src/HOL/Num.thy
author haftmann
Fri Jul 04 20:18:47 2014 +0200 (2014-07-04)
changeset 57512 cc97b347b301
parent 55974 c835a9379026
child 58112 8081087096ad
permissions -rw-r--r--
reduced name variants for assoc and commute on plus and mult
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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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header {* Binary Numerals *}
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theory Num
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imports Datatype BNF_LFP
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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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parse_translation {*
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  let
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    fun num_of_int n =
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      if n > 0 then
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        (case IntInf.quotRem (n, 2) of
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          (0, 1) => Syntax.const @{const_syntax One}
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        | (n, 0) => Syntax.const @{const_syntax Bit0} $ num_of_int n
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        | (n, 1) => Syntax.const @{const_syntax Bit1} $ num_of_int n)
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      else raise Match
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    val numeral = Syntax.const @{const_syntax numeral}
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    val uminus = Syntax.const @{const_syntax uminus}
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    val one = Syntax.const @{const_syntax Groups.one}
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    val zero = Syntax.const @{const_syntax Groups.zero}
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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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          let
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            val {value, ...} = Lexicon.read_xnum num;
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          in
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            if value = 0 then zero else
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            if value > 0
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            then numeral $ num_of_int value
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            else if value = ~1 then uminus $ one
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            else uminus $ (numeral $ num_of_int (~ value))
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          end
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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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   323
      in
wenzelm@52143
   324
        (case T of
wenzelm@52143
   325
          Type (@{type_name fun}, [_, T']) =>
wenzelm@52210
   326
            if Printer.type_emphasis ctxt T' then
wenzelm@52210
   327
              Syntax.const @{syntax_const "_constrain"} $ t' $
wenzelm@52210
   328
                Syntax_Phases.term_of_typ ctxt T'
wenzelm@52210
   329
            else t'
wenzelm@52187
   330
        | _ => if T = dummyT then t' else raise Match)
wenzelm@52143
   331
      end;
wenzelm@52143
   332
  in
haftmann@54489
   333
   [(@{const_syntax numeral}, num_tr')]
wenzelm@52143
   334
  end
huffman@47108
   335
*}
huffman@47108
   336
wenzelm@48891
   337
ML_file "Tools/numeral.ML"
huffman@47228
   338
huffman@47228
   339
huffman@47108
   340
subsection {* Class-specific numeral rules *}
huffman@47108
   341
huffman@47108
   342
text {*
huffman@47108
   343
  @{const numeral} is a morphism.
huffman@47108
   344
*}
huffman@47108
   345
huffman@47108
   346
subsubsection {* Structures with addition: class @{text numeral} *}
huffman@47108
   347
huffman@47108
   348
context numeral
huffman@47108
   349
begin
huffman@47108
   350
huffman@47108
   351
lemma numeral_add: "numeral (m + n) = numeral m + numeral n"
huffman@47108
   352
  by (induct n rule: num_induct)
haftmann@57512
   353
     (simp_all only: numeral_One add_One add_inc numeral_inc add.assoc)
huffman@47108
   354
huffman@47108
   355
lemma numeral_plus_numeral: "numeral m + numeral n = numeral (m + n)"
huffman@47108
   356
  by (rule numeral_add [symmetric])
huffman@47108
   357
huffman@47108
   358
lemma numeral_plus_one: "numeral n + 1 = numeral (n + One)"
huffman@47108
   359
  using numeral_add [of n One] by (simp add: numeral_One)
huffman@47108
   360
huffman@47108
   361
lemma one_plus_numeral: "1 + numeral n = numeral (One + n)"
huffman@47108
   362
  using numeral_add [of One n] by (simp add: numeral_One)
huffman@47108
   363
huffman@47108
   364
lemma one_add_one: "1 + 1 = 2"
huffman@47108
   365
  using numeral_add [of One One] by (simp add: numeral_One)
huffman@47108
   366
huffman@47108
   367
lemmas add_numeral_special =
huffman@47108
   368
  numeral_plus_one one_plus_numeral one_add_one
huffman@47108
   369
huffman@47108
   370
end
huffman@47108
   371
huffman@47108
   372
subsubsection {*
huffman@47108
   373
  Structures with negation: class @{text neg_numeral}
huffman@47108
   374
*}
huffman@47108
   375
haftmann@54489
   376
class neg_numeral = numeral + group_add
huffman@47108
   377
begin
huffman@47108
   378
haftmann@54489
   379
lemma uminus_numeral_One:
haftmann@54489
   380
  "- Numeral1 = - 1"
haftmann@54489
   381
  by (simp add: numeral_One)
haftmann@54489
   382
huffman@47108
   383
text {* Numerals form an abelian subgroup. *}
huffman@47108
   384
huffman@47108
   385
inductive is_num :: "'a \<Rightarrow> bool" where
huffman@47108
   386
  "is_num 1" |
huffman@47108
   387
  "is_num x \<Longrightarrow> is_num (- x)" |
huffman@47108
   388
  "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> is_num (x + y)"
huffman@47108
   389
huffman@47108
   390
lemma is_num_numeral: "is_num (numeral k)"
huffman@47108
   391
  by (induct k, simp_all add: numeral.simps is_num.intros)
huffman@47108
   392
huffman@47108
   393
lemma is_num_add_commute:
huffman@47108
   394
  "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + y = y + x"
huffman@47108
   395
  apply (induct x rule: is_num.induct)
huffman@47108
   396
  apply (induct y rule: is_num.induct)
huffman@47108
   397
  apply simp
huffman@47108
   398
  apply (rule_tac a=x in add_left_imp_eq)
huffman@47108
   399
  apply (rule_tac a=x in add_right_imp_eq)
haftmann@57512
   400
  apply (simp add: add.assoc)
haftmann@57512
   401
  apply (simp add: add.assoc [symmetric], simp add: add.assoc)
huffman@47108
   402
  apply (rule_tac a=x in add_left_imp_eq)
huffman@47108
   403
  apply (rule_tac a=x in add_right_imp_eq)
haftmann@57512
   404
  apply (simp add: add.assoc)
haftmann@57512
   405
  apply (simp add: add.assoc, simp add: add.assoc [symmetric])
huffman@47108
   406
  done
huffman@47108
   407
huffman@47108
   408
lemma is_num_add_left_commute:
huffman@47108
   409
  "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + (y + z) = y + (x + z)"
haftmann@57512
   410
  by (simp only: add.assoc [symmetric] is_num_add_commute)
huffman@47108
   411
huffman@47108
   412
lemmas is_num_normalize =
haftmann@57512
   413
  add.assoc is_num_add_commute is_num_add_left_commute
huffman@47108
   414
  is_num.intros is_num_numeral
haftmann@54230
   415
  minus_add
huffman@47108
   416
huffman@47108
   417
definition dbl :: "'a \<Rightarrow> 'a" where "dbl x = x + x"
huffman@47108
   418
definition dbl_inc :: "'a \<Rightarrow> 'a" where "dbl_inc x = x + x + 1"
huffman@47108
   419
definition dbl_dec :: "'a \<Rightarrow> 'a" where "dbl_dec x = x + x - 1"
huffman@47108
   420
huffman@47108
   421
definition sub :: "num \<Rightarrow> num \<Rightarrow> 'a" where
huffman@47108
   422
  "sub k l = numeral k - numeral l"
huffman@47108
   423
huffman@47108
   424
lemma numeral_BitM: "numeral (BitM n) = numeral (Bit0 n) - 1"
huffman@47108
   425
  by (simp only: BitM_plus_one [symmetric] numeral_add numeral_One eq_diff_eq)
huffman@47108
   426
huffman@47108
   427
lemma dbl_simps [simp]:
haftmann@54489
   428
  "dbl (- numeral k) = - dbl (numeral k)"
huffman@47108
   429
  "dbl 0 = 0"
huffman@47108
   430
  "dbl 1 = 2"
haftmann@54489
   431
  "dbl (- 1) = - 2"
huffman@47108
   432
  "dbl (numeral k) = numeral (Bit0 k)"
haftmann@54489
   433
  by (simp_all add: dbl_def numeral.simps minus_add)
huffman@47108
   434
huffman@47108
   435
lemma dbl_inc_simps [simp]:
haftmann@54489
   436
  "dbl_inc (- numeral k) = - dbl_dec (numeral k)"
huffman@47108
   437
  "dbl_inc 0 = 1"
huffman@47108
   438
  "dbl_inc 1 = 3"
haftmann@54489
   439
  "dbl_inc (- 1) = - 1"
huffman@47108
   440
  "dbl_inc (numeral k) = numeral (Bit1 k)"
haftmann@54489
   441
  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
   442
huffman@47108
   443
lemma dbl_dec_simps [simp]:
haftmann@54489
   444
  "dbl_dec (- numeral k) = - dbl_inc (numeral k)"
haftmann@54489
   445
  "dbl_dec 0 = - 1"
huffman@47108
   446
  "dbl_dec 1 = 1"
haftmann@54489
   447
  "dbl_dec (- 1) = - 3"
huffman@47108
   448
  "dbl_dec (numeral k) = numeral (BitM k)"
haftmann@54489
   449
  by (simp_all add: dbl_dec_def dbl_inc_def numeral.simps numeral_BitM is_num_normalize)
huffman@47108
   450
huffman@47108
   451
lemma sub_num_simps [simp]:
huffman@47108
   452
  "sub One One = 0"
haftmann@54489
   453
  "sub One (Bit0 l) = - numeral (BitM l)"
haftmann@54489
   454
  "sub One (Bit1 l) = - numeral (Bit0 l)"
huffman@47108
   455
  "sub (Bit0 k) One = numeral (BitM k)"
huffman@47108
   456
  "sub (Bit1 k) One = numeral (Bit0 k)"
huffman@47108
   457
  "sub (Bit0 k) (Bit0 l) = dbl (sub k l)"
huffman@47108
   458
  "sub (Bit0 k) (Bit1 l) = dbl_dec (sub k l)"
huffman@47108
   459
  "sub (Bit1 k) (Bit0 l) = dbl_inc (sub k l)"
huffman@47108
   460
  "sub (Bit1 k) (Bit1 l) = dbl (sub k l)"
haftmann@54489
   461
  by (simp_all add: dbl_def dbl_dec_def dbl_inc_def sub_def numeral.simps
haftmann@54230
   462
    numeral_BitM is_num_normalize del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   463
huffman@47108
   464
lemma add_neg_numeral_simps:
haftmann@54489
   465
  "numeral m + - numeral n = sub m n"
haftmann@54489
   466
  "- numeral m + numeral n = sub n m"
haftmann@54489
   467
  "- numeral m + - numeral n = - (numeral m + numeral n)"
haftmann@54489
   468
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
haftmann@54230
   469
    del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   470
huffman@47108
   471
lemma add_neg_numeral_special:
haftmann@54489
   472
  "1 + - numeral m = sub One m"
haftmann@54489
   473
  "- numeral m + 1 = sub One m"
haftmann@54489
   474
  "numeral m + - 1 = sub m One"
haftmann@54489
   475
  "- 1 + numeral n = sub n One"
haftmann@54489
   476
  "- 1 + - numeral n = - numeral (inc n)"
haftmann@54489
   477
  "- numeral m + - 1 = - numeral (inc m)"
haftmann@54489
   478
  "1 + - 1 = 0"
haftmann@54489
   479
  "- 1 + 1 = 0"
haftmann@54489
   480
  "- 1 + - 1 = - 2"
haftmann@54489
   481
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize right_minus numeral_inc
haftmann@54489
   482
    del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   483
huffman@47108
   484
lemma diff_numeral_simps:
huffman@47108
   485
  "numeral m - numeral n = sub m n"
haftmann@54489
   486
  "numeral m - - numeral n = numeral (m + n)"
haftmann@54489
   487
  "- numeral m - numeral n = - numeral (m + n)"
haftmann@54489
   488
  "- numeral m - - numeral n = sub n m"
haftmann@54489
   489
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
haftmann@54230
   490
    del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   491
huffman@47108
   492
lemma diff_numeral_special:
huffman@47108
   493
  "1 - numeral n = sub One n"
huffman@47108
   494
  "numeral m - 1 = sub m One"
haftmann@54489
   495
  "1 - - numeral n = numeral (One + n)"
haftmann@54489
   496
  "- numeral m - 1 = - numeral (m + One)"
haftmann@54489
   497
  "- 1 - numeral n = - numeral (inc n)"
haftmann@54489
   498
  "numeral m - - 1 = numeral (inc m)"
haftmann@54489
   499
  "- 1 - - numeral n = sub n One"
haftmann@54489
   500
  "- numeral m - - 1 = sub One m"
haftmann@54489
   501
  "1 - 1 = 0"
haftmann@54489
   502
  "- 1 - 1 = - 2"
haftmann@54489
   503
  "1 - - 1 = 2"
haftmann@54489
   504
  "- 1 - - 1 = 0"
haftmann@54489
   505
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize numeral_inc
haftmann@54489
   506
    del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   507
huffman@47108
   508
end
huffman@47108
   509
huffman@47108
   510
subsubsection {*
huffman@47108
   511
  Structures with multiplication: class @{text semiring_numeral}
huffman@47108
   512
*}
huffman@47108
   513
huffman@47108
   514
class semiring_numeral = semiring + monoid_mult
huffman@47108
   515
begin
huffman@47108
   516
huffman@47108
   517
subclass numeral ..
huffman@47108
   518
huffman@47108
   519
lemma numeral_mult: "numeral (m * n) = numeral m * numeral n"
huffman@47108
   520
  apply (induct n rule: num_induct)
huffman@47108
   521
  apply (simp add: numeral_One)
webertj@49962
   522
  apply (simp add: mult_inc numeral_inc numeral_add distrib_left)
huffman@47108
   523
  done
huffman@47108
   524
huffman@47108
   525
lemma numeral_times_numeral: "numeral m * numeral n = numeral (m * n)"
huffman@47108
   526
  by (rule numeral_mult [symmetric])
huffman@47108
   527
haftmann@53064
   528
lemma mult_2: "2 * z = z + z"
haftmann@53064
   529
  unfolding one_add_one [symmetric] distrib_right by simp
haftmann@53064
   530
haftmann@53064
   531
lemma mult_2_right: "z * 2 = z + z"
haftmann@53064
   532
  unfolding one_add_one [symmetric] distrib_left by simp
haftmann@53064
   533
huffman@47108
   534
end
huffman@47108
   535
huffman@47108
   536
subsubsection {*
huffman@47108
   537
  Structures with a zero: class @{text semiring_1}
huffman@47108
   538
*}
huffman@47108
   539
huffman@47108
   540
context semiring_1
huffman@47108
   541
begin
huffman@47108
   542
huffman@47108
   543
subclass semiring_numeral ..
huffman@47108
   544
huffman@47108
   545
lemma of_nat_numeral [simp]: "of_nat (numeral n) = numeral n"
huffman@47108
   546
  by (induct n,
huffman@47108
   547
    simp_all only: numeral.simps numeral_class.numeral.simps of_nat_add of_nat_1)
huffman@47108
   548
huffman@47108
   549
end
huffman@47108
   550
haftmann@51143
   551
lemma nat_of_num_numeral [code_abbrev]:
haftmann@51143
   552
  "nat_of_num = numeral"
huffman@47108
   553
proof
huffman@47108
   554
  fix n
huffman@47108
   555
  have "numeral n = nat_of_num n"
huffman@47108
   556
    by (induct n) (simp_all add: numeral.simps)
huffman@47108
   557
  then show "nat_of_num n = numeral n" by simp
huffman@47108
   558
qed
huffman@47108
   559
haftmann@51143
   560
lemma nat_of_num_code [code]:
haftmann@51143
   561
  "nat_of_num One = 1"
haftmann@51143
   562
  "nat_of_num (Bit0 n) = (let m = nat_of_num n in m + m)"
haftmann@51143
   563
  "nat_of_num (Bit1 n) = (let m = nat_of_num n in Suc (m + m))"
haftmann@51143
   564
  by (simp_all add: Let_def)
haftmann@51143
   565
huffman@47108
   566
subsubsection {*
huffman@47108
   567
  Equality: class @{text semiring_char_0}
huffman@47108
   568
*}
huffman@47108
   569
huffman@47108
   570
context semiring_char_0
huffman@47108
   571
begin
huffman@47108
   572
huffman@47108
   573
lemma numeral_eq_iff: "numeral m = numeral n \<longleftrightarrow> m = n"
huffman@47108
   574
  unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
huffman@47108
   575
    of_nat_eq_iff num_eq_iff ..
huffman@47108
   576
huffman@47108
   577
lemma numeral_eq_one_iff: "numeral n = 1 \<longleftrightarrow> n = One"
huffman@47108
   578
  by (rule numeral_eq_iff [of n One, unfolded numeral_One])
huffman@47108
   579
huffman@47108
   580
lemma one_eq_numeral_iff: "1 = numeral n \<longleftrightarrow> One = n"
huffman@47108
   581
  by (rule numeral_eq_iff [of One n, unfolded numeral_One])
huffman@47108
   582
huffman@47108
   583
lemma numeral_neq_zero: "numeral n \<noteq> 0"
huffman@47108
   584
  unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
huffman@47108
   585
  by (simp add: nat_of_num_pos)
huffman@47108
   586
huffman@47108
   587
lemma zero_neq_numeral: "0 \<noteq> numeral n"
huffman@47108
   588
  unfolding eq_commute [of 0] by (rule numeral_neq_zero)
huffman@47108
   589
huffman@47108
   590
lemmas eq_numeral_simps [simp] =
huffman@47108
   591
  numeral_eq_iff
huffman@47108
   592
  numeral_eq_one_iff
huffman@47108
   593
  one_eq_numeral_iff
huffman@47108
   594
  numeral_neq_zero
huffman@47108
   595
  zero_neq_numeral
huffman@47108
   596
huffman@47108
   597
end
huffman@47108
   598
huffman@47108
   599
subsubsection {*
huffman@47108
   600
  Comparisons: class @{text linordered_semidom}
huffman@47108
   601
*}
huffman@47108
   602
huffman@47108
   603
text {*  Could be perhaps more general than here. *}
huffman@47108
   604
huffman@47108
   605
context linordered_semidom
huffman@47108
   606
begin
huffman@47108
   607
huffman@47108
   608
lemma numeral_le_iff: "numeral m \<le> numeral n \<longleftrightarrow> m \<le> n"
huffman@47108
   609
proof -
huffman@47108
   610
  have "of_nat (numeral m) \<le> of_nat (numeral n) \<longleftrightarrow> m \<le> n"
huffman@47108
   611
    unfolding less_eq_num_def nat_of_num_numeral of_nat_le_iff ..
huffman@47108
   612
  then show ?thesis by simp
huffman@47108
   613
qed
huffman@47108
   614
huffman@47108
   615
lemma one_le_numeral: "1 \<le> numeral n"
huffman@47108
   616
using numeral_le_iff [of One n] by (simp add: numeral_One)
huffman@47108
   617
huffman@47108
   618
lemma numeral_le_one_iff: "numeral n \<le> 1 \<longleftrightarrow> n \<le> One"
huffman@47108
   619
using numeral_le_iff [of n One] by (simp add: numeral_One)
huffman@47108
   620
huffman@47108
   621
lemma numeral_less_iff: "numeral m < numeral n \<longleftrightarrow> m < n"
huffman@47108
   622
proof -
huffman@47108
   623
  have "of_nat (numeral m) < of_nat (numeral n) \<longleftrightarrow> m < n"
huffman@47108
   624
    unfolding less_num_def nat_of_num_numeral of_nat_less_iff ..
huffman@47108
   625
  then show ?thesis by simp
huffman@47108
   626
qed
huffman@47108
   627
huffman@47108
   628
lemma not_numeral_less_one: "\<not> numeral n < 1"
huffman@47108
   629
  using numeral_less_iff [of n One] by (simp add: numeral_One)
huffman@47108
   630
huffman@47108
   631
lemma one_less_numeral_iff: "1 < numeral n \<longleftrightarrow> One < n"
huffman@47108
   632
  using numeral_less_iff [of One n] by (simp add: numeral_One)
huffman@47108
   633
huffman@47108
   634
lemma zero_le_numeral: "0 \<le> numeral n"
huffman@47108
   635
  by (induct n) (simp_all add: numeral.simps)
huffman@47108
   636
huffman@47108
   637
lemma zero_less_numeral: "0 < numeral n"
huffman@47108
   638
  by (induct n) (simp_all add: numeral.simps add_pos_pos)
huffman@47108
   639
huffman@47108
   640
lemma not_numeral_le_zero: "\<not> numeral n \<le> 0"
huffman@47108
   641
  by (simp add: not_le zero_less_numeral)
huffman@47108
   642
huffman@47108
   643
lemma not_numeral_less_zero: "\<not> numeral n < 0"
huffman@47108
   644
  by (simp add: not_less zero_le_numeral)
huffman@47108
   645
huffman@47108
   646
lemmas le_numeral_extra =
huffman@47108
   647
  zero_le_one not_one_le_zero
huffman@47108
   648
  order_refl [of 0] order_refl [of 1]
huffman@47108
   649
huffman@47108
   650
lemmas less_numeral_extra =
huffman@47108
   651
  zero_less_one not_one_less_zero
huffman@47108
   652
  less_irrefl [of 0] less_irrefl [of 1]
huffman@47108
   653
huffman@47108
   654
lemmas le_numeral_simps [simp] =
huffman@47108
   655
  numeral_le_iff
huffman@47108
   656
  one_le_numeral
huffman@47108
   657
  numeral_le_one_iff
huffman@47108
   658
  zero_le_numeral
huffman@47108
   659
  not_numeral_le_zero
huffman@47108
   660
huffman@47108
   661
lemmas less_numeral_simps [simp] =
huffman@47108
   662
  numeral_less_iff
huffman@47108
   663
  one_less_numeral_iff
huffman@47108
   664
  not_numeral_less_one
huffman@47108
   665
  zero_less_numeral
huffman@47108
   666
  not_numeral_less_zero
huffman@47108
   667
huffman@47108
   668
end
huffman@47108
   669
huffman@47108
   670
subsubsection {*
huffman@47108
   671
  Multiplication and negation: class @{text ring_1}
huffman@47108
   672
*}
huffman@47108
   673
huffman@47108
   674
context ring_1
huffman@47108
   675
begin
huffman@47108
   676
huffman@47108
   677
subclass neg_numeral ..
huffman@47108
   678
huffman@47108
   679
lemma mult_neg_numeral_simps:
haftmann@54489
   680
  "- numeral m * - numeral n = numeral (m * n)"
haftmann@54489
   681
  "- numeral m * numeral n = - numeral (m * n)"
haftmann@54489
   682
  "numeral m * - numeral n = - numeral (m * n)"
haftmann@54489
   683
  unfolding mult_minus_left mult_minus_right
huffman@47108
   684
  by (simp_all only: minus_minus numeral_mult)
huffman@47108
   685
haftmann@54489
   686
lemma mult_minus1 [simp]: "- 1 * z = - z"
haftmann@54489
   687
  unfolding numeral.simps mult_minus_left by simp
huffman@47108
   688
haftmann@54489
   689
lemma mult_minus1_right [simp]: "z * - 1 = - z"
haftmann@54489
   690
  unfolding numeral.simps mult_minus_right by simp
huffman@47108
   691
huffman@47108
   692
end
huffman@47108
   693
huffman@47108
   694
subsubsection {*
huffman@47108
   695
  Equality using @{text iszero} for rings with non-zero characteristic
huffman@47108
   696
*}
huffman@47108
   697
huffman@47108
   698
context ring_1
huffman@47108
   699
begin
huffman@47108
   700
huffman@47108
   701
definition iszero :: "'a \<Rightarrow> bool"
huffman@47108
   702
  where "iszero z \<longleftrightarrow> z = 0"
huffman@47108
   703
huffman@47108
   704
lemma iszero_0 [simp]: "iszero 0"
huffman@47108
   705
  by (simp add: iszero_def)
huffman@47108
   706
huffman@47108
   707
lemma not_iszero_1 [simp]: "\<not> iszero 1"
huffman@47108
   708
  by (simp add: iszero_def)
huffman@47108
   709
huffman@47108
   710
lemma not_iszero_Numeral1: "\<not> iszero Numeral1"
huffman@47108
   711
  by (simp add: numeral_One)
huffman@47108
   712
haftmann@54489
   713
lemma not_iszero_neg_1 [simp]: "\<not> iszero (- 1)"
haftmann@54489
   714
  by (simp add: iszero_def)
haftmann@54489
   715
haftmann@54489
   716
lemma not_iszero_neg_Numeral1: "\<not> iszero (- Numeral1)"
haftmann@54489
   717
  by (simp add: numeral_One)
haftmann@54489
   718
huffman@47108
   719
lemma iszero_neg_numeral [simp]:
haftmann@54489
   720
  "iszero (- numeral w) \<longleftrightarrow> iszero (numeral w)"
haftmann@54489
   721
  unfolding iszero_def
huffman@47108
   722
  by (rule neg_equal_0_iff_equal)
huffman@47108
   723
huffman@47108
   724
lemma eq_iff_iszero_diff: "x = y \<longleftrightarrow> iszero (x - y)"
huffman@47108
   725
  unfolding iszero_def by (rule eq_iff_diff_eq_0)
huffman@47108
   726
huffman@47108
   727
text {* The @{text "eq_numeral_iff_iszero"} lemmas are not declared
huffman@47108
   728
@{text "[simp]"} by default, because for rings of characteristic zero,
huffman@47108
   729
better simp rules are possible. For a type like integers mod @{text
huffman@47108
   730
"n"}, type-instantiated versions of these rules should be added to the
huffman@47108
   731
simplifier, along with a type-specific rule for deciding propositions
huffman@47108
   732
of the form @{text "iszero (numeral w)"}.
huffman@47108
   733
huffman@47108
   734
bh: Maybe it would not be so bad to just declare these as simp
huffman@47108
   735
rules anyway? I should test whether these rules take precedence over
huffman@47108
   736
the @{text "ring_char_0"} rules in the simplifier.
huffman@47108
   737
*}
huffman@47108
   738
huffman@47108
   739
lemma eq_numeral_iff_iszero:
huffman@47108
   740
  "numeral x = numeral y \<longleftrightarrow> iszero (sub x y)"
haftmann@54489
   741
  "numeral x = - numeral y \<longleftrightarrow> iszero (numeral (x + y))"
haftmann@54489
   742
  "- numeral x = numeral y \<longleftrightarrow> iszero (numeral (x + y))"
haftmann@54489
   743
  "- numeral x = - numeral y \<longleftrightarrow> iszero (sub y x)"
huffman@47108
   744
  "numeral x = 1 \<longleftrightarrow> iszero (sub x One)"
huffman@47108
   745
  "1 = numeral y \<longleftrightarrow> iszero (sub One y)"
haftmann@54489
   746
  "- numeral x = 1 \<longleftrightarrow> iszero (numeral (x + One))"
haftmann@54489
   747
  "1 = - numeral y \<longleftrightarrow> iszero (numeral (One + y))"
huffman@47108
   748
  "numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
huffman@47108
   749
  "0 = numeral y \<longleftrightarrow> iszero (numeral y)"
haftmann@54489
   750
  "- numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
haftmann@54489
   751
  "0 = - numeral y \<longleftrightarrow> iszero (numeral y)"
huffman@47108
   752
  unfolding eq_iff_iszero_diff diff_numeral_simps diff_numeral_special
huffman@47108
   753
  by simp_all
huffman@47108
   754
huffman@47108
   755
end
huffman@47108
   756
huffman@47108
   757
subsubsection {*
huffman@47108
   758
  Equality and negation: class @{text ring_char_0}
huffman@47108
   759
*}
huffman@47108
   760
huffman@47108
   761
class ring_char_0 = ring_1 + semiring_char_0
huffman@47108
   762
begin
huffman@47108
   763
huffman@47108
   764
lemma not_iszero_numeral [simp]: "\<not> iszero (numeral w)"
huffman@47108
   765
  by (simp add: iszero_def)
huffman@47108
   766
haftmann@54489
   767
lemma neg_numeral_eq_iff: "- numeral m = - numeral n \<longleftrightarrow> m = n"
haftmann@54489
   768
  by simp
huffman@47108
   769
haftmann@54489
   770
lemma numeral_neq_neg_numeral: "numeral m \<noteq> - numeral n"
haftmann@54489
   771
  unfolding eq_neg_iff_add_eq_0
huffman@47108
   772
  by (simp add: numeral_plus_numeral)
huffman@47108
   773
haftmann@54489
   774
lemma neg_numeral_neq_numeral: "- numeral m \<noteq> numeral n"
huffman@47108
   775
  by (rule numeral_neq_neg_numeral [symmetric])
huffman@47108
   776
haftmann@54489
   777
lemma zero_neq_neg_numeral: "0 \<noteq> - numeral n"
haftmann@54489
   778
  unfolding neg_0_equal_iff_equal by simp
huffman@47108
   779
haftmann@54489
   780
lemma neg_numeral_neq_zero: "- numeral n \<noteq> 0"
haftmann@54489
   781
  unfolding neg_equal_0_iff_equal by simp
huffman@47108
   782
haftmann@54489
   783
lemma one_neq_neg_numeral: "1 \<noteq> - numeral n"
huffman@47108
   784
  using numeral_neq_neg_numeral [of One n] by (simp add: numeral_One)
huffman@47108
   785
haftmann@54489
   786
lemma neg_numeral_neq_one: "- numeral n \<noteq> 1"
huffman@47108
   787
  using neg_numeral_neq_numeral [of n One] by (simp add: numeral_One)
huffman@47108
   788
haftmann@54489
   789
lemma neg_one_neq_numeral:
haftmann@54489
   790
  "- 1 \<noteq> numeral n"
haftmann@54489
   791
  using neg_numeral_neq_numeral [of One n] by (simp add: numeral_One)
haftmann@54489
   792
haftmann@54489
   793
lemma numeral_neq_neg_one:
haftmann@54489
   794
  "numeral n \<noteq> - 1"
haftmann@54489
   795
  using numeral_neq_neg_numeral [of n One] by (simp add: numeral_One)
haftmann@54489
   796
haftmann@54489
   797
lemma neg_one_eq_numeral_iff:
haftmann@54489
   798
  "- 1 = - numeral n \<longleftrightarrow> n = One"
haftmann@54489
   799
  using neg_numeral_eq_iff [of One n] by (auto simp add: numeral_One)
haftmann@54489
   800
haftmann@54489
   801
lemma numeral_eq_neg_one_iff:
haftmann@54489
   802
  "- numeral n = - 1 \<longleftrightarrow> n = One"
haftmann@54489
   803
  using neg_numeral_eq_iff [of n One] by (auto simp add: numeral_One)
haftmann@54489
   804
haftmann@54489
   805
lemma neg_one_neq_zero:
haftmann@54489
   806
  "- 1 \<noteq> 0"
haftmann@54489
   807
  by simp
haftmann@54489
   808
haftmann@54489
   809
lemma zero_neq_neg_one:
haftmann@54489
   810
  "0 \<noteq> - 1"
haftmann@54489
   811
  by simp
haftmann@54489
   812
haftmann@54489
   813
lemma neg_one_neq_one:
haftmann@54489
   814
  "- 1 \<noteq> 1"
haftmann@54489
   815
  using neg_numeral_neq_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
haftmann@54489
   816
haftmann@54489
   817
lemma one_neq_neg_one:
haftmann@54489
   818
  "1 \<noteq> - 1"
haftmann@54489
   819
  using numeral_neq_neg_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
haftmann@54489
   820
huffman@47108
   821
lemmas eq_neg_numeral_simps [simp] =
huffman@47108
   822
  neg_numeral_eq_iff
huffman@47108
   823
  numeral_neq_neg_numeral neg_numeral_neq_numeral
huffman@47108
   824
  one_neq_neg_numeral neg_numeral_neq_one
huffman@47108
   825
  zero_neq_neg_numeral neg_numeral_neq_zero
haftmann@54489
   826
  neg_one_neq_numeral numeral_neq_neg_one
haftmann@54489
   827
  neg_one_eq_numeral_iff numeral_eq_neg_one_iff
haftmann@54489
   828
  neg_one_neq_zero zero_neq_neg_one
haftmann@54489
   829
  neg_one_neq_one one_neq_neg_one
huffman@47108
   830
huffman@47108
   831
end
huffman@47108
   832
huffman@47108
   833
subsubsection {*
huffman@47108
   834
  Structures with negation and order: class @{text linordered_idom}
huffman@47108
   835
*}
huffman@47108
   836
huffman@47108
   837
context linordered_idom
huffman@47108
   838
begin
huffman@47108
   839
huffman@47108
   840
subclass ring_char_0 ..
huffman@47108
   841
haftmann@54489
   842
lemma neg_numeral_le_iff: "- numeral m \<le> - numeral n \<longleftrightarrow> n \<le> m"
haftmann@54489
   843
  by (simp only: neg_le_iff_le numeral_le_iff)
huffman@47108
   844
haftmann@54489
   845
lemma neg_numeral_less_iff: "- numeral m < - numeral n \<longleftrightarrow> n < m"
haftmann@54489
   846
  by (simp only: neg_less_iff_less numeral_less_iff)
huffman@47108
   847
haftmann@54489
   848
lemma neg_numeral_less_zero: "- numeral n < 0"
haftmann@54489
   849
  by (simp only: neg_less_0_iff_less zero_less_numeral)
huffman@47108
   850
haftmann@54489
   851
lemma neg_numeral_le_zero: "- numeral n \<le> 0"
haftmann@54489
   852
  by (simp only: neg_le_0_iff_le zero_le_numeral)
huffman@47108
   853
haftmann@54489
   854
lemma not_zero_less_neg_numeral: "\<not> 0 < - numeral n"
huffman@47108
   855
  by (simp only: not_less neg_numeral_le_zero)
huffman@47108
   856
haftmann@54489
   857
lemma not_zero_le_neg_numeral: "\<not> 0 \<le> - numeral n"
huffman@47108
   858
  by (simp only: not_le neg_numeral_less_zero)
huffman@47108
   859
haftmann@54489
   860
lemma neg_numeral_less_numeral: "- numeral m < numeral n"
huffman@47108
   861
  using neg_numeral_less_zero zero_less_numeral by (rule less_trans)
huffman@47108
   862
haftmann@54489
   863
lemma neg_numeral_le_numeral: "- numeral m \<le> numeral n"
huffman@47108
   864
  by (simp only: less_imp_le neg_numeral_less_numeral)
huffman@47108
   865
haftmann@54489
   866
lemma not_numeral_less_neg_numeral: "\<not> numeral m < - numeral n"
huffman@47108
   867
  by (simp only: not_less neg_numeral_le_numeral)
huffman@47108
   868
haftmann@54489
   869
lemma not_numeral_le_neg_numeral: "\<not> numeral m \<le> - numeral n"
huffman@47108
   870
  by (simp only: not_le neg_numeral_less_numeral)
huffman@47108
   871
  
haftmann@54489
   872
lemma neg_numeral_less_one: "- numeral m < 1"
huffman@47108
   873
  by (rule neg_numeral_less_numeral [of m One, unfolded numeral_One])
huffman@47108
   874
haftmann@54489
   875
lemma neg_numeral_le_one: "- numeral m \<le> 1"
huffman@47108
   876
  by (rule neg_numeral_le_numeral [of m One, unfolded numeral_One])
huffman@47108
   877
haftmann@54489
   878
lemma not_one_less_neg_numeral: "\<not> 1 < - numeral m"
huffman@47108
   879
  by (simp only: not_less neg_numeral_le_one)
huffman@47108
   880
haftmann@54489
   881
lemma not_one_le_neg_numeral: "\<not> 1 \<le> - numeral m"
huffman@47108
   882
  by (simp only: not_le neg_numeral_less_one)
huffman@47108
   883
haftmann@54489
   884
lemma not_numeral_less_neg_one: "\<not> numeral m < - 1"
haftmann@54489
   885
  using not_numeral_less_neg_numeral [of m One] by (simp add: numeral_One)
haftmann@54489
   886
haftmann@54489
   887
lemma not_numeral_le_neg_one: "\<not> numeral m \<le> - 1"
haftmann@54489
   888
  using not_numeral_le_neg_numeral [of m One] by (simp add: numeral_One)
haftmann@54489
   889
haftmann@54489
   890
lemma neg_one_less_numeral: "- 1 < numeral m"
haftmann@54489
   891
  using neg_numeral_less_numeral [of One m] by (simp add: numeral_One)
haftmann@54489
   892
haftmann@54489
   893
lemma neg_one_le_numeral: "- 1 \<le> numeral m"
haftmann@54489
   894
  using neg_numeral_le_numeral [of One m] by (simp add: numeral_One)
haftmann@54489
   895
haftmann@54489
   896
lemma neg_numeral_less_neg_one_iff: "- numeral m < - 1 \<longleftrightarrow> m \<noteq> One"
haftmann@54489
   897
  by (cases m) simp_all
haftmann@54489
   898
haftmann@54489
   899
lemma neg_numeral_le_neg_one: "- numeral m \<le> - 1"
haftmann@54489
   900
  by simp
haftmann@54489
   901
haftmann@54489
   902
lemma not_neg_one_less_neg_numeral: "\<not> - 1 < - numeral m"
haftmann@54489
   903
  by simp
haftmann@54489
   904
haftmann@54489
   905
lemma not_neg_one_le_neg_numeral_iff: "\<not> - 1 \<le> - numeral m \<longleftrightarrow> m \<noteq> One"
haftmann@54489
   906
  by (cases m) simp_all
haftmann@54489
   907
huffman@47108
   908
lemma sub_non_negative:
huffman@47108
   909
  "sub n m \<ge> 0 \<longleftrightarrow> n \<ge> m"
huffman@47108
   910
  by (simp only: sub_def le_diff_eq) simp
huffman@47108
   911
huffman@47108
   912
lemma sub_positive:
huffman@47108
   913
  "sub n m > 0 \<longleftrightarrow> n > m"
huffman@47108
   914
  by (simp only: sub_def less_diff_eq) simp
huffman@47108
   915
huffman@47108
   916
lemma sub_non_positive:
huffman@47108
   917
  "sub n m \<le> 0 \<longleftrightarrow> n \<le> m"
huffman@47108
   918
  by (simp only: sub_def diff_le_eq) simp
huffman@47108
   919
huffman@47108
   920
lemma sub_negative:
huffman@47108
   921
  "sub n m < 0 \<longleftrightarrow> n < m"
huffman@47108
   922
  by (simp only: sub_def diff_less_eq) simp
huffman@47108
   923
huffman@47108
   924
lemmas le_neg_numeral_simps [simp] =
huffman@47108
   925
  neg_numeral_le_iff
huffman@47108
   926
  neg_numeral_le_numeral not_numeral_le_neg_numeral
huffman@47108
   927
  neg_numeral_le_zero not_zero_le_neg_numeral
huffman@47108
   928
  neg_numeral_le_one not_one_le_neg_numeral
haftmann@54489
   929
  neg_one_le_numeral not_numeral_le_neg_one
haftmann@54489
   930
  neg_numeral_le_neg_one not_neg_one_le_neg_numeral_iff
haftmann@54489
   931
haftmann@54489
   932
lemma le_minus_one_simps [simp]:
haftmann@54489
   933
  "- 1 \<le> 0"
haftmann@54489
   934
  "- 1 \<le> 1"
haftmann@54489
   935
  "\<not> 0 \<le> - 1"
haftmann@54489
   936
  "\<not> 1 \<le> - 1"
haftmann@54489
   937
  by simp_all
huffman@47108
   938
huffman@47108
   939
lemmas less_neg_numeral_simps [simp] =
huffman@47108
   940
  neg_numeral_less_iff
huffman@47108
   941
  neg_numeral_less_numeral not_numeral_less_neg_numeral
huffman@47108
   942
  neg_numeral_less_zero not_zero_less_neg_numeral
huffman@47108
   943
  neg_numeral_less_one not_one_less_neg_numeral
haftmann@54489
   944
  neg_one_less_numeral not_numeral_less_neg_one
haftmann@54489
   945
  neg_numeral_less_neg_one_iff not_neg_one_less_neg_numeral
haftmann@54489
   946
haftmann@54489
   947
lemma less_minus_one_simps [simp]:
haftmann@54489
   948
  "- 1 < 0"
haftmann@54489
   949
  "- 1 < 1"
haftmann@54489
   950
  "\<not> 0 < - 1"
haftmann@54489
   951
  "\<not> 1 < - 1"
haftmann@54489
   952
  by (simp_all add: less_le)
huffman@47108
   953
huffman@47108
   954
lemma abs_numeral [simp]: "abs (numeral n) = numeral n"
huffman@47108
   955
  by simp
huffman@47108
   956
haftmann@54489
   957
lemma abs_neg_numeral [simp]: "abs (- numeral n) = numeral n"
haftmann@54489
   958
  by (simp only: abs_minus_cancel abs_numeral)
haftmann@54489
   959
haftmann@54489
   960
lemma abs_neg_one [simp]:
haftmann@54489
   961
  "abs (- 1) = 1"
haftmann@54489
   962
  by simp
huffman@47108
   963
huffman@47108
   964
end
huffman@47108
   965
huffman@47108
   966
subsubsection {*
huffman@47108
   967
  Natural numbers
huffman@47108
   968
*}
huffman@47108
   969
huffman@47299
   970
lemma Suc_1 [simp]: "Suc 1 = 2"
huffman@47299
   971
  unfolding Suc_eq_plus1 by (rule one_add_one)
huffman@47299
   972
huffman@47108
   973
lemma Suc_numeral [simp]: "Suc (numeral n) = numeral (n + One)"
huffman@47299
   974
  unfolding Suc_eq_plus1 by (rule numeral_plus_one)
huffman@47108
   975
huffman@47209
   976
definition pred_numeral :: "num \<Rightarrow> nat"
huffman@47209
   977
  where [code del]: "pred_numeral k = numeral k - 1"
huffman@47209
   978
huffman@47209
   979
lemma numeral_eq_Suc: "numeral k = Suc (pred_numeral k)"
huffman@47209
   980
  unfolding pred_numeral_def by simp
huffman@47209
   981
huffman@47220
   982
lemma eval_nat_numeral:
huffman@47108
   983
  "numeral One = Suc 0"
huffman@47108
   984
  "numeral (Bit0 n) = Suc (numeral (BitM n))"
huffman@47108
   985
  "numeral (Bit1 n) = Suc (numeral (Bit0 n))"
huffman@47108
   986
  by (simp_all add: numeral.simps BitM_plus_one)
huffman@47108
   987
huffman@47209
   988
lemma pred_numeral_simps [simp]:
huffman@47300
   989
  "pred_numeral One = 0"
huffman@47300
   990
  "pred_numeral (Bit0 k) = numeral (BitM k)"
huffman@47300
   991
  "pred_numeral (Bit1 k) = numeral (Bit0 k)"
huffman@47220
   992
  unfolding pred_numeral_def eval_nat_numeral
huffman@47209
   993
  by (simp_all only: diff_Suc_Suc diff_0)
huffman@47209
   994
huffman@47192
   995
lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
huffman@47220
   996
  by (simp add: eval_nat_numeral)
huffman@47192
   997
huffman@47192
   998
lemma numeral_3_eq_3: "3 = Suc (Suc (Suc 0))"
huffman@47220
   999
  by (simp add: eval_nat_numeral)
huffman@47192
  1000
huffman@47207
  1001
lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
huffman@47207
  1002
  by (simp only: numeral_One One_nat_def)
huffman@47207
  1003
huffman@47207
  1004
lemma Suc_nat_number_of_add:
huffman@47300
  1005
  "Suc (numeral v + n) = numeral (v + One) + n"
huffman@47207
  1006
  by simp
huffman@47207
  1007
huffman@47207
  1008
(*Maps #n to n for n = 1, 2*)
huffman@47207
  1009
lemmas numerals = numeral_One [where 'a=nat] numeral_2_eq_2
huffman@47207
  1010
huffman@47209
  1011
text {* Comparisons involving @{term Suc}. *}
huffman@47209
  1012
huffman@47209
  1013
lemma eq_numeral_Suc [simp]: "numeral k = Suc n \<longleftrightarrow> pred_numeral k = n"
huffman@47209
  1014
  by (simp add: numeral_eq_Suc)
huffman@47209
  1015
huffman@47209
  1016
lemma Suc_eq_numeral [simp]: "Suc n = numeral k \<longleftrightarrow> n = pred_numeral k"
huffman@47209
  1017
  by (simp add: numeral_eq_Suc)
huffman@47209
  1018
huffman@47209
  1019
lemma less_numeral_Suc [simp]: "numeral k < Suc n \<longleftrightarrow> pred_numeral k < n"
huffman@47209
  1020
  by (simp add: numeral_eq_Suc)
huffman@47209
  1021
huffman@47209
  1022
lemma less_Suc_numeral [simp]: "Suc n < numeral k \<longleftrightarrow> n < pred_numeral k"
huffman@47209
  1023
  by (simp add: numeral_eq_Suc)
huffman@47209
  1024
huffman@47209
  1025
lemma le_numeral_Suc [simp]: "numeral k \<le> Suc n \<longleftrightarrow> pred_numeral k \<le> n"
huffman@47209
  1026
  by (simp add: numeral_eq_Suc)
huffman@47209
  1027
huffman@47209
  1028
lemma le_Suc_numeral [simp]: "Suc n \<le> numeral k \<longleftrightarrow> n \<le> pred_numeral k"
huffman@47209
  1029
  by (simp add: numeral_eq_Suc)
huffman@47209
  1030
huffman@47218
  1031
lemma diff_Suc_numeral [simp]: "Suc n - numeral k = n - pred_numeral k"
huffman@47218
  1032
  by (simp add: numeral_eq_Suc)
huffman@47218
  1033
huffman@47218
  1034
lemma diff_numeral_Suc [simp]: "numeral k - Suc n = pred_numeral k - n"
huffman@47218
  1035
  by (simp add: numeral_eq_Suc)
huffman@47218
  1036
huffman@47209
  1037
lemma max_Suc_numeral [simp]:
huffman@47209
  1038
  "max (Suc n) (numeral k) = Suc (max n (pred_numeral k))"
huffman@47209
  1039
  by (simp add: numeral_eq_Suc)
huffman@47209
  1040
huffman@47209
  1041
lemma max_numeral_Suc [simp]:
huffman@47209
  1042
  "max (numeral k) (Suc n) = Suc (max (pred_numeral k) n)"
huffman@47209
  1043
  by (simp add: numeral_eq_Suc)
huffman@47209
  1044
huffman@47209
  1045
lemma min_Suc_numeral [simp]:
huffman@47209
  1046
  "min (Suc n) (numeral k) = Suc (min n (pred_numeral k))"
huffman@47209
  1047
  by (simp add: numeral_eq_Suc)
huffman@47209
  1048
huffman@47209
  1049
lemma min_numeral_Suc [simp]:
huffman@47209
  1050
  "min (numeral k) (Suc n) = Suc (min (pred_numeral k) n)"
huffman@47209
  1051
  by (simp add: numeral_eq_Suc)
huffman@47209
  1052
blanchet@55415
  1053
text {* For @{term case_nat} and @{term rec_nat}. *}
huffman@47216
  1054
blanchet@55415
  1055
lemma case_nat_numeral [simp]:
blanchet@55415
  1056
  "case_nat a f (numeral v) = (let pv = pred_numeral v in f pv)"
huffman@47216
  1057
  by (simp add: numeral_eq_Suc)
huffman@47216
  1058
blanchet@55415
  1059
lemma case_nat_add_eq_if [simp]:
blanchet@55415
  1060
  "case_nat a f ((numeral v) + n) = (let pv = pred_numeral v in f (pv + n))"
huffman@47216
  1061
  by (simp add: numeral_eq_Suc)
huffman@47216
  1062
blanchet@55415
  1063
lemma rec_nat_numeral [simp]:
blanchet@55415
  1064
  "rec_nat a f (numeral v) =
blanchet@55415
  1065
    (let pv = pred_numeral v in f pv (rec_nat a f pv))"
huffman@47216
  1066
  by (simp add: numeral_eq_Suc Let_def)
huffman@47216
  1067
blanchet@55415
  1068
lemma rec_nat_add_eq_if [simp]:
blanchet@55415
  1069
  "rec_nat a f (numeral v + n) =
blanchet@55415
  1070
    (let pv = pred_numeral v in f (pv + n) (rec_nat a f (pv + n)))"
huffman@47216
  1071
  by (simp add: numeral_eq_Suc Let_def)
huffman@47216
  1072
huffman@47255
  1073
text {* Case analysis on @{term "n < 2"} *}
huffman@47255
  1074
huffman@47255
  1075
lemma less_2_cases: "n < 2 \<Longrightarrow> n = 0 \<or> n = Suc 0"
huffman@47255
  1076
  by (auto simp add: numeral_2_eq_2)
huffman@47255
  1077
huffman@47255
  1078
text {* Removal of Small Numerals: 0, 1 and (in additive positions) 2 *}
huffman@47255
  1079
text {* bh: Are these rules really a good idea? *}
huffman@47255
  1080
huffman@47255
  1081
lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
huffman@47255
  1082
  by simp
huffman@47255
  1083
huffman@47255
  1084
lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
huffman@47255
  1085
  by simp
huffman@47255
  1086
huffman@47255
  1087
text {* Can be used to eliminate long strings of Sucs, but not by default. *}
huffman@47255
  1088
huffman@47255
  1089
lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
huffman@47255
  1090
  by simp
huffman@47255
  1091
huffman@47255
  1092
lemmas nat_1_add_1 = one_add_one [where 'a=nat] (* legacy *)
huffman@47255
  1093
huffman@47108
  1094
huffman@47108
  1095
subsection {* Numeral equations as default simplification rules *}
huffman@47108
  1096
huffman@47108
  1097
declare (in numeral) numeral_One [simp]
huffman@47108
  1098
declare (in numeral) numeral_plus_numeral [simp]
huffman@47108
  1099
declare (in numeral) add_numeral_special [simp]
huffman@47108
  1100
declare (in neg_numeral) add_neg_numeral_simps [simp]
huffman@47108
  1101
declare (in neg_numeral) add_neg_numeral_special [simp]
huffman@47108
  1102
declare (in neg_numeral) diff_numeral_simps [simp]
huffman@47108
  1103
declare (in neg_numeral) diff_numeral_special [simp]
huffman@47108
  1104
declare (in semiring_numeral) numeral_times_numeral [simp]
huffman@47108
  1105
declare (in ring_1) mult_neg_numeral_simps [simp]
huffman@47108
  1106
huffman@47108
  1107
subsection {* Setting up simprocs *}
huffman@47108
  1108
huffman@47108
  1109
lemma mult_numeral_1: "Numeral1 * a = (a::'a::semiring_numeral)"
huffman@47108
  1110
  by simp
huffman@47108
  1111
huffman@47108
  1112
lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::semiring_numeral)"
huffman@47108
  1113
  by simp
huffman@47108
  1114
huffman@47108
  1115
lemma divide_numeral_1: "a / Numeral1 = (a::'a::field)"
huffman@47108
  1116
  by simp
huffman@47108
  1117
huffman@47108
  1118
lemma inverse_numeral_1:
huffman@47108
  1119
  "inverse Numeral1 = (Numeral1::'a::division_ring)"
huffman@47108
  1120
  by simp
huffman@47108
  1121
huffman@47211
  1122
text{*Theorem lists for the cancellation simprocs. The use of a binary
huffman@47108
  1123
numeral for 1 reduces the number of special cases.*}
huffman@47108
  1124
haftmann@54489
  1125
lemma mult_1s:
haftmann@54489
  1126
  fixes a :: "'a::semiring_numeral"
haftmann@54489
  1127
    and b :: "'b::ring_1"
haftmann@54489
  1128
  shows "Numeral1 * a = a"
haftmann@54489
  1129
    "a * Numeral1 = a"
haftmann@54489
  1130
    "- Numeral1 * b = - b"
haftmann@54489
  1131
    "b * - Numeral1 = - b"
haftmann@54489
  1132
  by simp_all
huffman@47108
  1133
huffman@47226
  1134
setup {*
huffman@47226
  1135
  Reorient_Proc.add
huffman@47226
  1136
    (fn Const (@{const_name numeral}, _) $ _ => true
haftmann@54489
  1137
    | Const (@{const_name uminus}, _) $ (Const (@{const_name numeral}, _) $ _) => true
huffman@47226
  1138
    | _ => false)
huffman@47226
  1139
*}
huffman@47226
  1140
huffman@47226
  1141
simproc_setup reorient_numeral
haftmann@54489
  1142
  ("numeral w = x" | "- numeral w = y") = Reorient_Proc.proc
huffman@47226
  1143
huffman@47108
  1144
huffman@47108
  1145
subsubsection {* Simplification of arithmetic operations on integer constants. *}
huffman@47108
  1146
huffman@47108
  1147
lemmas arith_special = (* already declared simp above *)
huffman@47108
  1148
  add_numeral_special add_neg_numeral_special
haftmann@54489
  1149
  diff_numeral_special
huffman@47108
  1150
huffman@47108
  1151
(* rules already in simpset *)
huffman@47108
  1152
lemmas arith_extra_simps =
huffman@47108
  1153
  numeral_plus_numeral add_neg_numeral_simps add_0_left add_0_right
haftmann@54489
  1154
  minus_zero
huffman@47108
  1155
  diff_numeral_simps diff_0 diff_0_right
huffman@47108
  1156
  numeral_times_numeral mult_neg_numeral_simps
huffman@47108
  1157
  mult_zero_left mult_zero_right
huffman@47108
  1158
  abs_numeral abs_neg_numeral
huffman@47108
  1159
huffman@47108
  1160
text {*
huffman@47108
  1161
  For making a minimal simpset, one must include these default simprules.
huffman@47108
  1162
  Also include @{text simp_thms}.
huffman@47108
  1163
*}
huffman@47108
  1164
huffman@47108
  1165
lemmas arith_simps =
huffman@47108
  1166
  add_num_simps mult_num_simps sub_num_simps
huffman@47108
  1167
  BitM.simps dbl_simps dbl_inc_simps dbl_dec_simps
huffman@47108
  1168
  abs_zero abs_one arith_extra_simps
huffman@47108
  1169
haftmann@54249
  1170
lemmas more_arith_simps =
haftmann@54249
  1171
  neg_le_iff_le
haftmann@54249
  1172
  minus_zero left_minus right_minus
haftmann@54249
  1173
  mult_1_left mult_1_right
haftmann@54249
  1174
  mult_minus_left mult_minus_right
haftmann@57512
  1175
  minus_add_distrib minus_minus mult.assoc
haftmann@54249
  1176
haftmann@54249
  1177
lemmas of_nat_simps =
haftmann@54249
  1178
  of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
haftmann@54249
  1179
huffman@47108
  1180
text {* Simplification of relational operations *}
huffman@47108
  1181
huffman@47108
  1182
lemmas eq_numeral_extra =
huffman@47108
  1183
  zero_neq_one one_neq_zero
huffman@47108
  1184
huffman@47108
  1185
lemmas rel_simps =
huffman@47108
  1186
  le_num_simps less_num_simps eq_num_simps
haftmann@54489
  1187
  le_numeral_simps le_neg_numeral_simps le_minus_one_simps le_numeral_extra
haftmann@54489
  1188
  less_numeral_simps less_neg_numeral_simps less_minus_one_simps less_numeral_extra
huffman@47108
  1189
  eq_numeral_simps eq_neg_numeral_simps eq_numeral_extra
huffman@47108
  1190
haftmann@54249
  1191
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
haftmann@54249
  1192
  -- {* Unfold all @{text let}s involving constants *}
haftmann@54249
  1193
  unfolding Let_def ..
haftmann@54249
  1194
haftmann@54489
  1195
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
haftmann@54249
  1196
  -- {* Unfold all @{text let}s involving constants *}
haftmann@54249
  1197
  unfolding Let_def ..
haftmann@54249
  1198
haftmann@54249
  1199
declaration {*
haftmann@54249
  1200
let 
haftmann@54249
  1201
  fun number_of thy T n =
haftmann@54249
  1202
    if not (Sign.of_sort thy (T, @{sort numeral}))
haftmann@54249
  1203
    then raise CTERM ("number_of", [])
haftmann@54249
  1204
    else Numeral.mk_cnumber (Thm.ctyp_of thy T) n;
haftmann@54249
  1205
in
haftmann@54249
  1206
  K (
haftmann@54249
  1207
    Lin_Arith.add_simps (@{thms arith_simps} @ @{thms more_arith_simps}
haftmann@54249
  1208
      @ @{thms rel_simps}
haftmann@54249
  1209
      @ @{thms pred_numeral_simps}
haftmann@54249
  1210
      @ @{thms arith_special numeral_One}
haftmann@54249
  1211
      @ @{thms of_nat_simps})
haftmann@54249
  1212
    #> Lin_Arith.add_simps [@{thm Suc_numeral},
haftmann@54249
  1213
      @{thm Let_numeral}, @{thm Let_neg_numeral}, @{thm Let_0}, @{thm Let_1},
haftmann@54249
  1214
      @{thm le_Suc_numeral}, @{thm le_numeral_Suc},
haftmann@54249
  1215
      @{thm less_Suc_numeral}, @{thm less_numeral_Suc},
haftmann@54249
  1216
      @{thm Suc_eq_numeral}, @{thm eq_numeral_Suc},
haftmann@54249
  1217
      @{thm mult_Suc}, @{thm mult_Suc_right},
haftmann@54249
  1218
      @{thm of_nat_numeral}]
haftmann@54249
  1219
    #> Lin_Arith.set_number_of number_of)
haftmann@54249
  1220
end
haftmann@54249
  1221
*}
haftmann@54249
  1222
huffman@47108
  1223
huffman@47108
  1224
subsubsection {* Simplification of arithmetic when nested to the right. *}
huffman@47108
  1225
huffman@47108
  1226
lemma add_numeral_left [simp]:
huffman@47108
  1227
  "numeral v + (numeral w + z) = (numeral(v + w) + z)"
haftmann@57512
  1228
  by (simp_all add: add.assoc [symmetric])
huffman@47108
  1229
huffman@47108
  1230
lemma add_neg_numeral_left [simp]:
haftmann@54489
  1231
  "numeral v + (- numeral w + y) = (sub v w + y)"
haftmann@54489
  1232
  "- numeral v + (numeral w + y) = (sub w v + y)"
haftmann@54489
  1233
  "- numeral v + (- numeral w + y) = (- numeral(v + w) + y)"
haftmann@57512
  1234
  by (simp_all add: add.assoc [symmetric])
huffman@47108
  1235
huffman@47108
  1236
lemma mult_numeral_left [simp]:
huffman@47108
  1237
  "numeral v * (numeral w * z) = (numeral(v * w) * z :: 'a::semiring_numeral)"
haftmann@54489
  1238
  "- numeral v * (numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
haftmann@54489
  1239
  "numeral v * (- numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
haftmann@54489
  1240
  "- numeral v * (- numeral w * y) = (numeral(v * w) * y :: 'b::ring_1)"
haftmann@57512
  1241
  by (simp_all add: mult.assoc [symmetric])
huffman@47108
  1242
huffman@47108
  1243
hide_const (open) One Bit0 Bit1 BitM inc pow sqr sub dbl dbl_inc dbl_dec
huffman@47108
  1244
haftmann@51143
  1245
huffman@47108
  1246
subsection {* code module namespace *}
huffman@47108
  1247
haftmann@52435
  1248
code_identifier
haftmann@52435
  1249
  code_module Num \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
huffman@47108
  1250
huffman@47108
  1251
end