src/HOL/ex/Numeral.thy
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(*  Title:      HOL/ex/Numeral.thy
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    ID:         $Id$
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    Author:     Florian Haftmann
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An experimental alternative numeral representation.
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*)
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theory Numeral
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imports Int Inductive
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begin
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subsection {* The @{text num} type *}
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datatype num = One | Dig0 num | Dig1 num
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text {* Increment function for type @{typ num} *}
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primrec
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  inc :: "num \<Rightarrow> num"
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where
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  "inc One = Dig0 One"
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| "inc (Dig0 x) = Dig1 x"
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| "inc (Dig1 x) = Dig0 (inc x)"
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text {* Converting between type @{typ num} and type @{typ nat} *}
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primrec
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  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 (Dig0 x) = nat_of_num x + nat_of_num x"
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| "nat_of_num (Dig1 x) = Suc (nat_of_num x + nat_of_num x)"
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primrec
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  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:
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  "0 < n \<Longrightarrow> num_of_nat (n + n) = Dig0 (num_of_nat n)"
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  by (induct n) simp_all
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text {*
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  Type @{typ num} is isomorphic to the strictly positive
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  natural numbers.
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*}
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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 show ?case using One 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 {*
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  From now on, there are two possible models for @{typ num}:
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  as positive naturals (rule @{text "num_induct"})
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  and as digit representation (rules @{text "num.induct"}, @{text "num.cases"}).
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  It is not entirely clear in which context it is better to use
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  the one or the other, or whether the construction should be reversed.
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*}
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subsection {* Numeral operations *}
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ML {*
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structure DigSimps =
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  NamedThmsFun(val name = "numeral"; val description = "Simplification rules for numerals")
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*}
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setup DigSimps.setup
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instantiation num :: "{plus,times,ord}"
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begin
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definition plus_num :: "num \<Rightarrow> num \<Rightarrow> num" where
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  [code del]: "m + n = num_of_nat (nat_of_num m + nat_of_num n)"
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definition times_num :: "num \<Rightarrow> num \<Rightarrow> num" where
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  [code del]: "m * n = num_of_nat (nat_of_num m * nat_of_num n)"
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definition less_eq_num :: "num \<Rightarrow> num \<Rightarrow> bool" where
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  [code del]: "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"
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definition less_num :: "num \<Rightarrow> num \<Rightarrow> bool" where
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  [code del]: "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"
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instance ..
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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 Dig_plus [numeral, simp, code]:
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  "One + One = Dig0 One"
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  "One + Dig0 m = Dig1 m"
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  "One + Dig1 m = Dig0 (m + One)"
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  "Dig0 n + One = Dig1 n"
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  "Dig0 n + Dig0 m = Dig0 (n + m)"
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  "Dig0 n + Dig1 m = Dig1 (n + m)"
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  "Dig1 n + One = Dig0 (n + One)"
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  "Dig1 n + Dig0 m = Dig1 (n + m)"
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  "Dig1 n + Dig1 m = Dig0 (n + m + One)"
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  by (simp_all add: num_eq_iff nat_of_num_add)
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lemma Dig_times [numeral, simp, code]:
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  "One * One = One"
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  "One * Dig0 n = Dig0 n"
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  "One * Dig1 n = Dig1 n"
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  "Dig0 n * One = Dig0 n"
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  "Dig0 n * Dig0 m = Dig0 (n * Dig0 m)"
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  "Dig0 n * Dig1 m = Dig0 (n * Dig1 m)"
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  "Dig1 n * One = Dig1 n"
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  "Dig1 n * Dig0 m = Dig0 (n * Dig0 m + m)"
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  "Dig1 n * Dig1 m = Dig1 (n * Dig1 m + m)"
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  by (simp_all add: num_eq_iff nat_of_num_add nat_of_num_mult
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                    left_distrib right_distrib)
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lemma less_eq_num_code [numeral, simp, code]:
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  "One \<le> n \<longleftrightarrow> True"
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  "Dig0 m \<le> One \<longleftrightarrow> False"
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  "Dig1 m \<le> One \<longleftrightarrow> False"
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  "Dig0 m \<le> Dig0 n \<longleftrightarrow> m \<le> n"
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  "Dig0 m \<le> Dig1 n \<longleftrightarrow> m \<le> n"
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  "Dig1 m \<le> Dig1 n \<longleftrightarrow> m \<le> n"
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  "Dig1 m \<le> Dig0 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_code [numeral, simp, code]:
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  "m < One \<longleftrightarrow> False"
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  "One < One \<longleftrightarrow> False"
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  "One < Dig0 n \<longleftrightarrow> True"
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  "One < Dig1 n \<longleftrightarrow> True"
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  "Dig0 m < Dig0 n \<longleftrightarrow> m < n"
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  "Dig0 m < Dig1 n \<longleftrightarrow> m \<le> n"
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  "Dig1 m < Dig1 n \<longleftrightarrow> m < n"
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  "Dig1 m < Dig0 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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text {* Rules using @{text One} and @{text inc} as constructors *}
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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_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_One: "x * One = x"
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  by (simp add: num_eq_iff nat_of_num_mult)
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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 {* A double-and-decrement function *}
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primrec DigM :: "num \<Rightarrow> num" where
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  "DigM One = One"
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  | "DigM (Dig0 n) = Dig1 (DigM n)"
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  | "DigM (Dig1 n) = Dig1 (Dig0 n)"
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lemma DigM_plus_one: "DigM n + One = Dig0 n"
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  by (induct n) simp_all
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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 one_plus_DigM: "One + DigM n = Dig0 n"
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  unfolding add_One_commute DigM_plus_one ..
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subsection {* Binary numerals *}
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text {*
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  We embed binary representations into a generic algebraic
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  structure using @{text of_num}
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*}
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class semiring_numeral = semiring + monoid_mult
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begin
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primrec of_num :: "num \<Rightarrow> 'a" where
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  of_num_one [numeral]: "of_num One = 1"
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  | "of_num (Dig0 n) = of_num n + of_num n"
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  | "of_num (Dig1 n) = of_num n + of_num n + 1"
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lemma of_num_inc: "of_num (inc x) = of_num x + 1"
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  by (induct x) (simp_all add: add_ac)
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declare of_num.simps [simp del]
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end
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text {*
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  ML stuff and syntax.
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*}
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ML {*
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fun mk_num 1 = @{term One}
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  | mk_num k =
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      let
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        val (l, b) = Integer.div_mod k 2;
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        val bit = (if b = 0 then @{term Dig0} else @{term Dig1});
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      in bit $ (mk_num l) end;
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fun dest_num @{term One} = 1
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  | dest_num (@{term Dig0} $ n) = 2 * dest_num n
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  | dest_num (@{term Dig1} $ n) = 2 * dest_num n + 1;
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(*FIXME these have to gain proper context via morphisms phi*)
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fun mk_numeral T k = Const (@{const_name of_num}, @{typ num} --> T)
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  $ mk_num k
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fun dest_numeral (Const (@{const_name of_num}, Type ("fun", [@{typ num}, T])) $ t) =
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  (T, dest_num t)
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*}
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syntax
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  "_Numerals" :: "xnum \<Rightarrow> 'a"    ("_")
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parse_translation {*
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let
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  fun num_of_int n = if n > 0 then case IntInf.quotRem (n, 2)
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     of (0, 1) => Const (@{const_name One}, dummyT)
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      | (n, 0) => Const (@{const_name Dig0}, dummyT) $ num_of_int n
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      | (n, 1) => Const (@{const_name Dig1}, dummyT) $ num_of_int n
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    else raise Match;
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  fun numeral_tr [Free (num, _)] =
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        let
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          val {leading_zeros, value, ...} = Syntax.read_xnum num;
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          val _ = leading_zeros = 0 andalso value > 0
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            orelse error ("Bad numeral: " ^ num);
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        in Const (@{const_name of_num}, @{typ num} --> dummyT) $ num_of_int value end
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    | numeral_tr ts = raise TERM ("numeral_tr", ts);
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in [("_Numerals", numeral_tr)] end
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*}
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typed_print_translation {*
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let
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  fun dig b n = b + 2 * n; 
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  fun int_of_num' (Const (@{const_syntax Dig0}, _) $ n) =
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        dig 0 (int_of_num' n)
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    | int_of_num' (Const (@{const_syntax Dig1}, _) $ n) =
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        dig 1 (int_of_num' n)
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    | int_of_num' (Const (@{const_syntax One}, _)) = 1;
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  fun num_tr' show_sorts T [n] =
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    let
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      val k = int_of_num' n;
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      val t' = Syntax.const "_Numerals" $ Syntax.free ("#" ^ string_of_int k);
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    in case T
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     of Type ("fun", [_, T']) =>
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         if not (! show_types) andalso can Term.dest_Type T' then t'
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         else Syntax.const Syntax.constrainC $ t' $ Syntax.term_of_typ show_sorts T'
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      | T' => if T' = dummyT then t' else raise Match
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    end;
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in [(@{const_syntax of_num}, num_tr')] end
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*}
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subsection {* Class-specific numeral rules *}
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text {*
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  @{const of_num} is a morphism.
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*}
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subsubsection {* Class @{text semiring_numeral} *}
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context semiring_numeral
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begin
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abbreviation "Num1 \<equiv> of_num One"
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text {*
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  Alas, there is still the duplication of @{term 1},
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  thought the duplicated @{term 0} has disappeared.
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  We could get rid of it by replacing the constructor
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  @{term 1} in @{typ num} by two constructors
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  @{text two} and @{text three}, resulting in a further
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  blow-up.  But it could be worth the effort.
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parents:
diff changeset
   323
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   324
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   325
lemma of_num_plus_one [numeral]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   326
  "of_num n + 1 = of_num (n + One)"
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   327
  by (rule sym, induct n) (simp_all add: of_num.simps add_ac)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   328
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   329
lemma of_num_one_plus [numeral]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   330
  "1 + of_num n = of_num (n + One)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   331
  unfolding of_num_plus_one [symmetric] add_commute ..
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   332
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   333
lemma of_num_plus [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   334
  "of_num m + of_num n = of_num (m + n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   335
  by (induct n rule: num_induct)
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   336
     (simp_all add: add_One add_inc of_num_one of_num_inc add_ac)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   337
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   338
lemma of_num_times_one [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   339
  "of_num n * 1 = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   340
  by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   341
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   342
lemma of_num_one_times [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   343
  "1 * of_num n = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   344
  by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   345
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   346
lemma of_num_times [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   347
  "of_num m * of_num n = of_num (m * n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   348
  by (induct n rule: num_induct)
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   349
    (simp_all add: of_num_plus [symmetric] mult_One mult_inc
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   350
    semiring_class.right_distrib right_distrib of_num_one of_num_inc)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   351
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   352
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   353
29945
75df553549b1 rearrange subsections
huffman
parents: 29944
diff changeset
   354
subsubsection {*
75df553549b1 rearrange subsections
huffman
parents: 29944
diff changeset
   355
  Structures with a @{term 0}: class @{text semiring_1}
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   356
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   357
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   358
context semiring_1
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   359
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   360
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   361
subclass semiring_numeral ..
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   362
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   363
lemma of_nat_of_num [numeral]: "of_nat (of_num n) = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   364
  by (induct n)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   365
    (simp_all add: semiring_numeral_class.of_num.simps of_num.simps add_ac)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   366
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   367
declare of_nat_1 [numeral]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   368
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   369
lemma Dig_plus_zero [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   370
  "0 + 1 = 1"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   371
  "0 + of_num n = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   372
  "1 + 0 = 1"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   373
  "of_num n + 0 = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   374
  by simp_all
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   375
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   376
lemma Dig_times_zero [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   377
  "0 * 1 = 0"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   378
  "0 * of_num n = 0"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   379
  "1 * 0 = 0"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   380
  "of_num n * 0 = 0"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   381
  by simp_all
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   382
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   383
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   384
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   385
lemma nat_of_num_of_num: "nat_of_num = of_num"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   386
proof
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   387
  fix n
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   388
  have "of_num n = nat_of_num n"
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   389
    by (induct n) (simp_all add: of_num.simps)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   390
  then show "nat_of_num n = of_num n" by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   391
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   392
29945
75df553549b1 rearrange subsections
huffman
parents: 29944
diff changeset
   393
subsubsection {*
75df553549b1 rearrange subsections
huffman
parents: 29944
diff changeset
   394
  Equality: class @{text semiring_char_0}
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   395
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   396
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   397
context semiring_char_0
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   398
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   399
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   400
lemma of_num_eq_iff [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   401
  "of_num m = of_num n \<longleftrightarrow> m = n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   402
  unfolding of_nat_of_num [symmetric] nat_of_num_of_num [symmetric]
29943
922b931fd2eb datatype num = One | Dig0 num | Dig1 num
huffman
parents: 29942
diff changeset
   403
    of_nat_eq_iff num_eq_iff ..
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   404
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   405
lemma of_num_eq_one_iff [numeral]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   406
  "of_num n = 1 \<longleftrightarrow> n = One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   407
proof -
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   408
  have "of_num n = of_num One \<longleftrightarrow> n = One" unfolding of_num_eq_iff ..
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   409
  then show ?thesis by (simp add: of_num_one)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   410
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   411
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   412
lemma one_eq_of_num_iff [numeral]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   413
  "1 = of_num n \<longleftrightarrow> n = One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   414
  unfolding of_num_eq_one_iff [symmetric] by auto
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   415
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   416
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   417
29945
75df553549b1 rearrange subsections
huffman
parents: 29944
diff changeset
   418
subsubsection {*
75df553549b1 rearrange subsections
huffman
parents: 29944
diff changeset
   419
  Comparisons: class @{text ordered_semidom}
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   420
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   421
29945
75df553549b1 rearrange subsections
huffman
parents: 29944
diff changeset
   422
text {*  Could be perhaps more general than here. *}
75df553549b1 rearrange subsections
huffman
parents: 29944
diff changeset
   423
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   424
lemma (in ordered_semidom) of_num_pos: "0 < of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   425
proof -
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   426
  have "(0::nat) < of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   427
    by (induct n) (simp_all add: semiring_numeral_class.of_num.simps)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   428
  then have "of_nat 0 \<noteq> of_nat (of_num n)" 
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   429
    by (cases n) (simp_all only: semiring_numeral_class.of_num.simps of_nat_eq_iff)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   430
  then have "0 \<noteq> of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   431
    by (simp add: of_nat_of_num)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   432
  moreover have "0 \<le> of_nat (of_num n)" by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   433
  ultimately show ?thesis by (simp add: of_nat_of_num)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   434
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   435
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   436
context ordered_semidom
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   437
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   438
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   439
lemma of_num_less_eq_iff [numeral]: "of_num m \<le> of_num n \<longleftrightarrow> m \<le> n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   440
proof -
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   441
  have "of_nat (of_num m) \<le> of_nat (of_num n) \<longleftrightarrow> m \<le> n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   442
    unfolding less_eq_num_def nat_of_num_of_num of_nat_le_iff ..
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   443
  then show ?thesis by (simp add: of_nat_of_num)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   444
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   445
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   446
lemma of_num_less_eq_one_iff [numeral]: "of_num n \<le> 1 \<longleftrightarrow> n = One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   447
proof -
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   448
  have "of_num n \<le> of_num One \<longleftrightarrow> n = One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   449
    by (cases n) (simp_all add: of_num_less_eq_iff)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   450
  then show ?thesis by (simp add: of_num_one)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   451
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   452
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   453
lemma one_less_eq_of_num_iff [numeral]: "1 \<le> of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   454
proof -
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   455
  have "of_num One \<le> of_num n"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   456
    by (cases n) (simp_all add: of_num_less_eq_iff)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   457
  then show ?thesis by (simp add: of_num_one)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   458
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   459
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   460
lemma of_num_less_iff [numeral]: "of_num m < of_num n \<longleftrightarrow> m < n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   461
proof -
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   462
  have "of_nat (of_num m) < of_nat (of_num n) \<longleftrightarrow> m < n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   463
    unfolding less_num_def nat_of_num_of_num of_nat_less_iff ..
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   464
  then show ?thesis by (simp add: of_nat_of_num)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   465
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   466
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   467
lemma of_num_less_one_iff [numeral]: "\<not> of_num n < 1"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   468
proof -
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   469
  have "\<not> of_num n < of_num One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   470
    by (cases n) (simp_all add: of_num_less_iff)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   471
  then show ?thesis by (simp add: of_num_one)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   472
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   473
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   474
lemma one_less_of_num_iff [numeral]: "1 < of_num n \<longleftrightarrow> n \<noteq> One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   475
proof -
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   476
  have "of_num One < of_num n \<longleftrightarrow> n \<noteq> One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   477
    by (cases n) (simp_all add: of_num_less_iff)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   478
  then show ?thesis by (simp add: of_num_one)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   479
qed
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   480
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   481
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   482
29945
75df553549b1 rearrange subsections
huffman
parents: 29944
diff changeset
   483
subsubsection {*
75df553549b1 rearrange subsections
huffman
parents: 29944
diff changeset
   484
  Structures with subtraction @{term "op -"}: class @{text semiring_1_minus}
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   485
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   486
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   487
class semiring_minus = semiring + minus + zero +
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   488
  assumes minus_inverts_plus1: "a + b = c \<Longrightarrow> c - b = a"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   489
  assumes minus_minus_zero_inverts_plus1: "a + b = c \<Longrightarrow> b - c = 0 - a"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   490
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   491
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   492
lemma minus_inverts_plus2: "a + b = c \<Longrightarrow> c - a = b"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   493
  by (simp add: add_ac minus_inverts_plus1 [of b a])
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   494
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   495
lemma minus_minus_zero_inverts_plus2: "a + b = c \<Longrightarrow> a - c = 0 - b"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   496
  by (simp add: add_ac minus_minus_zero_inverts_plus1 [of b a])
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   497
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   498
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   499
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   500
class semiring_1_minus = semiring_1 + semiring_minus
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   501
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   502
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   503
lemma Dig_of_num_pos:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   504
  assumes "k + n = m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   505
  shows "of_num m - of_num n = of_num k"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   506
  using assms by (simp add: of_num_plus minus_inverts_plus1)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   507
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   508
lemma Dig_of_num_zero:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   509
  shows "of_num n - of_num n = 0"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   510
  by (rule minus_inverts_plus1) simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   511
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   512
lemma Dig_of_num_neg:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   513
  assumes "k + m = n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   514
  shows "of_num m - of_num n = 0 - of_num k"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   515
  by (rule minus_minus_zero_inverts_plus1) (simp add: of_num_plus assms)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   516
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   517
lemmas Dig_plus_eval =
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   518
  of_num_plus of_num_eq_iff Dig_plus refl [of One, THEN eqTrueI] num.inject
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   519
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   520
simproc_setup numeral_minus ("of_num m - of_num n") = {*
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   521
  let
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   522
    (*TODO proper implicit use of morphism via pattern antiquotations*)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   523
    fun cdest_of_num ct = (snd o split_last o snd o Drule.strip_comb) ct;
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   524
    fun cdest_minus ct = case (rev o snd o Drule.strip_comb) ct of [n, m] => (m, n);
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   525
    fun attach_num ct = (dest_num (Thm.term_of ct), ct);
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   526
    fun cdifference t = (pairself (attach_num o cdest_of_num) o cdest_minus) t;
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   527
    val simplify = MetaSimplifier.rewrite false (map mk_meta_eq @{thms Dig_plus_eval});
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   528
    fun cert ck cl cj = @{thm eqTrueE} OF [@{thm meta_eq_to_obj_eq} OF [simplify (Drule.list_comb (@{cterm "op = :: num \<Rightarrow> _"},
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   529
      [Drule.list_comb (@{cterm "op + :: num \<Rightarrow> _"}, [ck, cl]), cj]))]];
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   530
  in fn phi => fn _ => fn ct => case try cdifference ct
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   531
   of NONE => (NONE)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   532
    | SOME ((k, ck), (l, cl)) => SOME (let val j = k - l in if j = 0
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   533
        then MetaSimplifier.rewrite false [mk_meta_eq (Morphism.thm phi @{thm Dig_of_num_zero})] ct
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   534
        else mk_meta_eq (let
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   535
          val cj = Thm.cterm_of (Thm.theory_of_cterm ct) (mk_num (abs j));
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   536
        in
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   537
          (if j > 0 then (Morphism.thm phi @{thm Dig_of_num_pos}) OF [cert cj cl ck]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   538
          else (Morphism.thm phi @{thm Dig_of_num_neg}) OF [cert cj ck cl])
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   539
        end) end)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   540
  end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   541
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   542
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   543
lemma Dig_of_num_minus_zero [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   544
  "of_num n - 0 = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   545
  by (simp add: minus_inverts_plus1)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   546
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   547
lemma Dig_one_minus_zero [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   548
  "1 - 0 = 1"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   549
  by (simp add: minus_inverts_plus1)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   550
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   551
lemma Dig_one_minus_one [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   552
  "1 - 1 = 0"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   553
  by (simp add: minus_inverts_plus1)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   554
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   555
lemma Dig_of_num_minus_one [numeral]:
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   556
  "of_num (Dig0 n) - 1 = of_num (DigM n)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   557
  "of_num (Dig1 n) - 1 = of_num (Dig0 n)"
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   558
  by (auto intro: minus_inverts_plus1 simp add: DigM_plus_one of_num.simps of_num_plus_one)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   559
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   560
lemma Dig_one_minus_of_num [numeral]:
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   561
  "1 - of_num (Dig0 n) = 0 - of_num (DigM n)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   562
  "1 - of_num (Dig1 n) = 0 - of_num (Dig0 n)"
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   563
  by (auto intro: minus_minus_zero_inverts_plus1 simp add: DigM_plus_one of_num.simps of_num_plus_one)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   564
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   565
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   566
29945
75df553549b1 rearrange subsections
huffman
parents: 29944
diff changeset
   567
subsubsection {*
75df553549b1 rearrange subsections
huffman
parents: 29944
diff changeset
   568
  Negation: class @{text ring_1}
75df553549b1 rearrange subsections
huffman
parents: 29944
diff changeset
   569
*}
75df553549b1 rearrange subsections
huffman
parents: 29944
diff changeset
   570
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   571
context ring_1
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   572
begin
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   573
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   574
subclass semiring_1_minus
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28823
diff changeset
   575
  proof qed (simp_all add: algebra_simps)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   576
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   577
lemma Dig_zero_minus_of_num [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   578
  "0 - of_num n = - of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   579
  by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   580
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   581
lemma Dig_zero_minus_one [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   582
  "0 - 1 = - 1"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   583
  by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   584
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   585
lemma Dig_uminus_uminus [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   586
  "- (- of_num n) = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   587
  by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   588
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   589
lemma Dig_plus_uminus [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   590
  "of_num m + - of_num n = of_num m - of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   591
  "- of_num m + of_num n = of_num n - of_num m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   592
  "- of_num m + - of_num n = - (of_num m + of_num n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   593
  "of_num m - - of_num n = of_num m + of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   594
  "- of_num m - of_num n = - (of_num m + of_num n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   595
  "- of_num m - - of_num n = of_num n - of_num m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   596
  by (simp_all add: diff_minus add_commute)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   597
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   598
lemma Dig_times_uminus [numeral]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   599
  "- of_num n * of_num m = - (of_num n * of_num m)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   600
  "of_num n * - of_num m = - (of_num n * of_num m)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   601
  "- of_num n * - of_num m = of_num n * of_num m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   602
  by (simp_all add: minus_mult_left [symmetric] minus_mult_right [symmetric])
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   603
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   604
lemma of_int_of_num [numeral]: "of_int (of_num n) = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   605
by (induct n)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   606
  (simp_all only: of_num.simps semiring_numeral_class.of_num.simps of_int_add, simp_all)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   607
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   608
declare of_int_1 [numeral]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   609
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   610
end
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   611
29945
75df553549b1 rearrange subsections
huffman
parents: 29944
diff changeset
   612
subsubsection {*
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   613
  Greetings to @{typ nat}.
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   614
*}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   615
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   616
instance nat :: semiring_1_minus proof qed simp_all
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   617
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   618
lemma Suc_of_num [numeral]: "Suc (of_num n) = of_num (n + One)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   619
  unfolding of_num_plus_one [symmetric] by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   620
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   621
lemma nat_number:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   622
  "1 = Suc 0"
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   623
  "of_num One = Suc 0"
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   624
  "of_num (Dig0 n) = Suc (of_num (DigM n))"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   625
  "of_num (Dig1 n) = Suc (of_num (Dig0 n))"
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   626
  by (simp_all add: of_num.simps DigM_plus_one Suc_of_num)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   627
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   628
declare diff_0_eq_0 [numeral]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   629
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   630
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   631
subsection {* Code generator setup for @{typ int} *}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   632
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   633
definition Pls :: "num \<Rightarrow> int" where
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   634
  [simp, code post]: "Pls n = of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   635
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   636
definition Mns :: "num \<Rightarrow> int" where
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   637
  [simp, code post]: "Mns n = - of_num n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   638
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   639
code_datatype "0::int" Pls Mns
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   640
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   641
lemmas [code inline] = Pls_def [symmetric] Mns_def [symmetric]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   642
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   643
definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28367
diff changeset
   644
  [simp, code del]: "sub m n = (of_num m - of_num n)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   645
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   646
definition dup :: "int \<Rightarrow> int" where
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28367
diff changeset
   647
  [code del]: "dup k = 2 * k"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   648
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   649
lemma Dig_sub [code]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   650
  "sub One One = 0"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   651
  "sub (Dig0 m) One = of_num (DigM m)"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   652
  "sub (Dig1 m) One = of_num (Dig0 m)"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   653
  "sub One (Dig0 n) = - of_num (DigM n)"
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   654
  "sub One (Dig1 n) = - of_num (Dig0 n)"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   655
  "sub (Dig0 m) (Dig0 n) = dup (sub m n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   656
  "sub (Dig1 m) (Dig1 n) = dup (sub m n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   657
  "sub (Dig1 m) (Dig0 n) = dup (sub m n) + 1"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   658
  "sub (Dig0 m) (Dig1 n) = dup (sub m n) - 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28823
diff changeset
   659
  apply (simp_all add: dup_def algebra_simps)
29941
b951d80774d5 replace dec with double-and-decrement function
huffman
parents: 29667
diff changeset
   660
  apply (simp_all add: of_num_plus one_plus_DigM)[4]
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   661
  apply (simp_all add: of_num.simps)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   662
  done
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   663
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   664
lemma dup_code [code]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   665
  "dup 0 = 0"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   666
  "dup (Pls n) = Pls (Dig0 n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   667
  "dup (Mns n) = Mns (Dig0 n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   668
  by (simp_all add: dup_def of_num.simps)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   669
  
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28367
diff changeset
   670
lemma [code, code del]:
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   671
  "(1 :: int) = 1"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   672
  "(op + :: int \<Rightarrow> int \<Rightarrow> int) = op +"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   673
  "(uminus :: int \<Rightarrow> int) = uminus"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   674
  "(op - :: int \<Rightarrow> int \<Rightarrow> int) = op -"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   675
  "(op * :: int \<Rightarrow> int \<Rightarrow> int) = op *"
28367
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   676
  "(eq_class.eq :: int \<Rightarrow> int \<Rightarrow> bool) = eq_class.eq"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   677
  "(op \<le> :: int \<Rightarrow> int \<Rightarrow> bool) = op \<le>"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   678
  "(op < :: int \<Rightarrow> int \<Rightarrow> bool) = op <"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   679
  by rule+
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   680
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   681
lemma one_int_code [code]:
29942
31069b8d39df replace 1::num with One; remove monoid_mult instance
huffman
parents: 29941
diff changeset
   682
  "1 = Pls One"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   683
  by (simp add: of_num_one)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   684
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   685
lemma plus_int_code [code]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   686
  "k + 0 = (k::int)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   687
  "0 + l = (l::int)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   688
  "Pls m + Pls n = Pls (m + n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   689
  "Pls m - Pls n = sub m n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   690
  "Mns m + Mns n = Mns (m + n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   691
  "Mns m - Mns n = sub n m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   692
  by (simp_all add: of_num_plus [symmetric])
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   693
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   694
lemma uminus_int_code [code]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   695
  "uminus 0 = (0::int)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   696
  "uminus (Pls m) = Mns m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   697
  "uminus (Mns m) = Pls m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   698
  by simp_all
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   699
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   700
lemma minus_int_code [code]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   701
  "k - 0 = (k::int)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   702
  "0 - l = uminus (l::int)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   703
  "Pls m - Pls n = sub m n"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   704
  "Pls m - Mns n = Pls (m + n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   705
  "Mns m - Pls n = Mns (m + n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   706
  "Mns m - Mns n = sub n m"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   707
  by (simp_all add: of_num_plus [symmetric])
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   708
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   709
lemma times_int_code [code]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   710
  "k * 0 = (0::int)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   711
  "0 * l = (0::int)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   712
  "Pls m * Pls n = Pls (m * n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   713
  "Pls m * Mns n = Mns (m * n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   714
  "Mns m * Pls n = Mns (m * n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   715
  "Mns m * Mns n = Pls (m * n)"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   716
  by (simp_all add: of_num_times [symmetric])
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   717
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   718
lemma eq_int_code [code]:
28367
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   719
  "eq_class.eq 0 (0::int) \<longleftrightarrow> True"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   720
  "eq_class.eq 0 (Pls l) \<longleftrightarrow> False"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   721
  "eq_class.eq 0 (Mns l) \<longleftrightarrow> False"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   722
  "eq_class.eq (Pls k) 0 \<longleftrightarrow> False"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   723
  "eq_class.eq (Pls k) (Pls l) \<longleftrightarrow> eq_class.eq k l"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   724
  "eq_class.eq (Pls k) (Mns l) \<longleftrightarrow> False"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   725
  "eq_class.eq (Mns k) 0 \<longleftrightarrow> False"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   726
  "eq_class.eq (Mns k) (Pls l) \<longleftrightarrow> False"
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   727
  "eq_class.eq (Mns k) (Mns l) \<longleftrightarrow> eq_class.eq k l"
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   728
  using of_num_pos [of l, where ?'a = int] of_num_pos [of k, where ?'a = int]
28367
10ea34297962 op = vs. eq
haftmann
parents: 28053
diff changeset
   729
  by (simp_all add: of_num_eq_iff eq)
28021
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   730
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   731
lemma less_eq_int_code [code]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   732
  "0 \<le> (0::int) \<longleftrightarrow> True"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   733
  "0 \<le> Pls l \<longleftrightarrow> True"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   734
  "0 \<le> Mns l \<longleftrightarrow> False"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   735
  "Pls k \<le> 0 \<longleftrightarrow> False"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   736
  "Pls k \<le> Pls l \<longleftrightarrow> k \<le> l"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   737
  "Pls k \<le> Mns l \<longleftrightarrow> False"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   738
  "Mns k \<le> 0 \<longleftrightarrow> True"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   739
  "Mns k \<le> Pls l \<longleftrightarrow> True"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   740
  "Mns k \<le> Mns l \<longleftrightarrow> l \<le> k"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   741
  using of_num_pos [of l, where ?'a = int] of_num_pos [of k, where ?'a = int]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   742
  by (simp_all add: of_num_less_eq_iff)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   743
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   744
lemma less_int_code [code]:
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   745
  "0 < (0::int) \<longleftrightarrow> False"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   746
  "0 < Pls l \<longleftrightarrow> True"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   747
  "0 < Mns l \<longleftrightarrow> False"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   748
  "Pls k < 0 \<longleftrightarrow> False"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   749
  "Pls k < Pls l \<longleftrightarrow> k < l"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   750
  "Pls k < Mns l \<longleftrightarrow> False"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   751
  "Mns k < 0 \<longleftrightarrow> True"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   752
  "Mns k < Pls l \<longleftrightarrow> True"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   753
  "Mns k < Mns l \<longleftrightarrow> l < k"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   754
  using of_num_pos [of l, where ?'a = int] of_num_pos [of k, where ?'a = int]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   755
  by (simp_all add: of_num_less_iff)
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   756
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   757
lemma [code inline del]: "(0::int) \<equiv> Numeral0" by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   758
lemma [code inline del]: "(1::int) \<equiv> Numeral1" by simp
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   759
declare zero_is_num_zero [code inline del]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   760
declare one_is_num_one [code inline del]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   761
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   762
hide (open) const sub dup
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   763
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   764
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   765
subsection {* Numeral equations as default simplification rules *}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   766
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   767
text {* TODO.  Be more precise here with respect to subsumed facts. *}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   768
declare (in semiring_numeral) numeral [simp]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   769
declare (in semiring_1) numeral [simp]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   770
declare (in semiring_char_0) numeral [simp]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   771
declare (in ring_1) numeral [simp]
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   772
thm numeral
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   773
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   774
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   775
text {* Toy examples *}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   776
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   777
definition "bar \<longleftrightarrow> #4 * #2 + #7 = (#8 :: nat) \<and> #4 * #2 + #7 \<ge> (#8 :: int) - #3"
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   778
code_thms bar
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   779
export_code bar in Haskell file -
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   780
export_code bar in OCaml module_name Foo file -
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   781
ML {* @{code bar} *}
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   782
32acf3c6cd12 added HOL/ex/Numeral.thy
haftmann
parents:
diff changeset
   783
end