> int) (min :: _ \<Rightarrow> _ \<Rightarrow> integer)"
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  by (unfold min_def [abs_def]) transfer_prover
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lemma [transfer_rule]:
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  "rel_fun pcr_integer (rel_fun pcr_integer pcr_integer) (max :: _ \<Rightarrow> _ \<Rightarrow> int) (max :: _ \<Rightarrow> _ \<Rightarrow> integer)"
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  by (unfold max_def [abs_def]) transfer_prover
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lemma int_of_integer_min [simp]:
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  "int_of_integer (min k l) = min (int_of_integer k) (int_of_integer l)"
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  by transfer rule
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lemma int_of_integer_max [simp]:
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  "int_of_integer (max k l) = max (int_of_integer k) (int_of_integer l)"
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  by transfer rule
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lemma nat_of_integer_non_positive [simp]:
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  "k \<le> 0 \<Longrightarrow> nat_of_integer k = 0"
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  by transfer simp
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lemma of_nat_of_integer [simp]:
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  "of_nat (nat_of_integer k) = max 0 k"
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  by transfer auto
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instantiation integer :: semiring_numeral_div
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begin
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definition divmod_integer :: "num \<Rightarrow> num \<Rightarrow> integer \<times> integer"
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where
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  divmod_integer'_def: "divmod_integer m n = (numeral m div numeral n, numeral m mod numeral n)"
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definition divmod_step_integer :: "num \<Rightarrow> integer \<times> integer \<Rightarrow> integer \<times> integer"
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where
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  "divmod_step_integer l qr = (let (q, r) = qr
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    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
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    else (2 * q, r))"
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instance proof
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  show "divmod m n = (numeral m div numeral n :: integer, numeral m mod numeral n)"
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    for m n by (fact divmod_integer'_def)
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  show "divmod_step l qr = (let (q, r) = qr
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    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
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    else (2 * q, r))" for l and qr :: "integer \<times> integer"
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    by (fact divmod_step_integer_def)
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qed (transfer,
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  fact le_add_diff_inverse2
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  semiring_numeral_div_class.div_less
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  semiring_numeral_div_class.mod_less
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  semiring_numeral_div_class.div_positive
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  semiring_numeral_div_class.mod_less_eq_dividend
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  semiring_numeral_div_class.pos_mod_bound
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  semiring_numeral_div_class.pos_mod_sign
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  semiring_numeral_div_class.mod_mult2_eq
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  semiring_numeral_div_class.div_mult2_eq
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  semiring_numeral_div_class.discrete)+
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end
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declare divmod_algorithm_code [where ?'a = integer,
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  unfolded numeral_unfold_integer_of_num, unfolded integer_of_num_triv, 
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  code]
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lemma integer_of_nat_0: "integer_of_nat 0 = 0"
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by transfer simp
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lemma integer_of_nat_1: "integer_of_nat 1 = 1"
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by transfer simp
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ff54d22fe357 make integer_of_char and char_of_integer work with NBE and code_simp
Andreas Lochbihler
parents: 54489
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   281
lemma integer_of_nat_numeral:
ff54d22fe357 make integer_of_char and char_of_integer work with NBE and code_simp
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parents: 54489
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   282
  "integer_of_nat (numeral n) = numeral n"
ff54d22fe357 make integer_of_char and char_of_integer work with NBE and code_simp
Andreas Lochbihler
parents: 54489
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   283
by transfer simp
26140
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haftmann
parents: 26086
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   284
60758
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parents: 60562
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   285
subsection \<open>Code theorems for target language integers\<close>
51143
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   286
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wenzelm
parents: 60562
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   287
text \<open>Constructors\<close>
26140
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   288
51143
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   289
definition Pos :: "num \<Rightarrow> integer"
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   290
where
61274
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   291
  [simp, code_post]: "Pos = numeral"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
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   292
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   293
lemma [transfer_rule]:
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55736
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   294
  "rel_fun HOL.eq pcr_integer numeral Pos"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
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   295
  by simp transfer_prover
30245
e67f42ac1157 consequent rewrite of index_size, size [index] to nat_of; support pseudo-primrec sepcifications with fun
haftmann
parents: 29823
diff changeset
   296
61274
0261eec37233 more selective preprocessing allows bare "numeral" occurences to be retained as real function in generated code
haftmann
parents: 61076
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   297
lemma Pos_fold [code_unfold]:
0261eec37233 more selective preprocessing allows bare "numeral" occurences to be retained as real function in generated code
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parents: 61076
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   298
  "numeral Num.One = Pos Num.One"
0261eec37233 more selective preprocessing allows bare "numeral" occurences to be retained as real function in generated code
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   299
  "numeral (Num.Bit0 k) = Pos (Num.Bit0 k)"
0261eec37233 more selective preprocessing allows bare "numeral" occurences to be retained as real function in generated code
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parents: 61076
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   300
  "numeral (Num.Bit1 k) = Pos (Num.Bit1 k)"
0261eec37233 more selective preprocessing allows bare "numeral" occurences to be retained as real function in generated code
haftmann
parents: 61076
diff changeset
   301
  by simp_all
0261eec37233 more selective preprocessing allows bare "numeral" occurences to be retained as real function in generated code
haftmann
parents: 61076
diff changeset
   302
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
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   303
definition Neg :: "num \<Rightarrow> integer"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
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   304
where
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 53069
diff changeset
   305
  [simp, code_abbrev]: "Neg n = - Pos n"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   306
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   307
lemma [transfer_rule]:
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55736
diff changeset
   308
  "rel_fun HOL.eq pcr_integer (\<lambda>n. - numeral n) Neg"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 53069
diff changeset
   309
  by (simp add: Neg_def [abs_def]) transfer_prover
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   310
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
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   311
code_datatype "0::integer" Pos Neg
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
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   312
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
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   313
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   314
text \<open>Auxiliary operations\<close>
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   315
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   316
lift_definition dup :: "integer \<Rightarrow> integer"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
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parents: 49962
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   317
  is "\<lambda>k::int. k + k"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   318
  .
26140
e45375135052 Zero/Suc recursion combinator for type index
haftmann
parents: 26086
diff changeset
   319
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   320
lemma dup_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   321
  "dup 0 = 0"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   322
  "dup (Pos n) = Pos (Num.Bit0 n)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   323
  "dup (Neg n) = Neg (Num.Bit0 n)"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 53069
diff changeset
   324
  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
   325
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   326
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
   327
  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
   328
  .
26140
e45375135052 Zero/Suc recursion combinator for type index
haftmann
parents: 26086
diff changeset
   329
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   330
lemma sub_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   331
  "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
   332
  "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
   333
  "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
   334
  "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
   335
  "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
   336
  "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
   337
  "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
   338
  "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
   339
  "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
   340
  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
   341
24999
haftmann
parents:
diff changeset
   342
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   343
text \<open>Implementations\<close>
24999
haftmann
parents:
diff changeset
   344
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   345
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
   346
  "1 = Pos Num.One"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   347
  by simp
24999
haftmann
parents:
diff changeset
   348
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   349
lemma plus_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   350
  "k + 0 = (k::integer)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   351
  "0 + l = (l::integer)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   352
  "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
   353
  "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
   354
  "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
   355
  "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
   356
  by (transfer, simp)+
24999
haftmann
parents:
diff changeset
   357
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   358
lemma uminus_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   359
  "uminus 0 = (0::integer)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   360
  "uminus (Pos m) = Neg m"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   361
  "uminus (Neg m) = Pos m"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   362
  by simp_all
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   363
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   364
lemma minus_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   365
  "k - 0 = (k::integer)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   366
  "0 - l = uminus (l::integer)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   367
  "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
   368
  "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
   369
  "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
   370
  "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
   371
  by (transfer, simp)+
46028
9f113cdf3d66 attribute code_abbrev superseedes code_unfold_post
haftmann
parents: 45694
diff changeset
   372
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   373
lemma abs_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   374
  "\<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
   375
  by simp
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46664
diff changeset
   376
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   377
lemma sgn_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   378
  "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
   379
  by simp
46028
9f113cdf3d66 attribute code_abbrev superseedes code_unfold_post
haftmann
parents: 45694
diff changeset
   380
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   381
lemma times_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   382
  "k * 0 = (0::integer)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   383
  "0 * l = (0::integer)"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   384
  "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
   385
  "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
   386
  "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
   387
  "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
   388
  by simp_all
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   389
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   390
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
   391
where
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   392
  "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
   393
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   394
lemma fst_divmod [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   395
  "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
   396
  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
   397
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   398
lemma snd_divmod [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   399
  "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
   400
  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
   401
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   402
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
   403
where
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   404
  "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
   405
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   406
lemma fst_divmod_abs [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   407
  "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
   408
  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
   409
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   410
lemma snd_divmod_abs [simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   411
  "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
   412
  by (simp add: divmod_abs_def)
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   413
53069
d165213e3924 execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents: 52435
diff changeset
   414
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
   415
  "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
   416
  "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
   417
  "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
   418
  "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
   419
  "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
   420
  "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
   421
  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
   422
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   423
lemma divmod_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   424
  "divmod_integer k l =
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   425
    (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
   426
    (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
   427
      then divmod_abs k l
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   428
      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
   429
        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
   430
proof -
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   431
  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
   432
    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
   433
  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
   434
  show ?thesis
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 54489
diff changeset
   435
    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
   436
      (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
   437
qed
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   438
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   439
lemma div_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   440
  "k div l = fst (divmod_integer k l)"
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   441
  by simp
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   442
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   443
lemma mod_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   444
  "k mod l = snd (divmod_integer k l)"
25767
852bce03412a index now a copy of nat rather than int
haftmann
parents: 25691
diff changeset
   445
  by simp
24999
haftmann
parents:
diff changeset
   446
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   447
lemma equal_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   448
  "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
   449
  "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
   450
  "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
   451
  "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
   452
  "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
   453
  "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
   454
  "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
   455
  "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
   456
  "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
   457
  by (simp_all add: equal)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   458
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   459
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
   460
  "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
   461
  by (fact equal_refl)
31266
55e70b6d812e added lemma about 0 - 1
haftmann
parents: 31205
diff changeset
   462
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   463
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
   464
  "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
   465
  "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
   466
  "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
   467
  "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
   468
  "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
   469
  "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
   470
  "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
   471
  "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
   472
  "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
   473
  by simp_all
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   474
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   475
lemma less_integer_code [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   476
  "0 < (0::integer) \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   477
  "0 < Pos l \<longleftrightarrow> True"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   478
  "0 < Neg l \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   479
  "Pos k < 0 \<longleftrightarrow> False"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   480
  "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
   481
  "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
   482
  "Neg k < 0 \<longleftrightarrow> True"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   483
  "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
   484
  "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
   485
  by simp_all
26140
e45375135052 Zero/Suc recursion combinator for type index
haftmann
parents: 26086
diff changeset
   486
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   487
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
   488
  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
   489
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   490
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   491
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
   492
  "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
   493
     else let
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   494
       (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
   495
       l' = num_of_integer l;
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   496
       l'' = l' + l'
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   497
     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
   498
proof -
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   499
  {
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   500
    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
   501
    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
   502
    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
   503
    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
   504
    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
   505
      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
   506
      by simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   507
    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
   508
      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
   509
      by (simp add: mult_2)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   510
    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
   511
      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
   512
      by simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   513
  }
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   514
  note aux = this
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   515
  show ?thesis
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 54489
diff changeset
   516
    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
   517
      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
   518
      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
   519
       mult_2 [where 'a=nat] aux add_One)
25918
haftmann
parents: 25767
diff changeset
   520
qed
haftmann
parents: 25767
diff changeset
   521
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   522
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
   523
  "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
   524
     else let
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   525
       (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
   526
       l' = nat_of_integer l;
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   527
       l'' = l' + l'
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   528
     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
   529
proof -
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   530
  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
   531
  proof
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   532
    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
   533
  qed
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   534
  moreover have "2 * (j div 2) = j - j mod 2"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56846
diff changeset
   535
    by (simp add: zmult_div_cancel mult.commute)
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   536
  ultimately show ?thesis
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   537
    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
   538
      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
   539
      (auto simp add: mult_2 [symmetric])
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   540
qed
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   541
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   542
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
   543
  "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
   544
     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
   545
     else let
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   546
       (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
   547
       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
   548
     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
   549
  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
   550
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   551
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
   552
  "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
   553
     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
   554
     else let
60868
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   555
       l = 2 * integer_of_int (k div 2);
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   556
       j = k mod 2
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   557
     in if j = 0 then l else l + 1)"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   558
  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
   559
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   560
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
   561
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   562
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   563
subsection \<open>Serializer setup for target language integers\<close>
24999
haftmann
parents:
diff changeset
   564
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   565
code_reserved Eval int Integer abs
25767
852bce03412a index now a copy of nat rather than int
haftmann
parents: 25691
diff changeset
   566
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   567
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
   568
  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
   569
    (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
   570
    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
   571
    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
   572
    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
   573
    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
   574
| 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
   575
    (Haskell) -
24999
haftmann
parents:
diff changeset
   576
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   577
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
   578
  constant "0::integer" \<rightharpoonup>
58400
d0d3c30806b4 always annotate potentially polymorphic Haskell numerals
haftmann
parents: 58399
diff changeset
   579
    (SML) "!(0/ :/ IntInf.int)"
52435
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 (OCaml) "Big'_int.zero'_big'_int"
58400
d0d3c30806b4 always annotate potentially polymorphic Haskell numerals
haftmann
parents: 58399
diff changeset
   581
    and (Haskell) "!(0/ ::/ Integer)"
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   582
    and (Scala) "BigInt(0)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46664
diff changeset
   583
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   584
setup \<open>
58399
haftmann
parents: 58390
diff changeset
   585
  fold (fn target =>
haftmann
parents: 58390
diff changeset
   586
    Numeral.add_code @{const_name Code_Numeral.Pos} I Code_Printer.literal_numeral target
haftmann
parents: 58390
diff changeset
   587
    #> Numeral.add_code @{const_name Code_Numeral.Neg} (op ~) Code_Printer.literal_numeral target)
haftmann
parents: 58390
diff changeset
   588
    ["SML", "OCaml", "Haskell", "Scala"]
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   589
\<close>
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   590
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   591
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
   592
  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
   593
    (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
   594
    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
   595
    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
   596
    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
   597
    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
   598
| 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
   599
    (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
   600
    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
   601
    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
   602
    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
   603
    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
   604
| 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
   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.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
   607
    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
   608
    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
   609
    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
   610
| 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
   611
    (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
   612
    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
   613
    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
   614
    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
   615
    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
   616
| 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
   617
    (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
   618
    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
   619
    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
   620
    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
   621
| 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
   622
    (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
   623
    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
   624
    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
   625
    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
   626
    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
   627
| 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
   628
    (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
   629
    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
   630
    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
   631
    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
   632
    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
   633
| 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
   634
    (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
   635
    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
   636
    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
   637
    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
   638
    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
   639
| 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
   640
    (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
   641
    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
   642
    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
   643
    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
   644
    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
   645
| 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
   646
    (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
   647
    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
   648
    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
   649
    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
   650
    and (Eval) infixl 6 "<"
61857
542f2c6da692 add serialisation for abs on integer to target language operation
Andreas Lochbihler
parents: 61275
diff changeset
   651
| constant "abs :: integer \<Rightarrow> _" \<rightharpoonup>
542f2c6da692 add serialisation for abs on integer to target language operation
Andreas Lochbihler
parents: 61275
diff changeset
   652
    (SML) "IntInf.abs"
542f2c6da692 add serialisation for abs on integer to target language operation
Andreas Lochbihler
parents: 61275
diff changeset
   653
    and (OCaml) "Big'_int.abs'_big'_int"
542f2c6da692 add serialisation for abs on integer to target language operation
Andreas Lochbihler
parents: 61275
diff changeset
   654
    and (Haskell) "Prelude.abs"
542f2c6da692 add serialisation for abs on integer to target language operation
Andreas Lochbihler
parents: 61275
diff changeset
   655
    and (Scala) "_.abs"
542f2c6da692 add serialisation for abs on integer to target language operation
Andreas Lochbihler
parents: 61275
diff changeset
   656
    and (Eval) "abs"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   657
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51375
diff changeset
   658
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
   659
  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
   660
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   661
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   662
subsection \<open>Type of target language naturals\<close>
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   663
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60868
diff changeset
   664
typedef natural = "UNIV :: nat set"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   665
  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
   666
59487
adaa430fc0f7 default abstypes and default abstract equations make technical (no_code) annotation superfluous
haftmann
parents: 58889
diff changeset
   667
setup_lifting type_definition_natural
51143
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
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
   670
  "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
   671
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   672
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   673
lemma natural_eqI:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   674
  "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
   675
  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
   676
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   677
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
   678
  "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
   679
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   680
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   681
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
   682
  "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
   683
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   684
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   685
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
   686
begin
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   687
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   688
lift_definition zero_natural :: natural
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   689
  is "0 :: nat"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   690
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   691
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   692
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
   693
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   694
lift_definition one_natural :: natural
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   695
  is "1 :: nat"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   696
  .
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
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
   699
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   700
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
   701
  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
   702
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   703
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   704
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
   705
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   706
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
   707
  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
   708
  .
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
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
   711
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   712
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
   713
  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
   714
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   715
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   716
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
   717
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   718
instance proof
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   719
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
   720
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   721
end
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
lemma [transfer_rule]:
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55736
diff changeset
   724
  "rel_fun 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
   725
proof -
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55736
diff changeset
   726
  have "rel_fun 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
   727
    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
   728
  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
   729
qed
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   730
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   731
lemma [transfer_rule]:
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55736
diff changeset
   732
  "rel_fun 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
   733
proof -
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55736
diff changeset
   734
  have "rel_fun 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
   735
    by transfer_prover
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   736
  then show ?thesis by simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   737
qed
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
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
   740
  "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
   741
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   742
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   743
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
   744
  "natural_of_nat = of_nat"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   745
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   746
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   747
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
   748
  "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
   749
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   750
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   751
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
   752
  "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
   753
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   754
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   755
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
   756
begin
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   757
60352
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59816
diff changeset
   758
lift_definition divide_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59816
diff changeset
   759
  is "divide :: nat \<Rightarrow> nat \<Rightarrow> nat"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   760
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   761
60352
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59816
diff changeset
   762
declare divide_natural.rep_eq [simp]
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   763
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   764
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
   765
  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
   766
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   767
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   768
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
   769
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   770
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
   771
  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
   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
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
   775
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   776
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
   777
  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
   778
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   779
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   780
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
   781
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   782
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
   783
  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
   784
  .
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
instance proof
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   787
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
   788
24999
haftmann
parents:
diff changeset
   789
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
   790
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   791
lemma [transfer_rule]:
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55736
diff changeset
   792
  "rel_fun pcr_natural (rel_fun 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
   793
  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
   794
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   795
lemma [transfer_rule]:
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55736
diff changeset
   796
  "rel_fun pcr_natural (rel_fun 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
   797
  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
   798
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   799
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
   800
  "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
   801
  by transfer rule
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
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
   804
  "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
   805
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   806
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   807
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
   808
  is "nat :: int \<Rightarrow> nat"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   809
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   810
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   811
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
   812
  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
   813
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   814
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   815
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
   816
  "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
   817
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   818
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   819
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
   820
  "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
   821
  by transfer auto
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   822
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   823
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
   824
  "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
   825
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   826
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   827
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
   828
  "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
   829
  by transfer rule
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   830
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   831
lemma [measure_function]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   832
  "is_measure nat_of_natural"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   833
  by (rule is_measure_trivial)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   834
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   835
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   836
subsection \<open>Inductive representation of target language naturals\<close>
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   837
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   838
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
   839
  is Nat.Suc
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   840
  .
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
declare Suc.rep_eq [simp]
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   843
58306
117ba6cbe414 renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
blanchet
parents: 57512
diff changeset
   844
old_rep_datatype "0::natural" Suc
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   845
  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
   846
55416
dd7992d4a61a adapted theories to 'xxx_case' to 'case_xxx'
blanchet
parents: 55414
diff changeset
   847
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
   848
  fixes m :: natural
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   849
  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
   850
  shows P
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   851
  using assms by transfer blast
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   852
58390
b74d8470b98e keep obsolete interpretations in Main, to avoid merge trouble
blanchet
parents: 58379
diff changeset
   853
lemma [simp, code]: "size_natural = nat_of_natural"
b74d8470b98e keep obsolete interpretations in Main, to avoid merge trouble
blanchet
parents: 58379
diff changeset
   854
proof (rule ext)
b74d8470b98e keep obsolete interpretations in Main, to avoid merge trouble
blanchet
parents: 58379
diff changeset
   855
  fix n
b74d8470b98e keep obsolete interpretations in Main, to avoid merge trouble
blanchet
parents: 58379
diff changeset
   856
  show "size_natural n = nat_of_natural n"
b74d8470b98e keep obsolete interpretations in Main, to avoid merge trouble
blanchet
parents: 58379
diff changeset
   857
    by (induct n) simp_all
b74d8470b98e keep obsolete interpretations in Main, to avoid merge trouble
blanchet
parents: 58379
diff changeset
   858
qed
58379
c044539a2bda reintroduced an instantiation of 'size' for 'numerals'
blanchet
parents: 58377
diff changeset
   859
58390
b74d8470b98e keep obsolete interpretations in Main, to avoid merge trouble
blanchet
parents: 58379
diff changeset
   860
lemma [simp, code]: "size = nat_of_natural"
b74d8470b98e keep obsolete interpretations in Main, to avoid merge trouble
blanchet
parents: 58379
diff changeset
   861
proof (rule ext)
b74d8470b98e keep obsolete interpretations in Main, to avoid merge trouble
blanchet
parents: 58379
diff changeset
   862
  fix n
b74d8470b98e keep obsolete interpretations in Main, to avoid merge trouble
blanchet
parents: 58379
diff changeset
   863
  show "size n = nat_of_natural n"
b74d8470b98e keep obsolete interpretations in Main, to avoid merge trouble
blanchet
parents: 58379
diff changeset
   864
    by (induct n) simp_all
b74d8470b98e keep obsolete interpretations in Main, to avoid merge trouble
blanchet
parents: 58379
diff changeset
   865
qed
58379
c044539a2bda reintroduced an instantiation of 'size' for 'numerals'
blanchet
parents: 58377
diff changeset
   866
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   867
lemma natural_decr [termination_simp]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   868
  "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
   869
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   870
58379
c044539a2bda reintroduced an instantiation of 'size' for 'numerals'
blanchet
parents: 58377
diff changeset
   871
lemma natural_zero_minus_one: "(0::natural) - 1 = 0"
c044539a2bda reintroduced an instantiation of 'size' for 'numerals'
blanchet
parents: 58377
diff changeset
   872
  by (rule zero_diff)
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   873
58379
c044539a2bda reintroduced an instantiation of 'size' for 'numerals'
blanchet
parents: 58377
diff changeset
   874
lemma Suc_natural_minus_one: "Suc n - 1 = n"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   875
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   876
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   877
hide_const (open) Suc
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
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60562
diff changeset
   880
subsection \<open>Code refinement for target language naturals\<close>
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   881
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   882
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
   883
  is nat
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   884
  .
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   885
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   886
lemma [code_post]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   887
  "Nat 0 = 0"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   888
  "Nat 1 = 1"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   889
  "Nat (numeral k) = numeral k"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   890
  by (transfer, simp)+
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   891
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   892
lemma [code abstype]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   893
  "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
   894
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   895
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   896
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   897
  "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
   898
  by simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   899
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   900
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   901
  "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
   902
  by simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   903
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   904
lemma [code_abbrev]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   905
  "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
   906
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   907
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   908
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   909
  "integer_of_natural 0 = 0"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   910
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   911
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   912
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   913
  "integer_of_natural 1 = 1"
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   914
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   915
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   916
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   917
  "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
   918
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   919
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   920
lemma [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   921
  "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
   922
  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
   923
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   924
lemma [code, code_unfold]:
55416
dd7992d4a61a adapted theories to 'xxx_case' to 'case_xxx'
blanchet
parents: 55414
diff changeset
   925
  "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
   926
  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
   927
55642
63beb38e9258 adapted to renaming of datatype 'cases' and 'recs' to 'case' and 'rec'
blanchet
parents: 55428
diff changeset
   928
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
   929
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   930
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   931
  "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
   932
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   933
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   934
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   935
  "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
   936
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   937
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   938
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   939
  "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
   940
  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
   941
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   942
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   943
  "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
   944
  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
   945
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   946
lemma [code abstract]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   947
  "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
   948
  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
   949
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   950
lemma [code]:
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   951
  "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
   952
  by transfer (simp add: equal)
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   953
58379
c044539a2bda reintroduced an instantiation of 'size' for 'numerals'
blanchet
parents: 58377
diff changeset
   954
lemma [code nbe]: "HOL.equal n (n::natural) \<longleftrightarrow> True"
c044539a2bda reintroduced an instantiation of 'size' for 'numerals'
blanchet
parents: 58377
diff changeset
   955
  by (rule equal_class.equal_refl)
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   956
58379
c044539a2bda reintroduced an instantiation of 'size' for 'numerals'
blanchet
parents: 58377
diff changeset
   957
lemma [code]: "m \<le> n \<longleftrightarrow> integer_of_natural m \<le> integer_of_natural n"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   958
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   959
58379
c044539a2bda reintroduced an instantiation of 'size' for 'numerals'
blanchet
parents: 58377
diff changeset
   960
lemma [code]: "m < n \<longleftrightarrow> integer_of_natural m < integer_of_natural n"
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   961
  by transfer simp
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   962
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   963
hide_const (open) Nat
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   964
55736
f1ed1e9cd080 unregister lifting setup following the best practice of Lifting
kuncar
parents: 55642
diff changeset
   965
lifting_update integer.lifting
f1ed1e9cd080 unregister lifting setup following the best practice of Lifting
kuncar
parents: 55642
diff changeset
   966
lifting_forget integer.lifting
f1ed1e9cd080 unregister lifting setup following the best practice of Lifting
kuncar
parents: 55642
diff changeset
   967
f1ed1e9cd080 unregister lifting setup following the best practice of Lifting
kuncar
parents: 55642
diff changeset
   968
lifting_update natural.lifting
f1ed1e9cd080 unregister lifting setup following the best practice of Lifting
kuncar
parents: 55642
diff changeset
   969
lifting_forget natural.lifting
51143
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   970
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   971
code_reflect Code_Numeral
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   972
  datatypes natural = _
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   973
  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
   974
0a2371e7ced3 two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents: 49962
diff changeset
   975
end