src/HOL/Num.thy
author wenzelm
Sun Nov 02 18:21:45 2014 +0100 (2014-11-02)
changeset 58889 5b7a9633cfa8
parent 58512 dc4d76dfa8f0
child 59621 291934bac95e
permissions -rw-r--r--
modernized header uniformly as section;
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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 {* Binary Numerals *}
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theory Num
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imports BNF_Least_Fixpoint
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begin
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subsection {* The @{text num} type *}
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datatype num = One | Bit0 num | Bit1 num
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text {* Increment function for type @{typ num} *}
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primrec inc :: "num \<Rightarrow> num" 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 {* Converting between type @{typ num} and type @{typ nat} *}
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primrec nat_of_num :: "num \<Rightarrow> nat" 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" 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) = Bit0 (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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*}
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subsection {* Numeral operations *}
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instantiation num :: "{plus,times,linorder}"
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begin
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definition [code del]:
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  "m + n = num_of_nat (nat_of_num m + nat_of_num n)"
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definition [code del]:
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  "m * n = num_of_nat (nat_of_num m * nat_of_num n)"
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definition [code del]:
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  "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"
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definition [code del]:
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  "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"
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instance
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  by (default, 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
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    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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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_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 {* The @{const num_of_nat} conversion *}
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lemma num_of_nat_One:
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  "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 {* A double-and-decrement function *}
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primrec BitM :: "num \<Rightarrow> num" 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 {* Squaring and exponentiation *}
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primrec sqr :: "num \<Rightarrow> num" 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" 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 {* 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 numeral}.
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*}
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class numeral = one + semigroup_add
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begin
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primrec numeral :: "num \<Rightarrow> 'a" 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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  apply (induct x)
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  apply simp
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  apply (simp add: add.assoc [symmetric], simp add: add.assoc)
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  apply (simp add: add.assoc [symmetric], simp add: add.assoc)
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  done
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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 (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 simp_all
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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 {* Numeral syntax. *}
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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 {*
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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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*}
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typed_print_translation {*
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  let
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    fun dest_num (Const (@{const_syntax Bit0}, _) $ n) = 2 * dest_num n
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      | dest_num (Const (@{const_syntax Bit1}, _) $ n) = 2 * dest_num n + 1
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      | dest_num (Const (@{const_syntax One}, _)) = 1;
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    fun num_tr' ctxt T [n] =
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      let
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        val k = dest_num 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
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*}
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subsection {* Class-specific numeral rules *}
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text {*
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  @{const numeral} is a morphism.
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*}
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subsubsection {* Structures with addition: class @{text numeral} *}
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context numeral
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begin
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lemma numeral_add: "numeral (m + n) = numeral m + numeral n"
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  by (induct n rule: num_induct)
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   334
     (simp_all only: numeral_One add_One add_inc numeral_inc add.assoc)
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   336
lemma numeral_plus_numeral: "numeral m + numeral n = numeral (m + n)"
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  by (rule numeral_add [symmetric])
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   338
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   339
lemma numeral_plus_one: "numeral n + 1 = numeral (n + One)"
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  using numeral_add [of n One] by (simp add: numeral_One)
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   341
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   342
lemma one_plus_numeral: "1 + numeral n = numeral (One + n)"
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   343
  using numeral_add [of One n] by (simp add: numeral_One)
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   345
lemma one_add_one: "1 + 1 = 2"
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  using numeral_add [of One One] by (simp add: numeral_One)
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   348
lemmas add_numeral_special =
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  numeral_plus_one one_plus_numeral one_add_one
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huffman@47108
   351
end
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   353
subsubsection {*
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  Structures with negation: class @{text neg_numeral}
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   355
*}
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   357
class neg_numeral = numeral + group_add
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   358
begin
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   359
haftmann@54489
   360
lemma uminus_numeral_One:
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  "- Numeral1 = - 1"
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   362
  by (simp add: numeral_One)
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   363
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   364
text {* Numerals form an abelian subgroup. *}
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   365
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   366
inductive is_num :: "'a \<Rightarrow> bool" where
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  "is_num 1" |
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   368
  "is_num x \<Longrightarrow> is_num (- x)" |
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   369
  "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> is_num (x + y)"
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   370
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   371
lemma is_num_numeral: "is_num (numeral k)"
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  by (induct k, simp_all add: numeral.simps is_num.intros)
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   373
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   374
lemma is_num_add_commute:
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  "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + y = y + x"
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   376
  apply (induct x rule: is_num.induct)
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   377
  apply (induct y rule: is_num.induct)
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  apply simp
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  apply (rule_tac a=x in add_left_imp_eq)
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  apply (rule_tac a=x in add_right_imp_eq)
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   381
  apply (simp add: add.assoc)
haftmann@57512
   382
  apply (simp add: add.assoc [symmetric], simp add: add.assoc)
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  apply (rule_tac a=x in add_left_imp_eq)
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  apply (rule_tac a=x in add_right_imp_eq)
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   385
  apply (simp add: add.assoc)
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   386
  apply (simp add: add.assoc, simp add: add.assoc [symmetric])
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   387
  done
huffman@47108
   388
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   389
lemma is_num_add_left_commute:
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   390
  "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + (y + z) = y + (x + z)"
haftmann@57512
   391
  by (simp only: add.assoc [symmetric] is_num_add_commute)
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   392
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   393
lemmas is_num_normalize =
haftmann@57512
   394
  add.assoc is_num_add_commute is_num_add_left_commute
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   395
  is_num.intros is_num_numeral
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  minus_add
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   397
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   398
definition dbl :: "'a \<Rightarrow> 'a" where "dbl x = x + x"
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definition dbl_inc :: "'a \<Rightarrow> 'a" where "dbl_inc x = x + x + 1"
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   400
definition dbl_dec :: "'a \<Rightarrow> 'a" where "dbl_dec x = x + x - 1"
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   401
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   402
definition sub :: "num \<Rightarrow> num \<Rightarrow> 'a" where
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   403
  "sub k l = numeral k - numeral l"
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   404
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   405
lemma numeral_BitM: "numeral (BitM n) = numeral (Bit0 n) - 1"
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   406
  by (simp only: BitM_plus_one [symmetric] numeral_add numeral_One eq_diff_eq)
huffman@47108
   407
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   408
lemma dbl_simps [simp]:
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   409
  "dbl (- numeral k) = - dbl (numeral k)"
huffman@47108
   410
  "dbl 0 = 0"
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   411
  "dbl 1 = 2"
haftmann@54489
   412
  "dbl (- 1) = - 2"
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   413
  "dbl (numeral k) = numeral (Bit0 k)"
haftmann@54489
   414
  by (simp_all add: dbl_def numeral.simps minus_add)
huffman@47108
   415
huffman@47108
   416
lemma dbl_inc_simps [simp]:
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   417
  "dbl_inc (- numeral k) = - dbl_dec (numeral k)"
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   418
  "dbl_inc 0 = 1"
huffman@47108
   419
  "dbl_inc 1 = 3"
haftmann@54489
   420
  "dbl_inc (- 1) = - 1"
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   421
  "dbl_inc (numeral k) = numeral (Bit1 k)"
haftmann@54489
   422
  by (simp_all add: dbl_inc_def dbl_dec_def numeral.simps numeral_BitM is_num_normalize algebra_simps del: add_uminus_conv_diff)
huffman@47108
   423
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   424
lemma dbl_dec_simps [simp]:
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   425
  "dbl_dec (- numeral k) = - dbl_inc (numeral k)"
haftmann@54489
   426
  "dbl_dec 0 = - 1"
huffman@47108
   427
  "dbl_dec 1 = 1"
haftmann@54489
   428
  "dbl_dec (- 1) = - 3"
huffman@47108
   429
  "dbl_dec (numeral k) = numeral (BitM k)"
haftmann@54489
   430
  by (simp_all add: dbl_dec_def dbl_inc_def numeral.simps numeral_BitM is_num_normalize)
huffman@47108
   431
huffman@47108
   432
lemma sub_num_simps [simp]:
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   433
  "sub One One = 0"
haftmann@54489
   434
  "sub One (Bit0 l) = - numeral (BitM l)"
haftmann@54489
   435
  "sub One (Bit1 l) = - numeral (Bit0 l)"
huffman@47108
   436
  "sub (Bit0 k) One = numeral (BitM k)"
huffman@47108
   437
  "sub (Bit1 k) One = numeral (Bit0 k)"
huffman@47108
   438
  "sub (Bit0 k) (Bit0 l) = dbl (sub k l)"
huffman@47108
   439
  "sub (Bit0 k) (Bit1 l) = dbl_dec (sub k l)"
huffman@47108
   440
  "sub (Bit1 k) (Bit0 l) = dbl_inc (sub k l)"
huffman@47108
   441
  "sub (Bit1 k) (Bit1 l) = dbl (sub k l)"
haftmann@54489
   442
  by (simp_all add: dbl_def dbl_dec_def dbl_inc_def sub_def numeral.simps
haftmann@54230
   443
    numeral_BitM is_num_normalize del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   444
huffman@47108
   445
lemma add_neg_numeral_simps:
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   446
  "numeral m + - numeral n = sub m n"
haftmann@54489
   447
  "- numeral m + numeral n = sub n m"
haftmann@54489
   448
  "- numeral m + - numeral n = - (numeral m + numeral n)"
haftmann@54489
   449
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
haftmann@54230
   450
    del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   451
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   452
lemma add_neg_numeral_special:
haftmann@54489
   453
  "1 + - numeral m = sub One m"
haftmann@54489
   454
  "- numeral m + 1 = sub One m"
haftmann@54489
   455
  "numeral m + - 1 = sub m One"
haftmann@54489
   456
  "- 1 + numeral n = sub n One"
haftmann@54489
   457
  "- 1 + - numeral n = - numeral (inc n)"
haftmann@54489
   458
  "- numeral m + - 1 = - numeral (inc m)"
haftmann@54489
   459
  "1 + - 1 = 0"
haftmann@54489
   460
  "- 1 + 1 = 0"
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   461
  "- 1 + - 1 = - 2"
haftmann@54489
   462
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize right_minus numeral_inc
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   463
    del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   464
huffman@47108
   465
lemma diff_numeral_simps:
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   466
  "numeral m - numeral n = sub m n"
haftmann@54489
   467
  "numeral m - - numeral n = numeral (m + n)"
haftmann@54489
   468
  "- numeral m - numeral n = - numeral (m + n)"
haftmann@54489
   469
  "- numeral m - - numeral n = sub n m"
haftmann@54489
   470
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
haftmann@54230
   471
    del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   472
huffman@47108
   473
lemma diff_numeral_special:
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   474
  "1 - numeral n = sub One n"
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   475
  "numeral m - 1 = sub m One"
haftmann@54489
   476
  "1 - - numeral n = numeral (One + n)"
haftmann@54489
   477
  "- numeral m - 1 = - numeral (m + One)"
haftmann@54489
   478
  "- 1 - numeral n = - numeral (inc n)"
haftmann@54489
   479
  "numeral m - - 1 = numeral (inc m)"
haftmann@54489
   480
  "- 1 - - numeral n = sub n One"
haftmann@54489
   481
  "- numeral m - - 1 = sub One m"
haftmann@54489
   482
  "1 - 1 = 0"
haftmann@54489
   483
  "- 1 - 1 = - 2"
haftmann@54489
   484
  "1 - - 1 = 2"
haftmann@54489
   485
  "- 1 - - 1 = 0"
haftmann@54489
   486
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize numeral_inc
haftmann@54489
   487
    del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   488
huffman@47108
   489
end
huffman@47108
   490
huffman@47108
   491
subsubsection {*
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   492
  Structures with multiplication: class @{text semiring_numeral}
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   493
*}
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   494
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   495
class semiring_numeral = semiring + monoid_mult
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   496
begin
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   497
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   498
subclass numeral ..
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   499
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   500
lemma numeral_mult: "numeral (m * n) = numeral m * numeral n"
huffman@47108
   501
  apply (induct n rule: num_induct)
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   502
  apply (simp add: numeral_One)
webertj@49962
   503
  apply (simp add: mult_inc numeral_inc numeral_add distrib_left)
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   504
  done
huffman@47108
   505
huffman@47108
   506
lemma numeral_times_numeral: "numeral m * numeral n = numeral (m * n)"
huffman@47108
   507
  by (rule numeral_mult [symmetric])
huffman@47108
   508
haftmann@53064
   509
lemma mult_2: "2 * z = z + z"
haftmann@53064
   510
  unfolding one_add_one [symmetric] distrib_right by simp
haftmann@53064
   511
haftmann@53064
   512
lemma mult_2_right: "z * 2 = z + z"
haftmann@53064
   513
  unfolding one_add_one [symmetric] distrib_left by simp
haftmann@53064
   514
huffman@47108
   515
end
huffman@47108
   516
huffman@47108
   517
subsubsection {*
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   518
  Structures with a zero: class @{text semiring_1}
huffman@47108
   519
*}
huffman@47108
   520
huffman@47108
   521
context semiring_1
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   522
begin
huffman@47108
   523
huffman@47108
   524
subclass semiring_numeral ..
huffman@47108
   525
huffman@47108
   526
lemma of_nat_numeral [simp]: "of_nat (numeral n) = numeral n"
huffman@47108
   527
  by (induct n,
huffman@47108
   528
    simp_all only: numeral.simps numeral_class.numeral.simps of_nat_add of_nat_1)
huffman@47108
   529
huffman@47108
   530
end
huffman@47108
   531
haftmann@51143
   532
lemma nat_of_num_numeral [code_abbrev]:
haftmann@51143
   533
  "nat_of_num = numeral"
huffman@47108
   534
proof
huffman@47108
   535
  fix n
huffman@47108
   536
  have "numeral n = nat_of_num n"
huffman@47108
   537
    by (induct n) (simp_all add: numeral.simps)
huffman@47108
   538
  then show "nat_of_num n = numeral n" by simp
huffman@47108
   539
qed
huffman@47108
   540
haftmann@51143
   541
lemma nat_of_num_code [code]:
haftmann@51143
   542
  "nat_of_num One = 1"
haftmann@51143
   543
  "nat_of_num (Bit0 n) = (let m = nat_of_num n in m + m)"
haftmann@51143
   544
  "nat_of_num (Bit1 n) = (let m = nat_of_num n in Suc (m + m))"
haftmann@51143
   545
  by (simp_all add: Let_def)
haftmann@51143
   546
huffman@47108
   547
subsubsection {*
huffman@47108
   548
  Equality: class @{text semiring_char_0}
huffman@47108
   549
*}
huffman@47108
   550
huffman@47108
   551
context semiring_char_0
huffman@47108
   552
begin
huffman@47108
   553
huffman@47108
   554
lemma numeral_eq_iff: "numeral m = numeral n \<longleftrightarrow> m = n"
huffman@47108
   555
  unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
huffman@47108
   556
    of_nat_eq_iff num_eq_iff ..
huffman@47108
   557
huffman@47108
   558
lemma numeral_eq_one_iff: "numeral n = 1 \<longleftrightarrow> n = One"
huffman@47108
   559
  by (rule numeral_eq_iff [of n One, unfolded numeral_One])
huffman@47108
   560
huffman@47108
   561
lemma one_eq_numeral_iff: "1 = numeral n \<longleftrightarrow> One = n"
huffman@47108
   562
  by (rule numeral_eq_iff [of One n, unfolded numeral_One])
huffman@47108
   563
huffman@47108
   564
lemma numeral_neq_zero: "numeral n \<noteq> 0"
huffman@47108
   565
  unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
huffman@47108
   566
  by (simp add: nat_of_num_pos)
huffman@47108
   567
huffman@47108
   568
lemma zero_neq_numeral: "0 \<noteq> numeral n"
huffman@47108
   569
  unfolding eq_commute [of 0] by (rule numeral_neq_zero)
huffman@47108
   570
huffman@47108
   571
lemmas eq_numeral_simps [simp] =
huffman@47108
   572
  numeral_eq_iff
huffman@47108
   573
  numeral_eq_one_iff
huffman@47108
   574
  one_eq_numeral_iff
huffman@47108
   575
  numeral_neq_zero
huffman@47108
   576
  zero_neq_numeral
huffman@47108
   577
huffman@47108
   578
end
huffman@47108
   579
huffman@47108
   580
subsubsection {*
huffman@47108
   581
  Comparisons: class @{text linordered_semidom}
huffman@47108
   582
*}
huffman@47108
   583
huffman@47108
   584
text {*  Could be perhaps more general than here. *}
huffman@47108
   585
huffman@47108
   586
context linordered_semidom
huffman@47108
   587
begin
huffman@47108
   588
huffman@47108
   589
lemma numeral_le_iff: "numeral m \<le> numeral n \<longleftrightarrow> m \<le> n"
huffman@47108
   590
proof -
huffman@47108
   591
  have "of_nat (numeral m) \<le> of_nat (numeral n) \<longleftrightarrow> m \<le> n"
huffman@47108
   592
    unfolding less_eq_num_def nat_of_num_numeral of_nat_le_iff ..
huffman@47108
   593
  then show ?thesis by simp
huffman@47108
   594
qed
huffman@47108
   595
huffman@47108
   596
lemma one_le_numeral: "1 \<le> numeral n"
huffman@47108
   597
using numeral_le_iff [of One n] by (simp add: numeral_One)
huffman@47108
   598
huffman@47108
   599
lemma numeral_le_one_iff: "numeral n \<le> 1 \<longleftrightarrow> n \<le> One"
huffman@47108
   600
using numeral_le_iff [of n One] by (simp add: numeral_One)
huffman@47108
   601
huffman@47108
   602
lemma numeral_less_iff: "numeral m < numeral n \<longleftrightarrow> m < n"
huffman@47108
   603
proof -
huffman@47108
   604
  have "of_nat (numeral m) < of_nat (numeral n) \<longleftrightarrow> m < n"
huffman@47108
   605
    unfolding less_num_def nat_of_num_numeral of_nat_less_iff ..
huffman@47108
   606
  then show ?thesis by simp
huffman@47108
   607
qed
huffman@47108
   608
huffman@47108
   609
lemma not_numeral_less_one: "\<not> numeral n < 1"
huffman@47108
   610
  using numeral_less_iff [of n One] by (simp add: numeral_One)
huffman@47108
   611
huffman@47108
   612
lemma one_less_numeral_iff: "1 < numeral n \<longleftrightarrow> One < n"
huffman@47108
   613
  using numeral_less_iff [of One n] by (simp add: numeral_One)
huffman@47108
   614
huffman@47108
   615
lemma zero_le_numeral: "0 \<le> numeral n"
huffman@47108
   616
  by (induct n) (simp_all add: numeral.simps)
huffman@47108
   617
huffman@47108
   618
lemma zero_less_numeral: "0 < numeral n"
huffman@47108
   619
  by (induct n) (simp_all add: numeral.simps add_pos_pos)
huffman@47108
   620
huffman@47108
   621
lemma not_numeral_le_zero: "\<not> numeral n \<le> 0"
huffman@47108
   622
  by (simp add: not_le zero_less_numeral)
huffman@47108
   623
huffman@47108
   624
lemma not_numeral_less_zero: "\<not> numeral n < 0"
huffman@47108
   625
  by (simp add: not_less zero_le_numeral)
huffman@47108
   626
huffman@47108
   627
lemmas le_numeral_extra =
huffman@47108
   628
  zero_le_one not_one_le_zero
huffman@47108
   629
  order_refl [of 0] order_refl [of 1]
huffman@47108
   630
huffman@47108
   631
lemmas less_numeral_extra =
huffman@47108
   632
  zero_less_one not_one_less_zero
huffman@47108
   633
  less_irrefl [of 0] less_irrefl [of 1]
huffman@47108
   634
huffman@47108
   635
lemmas le_numeral_simps [simp] =
huffman@47108
   636
  numeral_le_iff
huffman@47108
   637
  one_le_numeral
huffman@47108
   638
  numeral_le_one_iff
huffman@47108
   639
  zero_le_numeral
huffman@47108
   640
  not_numeral_le_zero
huffman@47108
   641
huffman@47108
   642
lemmas less_numeral_simps [simp] =
huffman@47108
   643
  numeral_less_iff
huffman@47108
   644
  one_less_numeral_iff
huffman@47108
   645
  not_numeral_less_one
huffman@47108
   646
  zero_less_numeral
huffman@47108
   647
  not_numeral_less_zero
huffman@47108
   648
huffman@47108
   649
end
huffman@47108
   650
huffman@47108
   651
subsubsection {*
huffman@47108
   652
  Multiplication and negation: class @{text ring_1}
huffman@47108
   653
*}
huffman@47108
   654
huffman@47108
   655
context ring_1
huffman@47108
   656
begin
huffman@47108
   657
huffman@47108
   658
subclass neg_numeral ..
huffman@47108
   659
huffman@47108
   660
lemma mult_neg_numeral_simps:
haftmann@54489
   661
  "- numeral m * - numeral n = numeral (m * n)"
haftmann@54489
   662
  "- numeral m * numeral n = - numeral (m * n)"
haftmann@54489
   663
  "numeral m * - numeral n = - numeral (m * n)"
haftmann@54489
   664
  unfolding mult_minus_left mult_minus_right
huffman@47108
   665
  by (simp_all only: minus_minus numeral_mult)
huffman@47108
   666
haftmann@54489
   667
lemma mult_minus1 [simp]: "- 1 * z = - z"
haftmann@54489
   668
  unfolding numeral.simps mult_minus_left by simp
huffman@47108
   669
haftmann@54489
   670
lemma mult_minus1_right [simp]: "z * - 1 = - z"
haftmann@54489
   671
  unfolding numeral.simps mult_minus_right by simp
huffman@47108
   672
huffman@47108
   673
end
huffman@47108
   674
huffman@47108
   675
subsubsection {*
huffman@47108
   676
  Equality using @{text iszero} for rings with non-zero characteristic
huffman@47108
   677
*}
huffman@47108
   678
huffman@47108
   679
context ring_1
huffman@47108
   680
begin
huffman@47108
   681
huffman@47108
   682
definition iszero :: "'a \<Rightarrow> bool"
huffman@47108
   683
  where "iszero z \<longleftrightarrow> z = 0"
huffman@47108
   684
huffman@47108
   685
lemma iszero_0 [simp]: "iszero 0"
huffman@47108
   686
  by (simp add: iszero_def)
huffman@47108
   687
huffman@47108
   688
lemma not_iszero_1 [simp]: "\<not> iszero 1"
huffman@47108
   689
  by (simp add: iszero_def)
huffman@47108
   690
huffman@47108
   691
lemma not_iszero_Numeral1: "\<not> iszero Numeral1"
huffman@47108
   692
  by (simp add: numeral_One)
huffman@47108
   693
haftmann@54489
   694
lemma not_iszero_neg_1 [simp]: "\<not> iszero (- 1)"
haftmann@54489
   695
  by (simp add: iszero_def)
haftmann@54489
   696
haftmann@54489
   697
lemma not_iszero_neg_Numeral1: "\<not> iszero (- Numeral1)"
haftmann@54489
   698
  by (simp add: numeral_One)
haftmann@54489
   699
huffman@47108
   700
lemma iszero_neg_numeral [simp]:
haftmann@54489
   701
  "iszero (- numeral w) \<longleftrightarrow> iszero (numeral w)"
haftmann@54489
   702
  unfolding iszero_def
huffman@47108
   703
  by (rule neg_equal_0_iff_equal)
huffman@47108
   704
huffman@47108
   705
lemma eq_iff_iszero_diff: "x = y \<longleftrightarrow> iszero (x - y)"
huffman@47108
   706
  unfolding iszero_def by (rule eq_iff_diff_eq_0)
huffman@47108
   707
huffman@47108
   708
text {* The @{text "eq_numeral_iff_iszero"} lemmas are not declared
huffman@47108
   709
@{text "[simp]"} by default, because for rings of characteristic zero,
huffman@47108
   710
better simp rules are possible. For a type like integers mod @{text
huffman@47108
   711
"n"}, type-instantiated versions of these rules should be added to the
huffman@47108
   712
simplifier, along with a type-specific rule for deciding propositions
huffman@47108
   713
of the form @{text "iszero (numeral w)"}.
huffman@47108
   714
huffman@47108
   715
bh: Maybe it would not be so bad to just declare these as simp
huffman@47108
   716
rules anyway? I should test whether these rules take precedence over
huffman@47108
   717
the @{text "ring_char_0"} rules in the simplifier.
huffman@47108
   718
*}
huffman@47108
   719
huffman@47108
   720
lemma eq_numeral_iff_iszero:
huffman@47108
   721
  "numeral x = numeral y \<longleftrightarrow> iszero (sub x y)"
haftmann@54489
   722
  "numeral x = - numeral y \<longleftrightarrow> iszero (numeral (x + y))"
haftmann@54489
   723
  "- numeral x = numeral y \<longleftrightarrow> iszero (numeral (x + y))"
haftmann@54489
   724
  "- numeral x = - numeral y \<longleftrightarrow> iszero (sub y x)"
huffman@47108
   725
  "numeral x = 1 \<longleftrightarrow> iszero (sub x One)"
huffman@47108
   726
  "1 = numeral y \<longleftrightarrow> iszero (sub One y)"
haftmann@54489
   727
  "- numeral x = 1 \<longleftrightarrow> iszero (numeral (x + One))"
haftmann@54489
   728
  "1 = - numeral y \<longleftrightarrow> iszero (numeral (One + y))"
huffman@47108
   729
  "numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
huffman@47108
   730
  "0 = numeral y \<longleftrightarrow> iszero (numeral y)"
haftmann@54489
   731
  "- numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
haftmann@54489
   732
  "0 = - numeral y \<longleftrightarrow> iszero (numeral y)"
huffman@47108
   733
  unfolding eq_iff_iszero_diff diff_numeral_simps diff_numeral_special
huffman@47108
   734
  by simp_all
huffman@47108
   735
huffman@47108
   736
end
huffman@47108
   737
huffman@47108
   738
subsubsection {*
huffman@47108
   739
  Equality and negation: class @{text ring_char_0}
huffman@47108
   740
*}
huffman@47108
   741
huffman@47108
   742
class ring_char_0 = ring_1 + semiring_char_0
huffman@47108
   743
begin
huffman@47108
   744
huffman@47108
   745
lemma not_iszero_numeral [simp]: "\<not> iszero (numeral w)"
huffman@47108
   746
  by (simp add: iszero_def)
huffman@47108
   747
haftmann@54489
   748
lemma neg_numeral_eq_iff: "- numeral m = - numeral n \<longleftrightarrow> m = n"
haftmann@54489
   749
  by simp
huffman@47108
   750
haftmann@54489
   751
lemma numeral_neq_neg_numeral: "numeral m \<noteq> - numeral n"
haftmann@54489
   752
  unfolding eq_neg_iff_add_eq_0
huffman@47108
   753
  by (simp add: numeral_plus_numeral)
huffman@47108
   754
haftmann@54489
   755
lemma neg_numeral_neq_numeral: "- numeral m \<noteq> numeral n"
huffman@47108
   756
  by (rule numeral_neq_neg_numeral [symmetric])
huffman@47108
   757
haftmann@54489
   758
lemma zero_neq_neg_numeral: "0 \<noteq> - numeral n"
haftmann@54489
   759
  unfolding neg_0_equal_iff_equal by simp
huffman@47108
   760
haftmann@54489
   761
lemma neg_numeral_neq_zero: "- numeral n \<noteq> 0"
haftmann@54489
   762
  unfolding neg_equal_0_iff_equal by simp
huffman@47108
   763
haftmann@54489
   764
lemma one_neq_neg_numeral: "1 \<noteq> - numeral n"
huffman@47108
   765
  using numeral_neq_neg_numeral [of One n] by (simp add: numeral_One)
huffman@47108
   766
haftmann@54489
   767
lemma neg_numeral_neq_one: "- numeral n \<noteq> 1"
huffman@47108
   768
  using neg_numeral_neq_numeral [of n One] by (simp add: numeral_One)
huffman@47108
   769
haftmann@54489
   770
lemma neg_one_neq_numeral:
haftmann@54489
   771
  "- 1 \<noteq> numeral n"
haftmann@54489
   772
  using neg_numeral_neq_numeral [of One n] by (simp add: numeral_One)
haftmann@54489
   773
haftmann@54489
   774
lemma numeral_neq_neg_one:
haftmann@54489
   775
  "numeral n \<noteq> - 1"
haftmann@54489
   776
  using numeral_neq_neg_numeral [of n One] by (simp add: numeral_One)
haftmann@54489
   777
haftmann@54489
   778
lemma neg_one_eq_numeral_iff:
haftmann@54489
   779
  "- 1 = - numeral n \<longleftrightarrow> n = One"
haftmann@54489
   780
  using neg_numeral_eq_iff [of One n] by (auto simp add: numeral_One)
haftmann@54489
   781
haftmann@54489
   782
lemma numeral_eq_neg_one_iff:
haftmann@54489
   783
  "- numeral n = - 1 \<longleftrightarrow> n = One"
haftmann@54489
   784
  using neg_numeral_eq_iff [of n One] by (auto simp add: numeral_One)
haftmann@54489
   785
haftmann@54489
   786
lemma neg_one_neq_zero:
haftmann@54489
   787
  "- 1 \<noteq> 0"
haftmann@54489
   788
  by simp
haftmann@54489
   789
haftmann@54489
   790
lemma zero_neq_neg_one:
haftmann@54489
   791
  "0 \<noteq> - 1"
haftmann@54489
   792
  by simp
haftmann@54489
   793
haftmann@54489
   794
lemma neg_one_neq_one:
haftmann@54489
   795
  "- 1 \<noteq> 1"
haftmann@54489
   796
  using neg_numeral_neq_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
haftmann@54489
   797
haftmann@54489
   798
lemma one_neq_neg_one:
haftmann@54489
   799
  "1 \<noteq> - 1"
haftmann@54489
   800
  using numeral_neq_neg_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
haftmann@54489
   801
huffman@47108
   802
lemmas eq_neg_numeral_simps [simp] =
huffman@47108
   803
  neg_numeral_eq_iff
huffman@47108
   804
  numeral_neq_neg_numeral neg_numeral_neq_numeral
huffman@47108
   805
  one_neq_neg_numeral neg_numeral_neq_one
huffman@47108
   806
  zero_neq_neg_numeral neg_numeral_neq_zero
haftmann@54489
   807
  neg_one_neq_numeral numeral_neq_neg_one
haftmann@54489
   808
  neg_one_eq_numeral_iff numeral_eq_neg_one_iff
haftmann@54489
   809
  neg_one_neq_zero zero_neq_neg_one
haftmann@54489
   810
  neg_one_neq_one one_neq_neg_one
huffman@47108
   811
huffman@47108
   812
end
huffman@47108
   813
huffman@47108
   814
subsubsection {*
huffman@47108
   815
  Structures with negation and order: class @{text linordered_idom}
huffman@47108
   816
*}
huffman@47108
   817
huffman@47108
   818
context linordered_idom
huffman@47108
   819
begin
huffman@47108
   820
huffman@47108
   821
subclass ring_char_0 ..
huffman@47108
   822
haftmann@54489
   823
lemma neg_numeral_le_iff: "- numeral m \<le> - numeral n \<longleftrightarrow> n \<le> m"
haftmann@54489
   824
  by (simp only: neg_le_iff_le numeral_le_iff)
huffman@47108
   825
haftmann@54489
   826
lemma neg_numeral_less_iff: "- numeral m < - numeral n \<longleftrightarrow> n < m"
haftmann@54489
   827
  by (simp only: neg_less_iff_less numeral_less_iff)
huffman@47108
   828
haftmann@54489
   829
lemma neg_numeral_less_zero: "- numeral n < 0"
haftmann@54489
   830
  by (simp only: neg_less_0_iff_less zero_less_numeral)
huffman@47108
   831
haftmann@54489
   832
lemma neg_numeral_le_zero: "- numeral n \<le> 0"
haftmann@54489
   833
  by (simp only: neg_le_0_iff_le zero_le_numeral)
huffman@47108
   834
haftmann@54489
   835
lemma not_zero_less_neg_numeral: "\<not> 0 < - numeral n"
huffman@47108
   836
  by (simp only: not_less neg_numeral_le_zero)
huffman@47108
   837
haftmann@54489
   838
lemma not_zero_le_neg_numeral: "\<not> 0 \<le> - numeral n"
huffman@47108
   839
  by (simp only: not_le neg_numeral_less_zero)
huffman@47108
   840
haftmann@54489
   841
lemma neg_numeral_less_numeral: "- numeral m < numeral n"
huffman@47108
   842
  using neg_numeral_less_zero zero_less_numeral by (rule less_trans)
huffman@47108
   843
haftmann@54489
   844
lemma neg_numeral_le_numeral: "- numeral m \<le> numeral n"
huffman@47108
   845
  by (simp only: less_imp_le neg_numeral_less_numeral)
huffman@47108
   846
haftmann@54489
   847
lemma not_numeral_less_neg_numeral: "\<not> numeral m < - numeral n"
huffman@47108
   848
  by (simp only: not_less neg_numeral_le_numeral)
huffman@47108
   849
haftmann@54489
   850
lemma not_numeral_le_neg_numeral: "\<not> numeral m \<le> - numeral n"
huffman@47108
   851
  by (simp only: not_le neg_numeral_less_numeral)
huffman@47108
   852
  
haftmann@54489
   853
lemma neg_numeral_less_one: "- numeral m < 1"
huffman@47108
   854
  by (rule neg_numeral_less_numeral [of m One, unfolded numeral_One])
huffman@47108
   855
haftmann@54489
   856
lemma neg_numeral_le_one: "- numeral m \<le> 1"
huffman@47108
   857
  by (rule neg_numeral_le_numeral [of m One, unfolded numeral_One])
huffman@47108
   858
haftmann@54489
   859
lemma not_one_less_neg_numeral: "\<not> 1 < - numeral m"
huffman@47108
   860
  by (simp only: not_less neg_numeral_le_one)
huffman@47108
   861
haftmann@54489
   862
lemma not_one_le_neg_numeral: "\<not> 1 \<le> - numeral m"
huffman@47108
   863
  by (simp only: not_le neg_numeral_less_one)
huffman@47108
   864
haftmann@54489
   865
lemma not_numeral_less_neg_one: "\<not> numeral m < - 1"
haftmann@54489
   866
  using not_numeral_less_neg_numeral [of m One] by (simp add: numeral_One)
haftmann@54489
   867
haftmann@54489
   868
lemma not_numeral_le_neg_one: "\<not> numeral m \<le> - 1"
haftmann@54489
   869
  using not_numeral_le_neg_numeral [of m One] by (simp add: numeral_One)
haftmann@54489
   870
haftmann@54489
   871
lemma neg_one_less_numeral: "- 1 < numeral m"
haftmann@54489
   872
  using neg_numeral_less_numeral [of One m] by (simp add: numeral_One)
haftmann@54489
   873
haftmann@54489
   874
lemma neg_one_le_numeral: "- 1 \<le> numeral m"
haftmann@54489
   875
  using neg_numeral_le_numeral [of One m] by (simp add: numeral_One)
haftmann@54489
   876
haftmann@54489
   877
lemma neg_numeral_less_neg_one_iff: "- numeral m < - 1 \<longleftrightarrow> m \<noteq> One"
haftmann@54489
   878
  by (cases m) simp_all
haftmann@54489
   879
haftmann@54489
   880
lemma neg_numeral_le_neg_one: "- numeral m \<le> - 1"
haftmann@54489
   881
  by simp
haftmann@54489
   882
haftmann@54489
   883
lemma not_neg_one_less_neg_numeral: "\<not> - 1 < - numeral m"
haftmann@54489
   884
  by simp
haftmann@54489
   885
haftmann@54489
   886
lemma not_neg_one_le_neg_numeral_iff: "\<not> - 1 \<le> - numeral m \<longleftrightarrow> m \<noteq> One"
haftmann@54489
   887
  by (cases m) simp_all
haftmann@54489
   888
huffman@47108
   889
lemma sub_non_negative:
huffman@47108
   890
  "sub n m \<ge> 0 \<longleftrightarrow> n \<ge> m"
huffman@47108
   891
  by (simp only: sub_def le_diff_eq) simp
huffman@47108
   892
huffman@47108
   893
lemma sub_positive:
huffman@47108
   894
  "sub n m > 0 \<longleftrightarrow> n > m"
huffman@47108
   895
  by (simp only: sub_def less_diff_eq) simp
huffman@47108
   896
huffman@47108
   897
lemma sub_non_positive:
huffman@47108
   898
  "sub n m \<le> 0 \<longleftrightarrow> n \<le> m"
huffman@47108
   899
  by (simp only: sub_def diff_le_eq) simp
huffman@47108
   900
huffman@47108
   901
lemma sub_negative:
huffman@47108
   902
  "sub n m < 0 \<longleftrightarrow> n < m"
huffman@47108
   903
  by (simp only: sub_def diff_less_eq) simp
huffman@47108
   904
huffman@47108
   905
lemmas le_neg_numeral_simps [simp] =
huffman@47108
   906
  neg_numeral_le_iff
huffman@47108
   907
  neg_numeral_le_numeral not_numeral_le_neg_numeral
huffman@47108
   908
  neg_numeral_le_zero not_zero_le_neg_numeral
huffman@47108
   909
  neg_numeral_le_one not_one_le_neg_numeral
haftmann@54489
   910
  neg_one_le_numeral not_numeral_le_neg_one
haftmann@54489
   911
  neg_numeral_le_neg_one not_neg_one_le_neg_numeral_iff
haftmann@54489
   912
haftmann@54489
   913
lemma le_minus_one_simps [simp]:
haftmann@54489
   914
  "- 1 \<le> 0"
haftmann@54489
   915
  "- 1 \<le> 1"
haftmann@54489
   916
  "\<not> 0 \<le> - 1"
haftmann@54489
   917
  "\<not> 1 \<le> - 1"
haftmann@54489
   918
  by simp_all
huffman@47108
   919
huffman@47108
   920
lemmas less_neg_numeral_simps [simp] =
huffman@47108
   921
  neg_numeral_less_iff
huffman@47108
   922
  neg_numeral_less_numeral not_numeral_less_neg_numeral
huffman@47108
   923
  neg_numeral_less_zero not_zero_less_neg_numeral
huffman@47108
   924
  neg_numeral_less_one not_one_less_neg_numeral
haftmann@54489
   925
  neg_one_less_numeral not_numeral_less_neg_one
haftmann@54489
   926
  neg_numeral_less_neg_one_iff not_neg_one_less_neg_numeral
haftmann@54489
   927
haftmann@54489
   928
lemma less_minus_one_simps [simp]:
haftmann@54489
   929
  "- 1 < 0"
haftmann@54489
   930
  "- 1 < 1"
haftmann@54489
   931
  "\<not> 0 < - 1"
haftmann@54489
   932
  "\<not> 1 < - 1"
haftmann@54489
   933
  by (simp_all add: less_le)
huffman@47108
   934
huffman@47108
   935
lemma abs_numeral [simp]: "abs (numeral n) = numeral n"
huffman@47108
   936
  by simp
huffman@47108
   937
haftmann@54489
   938
lemma abs_neg_numeral [simp]: "abs (- numeral n) = numeral n"
haftmann@54489
   939
  by (simp only: abs_minus_cancel abs_numeral)
haftmann@54489
   940
haftmann@54489
   941
lemma abs_neg_one [simp]:
haftmann@54489
   942
  "abs (- 1) = 1"
haftmann@54489
   943
  by simp
huffman@47108
   944
huffman@47108
   945
end
huffman@47108
   946
huffman@47108
   947
subsubsection {*
huffman@47108
   948
  Natural numbers
huffman@47108
   949
*}
huffman@47108
   950
huffman@47299
   951
lemma Suc_1 [simp]: "Suc 1 = 2"
huffman@47299
   952
  unfolding Suc_eq_plus1 by (rule one_add_one)
huffman@47299
   953
huffman@47108
   954
lemma Suc_numeral [simp]: "Suc (numeral n) = numeral (n + One)"
huffman@47299
   955
  unfolding Suc_eq_plus1 by (rule numeral_plus_one)
huffman@47108
   956
huffman@47209
   957
definition pred_numeral :: "num \<Rightarrow> nat"
huffman@47209
   958
  where [code del]: "pred_numeral k = numeral k - 1"
huffman@47209
   959
huffman@47209
   960
lemma numeral_eq_Suc: "numeral k = Suc (pred_numeral k)"
huffman@47209
   961
  unfolding pred_numeral_def by simp
huffman@47209
   962
huffman@47220
   963
lemma eval_nat_numeral:
huffman@47108
   964
  "numeral One = Suc 0"
huffman@47108
   965
  "numeral (Bit0 n) = Suc (numeral (BitM n))"
huffman@47108
   966
  "numeral (Bit1 n) = Suc (numeral (Bit0 n))"
huffman@47108
   967
  by (simp_all add: numeral.simps BitM_plus_one)
huffman@47108
   968
huffman@47209
   969
lemma pred_numeral_simps [simp]:
huffman@47300
   970
  "pred_numeral One = 0"
huffman@47300
   971
  "pred_numeral (Bit0 k) = numeral (BitM k)"
huffman@47300
   972
  "pred_numeral (Bit1 k) = numeral (Bit0 k)"
huffman@47220
   973
  unfolding pred_numeral_def eval_nat_numeral
huffman@47209
   974
  by (simp_all only: diff_Suc_Suc diff_0)
huffman@47209
   975
huffman@47192
   976
lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
huffman@47220
   977
  by (simp add: eval_nat_numeral)
huffman@47192
   978
huffman@47192
   979
lemma numeral_3_eq_3: "3 = Suc (Suc (Suc 0))"
huffman@47220
   980
  by (simp add: eval_nat_numeral)
huffman@47192
   981
huffman@47207
   982
lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
huffman@47207
   983
  by (simp only: numeral_One One_nat_def)
huffman@47207
   984
huffman@47207
   985
lemma Suc_nat_number_of_add:
huffman@47300
   986
  "Suc (numeral v + n) = numeral (v + One) + n"
huffman@47207
   987
  by simp
huffman@47207
   988
huffman@47207
   989
(*Maps #n to n for n = 1, 2*)
huffman@47207
   990
lemmas numerals = numeral_One [where 'a=nat] numeral_2_eq_2
huffman@47207
   991
huffman@47209
   992
text {* Comparisons involving @{term Suc}. *}
huffman@47209
   993
huffman@47209
   994
lemma eq_numeral_Suc [simp]: "numeral k = Suc n \<longleftrightarrow> pred_numeral k = n"
huffman@47209
   995
  by (simp add: numeral_eq_Suc)
huffman@47209
   996
huffman@47209
   997
lemma Suc_eq_numeral [simp]: "Suc n = numeral k \<longleftrightarrow> n = pred_numeral k"
huffman@47209
   998
  by (simp add: numeral_eq_Suc)
huffman@47209
   999
huffman@47209
  1000
lemma less_numeral_Suc [simp]: "numeral k < Suc n \<longleftrightarrow> pred_numeral k < n"
huffman@47209
  1001
  by (simp add: numeral_eq_Suc)
huffman@47209
  1002
huffman@47209
  1003
lemma less_Suc_numeral [simp]: "Suc n < numeral k \<longleftrightarrow> n < pred_numeral k"
huffman@47209
  1004
  by (simp add: numeral_eq_Suc)
huffman@47209
  1005
huffman@47209
  1006
lemma le_numeral_Suc [simp]: "numeral k \<le> Suc n \<longleftrightarrow> pred_numeral k \<le> n"
huffman@47209
  1007
  by (simp add: numeral_eq_Suc)
huffman@47209
  1008
huffman@47209
  1009
lemma le_Suc_numeral [simp]: "Suc n \<le> numeral k \<longleftrightarrow> n \<le> pred_numeral k"
huffman@47209
  1010
  by (simp add: numeral_eq_Suc)
huffman@47209
  1011
huffman@47218
  1012
lemma diff_Suc_numeral [simp]: "Suc n - numeral k = n - pred_numeral k"
huffman@47218
  1013
  by (simp add: numeral_eq_Suc)
huffman@47218
  1014
huffman@47218
  1015
lemma diff_numeral_Suc [simp]: "numeral k - Suc n = pred_numeral k - n"
huffman@47218
  1016
  by (simp add: numeral_eq_Suc)
huffman@47218
  1017
huffman@47209
  1018
lemma max_Suc_numeral [simp]:
huffman@47209
  1019
  "max (Suc n) (numeral k) = Suc (max n (pred_numeral k))"
huffman@47209
  1020
  by (simp add: numeral_eq_Suc)
huffman@47209
  1021
huffman@47209
  1022
lemma max_numeral_Suc [simp]:
huffman@47209
  1023
  "max (numeral k) (Suc n) = Suc (max (pred_numeral k) n)"
huffman@47209
  1024
  by (simp add: numeral_eq_Suc)
huffman@47209
  1025
huffman@47209
  1026
lemma min_Suc_numeral [simp]:
huffman@47209
  1027
  "min (Suc n) (numeral k) = Suc (min n (pred_numeral k))"
huffman@47209
  1028
  by (simp add: numeral_eq_Suc)
huffman@47209
  1029
huffman@47209
  1030
lemma min_numeral_Suc [simp]:
huffman@47209
  1031
  "min (numeral k) (Suc n) = Suc (min (pred_numeral k) n)"
huffman@47209
  1032
  by (simp add: numeral_eq_Suc)
huffman@47209
  1033
blanchet@55415
  1034
text {* For @{term case_nat} and @{term rec_nat}. *}
huffman@47216
  1035
blanchet@55415
  1036
lemma case_nat_numeral [simp]:
blanchet@55415
  1037
  "case_nat a f (numeral v) = (let pv = pred_numeral v in f pv)"
huffman@47216
  1038
  by (simp add: numeral_eq_Suc)
huffman@47216
  1039
blanchet@55415
  1040
lemma case_nat_add_eq_if [simp]:
blanchet@55415
  1041
  "case_nat a f ((numeral v) + n) = (let pv = pred_numeral v in f (pv + n))"
huffman@47216
  1042
  by (simp add: numeral_eq_Suc)
huffman@47216
  1043
blanchet@55415
  1044
lemma rec_nat_numeral [simp]:
blanchet@55415
  1045
  "rec_nat a f (numeral v) =
blanchet@55415
  1046
    (let pv = pred_numeral v in f pv (rec_nat a f pv))"
huffman@47216
  1047
  by (simp add: numeral_eq_Suc Let_def)
huffman@47216
  1048
blanchet@55415
  1049
lemma rec_nat_add_eq_if [simp]:
blanchet@55415
  1050
  "rec_nat a f (numeral v + n) =
blanchet@55415
  1051
    (let pv = pred_numeral v in f (pv + n) (rec_nat a f (pv + n)))"
huffman@47216
  1052
  by (simp add: numeral_eq_Suc Let_def)
huffman@47216
  1053
huffman@47255
  1054
text {* Case analysis on @{term "n < 2"} *}
huffman@47255
  1055
huffman@47255
  1056
lemma less_2_cases: "n < 2 \<Longrightarrow> n = 0 \<or> n = Suc 0"
huffman@47255
  1057
  by (auto simp add: numeral_2_eq_2)
huffman@47255
  1058
huffman@47255
  1059
text {* Removal of Small Numerals: 0, 1 and (in additive positions) 2 *}
huffman@47255
  1060
text {* bh: Are these rules really a good idea? *}
huffman@47255
  1061
huffman@47255
  1062
lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
huffman@47255
  1063
  by simp
huffman@47255
  1064
huffman@47255
  1065
lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
huffman@47255
  1066
  by simp
huffman@47255
  1067
huffman@47255
  1068
text {* Can be used to eliminate long strings of Sucs, but not by default. *}
huffman@47255
  1069
huffman@47255
  1070
lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
huffman@47255
  1071
  by simp
huffman@47255
  1072
huffman@47255
  1073
lemmas nat_1_add_1 = one_add_one [where 'a=nat] (* legacy *)
huffman@47255
  1074
huffman@47108
  1075
haftmann@58512
  1076
subsection {* Particular lemmas concerning @{term 2} *}
haftmann@58512
  1077
haftmann@58512
  1078
context linordered_field_inverse_zero
haftmann@58512
  1079
begin
haftmann@58512
  1080
haftmann@58512
  1081
lemma half_gt_zero_iff:
haftmann@58512
  1082
  "0 < a / 2 \<longleftrightarrow> 0 < a" (is "?P \<longleftrightarrow> ?Q")
haftmann@58512
  1083
  by (auto simp add: field_simps)
haftmann@58512
  1084
haftmann@58512
  1085
lemma half_gt_zero [simp]:
haftmann@58512
  1086
  "0 < a \<Longrightarrow> 0 < a / 2"
haftmann@58512
  1087
  by (simp add: half_gt_zero_iff)
haftmann@58512
  1088
haftmann@58512
  1089
end
haftmann@58512
  1090
haftmann@58512
  1091
huffman@47108
  1092
subsection {* Numeral equations as default simplification rules *}
huffman@47108
  1093
huffman@47108
  1094
declare (in numeral) numeral_One [simp]
huffman@47108
  1095
declare (in numeral) numeral_plus_numeral [simp]
huffman@47108
  1096
declare (in numeral) add_numeral_special [simp]
huffman@47108
  1097
declare (in neg_numeral) add_neg_numeral_simps [simp]
huffman@47108
  1098
declare (in neg_numeral) add_neg_numeral_special [simp]
huffman@47108
  1099
declare (in neg_numeral) diff_numeral_simps [simp]
huffman@47108
  1100
declare (in neg_numeral) diff_numeral_special [simp]
huffman@47108
  1101
declare (in semiring_numeral) numeral_times_numeral [simp]
huffman@47108
  1102
declare (in ring_1) mult_neg_numeral_simps [simp]
huffman@47108
  1103
huffman@47108
  1104
subsection {* Setting up simprocs *}
huffman@47108
  1105
huffman@47108
  1106
lemma mult_numeral_1: "Numeral1 * a = (a::'a::semiring_numeral)"
huffman@47108
  1107
  by simp
huffman@47108
  1108
huffman@47108
  1109
lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::semiring_numeral)"
huffman@47108
  1110
  by simp
huffman@47108
  1111
huffman@47108
  1112
lemma divide_numeral_1: "a / Numeral1 = (a::'a::field)"
huffman@47108
  1113
  by simp
huffman@47108
  1114
huffman@47108
  1115
lemma inverse_numeral_1:
huffman@47108
  1116
  "inverse Numeral1 = (Numeral1::'a::division_ring)"
huffman@47108
  1117
  by simp
huffman@47108
  1118
huffman@47211
  1119
text{*Theorem lists for the cancellation simprocs. The use of a binary
huffman@47108
  1120
numeral for 1 reduces the number of special cases.*}
huffman@47108
  1121
haftmann@54489
  1122
lemma mult_1s:
haftmann@54489
  1123
  fixes a :: "'a::semiring_numeral"
haftmann@54489
  1124
    and b :: "'b::ring_1"
haftmann@54489
  1125
  shows "Numeral1 * a = a"
haftmann@54489
  1126
    "a * Numeral1 = a"
haftmann@54489
  1127
    "- Numeral1 * b = - b"
haftmann@54489
  1128
    "b * - Numeral1 = - b"
haftmann@54489
  1129
  by simp_all
huffman@47108
  1130
huffman@47226
  1131
setup {*
huffman@47226
  1132
  Reorient_Proc.add
huffman@47226
  1133
    (fn Const (@{const_name numeral}, _) $ _ => true
haftmann@54489
  1134
    | Const (@{const_name uminus}, _) $ (Const (@{const_name numeral}, _) $ _) => true
huffman@47226
  1135
    | _ => false)
huffman@47226
  1136
*}
huffman@47226
  1137
huffman@47226
  1138
simproc_setup reorient_numeral
haftmann@54489
  1139
  ("numeral w = x" | "- numeral w = y") = Reorient_Proc.proc
huffman@47226
  1140
huffman@47108
  1141
huffman@47108
  1142
subsubsection {* Simplification of arithmetic operations on integer constants. *}
huffman@47108
  1143
huffman@47108
  1144
lemmas arith_special = (* already declared simp above *)
huffman@47108
  1145
  add_numeral_special add_neg_numeral_special
haftmann@54489
  1146
  diff_numeral_special
huffman@47108
  1147
huffman@47108
  1148
(* rules already in simpset *)
huffman@47108
  1149
lemmas arith_extra_simps =
huffman@47108
  1150
  numeral_plus_numeral add_neg_numeral_simps add_0_left add_0_right
haftmann@54489
  1151
  minus_zero
huffman@47108
  1152
  diff_numeral_simps diff_0 diff_0_right
huffman@47108
  1153
  numeral_times_numeral mult_neg_numeral_simps
huffman@47108
  1154
  mult_zero_left mult_zero_right
huffman@47108
  1155
  abs_numeral abs_neg_numeral
huffman@47108
  1156
huffman@47108
  1157
text {*
huffman@47108
  1158
  For making a minimal simpset, one must include these default simprules.
huffman@47108
  1159
  Also include @{text simp_thms}.
huffman@47108
  1160
*}
huffman@47108
  1161
huffman@47108
  1162
lemmas arith_simps =
huffman@47108
  1163
  add_num_simps mult_num_simps sub_num_simps
huffman@47108
  1164
  BitM.simps dbl_simps dbl_inc_simps dbl_dec_simps
huffman@47108
  1165
  abs_zero abs_one arith_extra_simps
huffman@47108
  1166
haftmann@54249
  1167
lemmas more_arith_simps =
haftmann@54249
  1168
  neg_le_iff_le
haftmann@54249
  1169
  minus_zero left_minus right_minus
haftmann@54249
  1170
  mult_1_left mult_1_right
haftmann@54249
  1171
  mult_minus_left mult_minus_right
haftmann@57512
  1172
  minus_add_distrib minus_minus mult.assoc
haftmann@54249
  1173
haftmann@54249
  1174
lemmas of_nat_simps =
haftmann@54249
  1175
  of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
haftmann@54249
  1176
huffman@47108
  1177
text {* Simplification of relational operations *}
huffman@47108
  1178
huffman@47108
  1179
lemmas eq_numeral_extra =
huffman@47108
  1180
  zero_neq_one one_neq_zero
huffman@47108
  1181
huffman@47108
  1182
lemmas rel_simps =
huffman@47108
  1183
  le_num_simps less_num_simps eq_num_simps
haftmann@54489
  1184
  le_numeral_simps le_neg_numeral_simps le_minus_one_simps le_numeral_extra
haftmann@54489
  1185
  less_numeral_simps less_neg_numeral_simps less_minus_one_simps less_numeral_extra
huffman@47108
  1186
  eq_numeral_simps eq_neg_numeral_simps eq_numeral_extra
huffman@47108
  1187
haftmann@54249
  1188
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
haftmann@54249
  1189
  -- {* Unfold all @{text let}s involving constants *}
haftmann@54249
  1190
  unfolding Let_def ..
haftmann@54249
  1191
haftmann@54489
  1192
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
haftmann@54249
  1193
  -- {* Unfold all @{text let}s involving constants *}
haftmann@54249
  1194
  unfolding Let_def ..
haftmann@54249
  1195
haftmann@54249
  1196
declaration {*
haftmann@54249
  1197
let 
haftmann@54249
  1198
  fun number_of thy T n =
haftmann@54249
  1199
    if not (Sign.of_sort thy (T, @{sort numeral}))
haftmann@54249
  1200
    then raise CTERM ("number_of", [])
haftmann@54249
  1201
    else Numeral.mk_cnumber (Thm.ctyp_of thy T) n;
haftmann@54249
  1202
in
haftmann@54249
  1203
  K (
haftmann@54249
  1204
    Lin_Arith.add_simps (@{thms arith_simps} @ @{thms more_arith_simps}
haftmann@54249
  1205
      @ @{thms rel_simps}
haftmann@54249
  1206
      @ @{thms pred_numeral_simps}
haftmann@54249
  1207
      @ @{thms arith_special numeral_One}
haftmann@54249
  1208
      @ @{thms of_nat_simps})
haftmann@54249
  1209
    #> Lin_Arith.add_simps [@{thm Suc_numeral},
haftmann@54249
  1210
      @{thm Let_numeral}, @{thm Let_neg_numeral}, @{thm Let_0}, @{thm Let_1},
haftmann@54249
  1211
      @{thm le_Suc_numeral}, @{thm le_numeral_Suc},
haftmann@54249
  1212
      @{thm less_Suc_numeral}, @{thm less_numeral_Suc},
haftmann@54249
  1213
      @{thm Suc_eq_numeral}, @{thm eq_numeral_Suc},
haftmann@54249
  1214
      @{thm mult_Suc}, @{thm mult_Suc_right},
haftmann@54249
  1215
      @{thm of_nat_numeral}]
haftmann@54249
  1216
    #> Lin_Arith.set_number_of number_of)
haftmann@54249
  1217
end
haftmann@54249
  1218
*}
haftmann@54249
  1219
huffman@47108
  1220
huffman@47108
  1221
subsubsection {* Simplification of arithmetic when nested to the right. *}
huffman@47108
  1222
huffman@47108
  1223
lemma add_numeral_left [simp]:
huffman@47108
  1224
  "numeral v + (numeral w + z) = (numeral(v + w) + z)"
haftmann@57512
  1225
  by (simp_all add: add.assoc [symmetric])
huffman@47108
  1226
huffman@47108
  1227
lemma add_neg_numeral_left [simp]:
haftmann@54489
  1228
  "numeral v + (- numeral w + y) = (sub v w + y)"
haftmann@54489
  1229
  "- numeral v + (numeral w + y) = (sub w v + y)"
haftmann@54489
  1230
  "- numeral v + (- numeral w + y) = (- numeral(v + w) + y)"
haftmann@57512
  1231
  by (simp_all add: add.assoc [symmetric])
huffman@47108
  1232
huffman@47108
  1233
lemma mult_numeral_left [simp]:
huffman@47108
  1234
  "numeral v * (numeral w * z) = (numeral(v * w) * z :: 'a::semiring_numeral)"
haftmann@54489
  1235
  "- numeral v * (numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
haftmann@54489
  1236
  "numeral v * (- numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
haftmann@54489
  1237
  "- numeral v * (- numeral w * y) = (numeral(v * w) * y :: 'b::ring_1)"
haftmann@57512
  1238
  by (simp_all add: mult.assoc [symmetric])
huffman@47108
  1239
huffman@47108
  1240
hide_const (open) One Bit0 Bit1 BitM inc pow sqr sub dbl dbl_inc dbl_dec
huffman@47108
  1241
haftmann@51143
  1242
huffman@47108
  1243
subsection {* code module namespace *}
huffman@47108
  1244
haftmann@52435
  1245
code_identifier
haftmann@52435
  1246
  code_module Num \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
huffman@47108
  1247
huffman@47108
  1248
end