src/HOL/Num.thy
author haftmann
Mon Jun 05 15:59:41 2017 +0200 (2017-06-05)
changeset 66010 2f7d39285a1a
parent 64238 b60a9752b6d0
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(*  Title:      HOL/Num.thy
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    Author:     Florian Haftmann
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    Author:     Brian Huffman
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*)
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section \<open>Binary Numerals\<close>
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theory Num
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  imports BNF_Least_Fixpoint Transfer
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begin
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subsection \<open>The \<open>num\<close> type\<close>
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datatype num = One | Bit0 num | Bit1 num
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text \<open>Increment function for type @{typ num}\<close>
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primrec inc :: "num \<Rightarrow> num"
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  where
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    "inc One = Bit0 One"
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  | "inc (Bit0 x) = Bit1 x"
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  | "inc (Bit1 x) = Bit0 (inc x)"
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text \<open>Converting between type @{typ num} and type @{typ nat}\<close>
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primrec nat_of_num :: "num \<Rightarrow> nat"
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  where
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    "nat_of_num One = Suc 0"
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  | "nat_of_num (Bit0 x) = nat_of_num x + nat_of_num x"
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  | "nat_of_num (Bit1 x) = Suc (nat_of_num x + nat_of_num x)"
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primrec num_of_nat :: "nat \<Rightarrow> num"
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  where
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    "num_of_nat 0 = One"
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  | "num_of_nat (Suc n) = (if 0 < n then inc (num_of_nat n) else One)"
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lemma nat_of_num_pos: "0 < nat_of_num x"
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  by (induct x) simp_all
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lemma nat_of_num_neq_0: " nat_of_num x \<noteq> 0"
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  by (induct x) simp_all
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lemma nat_of_num_inc: "nat_of_num (inc x) = Suc (nat_of_num x)"
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  by (induct x) simp_all
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lemma num_of_nat_double: "0 < n \<Longrightarrow> num_of_nat (n + n) = Bit0 (num_of_nat n)"
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  by (induct n) simp_all
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text \<open>Type @{typ num} is isomorphic to the strictly positive natural numbers.\<close>
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lemma nat_of_num_inverse: "num_of_nat (nat_of_num x) = x"
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  by (induct x) (simp_all add: num_of_nat_double nat_of_num_pos)
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lemma num_of_nat_inverse: "0 < n \<Longrightarrow> nat_of_num (num_of_nat n) = n"
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  by (induct n) (simp_all add: nat_of_num_inc)
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lemma num_eq_iff: "x = y \<longleftrightarrow> nat_of_num x = nat_of_num y"
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  apply safe
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  apply (drule arg_cong [where f=num_of_nat])
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  apply (simp add: nat_of_num_inverse)
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  done
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lemma num_induct [case_names One inc]:
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  fixes P :: "num \<Rightarrow> bool"
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  assumes One: "P One"
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    and inc: "\<And>x. P x \<Longrightarrow> P (inc x)"
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  shows "P x"
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proof -
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  obtain n where n: "Suc n = nat_of_num x"
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    by (cases "nat_of_num x") (simp_all add: nat_of_num_neq_0)
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  have "P (num_of_nat (Suc n))"
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  proof (induct n)
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    case 0
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    from One show ?case by simp
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  next
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    case (Suc n)
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    then have "P (inc (num_of_nat (Suc n)))" by (rule inc)
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    then show "P (num_of_nat (Suc (Suc n)))" by simp
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  qed
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  with n show "P x"
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    by (simp add: nat_of_num_inverse)
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qed
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text \<open>
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  From now on, there are two possible models for @{typ num}: as positive
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  naturals (rule \<open>num_induct\<close>) and as digit representation (rules
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  \<open>num.induct\<close>, \<open>num.cases\<close>).
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\<close>
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subsection \<open>Numeral operations\<close>
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instantiation num :: "{plus,times,linorder}"
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begin
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definition [code del]: "m + n = num_of_nat (nat_of_num m + nat_of_num n)"
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definition [code del]: "m * n = num_of_nat (nat_of_num m * nat_of_num n)"
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definition [code del]: "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"
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definition [code del]: "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"
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instance
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  by standard (auto simp add: less_num_def less_eq_num_def num_eq_iff)
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end
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lemma nat_of_num_add: "nat_of_num (x + y) = nat_of_num x + nat_of_num y"
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  unfolding plus_num_def
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  by (intro num_of_nat_inverse add_pos_pos nat_of_num_pos)
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lemma nat_of_num_mult: "nat_of_num (x * y) = nat_of_num x * nat_of_num y"
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  unfolding times_num_def
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  by (intro num_of_nat_inverse mult_pos_pos nat_of_num_pos)
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lemma add_num_simps [simp, code]:
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  "One + One = Bit0 One"
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  "One + Bit0 n = Bit1 n"
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  "One + Bit1 n = Bit0 (n + One)"
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  "Bit0 m + One = Bit1 m"
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  "Bit0 m + Bit0 n = Bit0 (m + n)"
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  "Bit0 m + Bit1 n = Bit1 (m + n)"
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  "Bit1 m + One = Bit0 (m + One)"
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  "Bit1 m + Bit0 n = Bit1 (m + n)"
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  "Bit1 m + Bit1 n = Bit0 (m + n + One)"
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  by (simp_all add: num_eq_iff nat_of_num_add)
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lemma mult_num_simps [simp, code]:
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  "m * One = m"
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  "One * n = n"
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  "Bit0 m * Bit0 n = Bit0 (Bit0 (m * n))"
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  "Bit0 m * Bit1 n = Bit0 (m * Bit1 n)"
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  "Bit1 m * Bit0 n = Bit0 (Bit1 m * n)"
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  "Bit1 m * Bit1 n = Bit1 (m + n + Bit0 (m * n))"
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  by (simp_all add: num_eq_iff nat_of_num_add nat_of_num_mult distrib_right distrib_left)
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lemma eq_num_simps:
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  "One = One \<longleftrightarrow> True"
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  "One = Bit0 n \<longleftrightarrow> False"
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  "One = Bit1 n \<longleftrightarrow> False"
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  "Bit0 m = One \<longleftrightarrow> False"
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  "Bit1 m = One \<longleftrightarrow> False"
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  "Bit0 m = Bit0 n \<longleftrightarrow> m = n"
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  "Bit0 m = Bit1 n \<longleftrightarrow> False"
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  "Bit1 m = Bit0 n \<longleftrightarrow> False"
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  "Bit1 m = Bit1 n \<longleftrightarrow> m = n"
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  by simp_all
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lemma le_num_simps [simp, code]:
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  "One \<le> n \<longleftrightarrow> True"
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  "Bit0 m \<le> One \<longleftrightarrow> False"
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  "Bit1 m \<le> One \<longleftrightarrow> False"
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  "Bit0 m \<le> Bit0 n \<longleftrightarrow> m \<le> n"
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  "Bit0 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"
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  "Bit1 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"
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  "Bit1 m \<le> Bit0 n \<longleftrightarrow> m < n"
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  using nat_of_num_pos [of n] nat_of_num_pos [of m]
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  by (auto simp add: less_eq_num_def less_num_def)
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lemma less_num_simps [simp, code]:
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  "m < One \<longleftrightarrow> False"
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  "One < Bit0 n \<longleftrightarrow> True"
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  "One < Bit1 n \<longleftrightarrow> True"
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  "Bit0 m < Bit0 n \<longleftrightarrow> m < n"
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  "Bit0 m < Bit1 n \<longleftrightarrow> m \<le> n"
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  "Bit1 m < Bit1 n \<longleftrightarrow> m < n"
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  "Bit1 m < Bit0 n \<longleftrightarrow> m < n"
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  using nat_of_num_pos [of n] nat_of_num_pos [of m]
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  by (auto simp add: less_eq_num_def less_num_def)
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lemma le_num_One_iff: "x \<le> num.One \<longleftrightarrow> x = num.One"
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  by (simp add: antisym_conv)
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text \<open>Rules using \<open>One\<close> and \<open>inc\<close> as constructors.\<close>
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lemma add_One: "x + One = inc x"
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  by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
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lemma add_One_commute: "One + n = n + One"
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  by (induct n) simp_all
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lemma add_inc: "x + inc y = inc (x + y)"
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  by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
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lemma mult_inc: "x * inc y = x * y + x"
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  by (simp add: num_eq_iff nat_of_num_mult nat_of_num_add nat_of_num_inc)
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text \<open>The @{const num_of_nat} conversion.\<close>
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lemma num_of_nat_One: "n \<le> 1 \<Longrightarrow> num_of_nat n = One"
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  by (cases n) simp_all
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lemma num_of_nat_plus_distrib:
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  "0 < m \<Longrightarrow> 0 < n \<Longrightarrow> num_of_nat (m + n) = num_of_nat m + num_of_nat n"
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  by (induct n) (auto simp add: add_One add_One_commute add_inc)
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text \<open>A double-and-decrement function.\<close>
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primrec BitM :: "num \<Rightarrow> num"
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  where
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    "BitM One = One"
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  | "BitM (Bit0 n) = Bit1 (BitM n)"
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  | "BitM (Bit1 n) = Bit1 (Bit0 n)"
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lemma BitM_plus_one: "BitM n + One = Bit0 n"
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  by (induct n) simp_all
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lemma one_plus_BitM: "One + BitM n = Bit0 n"
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  unfolding add_One_commute BitM_plus_one ..
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text \<open>Squaring and exponentiation.\<close>
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primrec sqr :: "num \<Rightarrow> num"
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  where
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    "sqr One = One"
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  | "sqr (Bit0 n) = Bit0 (Bit0 (sqr n))"
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  | "sqr (Bit1 n) = Bit1 (Bit0 (sqr n + n))"
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primrec pow :: "num \<Rightarrow> num \<Rightarrow> num"
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  where
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    "pow x One = x"
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  | "pow x (Bit0 y) = sqr (pow x y)"
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  | "pow x (Bit1 y) = sqr (pow x y) * x"
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lemma nat_of_num_sqr: "nat_of_num (sqr x) = nat_of_num x * nat_of_num x"
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  by (induct x) (simp_all add: algebra_simps nat_of_num_add)
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lemma sqr_conv_mult: "sqr x = x * x"
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  by (simp add: num_eq_iff nat_of_num_sqr nat_of_num_mult)
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subsection \<open>Binary numerals\<close>
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text \<open>
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  We embed binary representations into a generic algebraic
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  structure using \<open>numeral\<close>.
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\<close>
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class numeral = one + semigroup_add
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begin
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primrec numeral :: "num \<Rightarrow> 'a"
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  where
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    numeral_One: "numeral One = 1"
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  | numeral_Bit0: "numeral (Bit0 n) = numeral n + numeral n"
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  | numeral_Bit1: "numeral (Bit1 n) = numeral n + numeral n + 1"
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lemma numeral_code [code]:
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  "numeral One = 1"
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  "numeral (Bit0 n) = (let m = numeral n in m + m)"
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  "numeral (Bit1 n) = (let m = numeral n in m + m + 1)"
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  by (simp_all add: Let_def)
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lemma one_plus_numeral_commute: "1 + numeral x = numeral x + 1"
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proof (induct x)
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  case One
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  then show ?case by simp
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next
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  case Bit0
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  then show ?case by (simp add: add.assoc [symmetric]) (simp add: add.assoc)
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next
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  case Bit1
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  then show ?case by (simp add: add.assoc [symmetric]) (simp add: add.assoc)
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qed
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lemma numeral_inc: "numeral (inc x) = numeral x + 1"
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proof (induct x)
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  case One
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  then show ?case by simp
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next
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  case Bit0
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  then show ?case by simp
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next
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  case (Bit1 x)
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  have "numeral x + (1 + numeral x) + 1 = numeral x + (numeral x + 1) + 1"
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    by (simp only: one_plus_numeral_commute)
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  with Bit1 show ?case
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    by (simp add: add.assoc)
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qed
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declare numeral.simps [simp del]
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abbreviation "Numeral1 \<equiv> numeral One"
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declare numeral_One [code_post]
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end
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text \<open>Numeral syntax.\<close>
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syntax
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  "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
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ML_file "Tools/numeral.ML"
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parse_translation \<open>
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  let
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    fun numeral_tr [(c as Const (@{syntax_const "_constrain"}, _)) $ t $ u] =
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          c $ numeral_tr [t] $ u
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      | numeral_tr [Const (num, _)] =
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          (Numeral.mk_number_syntax o #value o Lexicon.read_num) num
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      | numeral_tr ts = raise TERM ("numeral_tr", ts);
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  in [(@{syntax_const "_Numeral"}, K numeral_tr)] end
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\<close>
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typed_print_translation \<open>
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  let
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    fun num_tr' ctxt T [n] =
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      let
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        val k = Numeral.dest_num_syntax n;
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        val t' =
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          Syntax.const @{syntax_const "_Numeral"} $
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            Syntax.free (string_of_int k);
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      in
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        (case T of
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          Type (@{type_name fun}, [_, T']) =>
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            if Printer.type_emphasis ctxt T' then
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              Syntax.const @{syntax_const "_constrain"} $ t' $
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                Syntax_Phases.term_of_typ ctxt T'
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            else t'
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        | _ => if T = dummyT then t' else raise Match)
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      end;
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  in
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   [(@{const_syntax numeral}, num_tr')]
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  end
wenzelm@60758
   327
\<close>
huffman@47108
   328
huffman@47228
   329
wenzelm@60758
   330
subsection \<open>Class-specific numeral rules\<close>
huffman@47108
   331
wenzelm@63654
   332
text \<open>@{const numeral} is a morphism.\<close>
wenzelm@63654
   333
huffman@47108
   334
wenzelm@61799
   335
subsubsection \<open>Structures with addition: class \<open>numeral\<close>\<close>
huffman@47108
   336
huffman@47108
   337
context numeral
huffman@47108
   338
begin
huffman@47108
   339
huffman@47108
   340
lemma numeral_add: "numeral (m + n) = numeral m + numeral n"
huffman@47108
   341
  by (induct n rule: num_induct)
wenzelm@63654
   342
    (simp_all only: numeral_One add_One add_inc numeral_inc add.assoc)
huffman@47108
   343
huffman@47108
   344
lemma numeral_plus_numeral: "numeral m + numeral n = numeral (m + n)"
huffman@47108
   345
  by (rule numeral_add [symmetric])
huffman@47108
   346
huffman@47108
   347
lemma numeral_plus_one: "numeral n + 1 = numeral (n + One)"
huffman@47108
   348
  using numeral_add [of n One] by (simp add: numeral_One)
huffman@47108
   349
huffman@47108
   350
lemma one_plus_numeral: "1 + numeral n = numeral (One + n)"
huffman@47108
   351
  using numeral_add [of One n] by (simp add: numeral_One)
huffman@47108
   352
huffman@47108
   353
lemma one_add_one: "1 + 1 = 2"
huffman@47108
   354
  using numeral_add [of One One] by (simp add: numeral_One)
huffman@47108
   355
huffman@47108
   356
lemmas add_numeral_special =
huffman@47108
   357
  numeral_plus_one one_plus_numeral one_add_one
huffman@47108
   358
huffman@47108
   359
end
huffman@47108
   360
wenzelm@63654
   361
wenzelm@63654
   362
subsubsection \<open>Structures with negation: class \<open>neg_numeral\<close>\<close>
huffman@47108
   363
haftmann@54489
   364
class neg_numeral = numeral + group_add
huffman@47108
   365
begin
huffman@47108
   366
wenzelm@63654
   367
lemma uminus_numeral_One: "- Numeral1 = - 1"
haftmann@54489
   368
  by (simp add: numeral_One)
haftmann@54489
   369
wenzelm@60758
   370
text \<open>Numerals form an abelian subgroup.\<close>
huffman@47108
   371
wenzelm@63654
   372
inductive is_num :: "'a \<Rightarrow> bool"
wenzelm@63654
   373
  where
wenzelm@63654
   374
    "is_num 1"
wenzelm@63654
   375
  | "is_num x \<Longrightarrow> is_num (- x)"
wenzelm@63654
   376
  | "is_num x \<Longrightarrow> is_num y \<Longrightarrow> is_num (x + y)"
huffman@47108
   377
huffman@47108
   378
lemma is_num_numeral: "is_num (numeral k)"
wenzelm@63654
   379
  by (induct k) (simp_all add: numeral.simps is_num.intros)
huffman@47108
   380
wenzelm@63654
   381
lemma is_num_add_commute: "is_num x \<Longrightarrow> is_num y \<Longrightarrow> x + y = y + x"
huffman@47108
   382
  apply (induct x rule: is_num.induct)
wenzelm@63654
   383
    apply (induct y rule: is_num.induct)
wenzelm@63654
   384
      apply simp
wenzelm@63654
   385
     apply (rule_tac a=x in add_left_imp_eq)
wenzelm@63654
   386
     apply (rule_tac a=x in add_right_imp_eq)
wenzelm@63654
   387
     apply (simp add: add.assoc)
wenzelm@63654
   388
    apply (simp add: add.assoc [symmetric])
wenzelm@63654
   389
    apply (simp add: add.assoc)
wenzelm@63654
   390
   apply (rule_tac a=x in add_left_imp_eq)
wenzelm@63654
   391
   apply (rule_tac a=x in add_right_imp_eq)
wenzelm@63654
   392
   apply (simp add: add.assoc)
haftmann@57512
   393
  apply (simp add: add.assoc)
wenzelm@63654
   394
  apply (simp add: add.assoc [symmetric])
huffman@47108
   395
  done
huffman@47108
   396
wenzelm@63654
   397
lemma is_num_add_left_commute: "is_num x \<Longrightarrow> is_num y \<Longrightarrow> x + (y + z) = y + (x + z)"
haftmann@57512
   398
  by (simp only: add.assoc [symmetric] is_num_add_commute)
huffman@47108
   399
huffman@47108
   400
lemmas is_num_normalize =
haftmann@57512
   401
  add.assoc is_num_add_commute is_num_add_left_commute
huffman@47108
   402
  is_num.intros is_num_numeral
haftmann@54230
   403
  minus_add
huffman@47108
   404
wenzelm@63654
   405
definition dbl :: "'a \<Rightarrow> 'a"
wenzelm@63654
   406
  where "dbl x = x + x"
wenzelm@63654
   407
wenzelm@63654
   408
definition dbl_inc :: "'a \<Rightarrow> 'a"
wenzelm@63654
   409
  where "dbl_inc x = x + x + 1"
huffman@47108
   410
wenzelm@63654
   411
definition dbl_dec :: "'a \<Rightarrow> 'a"
wenzelm@63654
   412
  where "dbl_dec x = x + x - 1"
wenzelm@63654
   413
wenzelm@63654
   414
definition sub :: "num \<Rightarrow> num \<Rightarrow> 'a"
wenzelm@63654
   415
  where "sub k l = numeral k - numeral l"
huffman@47108
   416
huffman@47108
   417
lemma numeral_BitM: "numeral (BitM n) = numeral (Bit0 n) - 1"
huffman@47108
   418
  by (simp only: BitM_plus_one [symmetric] numeral_add numeral_One eq_diff_eq)
huffman@47108
   419
huffman@47108
   420
lemma dbl_simps [simp]:
haftmann@54489
   421
  "dbl (- numeral k) = - dbl (numeral k)"
huffman@47108
   422
  "dbl 0 = 0"
huffman@47108
   423
  "dbl 1 = 2"
haftmann@54489
   424
  "dbl (- 1) = - 2"
huffman@47108
   425
  "dbl (numeral k) = numeral (Bit0 k)"
haftmann@54489
   426
  by (simp_all add: dbl_def numeral.simps minus_add)
huffman@47108
   427
huffman@47108
   428
lemma dbl_inc_simps [simp]:
haftmann@54489
   429
  "dbl_inc (- numeral k) = - dbl_dec (numeral k)"
huffman@47108
   430
  "dbl_inc 0 = 1"
huffman@47108
   431
  "dbl_inc 1 = 3"
haftmann@54489
   432
  "dbl_inc (- 1) = - 1"
huffman@47108
   433
  "dbl_inc (numeral k) = numeral (Bit1 k)"
wenzelm@63654
   434
  by (simp_all add: dbl_inc_def dbl_dec_def numeral.simps numeral_BitM is_num_normalize algebra_simps
wenzelm@63654
   435
      del: add_uminus_conv_diff)
huffman@47108
   436
huffman@47108
   437
lemma dbl_dec_simps [simp]:
haftmann@54489
   438
  "dbl_dec (- numeral k) = - dbl_inc (numeral k)"
haftmann@54489
   439
  "dbl_dec 0 = - 1"
huffman@47108
   440
  "dbl_dec 1 = 1"
haftmann@54489
   441
  "dbl_dec (- 1) = - 3"
huffman@47108
   442
  "dbl_dec (numeral k) = numeral (BitM k)"
haftmann@54489
   443
  by (simp_all add: dbl_dec_def dbl_inc_def numeral.simps numeral_BitM is_num_normalize)
huffman@47108
   444
huffman@47108
   445
lemma sub_num_simps [simp]:
huffman@47108
   446
  "sub One One = 0"
haftmann@54489
   447
  "sub One (Bit0 l) = - numeral (BitM l)"
haftmann@54489
   448
  "sub One (Bit1 l) = - numeral (Bit0 l)"
huffman@47108
   449
  "sub (Bit0 k) One = numeral (BitM k)"
huffman@47108
   450
  "sub (Bit1 k) One = numeral (Bit0 k)"
huffman@47108
   451
  "sub (Bit0 k) (Bit0 l) = dbl (sub k l)"
huffman@47108
   452
  "sub (Bit0 k) (Bit1 l) = dbl_dec (sub k l)"
huffman@47108
   453
  "sub (Bit1 k) (Bit0 l) = dbl_inc (sub k l)"
huffman@47108
   454
  "sub (Bit1 k) (Bit1 l) = dbl (sub k l)"
haftmann@54489
   455
  by (simp_all add: dbl_def dbl_dec_def dbl_inc_def sub_def numeral.simps
haftmann@54230
   456
    numeral_BitM is_num_normalize del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   457
huffman@47108
   458
lemma add_neg_numeral_simps:
haftmann@54489
   459
  "numeral m + - numeral n = sub m n"
haftmann@54489
   460
  "- numeral m + numeral n = sub n m"
haftmann@54489
   461
  "- numeral m + - numeral n = - (numeral m + numeral n)"
haftmann@54489
   462
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
wenzelm@63654
   463
      del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   464
huffman@47108
   465
lemma add_neg_numeral_special:
haftmann@54489
   466
  "1 + - numeral m = sub One m"
haftmann@54489
   467
  "- numeral m + 1 = sub One m"
haftmann@54489
   468
  "numeral m + - 1 = sub m One"
haftmann@54489
   469
  "- 1 + numeral n = sub n One"
haftmann@54489
   470
  "- 1 + - numeral n = - numeral (inc n)"
haftmann@54489
   471
  "- numeral m + - 1 = - numeral (inc m)"
haftmann@54489
   472
  "1 + - 1 = 0"
haftmann@54489
   473
  "- 1 + 1 = 0"
haftmann@54489
   474
  "- 1 + - 1 = - 2"
haftmann@54489
   475
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize right_minus numeral_inc
wenzelm@63654
   476
      del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   477
huffman@47108
   478
lemma diff_numeral_simps:
huffman@47108
   479
  "numeral m - numeral n = sub m n"
haftmann@54489
   480
  "numeral m - - numeral n = numeral (m + n)"
haftmann@54489
   481
  "- numeral m - numeral n = - numeral (m + n)"
haftmann@54489
   482
  "- numeral m - - numeral n = sub n m"
haftmann@54489
   483
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
wenzelm@63654
   484
      del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   485
huffman@47108
   486
lemma diff_numeral_special:
huffman@47108
   487
  "1 - numeral n = sub One n"
huffman@47108
   488
  "numeral m - 1 = sub m One"
haftmann@54489
   489
  "1 - - numeral n = numeral (One + n)"
haftmann@54489
   490
  "- numeral m - 1 = - numeral (m + One)"
haftmann@54489
   491
  "- 1 - numeral n = - numeral (inc n)"
haftmann@54489
   492
  "numeral m - - 1 = numeral (inc m)"
haftmann@54489
   493
  "- 1 - - numeral n = sub n One"
haftmann@54489
   494
  "- numeral m - - 1 = sub One m"
haftmann@54489
   495
  "1 - 1 = 0"
haftmann@54489
   496
  "- 1 - 1 = - 2"
haftmann@54489
   497
  "1 - - 1 = 2"
haftmann@54489
   498
  "- 1 - - 1 = 0"
haftmann@54489
   499
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize numeral_inc
wenzelm@63654
   500
      del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   501
huffman@47108
   502
end
huffman@47108
   503
wenzelm@63654
   504
wenzelm@63654
   505
subsubsection \<open>Structures with multiplication: class \<open>semiring_numeral\<close>\<close>
huffman@47108
   506
huffman@47108
   507
class semiring_numeral = semiring + monoid_mult
huffman@47108
   508
begin
huffman@47108
   509
huffman@47108
   510
subclass numeral ..
huffman@47108
   511
huffman@47108
   512
lemma numeral_mult: "numeral (m * n) = numeral m * numeral n"
wenzelm@63654
   513
  by (induct n rule: num_induct)
wenzelm@63654
   514
    (simp_all add: numeral_One mult_inc numeral_inc numeral_add distrib_left)
huffman@47108
   515
huffman@47108
   516
lemma numeral_times_numeral: "numeral m * numeral n = numeral (m * n)"
huffman@47108
   517
  by (rule numeral_mult [symmetric])
huffman@47108
   518
haftmann@53064
   519
lemma mult_2: "2 * z = z + z"
wenzelm@63654
   520
  by (simp add: one_add_one [symmetric] distrib_right)
haftmann@53064
   521
haftmann@53064
   522
lemma mult_2_right: "z * 2 = z + z"
wenzelm@63654
   523
  by (simp add: one_add_one [symmetric] distrib_left)
haftmann@53064
   524
huffman@47108
   525
end
huffman@47108
   526
wenzelm@63654
   527
wenzelm@63654
   528
subsubsection \<open>Structures with a zero: class \<open>semiring_1\<close>\<close>
huffman@47108
   529
huffman@47108
   530
context semiring_1
huffman@47108
   531
begin
huffman@47108
   532
huffman@47108
   533
subclass semiring_numeral ..
huffman@47108
   534
huffman@47108
   535
lemma of_nat_numeral [simp]: "of_nat (numeral n) = numeral n"
wenzelm@63654
   536
  by (induct n) (simp_all only: numeral.simps numeral_class.numeral.simps of_nat_add of_nat_1)
huffman@47108
   537
haftmann@64178
   538
lemma numeral_unfold_funpow:
haftmann@64178
   539
  "numeral k = (op + 1 ^^ numeral k) 0"
haftmann@64178
   540
  unfolding of_nat_def [symmetric] by simp
haftmann@64178
   541
huffman@47108
   542
end
huffman@47108
   543
haftmann@64178
   544
lemma transfer_rule_numeral:
haftmann@64178
   545
  fixes R :: "'a::semiring_1 \<Rightarrow> 'b::semiring_1 \<Rightarrow> bool"
haftmann@64178
   546
  assumes [transfer_rule]: "R 0 0" "R 1 1"
haftmann@64178
   547
    "rel_fun R (rel_fun R R) plus plus"
haftmann@64178
   548
  shows "rel_fun HOL.eq R numeral numeral"
haftmann@64178
   549
  apply (subst (2) numeral_unfold_funpow [abs_def])
haftmann@64178
   550
  apply (subst (1) numeral_unfold_funpow [abs_def])
haftmann@64178
   551
  apply transfer_prover
haftmann@64178
   552
  done
haftmann@64178
   553
wenzelm@63654
   554
lemma nat_of_num_numeral [code_abbrev]: "nat_of_num = numeral"
huffman@47108
   555
proof
huffman@47108
   556
  fix n
huffman@47108
   557
  have "numeral n = nat_of_num n"
huffman@47108
   558
    by (induct n) (simp_all add: numeral.simps)
wenzelm@63654
   559
  then show "nat_of_num n = numeral n"
wenzelm@63654
   560
    by simp
huffman@47108
   561
qed
huffman@47108
   562
haftmann@51143
   563
lemma nat_of_num_code [code]:
haftmann@51143
   564
  "nat_of_num One = 1"
haftmann@51143
   565
  "nat_of_num (Bit0 n) = (let m = nat_of_num n in m + m)"
haftmann@51143
   566
  "nat_of_num (Bit1 n) = (let m = nat_of_num n in Suc (m + m))"
haftmann@51143
   567
  by (simp_all add: Let_def)
haftmann@51143
   568
wenzelm@63654
   569
wenzelm@63654
   570
subsubsection \<open>Equality: class \<open>semiring_char_0\<close>\<close>
huffman@47108
   571
huffman@47108
   572
context semiring_char_0
huffman@47108
   573
begin
huffman@47108
   574
huffman@47108
   575
lemma numeral_eq_iff: "numeral m = numeral n \<longleftrightarrow> m = n"
wenzelm@63654
   576
  by (simp only: of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
wenzelm@63654
   577
    of_nat_eq_iff num_eq_iff)
huffman@47108
   578
huffman@47108
   579
lemma numeral_eq_one_iff: "numeral n = 1 \<longleftrightarrow> n = One"
huffman@47108
   580
  by (rule numeral_eq_iff [of n One, unfolded numeral_One])
huffman@47108
   581
huffman@47108
   582
lemma one_eq_numeral_iff: "1 = numeral n \<longleftrightarrow> One = n"
huffman@47108
   583
  by (rule numeral_eq_iff [of One n, unfolded numeral_One])
huffman@47108
   584
huffman@47108
   585
lemma numeral_neq_zero: "numeral n \<noteq> 0"
wenzelm@63654
   586
  by (simp add: of_nat_numeral [symmetric] nat_of_num_numeral [symmetric] nat_of_num_pos)
huffman@47108
   587
huffman@47108
   588
lemma zero_neq_numeral: "0 \<noteq> numeral n"
huffman@47108
   589
  unfolding eq_commute [of 0] by (rule numeral_neq_zero)
huffman@47108
   590
huffman@47108
   591
lemmas eq_numeral_simps [simp] =
huffman@47108
   592
  numeral_eq_iff
huffman@47108
   593
  numeral_eq_one_iff
huffman@47108
   594
  one_eq_numeral_iff
huffman@47108
   595
  numeral_neq_zero
huffman@47108
   596
  zero_neq_numeral
huffman@47108
   597
huffman@47108
   598
end
huffman@47108
   599
wenzelm@63654
   600
wenzelm@63654
   601
subsubsection \<open>Comparisons: class \<open>linordered_semidom\<close>\<close>
huffman@47108
   602
wenzelm@60758
   603
text \<open>Could be perhaps more general than here.\<close>
huffman@47108
   604
huffman@47108
   605
context linordered_semidom
huffman@47108
   606
begin
huffman@47108
   607
huffman@47108
   608
lemma numeral_le_iff: "numeral m \<le> numeral n \<longleftrightarrow> m \<le> n"
huffman@47108
   609
proof -
huffman@47108
   610
  have "of_nat (numeral m) \<le> of_nat (numeral n) \<longleftrightarrow> m \<le> n"
wenzelm@63654
   611
    by (simp only: less_eq_num_def nat_of_num_numeral of_nat_le_iff)
huffman@47108
   612
  then show ?thesis by simp
huffman@47108
   613
qed
huffman@47108
   614
huffman@47108
   615
lemma one_le_numeral: "1 \<le> numeral n"
wenzelm@63654
   616
  using numeral_le_iff [of One n] by (simp add: numeral_One)
huffman@47108
   617
huffman@47108
   618
lemma numeral_le_one_iff: "numeral n \<le> 1 \<longleftrightarrow> n \<le> One"
wenzelm@63654
   619
  using numeral_le_iff [of n One] by (simp add: numeral_One)
huffman@47108
   620
huffman@47108
   621
lemma numeral_less_iff: "numeral m < numeral n \<longleftrightarrow> m < n"
huffman@47108
   622
proof -
huffman@47108
   623
  have "of_nat (numeral m) < of_nat (numeral n) \<longleftrightarrow> m < n"
huffman@47108
   624
    unfolding less_num_def nat_of_num_numeral of_nat_less_iff ..
huffman@47108
   625
  then show ?thesis by simp
huffman@47108
   626
qed
huffman@47108
   627
huffman@47108
   628
lemma not_numeral_less_one: "\<not> numeral n < 1"
huffman@47108
   629
  using numeral_less_iff [of n One] by (simp add: numeral_One)
huffman@47108
   630
huffman@47108
   631
lemma one_less_numeral_iff: "1 < numeral n \<longleftrightarrow> One < n"
huffman@47108
   632
  using numeral_less_iff [of One n] by (simp add: numeral_One)
huffman@47108
   633
huffman@47108
   634
lemma zero_le_numeral: "0 \<le> numeral n"
huffman@47108
   635
  by (induct n) (simp_all add: numeral.simps)
huffman@47108
   636
huffman@47108
   637
lemma zero_less_numeral: "0 < numeral n"
huffman@47108
   638
  by (induct n) (simp_all add: numeral.simps add_pos_pos)
huffman@47108
   639
huffman@47108
   640
lemma not_numeral_le_zero: "\<not> numeral n \<le> 0"
huffman@47108
   641
  by (simp add: not_le zero_less_numeral)
huffman@47108
   642
huffman@47108
   643
lemma not_numeral_less_zero: "\<not> numeral n < 0"
huffman@47108
   644
  by (simp add: not_less zero_le_numeral)
huffman@47108
   645
huffman@47108
   646
lemmas le_numeral_extra =
huffman@47108
   647
  zero_le_one not_one_le_zero
huffman@47108
   648
  order_refl [of 0] order_refl [of 1]
huffman@47108
   649
huffman@47108
   650
lemmas less_numeral_extra =
huffman@47108
   651
  zero_less_one not_one_less_zero
huffman@47108
   652
  less_irrefl [of 0] less_irrefl [of 1]
huffman@47108
   653
huffman@47108
   654
lemmas le_numeral_simps [simp] =
huffman@47108
   655
  numeral_le_iff
huffman@47108
   656
  one_le_numeral
huffman@47108
   657
  numeral_le_one_iff
huffman@47108
   658
  zero_le_numeral
huffman@47108
   659
  not_numeral_le_zero
huffman@47108
   660
huffman@47108
   661
lemmas less_numeral_simps [simp] =
huffman@47108
   662
  numeral_less_iff
huffman@47108
   663
  one_less_numeral_iff
huffman@47108
   664
  not_numeral_less_one
huffman@47108
   665
  zero_less_numeral
huffman@47108
   666
  not_numeral_less_zero
huffman@47108
   667
Andreas@61630
   668
lemma min_0_1 [simp]:
wenzelm@63654
   669
  fixes min' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
wenzelm@63654
   670
  defines "min' \<equiv> min"
wenzelm@63654
   671
  shows
wenzelm@63654
   672
    "min' 0 1 = 0"
wenzelm@63654
   673
    "min' 1 0 = 0"
wenzelm@63654
   674
    "min' 0 (numeral x) = 0"
wenzelm@63654
   675
    "min' (numeral x) 0 = 0"
wenzelm@63654
   676
    "min' 1 (numeral x) = 1"
wenzelm@63654
   677
    "min' (numeral x) 1 = 1"
wenzelm@63654
   678
  by (simp_all add: min'_def min_def le_num_One_iff)
Andreas@61630
   679
wenzelm@63654
   680
lemma max_0_1 [simp]:
wenzelm@63654
   681
  fixes max' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
wenzelm@63654
   682
  defines "max' \<equiv> max"
wenzelm@63654
   683
  shows
wenzelm@63654
   684
    "max' 0 1 = 1"
wenzelm@63654
   685
    "max' 1 0 = 1"
wenzelm@63654
   686
    "max' 0 (numeral x) = numeral x"
wenzelm@63654
   687
    "max' (numeral x) 0 = numeral x"
wenzelm@63654
   688
    "max' 1 (numeral x) = numeral x"
wenzelm@63654
   689
    "max' (numeral x) 1 = numeral x"
wenzelm@63654
   690
  by (simp_all add: max'_def max_def le_num_One_iff)
Andreas@61630
   691
huffman@47108
   692
end
huffman@47108
   693
wenzelm@63654
   694
wenzelm@63654
   695
subsubsection \<open>Multiplication and negation: class \<open>ring_1\<close>\<close>
huffman@47108
   696
huffman@47108
   697
context ring_1
huffman@47108
   698
begin
huffman@47108
   699
huffman@47108
   700
subclass neg_numeral ..
huffman@47108
   701
huffman@47108
   702
lemma mult_neg_numeral_simps:
haftmann@54489
   703
  "- numeral m * - numeral n = numeral (m * n)"
haftmann@54489
   704
  "- numeral m * numeral n = - numeral (m * n)"
haftmann@54489
   705
  "numeral m * - numeral n = - numeral (m * n)"
wenzelm@63654
   706
  by (simp_all only: mult_minus_left mult_minus_right minus_minus numeral_mult)
huffman@47108
   707
haftmann@54489
   708
lemma mult_minus1 [simp]: "- 1 * z = - z"
wenzelm@63654
   709
  by (simp add: numeral.simps)
huffman@47108
   710
haftmann@54489
   711
lemma mult_minus1_right [simp]: "z * - 1 = - z"
wenzelm@63654
   712
  by (simp add: numeral.simps)
huffman@47108
   713
huffman@47108
   714
end
huffman@47108
   715
wenzelm@63654
   716
wenzelm@63654
   717
subsubsection \<open>Equality using \<open>iszero\<close> for rings with non-zero characteristic\<close>
huffman@47108
   718
huffman@47108
   719
context ring_1
huffman@47108
   720
begin
huffman@47108
   721
huffman@47108
   722
definition iszero :: "'a \<Rightarrow> bool"
huffman@47108
   723
  where "iszero z \<longleftrightarrow> z = 0"
huffman@47108
   724
huffman@47108
   725
lemma iszero_0 [simp]: "iszero 0"
huffman@47108
   726
  by (simp add: iszero_def)
huffman@47108
   727
huffman@47108
   728
lemma not_iszero_1 [simp]: "\<not> iszero 1"
huffman@47108
   729
  by (simp add: iszero_def)
huffman@47108
   730
huffman@47108
   731
lemma not_iszero_Numeral1: "\<not> iszero Numeral1"
huffman@47108
   732
  by (simp add: numeral_One)
huffman@47108
   733
haftmann@54489
   734
lemma not_iszero_neg_1 [simp]: "\<not> iszero (- 1)"
haftmann@54489
   735
  by (simp add: iszero_def)
haftmann@54489
   736
haftmann@54489
   737
lemma not_iszero_neg_Numeral1: "\<not> iszero (- Numeral1)"
haftmann@54489
   738
  by (simp add: numeral_One)
haftmann@54489
   739
wenzelm@63654
   740
lemma iszero_neg_numeral [simp]: "iszero (- numeral w) \<longleftrightarrow> iszero (numeral w)"
wenzelm@63654
   741
  unfolding iszero_def by (rule neg_equal_0_iff_equal)
huffman@47108
   742
huffman@47108
   743
lemma eq_iff_iszero_diff: "x = y \<longleftrightarrow> iszero (x - y)"
huffman@47108
   744
  unfolding iszero_def by (rule eq_iff_diff_eq_0)
huffman@47108
   745
wenzelm@63654
   746
text \<open>
wenzelm@63654
   747
  The \<open>eq_numeral_iff_iszero\<close> lemmas are not declared \<open>[simp]\<close> by default,
wenzelm@63654
   748
  because for rings of characteristic zero, better simp rules are possible.
wenzelm@63654
   749
  For a type like integers mod \<open>n\<close>, type-instantiated versions of these rules
wenzelm@63654
   750
  should be added to the simplifier, along with a type-specific rule for
wenzelm@63654
   751
  deciding propositions of the form \<open>iszero (numeral w)\<close>.
huffman@47108
   752
wenzelm@63654
   753
  bh: Maybe it would not be so bad to just declare these as simp rules anyway?
wenzelm@63654
   754
  I should test whether these rules take precedence over the \<open>ring_char_0\<close>
wenzelm@63654
   755
  rules in the simplifier.
wenzelm@60758
   756
\<close>
huffman@47108
   757
huffman@47108
   758
lemma eq_numeral_iff_iszero:
huffman@47108
   759
  "numeral x = numeral y \<longleftrightarrow> iszero (sub x y)"
haftmann@54489
   760
  "numeral x = - numeral y \<longleftrightarrow> iszero (numeral (x + y))"
haftmann@54489
   761
  "- numeral x = numeral y \<longleftrightarrow> iszero (numeral (x + y))"
haftmann@54489
   762
  "- numeral x = - numeral y \<longleftrightarrow> iszero (sub y x)"
huffman@47108
   763
  "numeral x = 1 \<longleftrightarrow> iszero (sub x One)"
huffman@47108
   764
  "1 = numeral y \<longleftrightarrow> iszero (sub One y)"
haftmann@54489
   765
  "- numeral x = 1 \<longleftrightarrow> iszero (numeral (x + One))"
haftmann@54489
   766
  "1 = - numeral y \<longleftrightarrow> iszero (numeral (One + y))"
huffman@47108
   767
  "numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
huffman@47108
   768
  "0 = numeral y \<longleftrightarrow> iszero (numeral y)"
haftmann@54489
   769
  "- numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
haftmann@54489
   770
  "0 = - numeral y \<longleftrightarrow> iszero (numeral y)"
huffman@47108
   771
  unfolding eq_iff_iszero_diff diff_numeral_simps diff_numeral_special
huffman@47108
   772
  by simp_all
huffman@47108
   773
huffman@47108
   774
end
huffman@47108
   775
wenzelm@63654
   776
wenzelm@63654
   777
subsubsection \<open>Equality and negation: class \<open>ring_char_0\<close>\<close>
huffman@47108
   778
haftmann@62481
   779
context ring_char_0
huffman@47108
   780
begin
huffman@47108
   781
huffman@47108
   782
lemma not_iszero_numeral [simp]: "\<not> iszero (numeral w)"
huffman@47108
   783
  by (simp add: iszero_def)
huffman@47108
   784
haftmann@54489
   785
lemma neg_numeral_eq_iff: "- numeral m = - numeral n \<longleftrightarrow> m = n"
haftmann@54489
   786
  by simp
huffman@47108
   787
haftmann@54489
   788
lemma numeral_neq_neg_numeral: "numeral m \<noteq> - numeral n"
wenzelm@63654
   789
  by (simp add: eq_neg_iff_add_eq_0 numeral_plus_numeral)
huffman@47108
   790
haftmann@54489
   791
lemma neg_numeral_neq_numeral: "- numeral m \<noteq> numeral n"
huffman@47108
   792
  by (rule numeral_neq_neg_numeral [symmetric])
huffman@47108
   793
haftmann@54489
   794
lemma zero_neq_neg_numeral: "0 \<noteq> - numeral n"
wenzelm@63654
   795
  by simp
huffman@47108
   796
haftmann@54489
   797
lemma neg_numeral_neq_zero: "- numeral n \<noteq> 0"
wenzelm@63654
   798
  by simp
huffman@47108
   799
haftmann@54489
   800
lemma one_neq_neg_numeral: "1 \<noteq> - numeral n"
huffman@47108
   801
  using numeral_neq_neg_numeral [of One n] by (simp add: numeral_One)
huffman@47108
   802
haftmann@54489
   803
lemma neg_numeral_neq_one: "- numeral n \<noteq> 1"
huffman@47108
   804
  using neg_numeral_neq_numeral [of n One] by (simp add: numeral_One)
huffman@47108
   805
wenzelm@63654
   806
lemma neg_one_neq_numeral: "- 1 \<noteq> numeral n"
haftmann@54489
   807
  using neg_numeral_neq_numeral [of One n] by (simp add: numeral_One)
haftmann@54489
   808
wenzelm@63654
   809
lemma numeral_neq_neg_one: "numeral n \<noteq> - 1"
haftmann@54489
   810
  using numeral_neq_neg_numeral [of n One] by (simp add: numeral_One)
haftmann@54489
   811
wenzelm@63654
   812
lemma neg_one_eq_numeral_iff: "- 1 = - numeral n \<longleftrightarrow> n = One"
haftmann@54489
   813
  using neg_numeral_eq_iff [of One n] by (auto simp add: numeral_One)
haftmann@54489
   814
wenzelm@63654
   815
lemma numeral_eq_neg_one_iff: "- numeral n = - 1 \<longleftrightarrow> n = One"
haftmann@54489
   816
  using neg_numeral_eq_iff [of n One] by (auto simp add: numeral_One)
haftmann@54489
   817
wenzelm@63654
   818
lemma neg_one_neq_zero: "- 1 \<noteq> 0"
haftmann@54489
   819
  by simp
haftmann@54489
   820
wenzelm@63654
   821
lemma zero_neq_neg_one: "0 \<noteq> - 1"
haftmann@54489
   822
  by simp
haftmann@54489
   823
wenzelm@63654
   824
lemma neg_one_neq_one: "- 1 \<noteq> 1"
haftmann@54489
   825
  using neg_numeral_neq_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
haftmann@54489
   826
wenzelm@63654
   827
lemma one_neq_neg_one: "1 \<noteq> - 1"
haftmann@54489
   828
  using numeral_neq_neg_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
haftmann@54489
   829
huffman@47108
   830
lemmas eq_neg_numeral_simps [simp] =
huffman@47108
   831
  neg_numeral_eq_iff
huffman@47108
   832
  numeral_neq_neg_numeral neg_numeral_neq_numeral
huffman@47108
   833
  one_neq_neg_numeral neg_numeral_neq_one
huffman@47108
   834
  zero_neq_neg_numeral neg_numeral_neq_zero
haftmann@54489
   835
  neg_one_neq_numeral numeral_neq_neg_one
haftmann@54489
   836
  neg_one_eq_numeral_iff numeral_eq_neg_one_iff
haftmann@54489
   837
  neg_one_neq_zero zero_neq_neg_one
haftmann@54489
   838
  neg_one_neq_one one_neq_neg_one
huffman@47108
   839
huffman@47108
   840
end
huffman@47108
   841
haftmann@62348
   842
wenzelm@63654
   843
subsubsection \<open>Structures with negation and order: class \<open>linordered_idom\<close>\<close>
huffman@47108
   844
huffman@47108
   845
context linordered_idom
huffman@47108
   846
begin
huffman@47108
   847
huffman@47108
   848
subclass ring_char_0 ..
huffman@47108
   849
haftmann@54489
   850
lemma neg_numeral_le_iff: "- numeral m \<le> - numeral n \<longleftrightarrow> n \<le> m"
haftmann@54489
   851
  by (simp only: neg_le_iff_le numeral_le_iff)
huffman@47108
   852
haftmann@54489
   853
lemma neg_numeral_less_iff: "- numeral m < - numeral n \<longleftrightarrow> n < m"
haftmann@54489
   854
  by (simp only: neg_less_iff_less numeral_less_iff)
huffman@47108
   855
haftmann@54489
   856
lemma neg_numeral_less_zero: "- numeral n < 0"
haftmann@54489
   857
  by (simp only: neg_less_0_iff_less zero_less_numeral)
huffman@47108
   858
haftmann@54489
   859
lemma neg_numeral_le_zero: "- numeral n \<le> 0"
haftmann@54489
   860
  by (simp only: neg_le_0_iff_le zero_le_numeral)
huffman@47108
   861
haftmann@54489
   862
lemma not_zero_less_neg_numeral: "\<not> 0 < - numeral n"
huffman@47108
   863
  by (simp only: not_less neg_numeral_le_zero)
huffman@47108
   864
haftmann@54489
   865
lemma not_zero_le_neg_numeral: "\<not> 0 \<le> - numeral n"
huffman@47108
   866
  by (simp only: not_le neg_numeral_less_zero)
huffman@47108
   867
haftmann@54489
   868
lemma neg_numeral_less_numeral: "- numeral m < numeral n"
huffman@47108
   869
  using neg_numeral_less_zero zero_less_numeral by (rule less_trans)
huffman@47108
   870
haftmann@54489
   871
lemma neg_numeral_le_numeral: "- numeral m \<le> numeral n"
huffman@47108
   872
  by (simp only: less_imp_le neg_numeral_less_numeral)
huffman@47108
   873
haftmann@54489
   874
lemma not_numeral_less_neg_numeral: "\<not> numeral m < - numeral n"
huffman@47108
   875
  by (simp only: not_less neg_numeral_le_numeral)
huffman@47108
   876
haftmann@54489
   877
lemma not_numeral_le_neg_numeral: "\<not> numeral m \<le> - numeral n"
huffman@47108
   878
  by (simp only: not_le neg_numeral_less_numeral)
wenzelm@63654
   879
haftmann@54489
   880
lemma neg_numeral_less_one: "- numeral m < 1"
huffman@47108
   881
  by (rule neg_numeral_less_numeral [of m One, unfolded numeral_One])
huffman@47108
   882
haftmann@54489
   883
lemma neg_numeral_le_one: "- numeral m \<le> 1"
huffman@47108
   884
  by (rule neg_numeral_le_numeral [of m One, unfolded numeral_One])
huffman@47108
   885
haftmann@54489
   886
lemma not_one_less_neg_numeral: "\<not> 1 < - numeral m"
huffman@47108
   887
  by (simp only: not_less neg_numeral_le_one)
huffman@47108
   888
haftmann@54489
   889
lemma not_one_le_neg_numeral: "\<not> 1 \<le> - numeral m"
huffman@47108
   890
  by (simp only: not_le neg_numeral_less_one)
huffman@47108
   891
haftmann@54489
   892
lemma not_numeral_less_neg_one: "\<not> numeral m < - 1"
haftmann@54489
   893
  using not_numeral_less_neg_numeral [of m One] by (simp add: numeral_One)
haftmann@54489
   894
haftmann@54489
   895
lemma not_numeral_le_neg_one: "\<not> numeral m \<le> - 1"
haftmann@54489
   896
  using not_numeral_le_neg_numeral [of m One] by (simp add: numeral_One)
haftmann@54489
   897
haftmann@54489
   898
lemma neg_one_less_numeral: "- 1 < numeral m"
haftmann@54489
   899
  using neg_numeral_less_numeral [of One m] by (simp add: numeral_One)
haftmann@54489
   900
haftmann@54489
   901
lemma neg_one_le_numeral: "- 1 \<le> numeral m"
haftmann@54489
   902
  using neg_numeral_le_numeral [of One m] by (simp add: numeral_One)
haftmann@54489
   903
haftmann@54489
   904
lemma neg_numeral_less_neg_one_iff: "- numeral m < - 1 \<longleftrightarrow> m \<noteq> One"
haftmann@54489
   905
  by (cases m) simp_all
haftmann@54489
   906
haftmann@54489
   907
lemma neg_numeral_le_neg_one: "- numeral m \<le> - 1"
haftmann@54489
   908
  by simp
haftmann@54489
   909
haftmann@54489
   910
lemma not_neg_one_less_neg_numeral: "\<not> - 1 < - numeral m"
haftmann@54489
   911
  by simp
haftmann@54489
   912
haftmann@54489
   913
lemma not_neg_one_le_neg_numeral_iff: "\<not> - 1 \<le> - numeral m \<longleftrightarrow> m \<noteq> One"
haftmann@54489
   914
  by (cases m) simp_all
haftmann@54489
   915
wenzelm@63654
   916
lemma sub_non_negative: "sub n m \<ge> 0 \<longleftrightarrow> n \<ge> m"
huffman@47108
   917
  by (simp only: sub_def le_diff_eq) simp
huffman@47108
   918
wenzelm@63654
   919
lemma sub_positive: "sub n m > 0 \<longleftrightarrow> n > m"
huffman@47108
   920
  by (simp only: sub_def less_diff_eq) simp
huffman@47108
   921
wenzelm@63654
   922
lemma sub_non_positive: "sub n m \<le> 0 \<longleftrightarrow> n \<le> m"
huffman@47108
   923
  by (simp only: sub_def diff_le_eq) simp
huffman@47108
   924
wenzelm@63654
   925
lemma sub_negative: "sub n m < 0 \<longleftrightarrow> n < m"
huffman@47108
   926
  by (simp only: sub_def diff_less_eq) simp
huffman@47108
   927
huffman@47108
   928
lemmas le_neg_numeral_simps [simp] =
huffman@47108
   929
  neg_numeral_le_iff
huffman@47108
   930
  neg_numeral_le_numeral not_numeral_le_neg_numeral
huffman@47108
   931
  neg_numeral_le_zero not_zero_le_neg_numeral
huffman@47108
   932
  neg_numeral_le_one not_one_le_neg_numeral
haftmann@54489
   933
  neg_one_le_numeral not_numeral_le_neg_one
haftmann@54489
   934
  neg_numeral_le_neg_one not_neg_one_le_neg_numeral_iff
haftmann@54489
   935
haftmann@54489
   936
lemma le_minus_one_simps [simp]:
haftmann@54489
   937
  "- 1 \<le> 0"
haftmann@54489
   938
  "- 1 \<le> 1"
haftmann@54489
   939
  "\<not> 0 \<le> - 1"
haftmann@54489
   940
  "\<not> 1 \<le> - 1"
haftmann@54489
   941
  by simp_all
huffman@47108
   942
huffman@47108
   943
lemmas less_neg_numeral_simps [simp] =
huffman@47108
   944
  neg_numeral_less_iff
huffman@47108
   945
  neg_numeral_less_numeral not_numeral_less_neg_numeral
huffman@47108
   946
  neg_numeral_less_zero not_zero_less_neg_numeral
huffman@47108
   947
  neg_numeral_less_one not_one_less_neg_numeral
haftmann@54489
   948
  neg_one_less_numeral not_numeral_less_neg_one
haftmann@54489
   949
  neg_numeral_less_neg_one_iff not_neg_one_less_neg_numeral
haftmann@54489
   950
haftmann@54489
   951
lemma less_minus_one_simps [simp]:
haftmann@54489
   952
  "- 1 < 0"
haftmann@54489
   953
  "- 1 < 1"
haftmann@54489
   954
  "\<not> 0 < - 1"
haftmann@54489
   955
  "\<not> 1 < - 1"
haftmann@54489
   956
  by (simp_all add: less_le)
huffman@47108
   957
wenzelm@61944
   958
lemma abs_numeral [simp]: "\<bar>numeral n\<bar> = numeral n"
huffman@47108
   959
  by simp
huffman@47108
   960
wenzelm@61944
   961
lemma abs_neg_numeral [simp]: "\<bar>- numeral n\<bar> = numeral n"
haftmann@54489
   962
  by (simp only: abs_minus_cancel abs_numeral)
haftmann@54489
   963
wenzelm@61944
   964
lemma abs_neg_one [simp]: "\<bar>- 1\<bar> = 1"
haftmann@54489
   965
  by simp
huffman@47108
   966
huffman@47108
   967
end
huffman@47108
   968
wenzelm@63654
   969
wenzelm@63654
   970
subsubsection \<open>Natural numbers\<close>
huffman@47108
   971
huffman@47299
   972
lemma Suc_1 [simp]: "Suc 1 = 2"
huffman@47299
   973
  unfolding Suc_eq_plus1 by (rule one_add_one)
huffman@47299
   974
huffman@47108
   975
lemma Suc_numeral [simp]: "Suc (numeral n) = numeral (n + One)"
huffman@47299
   976
  unfolding Suc_eq_plus1 by (rule numeral_plus_one)
huffman@47108
   977
huffman@47209
   978
definition pred_numeral :: "num \<Rightarrow> nat"
huffman@47209
   979
  where [code del]: "pred_numeral k = numeral k - 1"
huffman@47209
   980
huffman@47209
   981
lemma numeral_eq_Suc: "numeral k = Suc (pred_numeral k)"
wenzelm@63654
   982
  by (simp add: pred_numeral_def)
huffman@47209
   983
huffman@47220
   984
lemma eval_nat_numeral:
huffman@47108
   985
  "numeral One = Suc 0"
huffman@47108
   986
  "numeral (Bit0 n) = Suc (numeral (BitM n))"
huffman@47108
   987
  "numeral (Bit1 n) = Suc (numeral (Bit0 n))"
huffman@47108
   988
  by (simp_all add: numeral.simps BitM_plus_one)
huffman@47108
   989
huffman@47209
   990
lemma pred_numeral_simps [simp]:
huffman@47300
   991
  "pred_numeral One = 0"
huffman@47300
   992
  "pred_numeral (Bit0 k) = numeral (BitM k)"
huffman@47300
   993
  "pred_numeral (Bit1 k) = numeral (Bit0 k)"
wenzelm@63654
   994
  by (simp_all only: pred_numeral_def eval_nat_numeral diff_Suc_Suc diff_0)
huffman@47209
   995
huffman@47192
   996
lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
huffman@47220
   997
  by (simp add: eval_nat_numeral)
huffman@47192
   998
huffman@47192
   999
lemma numeral_3_eq_3: "3 = Suc (Suc (Suc 0))"
huffman@47220
  1000
  by (simp add: eval_nat_numeral)
huffman@47192
  1001
huffman@47207
  1002
lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
huffman@47207
  1003
  by (simp only: numeral_One One_nat_def)
huffman@47207
  1004
wenzelm@63654
  1005
lemma Suc_nat_number_of_add: "Suc (numeral v + n) = numeral (v + One) + n"
huffman@47207
  1006
  by simp
huffman@47207
  1007
wenzelm@63654
  1008
lemma numerals: "Numeral1 = (1::nat)" "2 = Suc (Suc 0)"
wenzelm@63654
  1009
  by (rule numeral_One) (rule numeral_2_eq_2)
huffman@47207
  1010
lars@63913
  1011
lemmas numeral_nat = eval_nat_numeral BitM.simps One_nat_def
lars@63913
  1012
wenzelm@60758
  1013
text \<open>Comparisons involving @{term Suc}.\<close>
huffman@47209
  1014
huffman@47209
  1015
lemma eq_numeral_Suc [simp]: "numeral k = Suc n \<longleftrightarrow> pred_numeral k = n"
huffman@47209
  1016
  by (simp add: numeral_eq_Suc)
huffman@47209
  1017
huffman@47209
  1018
lemma Suc_eq_numeral [simp]: "Suc n = numeral k \<longleftrightarrow> n = pred_numeral k"
huffman@47209
  1019
  by (simp add: numeral_eq_Suc)
huffman@47209
  1020
huffman@47209
  1021
lemma less_numeral_Suc [simp]: "numeral k < Suc n \<longleftrightarrow> pred_numeral k < n"
huffman@47209
  1022
  by (simp add: numeral_eq_Suc)
huffman@47209
  1023
huffman@47209
  1024
lemma less_Suc_numeral [simp]: "Suc n < numeral k \<longleftrightarrow> n < pred_numeral k"
huffman@47209
  1025
  by (simp add: numeral_eq_Suc)
huffman@47209
  1026
huffman@47209
  1027
lemma le_numeral_Suc [simp]: "numeral k \<le> Suc n \<longleftrightarrow> pred_numeral k \<le> n"
huffman@47209
  1028
  by (simp add: numeral_eq_Suc)
huffman@47209
  1029
huffman@47209
  1030
lemma le_Suc_numeral [simp]: "Suc n \<le> numeral k \<longleftrightarrow> n \<le> pred_numeral k"
huffman@47209
  1031
  by (simp add: numeral_eq_Suc)
huffman@47209
  1032
huffman@47218
  1033
lemma diff_Suc_numeral [simp]: "Suc n - numeral k = n - pred_numeral k"
huffman@47218
  1034
  by (simp add: numeral_eq_Suc)
huffman@47218
  1035
huffman@47218
  1036
lemma diff_numeral_Suc [simp]: "numeral k - Suc n = pred_numeral k - n"
huffman@47218
  1037
  by (simp add: numeral_eq_Suc)
huffman@47218
  1038
wenzelm@63654
  1039
lemma max_Suc_numeral [simp]: "max (Suc n) (numeral k) = Suc (max n (pred_numeral k))"
huffman@47209
  1040
  by (simp add: numeral_eq_Suc)
huffman@47209
  1041
wenzelm@63654
  1042
lemma max_numeral_Suc [simp]: "max (numeral k) (Suc n) = Suc (max (pred_numeral k) n)"
huffman@47209
  1043
  by (simp add: numeral_eq_Suc)
huffman@47209
  1044
wenzelm@63654
  1045
lemma min_Suc_numeral [simp]: "min (Suc n) (numeral k) = Suc (min n (pred_numeral k))"
huffman@47209
  1046
  by (simp add: numeral_eq_Suc)
huffman@47209
  1047
wenzelm@63654
  1048
lemma min_numeral_Suc [simp]: "min (numeral k) (Suc n) = Suc (min (pred_numeral k) n)"
huffman@47209
  1049
  by (simp add: numeral_eq_Suc)
huffman@47209
  1050
wenzelm@60758
  1051
text \<open>For @{term case_nat} and @{term rec_nat}.\<close>
huffman@47216
  1052
wenzelm@63654
  1053
lemma case_nat_numeral [simp]: "case_nat a f (numeral v) = (let pv = pred_numeral v in f pv)"
huffman@47216
  1054
  by (simp add: numeral_eq_Suc)
huffman@47216
  1055
blanchet@55415
  1056
lemma case_nat_add_eq_if [simp]:
blanchet@55415
  1057
  "case_nat a f ((numeral v) + n) = (let pv = pred_numeral v in f (pv + n))"
huffman@47216
  1058
  by (simp add: numeral_eq_Suc)
huffman@47216
  1059
blanchet@55415
  1060
lemma rec_nat_numeral [simp]:
wenzelm@63654
  1061
  "rec_nat a f (numeral v) = (let pv = pred_numeral v in f pv (rec_nat a f pv))"
huffman@47216
  1062
  by (simp add: numeral_eq_Suc Let_def)
huffman@47216
  1063
blanchet@55415
  1064
lemma rec_nat_add_eq_if [simp]:
wenzelm@63654
  1065
  "rec_nat a f (numeral v + n) = (let pv = pred_numeral v in f (pv + n) (rec_nat a f (pv + n)))"
huffman@47216
  1066
  by (simp add: numeral_eq_Suc Let_def)
huffman@47216
  1067
wenzelm@63654
  1068
text \<open>Case analysis on @{term "n < 2"}.\<close>
huffman@47255
  1069
lemma less_2_cases: "n < 2 \<Longrightarrow> n = 0 \<or> n = Suc 0"
huffman@47255
  1070
  by (auto simp add: numeral_2_eq_2)
huffman@47255
  1071
wenzelm@63654
  1072
text \<open>Removal of Small Numerals: 0, 1 and (in additive positions) 2.\<close>
wenzelm@60758
  1073
text \<open>bh: Are these rules really a good idea?\<close>
huffman@47255
  1074
huffman@47255
  1075
lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
huffman@47255
  1076
  by simp
huffman@47255
  1077
huffman@47255
  1078
lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
huffman@47255
  1079
  by simp
huffman@47255
  1080
wenzelm@60758
  1081
text \<open>Can be used to eliminate long strings of Sucs, but not by default.\<close>
huffman@47255
  1082
lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
huffman@47255
  1083
  by simp
huffman@47255
  1084
huffman@47255
  1085
lemmas nat_1_add_1 = one_add_one [where 'a=nat] (* legacy *)
huffman@47255
  1086
huffman@47108
  1087
wenzelm@60758
  1088
subsection \<open>Particular lemmas concerning @{term 2}\<close>
haftmann@58512
  1089
haftmann@59867
  1090
context linordered_field
haftmann@58512
  1091
begin
haftmann@58512
  1092
haftmann@62348
  1093
subclass field_char_0 ..
haftmann@62348
  1094
wenzelm@63654
  1095
lemma half_gt_zero_iff: "0 < a / 2 \<longleftrightarrow> 0 < a"
haftmann@58512
  1096
  by (auto simp add: field_simps)
haftmann@58512
  1097
wenzelm@63654
  1098
lemma half_gt_zero [simp]: "0 < a \<Longrightarrow> 0 < a / 2"
haftmann@58512
  1099
  by (simp add: half_gt_zero_iff)
haftmann@58512
  1100
haftmann@58512
  1101
end
haftmann@58512
  1102
haftmann@58512
  1103
wenzelm@60758
  1104
subsection \<open>Numeral equations as default simplification rules\<close>
huffman@47108
  1105
huffman@47108
  1106
declare (in numeral) numeral_One [simp]
huffman@47108
  1107
declare (in numeral) numeral_plus_numeral [simp]
huffman@47108
  1108
declare (in numeral) add_numeral_special [simp]
huffman@47108
  1109
declare (in neg_numeral) add_neg_numeral_simps [simp]
huffman@47108
  1110
declare (in neg_numeral) add_neg_numeral_special [simp]
huffman@47108
  1111
declare (in neg_numeral) diff_numeral_simps [simp]
huffman@47108
  1112
declare (in neg_numeral) diff_numeral_special [simp]
huffman@47108
  1113
declare (in semiring_numeral) numeral_times_numeral [simp]
huffman@47108
  1114
declare (in ring_1) mult_neg_numeral_simps [simp]
huffman@47108
  1115
wenzelm@60758
  1116
subsection \<open>Setting up simprocs\<close>
huffman@47108
  1117
wenzelm@63654
  1118
lemma mult_numeral_1: "Numeral1 * a = a"
wenzelm@63654
  1119
  for a :: "'a::semiring_numeral"
huffman@47108
  1120
  by simp
huffman@47108
  1121
wenzelm@63654
  1122
lemma mult_numeral_1_right: "a * Numeral1 = a"
wenzelm@63654
  1123
  for a :: "'a::semiring_numeral"
huffman@47108
  1124
  by simp
huffman@47108
  1125
wenzelm@63654
  1126
lemma divide_numeral_1: "a / Numeral1 = a"
wenzelm@63654
  1127
  for a :: "'a::field"
huffman@47108
  1128
  by simp
huffman@47108
  1129
wenzelm@63654
  1130
lemma inverse_numeral_1: "inverse Numeral1 = (Numeral1::'a::division_ring)"
huffman@47108
  1131
  by simp
huffman@47108
  1132
wenzelm@63654
  1133
text \<open>
wenzelm@63654
  1134
  Theorem lists for the cancellation simprocs. The use of a binary
wenzelm@63654
  1135
  numeral for 1 reduces the number of special cases.
wenzelm@63654
  1136
\<close>
huffman@47108
  1137
haftmann@54489
  1138
lemma mult_1s:
wenzelm@63654
  1139
  "Numeral1 * a = a"
wenzelm@63654
  1140
  "a * Numeral1 = a"
wenzelm@63654
  1141
  "- Numeral1 * b = - b"
wenzelm@63654
  1142
  "b * - Numeral1 = - b"
wenzelm@63654
  1143
  for a :: "'a::semiring_numeral" and b :: "'b::ring_1"
haftmann@54489
  1144
  by simp_all
huffman@47108
  1145
wenzelm@60758
  1146
setup \<open>
huffman@47226
  1147
  Reorient_Proc.add
huffman@47226
  1148
    (fn Const (@{const_name numeral}, _) $ _ => true
wenzelm@63654
  1149
      | Const (@{const_name uminus}, _) $ (Const (@{const_name numeral}, _) $ _) => true
wenzelm@63654
  1150
      | _ => false)
wenzelm@60758
  1151
\<close>
huffman@47226
  1152
wenzelm@63654
  1153
simproc_setup reorient_numeral ("numeral w = x" | "- numeral w = y") =
wenzelm@63654
  1154
  Reorient_Proc.proc
huffman@47226
  1155
huffman@47108
  1156
wenzelm@63654
  1157
subsubsection \<open>Simplification of arithmetic operations on integer constants\<close>
huffman@47108
  1158
huffman@47108
  1159
lemmas arith_special = (* already declared simp above *)
huffman@47108
  1160
  add_numeral_special add_neg_numeral_special
haftmann@54489
  1161
  diff_numeral_special
huffman@47108
  1162
wenzelm@63654
  1163
lemmas arith_extra_simps = (* rules already in simpset *)
huffman@47108
  1164
  numeral_plus_numeral add_neg_numeral_simps add_0_left add_0_right
haftmann@54489
  1165
  minus_zero
huffman@47108
  1166
  diff_numeral_simps diff_0 diff_0_right
huffman@47108
  1167
  numeral_times_numeral mult_neg_numeral_simps
huffman@47108
  1168
  mult_zero_left mult_zero_right
huffman@47108
  1169
  abs_numeral abs_neg_numeral
huffman@47108
  1170
wenzelm@60758
  1171
text \<open>
huffman@47108
  1172
  For making a minimal simpset, one must include these default simprules.
wenzelm@61799
  1173
  Also include \<open>simp_thms\<close>.
wenzelm@60758
  1174
\<close>
huffman@47108
  1175
huffman@47108
  1176
lemmas arith_simps =
huffman@47108
  1177
  add_num_simps mult_num_simps sub_num_simps
huffman@47108
  1178
  BitM.simps dbl_simps dbl_inc_simps dbl_dec_simps
huffman@47108
  1179
  abs_zero abs_one arith_extra_simps
huffman@47108
  1180
haftmann@54249
  1181
lemmas more_arith_simps =
haftmann@54249
  1182
  neg_le_iff_le
haftmann@54249
  1183
  minus_zero left_minus right_minus
haftmann@54249
  1184
  mult_1_left mult_1_right
haftmann@54249
  1185
  mult_minus_left mult_minus_right
haftmann@57512
  1186
  minus_add_distrib minus_minus mult.assoc
haftmann@54249
  1187
haftmann@54249
  1188
lemmas of_nat_simps =
haftmann@54249
  1189
  of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
haftmann@54249
  1190
wenzelm@63654
  1191
text \<open>Simplification of relational operations.\<close>
huffman@47108
  1192
huffman@47108
  1193
lemmas eq_numeral_extra =
huffman@47108
  1194
  zero_neq_one one_neq_zero
huffman@47108
  1195
huffman@47108
  1196
lemmas rel_simps =
huffman@47108
  1197
  le_num_simps less_num_simps eq_num_simps
haftmann@54489
  1198
  le_numeral_simps le_neg_numeral_simps le_minus_one_simps le_numeral_extra
haftmann@54489
  1199
  less_numeral_simps less_neg_numeral_simps less_minus_one_simps less_numeral_extra
huffman@47108
  1200
  eq_numeral_simps eq_neg_numeral_simps eq_numeral_extra
huffman@47108
  1201
haftmann@54249
  1202
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
wenzelm@61799
  1203
  \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
haftmann@54249
  1204
  unfolding Let_def ..
haftmann@54249
  1205
haftmann@54489
  1206
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
wenzelm@61799
  1207
  \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
haftmann@54249
  1208
  unfolding Let_def ..
haftmann@54249
  1209
wenzelm@60758
  1210
declaration \<open>
wenzelm@63654
  1211
let
wenzelm@59996
  1212
  fun number_of ctxt T n =
wenzelm@59996
  1213
    if not (Sign.of_sort (Proof_Context.theory_of ctxt) (T, @{sort numeral}))
haftmann@54249
  1214
    then raise CTERM ("number_of", [])
wenzelm@59996
  1215
    else Numeral.mk_cnumber (Thm.ctyp_of ctxt T) n;
haftmann@54249
  1216
in
haftmann@54249
  1217
  K (
wenzelm@63654
  1218
    Lin_Arith.add_simps
wenzelm@63654
  1219
      @{thms arith_simps more_arith_simps rel_simps pred_numeral_simps
haftmann@64238
  1220
        arith_special numeral_One of_nat_simps uminus_numeral_One}
wenzelm@63654
  1221
    #> Lin_Arith.add_simps
wenzelm@63654
  1222
      @{thms Suc_numeral Let_numeral Let_neg_numeral Let_0 Let_1
wenzelm@63654
  1223
        le_Suc_numeral le_numeral_Suc less_Suc_numeral less_numeral_Suc
wenzelm@63654
  1224
        Suc_eq_numeral eq_numeral_Suc mult_Suc mult_Suc_right of_nat_numeral}
haftmann@54249
  1225
    #> Lin_Arith.set_number_of number_of)
haftmann@54249
  1226
end
wenzelm@60758
  1227
\<close>
haftmann@54249
  1228
huffman@47108
  1229
wenzelm@63654
  1230
subsubsection \<open>Simplification of arithmetic when nested to the right\<close>
huffman@47108
  1231
wenzelm@63654
  1232
lemma add_numeral_left [simp]: "numeral v + (numeral w + z) = (numeral(v + w) + z)"
haftmann@57512
  1233
  by (simp_all add: add.assoc [symmetric])
huffman@47108
  1234
huffman@47108
  1235
lemma add_neg_numeral_left [simp]:
haftmann@54489
  1236
  "numeral v + (- numeral w + y) = (sub v w + y)"
haftmann@54489
  1237
  "- numeral v + (numeral w + y) = (sub w v + y)"
haftmann@54489
  1238
  "- numeral v + (- numeral w + y) = (- numeral(v + w) + y)"
haftmann@57512
  1239
  by (simp_all add: add.assoc [symmetric])
huffman@47108
  1240
huffman@47108
  1241
lemma mult_numeral_left [simp]:
huffman@47108
  1242
  "numeral v * (numeral w * z) = (numeral(v * w) * z :: 'a::semiring_numeral)"
haftmann@54489
  1243
  "- numeral v * (numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
haftmann@54489
  1244
  "numeral v * (- numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
haftmann@54489
  1245
  "- numeral v * (- numeral w * y) = (numeral(v * w) * y :: 'b::ring_1)"
haftmann@57512
  1246
  by (simp_all add: mult.assoc [symmetric])
huffman@47108
  1247
huffman@47108
  1248
hide_const (open) One Bit0 Bit1 BitM inc pow sqr sub dbl dbl_inc dbl_dec
huffman@47108
  1249
haftmann@51143
  1250
wenzelm@63654
  1251
subsection \<open>Code module namespace\<close>
huffman@47108
  1252
haftmann@52435
  1253
code_identifier
haftmann@52435
  1254
  code_module Num \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
huffman@47108
  1255
huffman@47108
  1256
end