huffman@47108: (*  Title:      HOL/Num.thy
huffman@47108:     Author:     Florian Haftmann
huffman@47108:     Author:     Brian Huffman
huffman@47108: *)
huffman@47108: 
huffman@47108: header {* Binary Numerals *}
huffman@47108: 
huffman@47108: theory Num
huffman@47191: imports Datatype
huffman@47108: begin
huffman@47108: 
huffman@47108: subsection {* The @{text num} type *}
huffman@47108: 
huffman@47108: datatype num = One | Bit0 num | Bit1 num
huffman@47108: 
huffman@47108: text {* Increment function for type @{typ num} *}
huffman@47108: 
huffman@47108: primrec inc :: "num \<Rightarrow> num" where
huffman@47108:   "inc One = Bit0 One" |
huffman@47108:   "inc (Bit0 x) = Bit1 x" |
huffman@47108:   "inc (Bit1 x) = Bit0 (inc x)"
huffman@47108: 
huffman@47108: text {* Converting between type @{typ num} and type @{typ nat} *}
huffman@47108: 
huffman@47108: primrec nat_of_num :: "num \<Rightarrow> nat" where
huffman@47108:   "nat_of_num One = Suc 0" |
huffman@47108:   "nat_of_num (Bit0 x) = nat_of_num x + nat_of_num x" |
huffman@47108:   "nat_of_num (Bit1 x) = Suc (nat_of_num x + nat_of_num x)"
huffman@47108: 
huffman@47108: primrec num_of_nat :: "nat \<Rightarrow> num" where
huffman@47108:   "num_of_nat 0 = One" |
huffman@47108:   "num_of_nat (Suc n) = (if 0 < n then inc (num_of_nat n) else One)"
huffman@47108: 
huffman@47108: lemma nat_of_num_pos: "0 < nat_of_num x"
huffman@47108:   by (induct x) simp_all
huffman@47108: 
huffman@47108: lemma nat_of_num_neq_0: " nat_of_num x \<noteq> 0"
huffman@47108:   by (induct x) simp_all
huffman@47108: 
huffman@47108: lemma nat_of_num_inc: "nat_of_num (inc x) = Suc (nat_of_num x)"
huffman@47108:   by (induct x) simp_all
huffman@47108: 
huffman@47108: lemma num_of_nat_double:
huffman@47108:   "0 < n \<Longrightarrow> num_of_nat (n + n) = Bit0 (num_of_nat n)"
huffman@47108:   by (induct n) simp_all
huffman@47108: 
huffman@47108: text {*
huffman@47108:   Type @{typ num} is isomorphic to the strictly positive
huffman@47108:   natural numbers.
huffman@47108: *}
huffman@47108: 
huffman@47108: lemma nat_of_num_inverse: "num_of_nat (nat_of_num x) = x"
huffman@47108:   by (induct x) (simp_all add: num_of_nat_double nat_of_num_pos)
huffman@47108: 
huffman@47108: lemma num_of_nat_inverse: "0 < n \<Longrightarrow> nat_of_num (num_of_nat n) = n"
huffman@47108:   by (induct n) (simp_all add: nat_of_num_inc)
huffman@47108: 
huffman@47108: lemma num_eq_iff: "x = y \<longleftrightarrow> nat_of_num x = nat_of_num y"
huffman@47108:   apply safe
huffman@47108:   apply (drule arg_cong [where f=num_of_nat])
huffman@47108:   apply (simp add: nat_of_num_inverse)
huffman@47108:   done
huffman@47108: 
huffman@47108: lemma num_induct [case_names One inc]:
huffman@47108:   fixes P :: "num \<Rightarrow> bool"
huffman@47108:   assumes One: "P One"
huffman@47108:     and inc: "\<And>x. P x \<Longrightarrow> P (inc x)"
huffman@47108:   shows "P x"
huffman@47108: proof -
huffman@47108:   obtain n where n: "Suc n = nat_of_num x"
huffman@47108:     by (cases "nat_of_num x", simp_all add: nat_of_num_neq_0)
huffman@47108:   have "P (num_of_nat (Suc n))"
huffman@47108:   proof (induct n)
huffman@47108:     case 0 show ?case using One by simp
huffman@47108:   next
huffman@47108:     case (Suc n)
huffman@47108:     then have "P (inc (num_of_nat (Suc n)))" by (rule inc)
huffman@47108:     then show "P (num_of_nat (Suc (Suc n)))" by simp
huffman@47108:   qed
huffman@47108:   with n show "P x"
huffman@47108:     by (simp add: nat_of_num_inverse)
huffman@47108: qed
huffman@47108: 
huffman@47108: text {*
huffman@47108:   From now on, there are two possible models for @{typ num}:
huffman@47108:   as positive naturals (rule @{text "num_induct"})
huffman@47108:   and as digit representation (rules @{text "num.induct"}, @{text "num.cases"}).
huffman@47108: *}
huffman@47108: 
huffman@47108: 
huffman@47108: subsection {* Numeral operations *}
huffman@47108: 
huffman@47108: instantiation num :: "{plus,times,linorder}"
huffman@47108: begin
huffman@47108: 
huffman@47108: definition [code del]:
huffman@47108:   "m + n = num_of_nat (nat_of_num m + nat_of_num n)"
huffman@47108: 
huffman@47108: definition [code del]:
huffman@47108:   "m * n = num_of_nat (nat_of_num m * nat_of_num n)"
huffman@47108: 
huffman@47108: definition [code del]:
huffman@47108:   "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"
huffman@47108: 
huffman@47108: definition [code del]:
huffman@47108:   "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"
huffman@47108: 
huffman@47108: instance
huffman@47108:   by (default, auto simp add: less_num_def less_eq_num_def num_eq_iff)
huffman@47108: 
huffman@47108: end
huffman@47108: 
huffman@47108: lemma nat_of_num_add: "nat_of_num (x + y) = nat_of_num x + nat_of_num y"
huffman@47108:   unfolding plus_num_def
huffman@47108:   by (intro num_of_nat_inverse add_pos_pos nat_of_num_pos)
huffman@47108: 
huffman@47108: lemma nat_of_num_mult: "nat_of_num (x * y) = nat_of_num x * nat_of_num y"
huffman@47108:   unfolding times_num_def
huffman@47108:   by (intro num_of_nat_inverse mult_pos_pos nat_of_num_pos)
huffman@47108: 
huffman@47108: lemma add_num_simps [simp, code]:
huffman@47108:   "One + One = Bit0 One"
huffman@47108:   "One + Bit0 n = Bit1 n"
huffman@47108:   "One + Bit1 n = Bit0 (n + One)"
huffman@47108:   "Bit0 m + One = Bit1 m"
huffman@47108:   "Bit0 m + Bit0 n = Bit0 (m + n)"
huffman@47108:   "Bit0 m + Bit1 n = Bit1 (m + n)"
huffman@47108:   "Bit1 m + One = Bit0 (m + One)"
huffman@47108:   "Bit1 m + Bit0 n = Bit1 (m + n)"
huffman@47108:   "Bit1 m + Bit1 n = Bit0 (m + n + One)"
huffman@47108:   by (simp_all add: num_eq_iff nat_of_num_add)
huffman@47108: 
huffman@47108: lemma mult_num_simps [simp, code]:
huffman@47108:   "m * One = m"
huffman@47108:   "One * n = n"
huffman@47108:   "Bit0 m * Bit0 n = Bit0 (Bit0 (m * n))"
huffman@47108:   "Bit0 m * Bit1 n = Bit0 (m * Bit1 n)"
huffman@47108:   "Bit1 m * Bit0 n = Bit0 (Bit1 m * n)"
huffman@47108:   "Bit1 m * Bit1 n = Bit1 (m + n + Bit0 (m * n))"
huffman@47108:   by (simp_all add: num_eq_iff nat_of_num_add
webertj@49962:     nat_of_num_mult distrib_right distrib_left)
huffman@47108: 
huffman@47108: lemma eq_num_simps:
huffman@47108:   "One = One \<longleftrightarrow> True"
huffman@47108:   "One = Bit0 n \<longleftrightarrow> False"
huffman@47108:   "One = Bit1 n \<longleftrightarrow> False"
huffman@47108:   "Bit0 m = One \<longleftrightarrow> False"
huffman@47108:   "Bit1 m = One \<longleftrightarrow> False"
huffman@47108:   "Bit0 m = Bit0 n \<longleftrightarrow> m = n"
huffman@47108:   "Bit0 m = Bit1 n \<longleftrightarrow> False"
huffman@47108:   "Bit1 m = Bit0 n \<longleftrightarrow> False"
huffman@47108:   "Bit1 m = Bit1 n \<longleftrightarrow> m = n"
huffman@47108:   by simp_all
huffman@47108: 
huffman@47108: lemma le_num_simps [simp, code]:
huffman@47108:   "One \<le> n \<longleftrightarrow> True"
huffman@47108:   "Bit0 m \<le> One \<longleftrightarrow> False"
huffman@47108:   "Bit1 m \<le> One \<longleftrightarrow> False"
huffman@47108:   "Bit0 m \<le> Bit0 n \<longleftrightarrow> m \<le> n"
huffman@47108:   "Bit0 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"
huffman@47108:   "Bit1 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"
huffman@47108:   "Bit1 m \<le> Bit0 n \<longleftrightarrow> m < n"
huffman@47108:   using nat_of_num_pos [of n] nat_of_num_pos [of m]
huffman@47108:   by (auto simp add: less_eq_num_def less_num_def)
huffman@47108: 
huffman@47108: lemma less_num_simps [simp, code]:
huffman@47108:   "m < One \<longleftrightarrow> False"
huffman@47108:   "One < Bit0 n \<longleftrightarrow> True"
huffman@47108:   "One < Bit1 n \<longleftrightarrow> True"
huffman@47108:   "Bit0 m < Bit0 n \<longleftrightarrow> m < n"
huffman@47108:   "Bit0 m < Bit1 n \<longleftrightarrow> m \<le> n"
huffman@47108:   "Bit1 m < Bit1 n \<longleftrightarrow> m < n"
huffman@47108:   "Bit1 m < Bit0 n \<longleftrightarrow> m < n"
huffman@47108:   using nat_of_num_pos [of n] nat_of_num_pos [of m]
huffman@47108:   by (auto simp add: less_eq_num_def less_num_def)
huffman@47108: 
huffman@47108: text {* Rules using @{text One} and @{text inc} as constructors *}
huffman@47108: 
huffman@47108: lemma add_One: "x + One = inc x"
huffman@47108:   by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
huffman@47108: 
huffman@47108: lemma add_One_commute: "One + n = n + One"
huffman@47108:   by (induct n) simp_all
huffman@47108: 
huffman@47108: lemma add_inc: "x + inc y = inc (x + y)"
huffman@47108:   by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
huffman@47108: 
huffman@47108: lemma mult_inc: "x * inc y = x * y + x"
huffman@47108:   by (simp add: num_eq_iff nat_of_num_mult nat_of_num_add nat_of_num_inc)
huffman@47108: 
huffman@47108: text {* The @{const num_of_nat} conversion *}
huffman@47108: 
huffman@47108: lemma num_of_nat_One:
huffman@47300:   "n \<le> 1 \<Longrightarrow> num_of_nat n = One"
huffman@47108:   by (cases n) simp_all
huffman@47108: 
huffman@47108: lemma num_of_nat_plus_distrib:
huffman@47108:   "0 < m \<Longrightarrow> 0 < n \<Longrightarrow> num_of_nat (m + n) = num_of_nat m + num_of_nat n"
huffman@47108:   by (induct n) (auto simp add: add_One add_One_commute add_inc)
huffman@47108: 
huffman@47108: text {* A double-and-decrement function *}
huffman@47108: 
huffman@47108: primrec BitM :: "num \<Rightarrow> num" where
huffman@47108:   "BitM One = One" |
huffman@47108:   "BitM (Bit0 n) = Bit1 (BitM n)" |
huffman@47108:   "BitM (Bit1 n) = Bit1 (Bit0 n)"
huffman@47108: 
huffman@47108: lemma BitM_plus_one: "BitM n + One = Bit0 n"
huffman@47108:   by (induct n) simp_all
huffman@47108: 
huffman@47108: lemma one_plus_BitM: "One + BitM n = Bit0 n"
huffman@47108:   unfolding add_One_commute BitM_plus_one ..
huffman@47108: 
huffman@47108: text {* Squaring and exponentiation *}
huffman@47108: 
huffman@47108: primrec sqr :: "num \<Rightarrow> num" where
huffman@47108:   "sqr One = One" |
huffman@47108:   "sqr (Bit0 n) = Bit0 (Bit0 (sqr n))" |
huffman@47108:   "sqr (Bit1 n) = Bit1 (Bit0 (sqr n + n))"
huffman@47108: 
huffman@47108: primrec pow :: "num \<Rightarrow> num \<Rightarrow> num" where
huffman@47108:   "pow x One = x" |
huffman@47108:   "pow x (Bit0 y) = sqr (pow x y)" |
huffman@47191:   "pow x (Bit1 y) = sqr (pow x y) * x"
huffman@47108: 
huffman@47108: lemma nat_of_num_sqr: "nat_of_num (sqr x) = nat_of_num x * nat_of_num x"
huffman@47108:   by (induct x, simp_all add: algebra_simps nat_of_num_add)
huffman@47108: 
huffman@47108: lemma sqr_conv_mult: "sqr x = x * x"
huffman@47108:   by (simp add: num_eq_iff nat_of_num_sqr nat_of_num_mult)
huffman@47108: 
huffman@47108: 
huffman@47211: subsection {* Binary numerals *}
huffman@47108: 
huffman@47108: text {*
huffman@47211:   We embed binary representations into a generic algebraic
huffman@47108:   structure using @{text numeral}.
huffman@47108: *}
huffman@47108: 
huffman@47108: class numeral = one + semigroup_add
huffman@47108: begin
huffman@47108: 
huffman@47108: primrec numeral :: "num \<Rightarrow> 'a" where
huffman@47108:   numeral_One: "numeral One = 1" |
huffman@47108:   numeral_Bit0: "numeral (Bit0 n) = numeral n + numeral n" |
huffman@47108:   numeral_Bit1: "numeral (Bit1 n) = numeral n + numeral n + 1"
huffman@47108: 
haftmann@50817: lemma numeral_code [code]:
haftmann@50817:   "numeral One = 1"
haftmann@50817:   "numeral (Bit0 n) = (let m = numeral n in m + m)"
haftmann@50817:   "numeral (Bit1 n) = (let m = numeral n in m + m + 1)"
haftmann@50817:   by (simp_all add: Let_def)
haftmann@50817:   
huffman@47108: lemma one_plus_numeral_commute: "1 + numeral x = numeral x + 1"
huffman@47108:   apply (induct x)
huffman@47108:   apply simp
huffman@47108:   apply (simp add: add_assoc [symmetric], simp add: add_assoc)
huffman@47108:   apply (simp add: add_assoc [symmetric], simp add: add_assoc)
huffman@47108:   done
huffman@47108: 
huffman@47108: lemma numeral_inc: "numeral (inc x) = numeral x + 1"
huffman@47108: proof (induct x)
huffman@47108:   case (Bit1 x)
huffman@47108:   have "numeral x + (1 + numeral x) + 1 = numeral x + (numeral x + 1) + 1"
huffman@47108:     by (simp only: one_plus_numeral_commute)
huffman@47108:   with Bit1 show ?case
huffman@47108:     by (simp add: add_assoc)
huffman@47108: qed simp_all
huffman@47108: 
huffman@47108: declare numeral.simps [simp del]
huffman@47108: 
huffman@47108: abbreviation "Numeral1 \<equiv> numeral One"
huffman@47108: 
huffman@47108: declare numeral_One [code_post]
huffman@47108: 
huffman@47108: end
huffman@47108: 
huffman@47108: text {* Negative numerals. *}
huffman@47108: 
huffman@47108: class neg_numeral = numeral + group_add
huffman@47108: begin
huffman@47108: 
huffman@47108: definition neg_numeral :: "num \<Rightarrow> 'a" where
huffman@47108:   "neg_numeral k = - numeral k"
huffman@47108: 
huffman@47108: end
huffman@47108: 
huffman@47108: text {* Numeral syntax. *}
huffman@47108: 
huffman@47108: syntax
huffman@47108:   "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
huffman@47108: 
huffman@47108: parse_translation {*
wenzelm@52143:   let
wenzelm@52143:     fun num_of_int n =
wenzelm@52143:       if n > 0 then
wenzelm@52143:         (case IntInf.quotRem (n, 2) of
wenzelm@52143:           (0, 1) => Syntax.const @{const_name One}
wenzelm@52143:         | (n, 0) => Syntax.const @{const_name Bit0} $ num_of_int n
wenzelm@52143:         | (n, 1) => Syntax.const @{const_name Bit1} $ num_of_int n)
wenzelm@52143:       else raise Match
wenzelm@52143:     val pos = Syntax.const @{const_name numeral}
wenzelm@52143:     val neg = Syntax.const @{const_name neg_numeral}
wenzelm@52143:     val one = Syntax.const @{const_name Groups.one}
wenzelm@52143:     val zero = Syntax.const @{const_name Groups.zero}
wenzelm@52143:     fun numeral_tr [(c as Const (@{syntax_const "_constrain"}, _)) $ t $ u] =
wenzelm@52143:           c $ numeral_tr [t] $ u
wenzelm@52143:       | numeral_tr [Const (num, _)] =
wenzelm@52143:           let
wenzelm@52143:             val {value, ...} = Lexicon.read_xnum num;
wenzelm@52143:           in
wenzelm@52143:             if value = 0 then zero else
wenzelm@52143:             if value > 0
wenzelm@52143:             then pos $ num_of_int value
wenzelm@52143:             else neg $ num_of_int (~value)
wenzelm@52143:           end
wenzelm@52143:       | numeral_tr ts = raise TERM ("numeral_tr", ts);
wenzelm@52143:   in [("_Numeral", K numeral_tr)] end
huffman@47108: *}
huffman@47108: 
wenzelm@52143: typed_print_translation {*
wenzelm@52143:   let
wenzelm@52143:     fun dest_num (Const (@{const_syntax Bit0}, _) $ n) = 2 * dest_num n
wenzelm@52143:       | dest_num (Const (@{const_syntax Bit1}, _) $ n) = 2 * dest_num n + 1
wenzelm@52143:       | dest_num (Const (@{const_syntax One}, _)) = 1;
wenzelm@52143:     fun num_tr' sign ctxt T [n] =
wenzelm@52143:       let
wenzelm@52143:         val k = dest_num n;
wenzelm@52187:         val t' =
wenzelm@52187:           Syntax.const @{syntax_const "_Numeral"} $
wenzelm@52187:             Syntax.free (sign ^ string_of_int k);
wenzelm@52143:       in
wenzelm@52143:         (case T of
wenzelm@52143:           Type (@{type_name fun}, [_, T']) =>
wenzelm@52210:             if Printer.type_emphasis ctxt T' then
wenzelm@52210:               Syntax.const @{syntax_const "_constrain"} $ t' $
wenzelm@52210:                 Syntax_Phases.term_of_typ ctxt T'
wenzelm@52210:             else t'
wenzelm@52187:         | _ => if T = dummyT then t' else raise Match)
wenzelm@52143:       end;
wenzelm@52143:   in
wenzelm@52143:    [(@{const_syntax numeral}, num_tr' ""),
wenzelm@52143:     (@{const_syntax neg_numeral}, num_tr' "-")]
wenzelm@52143:   end
huffman@47108: *}
huffman@47108: 
wenzelm@48891: ML_file "Tools/numeral.ML"
huffman@47228: 
huffman@47228: 
huffman@47108: subsection {* Class-specific numeral rules *}
huffman@47108: 
huffman@47108: text {*
huffman@47108:   @{const numeral} is a morphism.
huffman@47108: *}
huffman@47108: 
huffman@47108: subsubsection {* Structures with addition: class @{text numeral} *}
huffman@47108: 
huffman@47108: context numeral
huffman@47108: begin
huffman@47108: 
huffman@47108: lemma numeral_add: "numeral (m + n) = numeral m + numeral n"
huffman@47108:   by (induct n rule: num_induct)
huffman@47108:      (simp_all only: numeral_One add_One add_inc numeral_inc add_assoc)
huffman@47108: 
huffman@47108: lemma numeral_plus_numeral: "numeral m + numeral n = numeral (m + n)"
huffman@47108:   by (rule numeral_add [symmetric])
huffman@47108: 
huffman@47108: lemma numeral_plus_one: "numeral n + 1 = numeral (n + One)"
huffman@47108:   using numeral_add [of n One] by (simp add: numeral_One)
huffman@47108: 
huffman@47108: lemma one_plus_numeral: "1 + numeral n = numeral (One + n)"
huffman@47108:   using numeral_add [of One n] by (simp add: numeral_One)
huffman@47108: 
huffman@47108: lemma one_add_one: "1 + 1 = 2"
huffman@47108:   using numeral_add [of One One] by (simp add: numeral_One)
huffman@47108: 
huffman@47108: lemmas add_numeral_special =
huffman@47108:   numeral_plus_one one_plus_numeral one_add_one
huffman@47108: 
huffman@47108: end
huffman@47108: 
huffman@47108: subsubsection {*
huffman@47108:   Structures with negation: class @{text neg_numeral}
huffman@47108: *}
huffman@47108: 
huffman@47108: context neg_numeral
huffman@47108: begin
huffman@47108: 
huffman@47108: text {* Numerals form an abelian subgroup. *}
huffman@47108: 
huffman@47108: inductive is_num :: "'a \<Rightarrow> bool" where
huffman@47108:   "is_num 1" |
huffman@47108:   "is_num x \<Longrightarrow> is_num (- x)" |
huffman@47108:   "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> is_num (x + y)"
huffman@47108: 
huffman@47108: lemma is_num_numeral: "is_num (numeral k)"
huffman@47108:   by (induct k, simp_all add: numeral.simps is_num.intros)
huffman@47108: 
huffman@47108: lemma is_num_add_commute:
huffman@47108:   "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + y = y + x"
huffman@47108:   apply (induct x rule: is_num.induct)
huffman@47108:   apply (induct y rule: is_num.induct)
huffman@47108:   apply simp
huffman@47108:   apply (rule_tac a=x in add_left_imp_eq)
huffman@47108:   apply (rule_tac a=x in add_right_imp_eq)
huffman@47108:   apply (simp add: add_assoc minus_add_cancel)
huffman@47108:   apply (simp add: add_assoc [symmetric], simp add: add_assoc)
huffman@47108:   apply (rule_tac a=x in add_left_imp_eq)
huffman@47108:   apply (rule_tac a=x in add_right_imp_eq)
haftmann@54230:   apply (simp add: add_assoc)
huffman@47108:   apply (simp add: add_assoc, simp add: add_assoc [symmetric])
huffman@47108:   done
huffman@47108: 
huffman@47108: lemma is_num_add_left_commute:
huffman@47108:   "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + (y + z) = y + (x + z)"
huffman@47108:   by (simp only: add_assoc [symmetric] is_num_add_commute)
huffman@47108: 
huffman@47108: lemmas is_num_normalize =
huffman@47108:   add_assoc is_num_add_commute is_num_add_left_commute
huffman@47108:   is_num.intros is_num_numeral
haftmann@54230:   minus_add
huffman@47108: 
huffman@47108: definition dbl :: "'a \<Rightarrow> 'a" where "dbl x = x + x"
huffman@47108: definition dbl_inc :: "'a \<Rightarrow> 'a" where "dbl_inc x = x + x + 1"
huffman@47108: definition dbl_dec :: "'a \<Rightarrow> 'a" where "dbl_dec x = x + x - 1"
huffman@47108: 
huffman@47108: definition sub :: "num \<Rightarrow> num \<Rightarrow> 'a" where
huffman@47108:   "sub k l = numeral k - numeral l"
huffman@47108: 
huffman@47108: lemma numeral_BitM: "numeral (BitM n) = numeral (Bit0 n) - 1"
huffman@47108:   by (simp only: BitM_plus_one [symmetric] numeral_add numeral_One eq_diff_eq)
huffman@47108: 
huffman@47108: lemma dbl_simps [simp]:
huffman@47108:   "dbl (neg_numeral k) = neg_numeral (Bit0 k)"
huffman@47108:   "dbl 0 = 0"
huffman@47108:   "dbl 1 = 2"
huffman@47108:   "dbl (numeral k) = numeral (Bit0 k)"
haftmann@54230:   by (simp_all add: dbl_def neg_numeral_def numeral.simps minus_add)
huffman@47108: 
huffman@47108: lemma dbl_inc_simps [simp]:
huffman@47108:   "dbl_inc (neg_numeral k) = neg_numeral (BitM k)"
huffman@47108:   "dbl_inc 0 = 1"
huffman@47108:   "dbl_inc 1 = 3"
huffman@47108:   "dbl_inc (numeral k) = numeral (Bit1 k)"
haftmann@54230:   by (simp_all add: dbl_inc_def neg_numeral_def numeral.simps numeral_BitM is_num_normalize algebra_simps del: add_uminus_conv_diff)
huffman@47108: 
huffman@47108: lemma dbl_dec_simps [simp]:
huffman@47108:   "dbl_dec (neg_numeral k) = neg_numeral (Bit1 k)"
huffman@47108:   "dbl_dec 0 = -1"
huffman@47108:   "dbl_dec 1 = 1"
huffman@47108:   "dbl_dec (numeral k) = numeral (BitM k)"
haftmann@54230:   by (simp_all add: dbl_dec_def neg_numeral_def numeral.simps numeral_BitM is_num_normalize)
huffman@47108: 
huffman@47108: lemma sub_num_simps [simp]:
huffman@47108:   "sub One One = 0"
huffman@47108:   "sub One (Bit0 l) = neg_numeral (BitM l)"
huffman@47108:   "sub One (Bit1 l) = neg_numeral (Bit0 l)"
huffman@47108:   "sub (Bit0 k) One = numeral (BitM k)"
huffman@47108:   "sub (Bit1 k) One = numeral (Bit0 k)"
huffman@47108:   "sub (Bit0 k) (Bit0 l) = dbl (sub k l)"
huffman@47108:   "sub (Bit0 k) (Bit1 l) = dbl_dec (sub k l)"
huffman@47108:   "sub (Bit1 k) (Bit0 l) = dbl_inc (sub k l)"
huffman@47108:   "sub (Bit1 k) (Bit1 l) = dbl (sub k l)"
haftmann@54230:   by (simp_all add: dbl_def dbl_dec_def dbl_inc_def sub_def neg_numeral_def numeral.simps
haftmann@54230:     numeral_BitM is_num_normalize del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108: 
huffman@47108: lemma add_neg_numeral_simps:
huffman@47108:   "numeral m + neg_numeral n = sub m n"
huffman@47108:   "neg_numeral m + numeral n = sub n m"
huffman@47108:   "neg_numeral m + neg_numeral n = neg_numeral (m + n)"
haftmann@54230:   by (simp_all add: sub_def neg_numeral_def numeral_add numeral.simps is_num_normalize
haftmann@54230:     del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108: 
huffman@47108: lemma add_neg_numeral_special:
huffman@47108:   "1 + neg_numeral m = sub One m"
huffman@47108:   "neg_numeral m + 1 = sub One m"
haftmann@54230:   by (simp_all add: sub_def neg_numeral_def numeral_add numeral.simps is_num_normalize)
huffman@47108: 
huffman@47108: lemma diff_numeral_simps:
huffman@47108:   "numeral m - numeral n = sub m n"
huffman@47108:   "numeral m - neg_numeral n = numeral (m + n)"
huffman@47108:   "neg_numeral m - numeral n = neg_numeral (m + n)"
huffman@47108:   "neg_numeral m - neg_numeral n = sub n m"
haftmann@54230:   by (simp_all add: neg_numeral_def sub_def numeral_add numeral.simps is_num_normalize
haftmann@54230:     del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108: 
huffman@47108: lemma diff_numeral_special:
huffman@47108:   "1 - numeral n = sub One n"
huffman@47108:   "1 - neg_numeral n = numeral (One + n)"
huffman@47108:   "numeral m - 1 = sub m One"
huffman@47108:   "neg_numeral m - 1 = neg_numeral (m + One)"
haftmann@54230:   by (simp_all add: neg_numeral_def sub_def numeral_add numeral.simps add: is_num_normalize)
huffman@47108: 
huffman@47108: lemma minus_one: "- 1 = -1"
huffman@47108:   unfolding neg_numeral_def numeral.simps ..
huffman@47108: 
huffman@47108: lemma minus_numeral: "- numeral n = neg_numeral n"
huffman@47108:   unfolding neg_numeral_def ..
huffman@47108: 
huffman@47108: lemma minus_neg_numeral: "- neg_numeral n = numeral n"
huffman@47108:   unfolding neg_numeral_def by simp
huffman@47108: 
huffman@47108: lemmas minus_numeral_simps [simp] =
huffman@47108:   minus_one minus_numeral minus_neg_numeral
huffman@47108: 
huffman@47108: end
huffman@47108: 
huffman@47108: subsubsection {*
huffman@47108:   Structures with multiplication: class @{text semiring_numeral}
huffman@47108: *}
huffman@47108: 
huffman@47108: class semiring_numeral = semiring + monoid_mult
huffman@47108: begin
huffman@47108: 
huffman@47108: subclass numeral ..
huffman@47108: 
huffman@47108: lemma numeral_mult: "numeral (m * n) = numeral m * numeral n"
huffman@47108:   apply (induct n rule: num_induct)
huffman@47108:   apply (simp add: numeral_One)
webertj@49962:   apply (simp add: mult_inc numeral_inc numeral_add distrib_left)
huffman@47108:   done
huffman@47108: 
huffman@47108: lemma numeral_times_numeral: "numeral m * numeral n = numeral (m * n)"
huffman@47108:   by (rule numeral_mult [symmetric])
huffman@47108: 
haftmann@53064: lemma mult_2: "2 * z = z + z"
haftmann@53064:   unfolding one_add_one [symmetric] distrib_right by simp
haftmann@53064: 
haftmann@53064: lemma mult_2_right: "z * 2 = z + z"
haftmann@53064:   unfolding one_add_one [symmetric] distrib_left by simp
haftmann@53064: 
huffman@47108: end
huffman@47108: 
huffman@47108: subsubsection {*
huffman@47108:   Structures with a zero: class @{text semiring_1}
huffman@47108: *}
huffman@47108: 
huffman@47108: context semiring_1
huffman@47108: begin
huffman@47108: 
huffman@47108: subclass semiring_numeral ..
huffman@47108: 
huffman@47108: lemma of_nat_numeral [simp]: "of_nat (numeral n) = numeral n"
huffman@47108:   by (induct n,
huffman@47108:     simp_all only: numeral.simps numeral_class.numeral.simps of_nat_add of_nat_1)
huffman@47108: 
huffman@47108: end
huffman@47108: 
haftmann@51143: lemma nat_of_num_numeral [code_abbrev]:
haftmann@51143:   "nat_of_num = numeral"
huffman@47108: proof
huffman@47108:   fix n
huffman@47108:   have "numeral n = nat_of_num n"
huffman@47108:     by (induct n) (simp_all add: numeral.simps)
huffman@47108:   then show "nat_of_num n = numeral n" by simp
huffman@47108: qed
huffman@47108: 
haftmann@51143: lemma nat_of_num_code [code]:
haftmann@51143:   "nat_of_num One = 1"
haftmann@51143:   "nat_of_num (Bit0 n) = (let m = nat_of_num n in m + m)"
haftmann@51143:   "nat_of_num (Bit1 n) = (let m = nat_of_num n in Suc (m + m))"
haftmann@51143:   by (simp_all add: Let_def)
haftmann@51143: 
huffman@47108: subsubsection {*
huffman@47108:   Equality: class @{text semiring_char_0}
huffman@47108: *}
huffman@47108: 
huffman@47108: context semiring_char_0
huffman@47108: begin
huffman@47108: 
huffman@47108: lemma numeral_eq_iff: "numeral m = numeral n \<longleftrightarrow> m = n"
huffman@47108:   unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
huffman@47108:     of_nat_eq_iff num_eq_iff ..
huffman@47108: 
huffman@47108: lemma numeral_eq_one_iff: "numeral n = 1 \<longleftrightarrow> n = One"
huffman@47108:   by (rule numeral_eq_iff [of n One, unfolded numeral_One])
huffman@47108: 
huffman@47108: lemma one_eq_numeral_iff: "1 = numeral n \<longleftrightarrow> One = n"
huffman@47108:   by (rule numeral_eq_iff [of One n, unfolded numeral_One])
huffman@47108: 
huffman@47108: lemma numeral_neq_zero: "numeral n \<noteq> 0"
huffman@47108:   unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
huffman@47108:   by (simp add: nat_of_num_pos)
huffman@47108: 
huffman@47108: lemma zero_neq_numeral: "0 \<noteq> numeral n"
huffman@47108:   unfolding eq_commute [of 0] by (rule numeral_neq_zero)
huffman@47108: 
huffman@47108: lemmas eq_numeral_simps [simp] =
huffman@47108:   numeral_eq_iff
huffman@47108:   numeral_eq_one_iff
huffman@47108:   one_eq_numeral_iff
huffman@47108:   numeral_neq_zero
huffman@47108:   zero_neq_numeral
huffman@47108: 
huffman@47108: end
huffman@47108: 
huffman@47108: subsubsection {*
huffman@47108:   Comparisons: class @{text linordered_semidom}
huffman@47108: *}
huffman@47108: 
huffman@47108: text {*  Could be perhaps more general than here. *}
huffman@47108: 
huffman@47108: context linordered_semidom
huffman@47108: begin
huffman@47108: 
huffman@47108: lemma numeral_le_iff: "numeral m \<le> numeral n \<longleftrightarrow> m \<le> n"
huffman@47108: proof -
huffman@47108:   have "of_nat (numeral m) \<le> of_nat (numeral n) \<longleftrightarrow> m \<le> n"
huffman@47108:     unfolding less_eq_num_def nat_of_num_numeral of_nat_le_iff ..
huffman@47108:   then show ?thesis by simp
huffman@47108: qed
huffman@47108: 
huffman@47108: lemma one_le_numeral: "1 \<le> numeral n"
huffman@47108: using numeral_le_iff [of One n] by (simp add: numeral_One)
huffman@47108: 
huffman@47108: lemma numeral_le_one_iff: "numeral n \<le> 1 \<longleftrightarrow> n \<le> One"
huffman@47108: using numeral_le_iff [of n One] by (simp add: numeral_One)
huffman@47108: 
huffman@47108: lemma numeral_less_iff: "numeral m < numeral n \<longleftrightarrow> m < n"
huffman@47108: proof -
huffman@47108:   have "of_nat (numeral m) < of_nat (numeral n) \<longleftrightarrow> m < n"
huffman@47108:     unfolding less_num_def nat_of_num_numeral of_nat_less_iff ..
huffman@47108:   then show ?thesis by simp
huffman@47108: qed
huffman@47108: 
huffman@47108: lemma not_numeral_less_one: "\<not> numeral n < 1"
huffman@47108:   using numeral_less_iff [of n One] by (simp add: numeral_One)
huffman@47108: 
huffman@47108: lemma one_less_numeral_iff: "1 < numeral n \<longleftrightarrow> One < n"
huffman@47108:   using numeral_less_iff [of One n] by (simp add: numeral_One)
huffman@47108: 
huffman@47108: lemma zero_le_numeral: "0 \<le> numeral n"
huffman@47108:   by (induct n) (simp_all add: numeral.simps)
huffman@47108: 
huffman@47108: lemma zero_less_numeral: "0 < numeral n"
huffman@47108:   by (induct n) (simp_all add: numeral.simps add_pos_pos)
huffman@47108: 
huffman@47108: lemma not_numeral_le_zero: "\<not> numeral n \<le> 0"
huffman@47108:   by (simp add: not_le zero_less_numeral)
huffman@47108: 
huffman@47108: lemma not_numeral_less_zero: "\<not> numeral n < 0"
huffman@47108:   by (simp add: not_less zero_le_numeral)
huffman@47108: 
huffman@47108: lemmas le_numeral_extra =
huffman@47108:   zero_le_one not_one_le_zero
huffman@47108:   order_refl [of 0] order_refl [of 1]
huffman@47108: 
huffman@47108: lemmas less_numeral_extra =
huffman@47108:   zero_less_one not_one_less_zero
huffman@47108:   less_irrefl [of 0] less_irrefl [of 1]
huffman@47108: 
huffman@47108: lemmas le_numeral_simps [simp] =
huffman@47108:   numeral_le_iff
huffman@47108:   one_le_numeral
huffman@47108:   numeral_le_one_iff
huffman@47108:   zero_le_numeral
huffman@47108:   not_numeral_le_zero
huffman@47108: 
huffman@47108: lemmas less_numeral_simps [simp] =
huffman@47108:   numeral_less_iff
huffman@47108:   one_less_numeral_iff
huffman@47108:   not_numeral_less_one
huffman@47108:   zero_less_numeral
huffman@47108:   not_numeral_less_zero
huffman@47108: 
huffman@47108: end
huffman@47108: 
huffman@47108: subsubsection {*
huffman@47108:   Multiplication and negation: class @{text ring_1}
huffman@47108: *}
huffman@47108: 
huffman@47108: context ring_1
huffman@47108: begin
huffman@47108: 
huffman@47108: subclass neg_numeral ..
huffman@47108: 
huffman@47108: lemma mult_neg_numeral_simps:
huffman@47108:   "neg_numeral m * neg_numeral n = numeral (m * n)"
huffman@47108:   "neg_numeral m * numeral n = neg_numeral (m * n)"
huffman@47108:   "numeral m * neg_numeral n = neg_numeral (m * n)"
huffman@47108:   unfolding neg_numeral_def mult_minus_left mult_minus_right
huffman@47108:   by (simp_all only: minus_minus numeral_mult)
huffman@47108: 
huffman@47108: lemma mult_minus1 [simp]: "-1 * z = - z"
huffman@47108:   unfolding neg_numeral_def numeral.simps mult_minus_left by simp
huffman@47108: 
huffman@47108: lemma mult_minus1_right [simp]: "z * -1 = - z"
huffman@47108:   unfolding neg_numeral_def numeral.simps mult_minus_right by simp
huffman@47108: 
huffman@47108: end
huffman@47108: 
huffman@47108: subsubsection {*
huffman@47108:   Equality using @{text iszero} for rings with non-zero characteristic
huffman@47108: *}
huffman@47108: 
huffman@47108: context ring_1
huffman@47108: begin
huffman@47108: 
huffman@47108: definition iszero :: "'a \<Rightarrow> bool"
huffman@47108:   where "iszero z \<longleftrightarrow> z = 0"
huffman@47108: 
huffman@47108: lemma iszero_0 [simp]: "iszero 0"
huffman@47108:   by (simp add: iszero_def)
huffman@47108: 
huffman@47108: lemma not_iszero_1 [simp]: "\<not> iszero 1"
huffman@47108:   by (simp add: iszero_def)
huffman@47108: 
huffman@47108: lemma not_iszero_Numeral1: "\<not> iszero Numeral1"
huffman@47108:   by (simp add: numeral_One)
huffman@47108: 
huffman@47108: lemma iszero_neg_numeral [simp]:
huffman@47108:   "iszero (neg_numeral w) \<longleftrightarrow> iszero (numeral w)"
huffman@47108:   unfolding iszero_def neg_numeral_def
huffman@47108:   by (rule neg_equal_0_iff_equal)
huffman@47108: 
huffman@47108: lemma eq_iff_iszero_diff: "x = y \<longleftrightarrow> iszero (x - y)"
huffman@47108:   unfolding iszero_def by (rule eq_iff_diff_eq_0)
huffman@47108: 
huffman@47108: text {* The @{text "eq_numeral_iff_iszero"} lemmas are not declared
huffman@47108: @{text "[simp]"} by default, because for rings of characteristic zero,
huffman@47108: better simp rules are possible. For a type like integers mod @{text
huffman@47108: "n"}, type-instantiated versions of these rules should be added to the
huffman@47108: simplifier, along with a type-specific rule for deciding propositions
huffman@47108: of the form @{text "iszero (numeral w)"}.
huffman@47108: 
huffman@47108: bh: Maybe it would not be so bad to just declare these as simp
huffman@47108: rules anyway? I should test whether these rules take precedence over
huffman@47108: the @{text "ring_char_0"} rules in the simplifier.
huffman@47108: *}
huffman@47108: 
huffman@47108: lemma eq_numeral_iff_iszero:
huffman@47108:   "numeral x = numeral y \<longleftrightarrow> iszero (sub x y)"
huffman@47108:   "numeral x = neg_numeral y \<longleftrightarrow> iszero (numeral (x + y))"
huffman@47108:   "neg_numeral x = numeral y \<longleftrightarrow> iszero (numeral (x + y))"
huffman@47108:   "neg_numeral x = neg_numeral y \<longleftrightarrow> iszero (sub y x)"
huffman@47108:   "numeral x = 1 \<longleftrightarrow> iszero (sub x One)"
huffman@47108:   "1 = numeral y \<longleftrightarrow> iszero (sub One y)"
huffman@47108:   "neg_numeral x = 1 \<longleftrightarrow> iszero (numeral (x + One))"
huffman@47108:   "1 = neg_numeral y \<longleftrightarrow> iszero (numeral (One + y))"
huffman@47108:   "numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
huffman@47108:   "0 = numeral y \<longleftrightarrow> iszero (numeral y)"
huffman@47108:   "neg_numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
huffman@47108:   "0 = neg_numeral y \<longleftrightarrow> iszero (numeral y)"
huffman@47108:   unfolding eq_iff_iszero_diff diff_numeral_simps diff_numeral_special
huffman@47108:   by simp_all
huffman@47108: 
huffman@47108: end
huffman@47108: 
huffman@47108: subsubsection {*
huffman@47108:   Equality and negation: class @{text ring_char_0}
huffman@47108: *}
huffman@47108: 
huffman@47108: class ring_char_0 = ring_1 + semiring_char_0
huffman@47108: begin
huffman@47108: 
huffman@47108: lemma not_iszero_numeral [simp]: "\<not> iszero (numeral w)"
huffman@47108:   by (simp add: iszero_def)
huffman@47108: 
huffman@47108: lemma neg_numeral_eq_iff: "neg_numeral m = neg_numeral n \<longleftrightarrow> m = n"
huffman@47108:   by (simp only: neg_numeral_def neg_equal_iff_equal numeral_eq_iff)
huffman@47108: 
huffman@47108: lemma numeral_neq_neg_numeral: "numeral m \<noteq> neg_numeral n"
huffman@47108:   unfolding neg_numeral_def eq_neg_iff_add_eq_0
huffman@47108:   by (simp add: numeral_plus_numeral)
huffman@47108: 
huffman@47108: lemma neg_numeral_neq_numeral: "neg_numeral m \<noteq> numeral n"
huffman@47108:   by (rule numeral_neq_neg_numeral [symmetric])
huffman@47108: 
huffman@47108: lemma zero_neq_neg_numeral: "0 \<noteq> neg_numeral n"
huffman@47108:   unfolding neg_numeral_def neg_0_equal_iff_equal by simp
huffman@47108: 
huffman@47108: lemma neg_numeral_neq_zero: "neg_numeral n \<noteq> 0"
huffman@47108:   unfolding neg_numeral_def neg_equal_0_iff_equal by simp
huffman@47108: 
huffman@47108: lemma one_neq_neg_numeral: "1 \<noteq> neg_numeral n"
huffman@47108:   using numeral_neq_neg_numeral [of One n] by (simp add: numeral_One)
huffman@47108: 
huffman@47108: lemma neg_numeral_neq_one: "neg_numeral n \<noteq> 1"
huffman@47108:   using neg_numeral_neq_numeral [of n One] by (simp add: numeral_One)
huffman@47108: 
huffman@47108: lemmas eq_neg_numeral_simps [simp] =
huffman@47108:   neg_numeral_eq_iff
huffman@47108:   numeral_neq_neg_numeral neg_numeral_neq_numeral
huffman@47108:   one_neq_neg_numeral neg_numeral_neq_one
huffman@47108:   zero_neq_neg_numeral neg_numeral_neq_zero
huffman@47108: 
huffman@47108: end
huffman@47108: 
huffman@47108: subsubsection {*
huffman@47108:   Structures with negation and order: class @{text linordered_idom}
huffman@47108: *}
huffman@47108: 
huffman@47108: context linordered_idom
huffman@47108: begin
huffman@47108: 
huffman@47108: subclass ring_char_0 ..
huffman@47108: 
huffman@47108: lemma neg_numeral_le_iff: "neg_numeral m \<le> neg_numeral n \<longleftrightarrow> n \<le> m"
huffman@47108:   by (simp only: neg_numeral_def neg_le_iff_le numeral_le_iff)
huffman@47108: 
huffman@47108: lemma neg_numeral_less_iff: "neg_numeral m < neg_numeral n \<longleftrightarrow> n < m"
huffman@47108:   by (simp only: neg_numeral_def neg_less_iff_less numeral_less_iff)
huffman@47108: 
huffman@47108: lemma neg_numeral_less_zero: "neg_numeral n < 0"
huffman@47108:   by (simp only: neg_numeral_def neg_less_0_iff_less zero_less_numeral)
huffman@47108: 
huffman@47108: lemma neg_numeral_le_zero: "neg_numeral n \<le> 0"
huffman@47108:   by (simp only: neg_numeral_def neg_le_0_iff_le zero_le_numeral)
huffman@47108: 
huffman@47108: lemma not_zero_less_neg_numeral: "\<not> 0 < neg_numeral n"
huffman@47108:   by (simp only: not_less neg_numeral_le_zero)
huffman@47108: 
huffman@47108: lemma not_zero_le_neg_numeral: "\<not> 0 \<le> neg_numeral n"
huffman@47108:   by (simp only: not_le neg_numeral_less_zero)
huffman@47108: 
huffman@47108: lemma neg_numeral_less_numeral: "neg_numeral m < numeral n"
huffman@47108:   using neg_numeral_less_zero zero_less_numeral by (rule less_trans)
huffman@47108: 
huffman@47108: lemma neg_numeral_le_numeral: "neg_numeral m \<le> numeral n"
huffman@47108:   by (simp only: less_imp_le neg_numeral_less_numeral)
huffman@47108: 
huffman@47108: lemma not_numeral_less_neg_numeral: "\<not> numeral m < neg_numeral n"
huffman@47108:   by (simp only: not_less neg_numeral_le_numeral)
huffman@47108: 
huffman@47108: lemma not_numeral_le_neg_numeral: "\<not> numeral m \<le> neg_numeral n"
huffman@47108:   by (simp only: not_le neg_numeral_less_numeral)
huffman@47108:   
huffman@47108: lemma neg_numeral_less_one: "neg_numeral m < 1"
huffman@47108:   by (rule neg_numeral_less_numeral [of m One, unfolded numeral_One])
huffman@47108: 
huffman@47108: lemma neg_numeral_le_one: "neg_numeral m \<le> 1"
huffman@47108:   by (rule neg_numeral_le_numeral [of m One, unfolded numeral_One])
huffman@47108: 
huffman@47108: lemma not_one_less_neg_numeral: "\<not> 1 < neg_numeral m"
huffman@47108:   by (simp only: not_less neg_numeral_le_one)
huffman@47108: 
huffman@47108: lemma not_one_le_neg_numeral: "\<not> 1 \<le> neg_numeral m"
huffman@47108:   by (simp only: not_le neg_numeral_less_one)
huffman@47108: 
huffman@47108: lemma sub_non_negative:
huffman@47108:   "sub n m \<ge> 0 \<longleftrightarrow> n \<ge> m"
huffman@47108:   by (simp only: sub_def le_diff_eq) simp
huffman@47108: 
huffman@47108: lemma sub_positive:
huffman@47108:   "sub n m > 0 \<longleftrightarrow> n > m"
huffman@47108:   by (simp only: sub_def less_diff_eq) simp
huffman@47108: 
huffman@47108: lemma sub_non_positive:
huffman@47108:   "sub n m \<le> 0 \<longleftrightarrow> n \<le> m"
huffman@47108:   by (simp only: sub_def diff_le_eq) simp
huffman@47108: 
huffman@47108: lemma sub_negative:
huffman@47108:   "sub n m < 0 \<longleftrightarrow> n < m"
huffman@47108:   by (simp only: sub_def diff_less_eq) simp
huffman@47108: 
huffman@47108: lemmas le_neg_numeral_simps [simp] =
huffman@47108:   neg_numeral_le_iff
huffman@47108:   neg_numeral_le_numeral not_numeral_le_neg_numeral
huffman@47108:   neg_numeral_le_zero not_zero_le_neg_numeral
huffman@47108:   neg_numeral_le_one not_one_le_neg_numeral
huffman@47108: 
huffman@47108: lemmas less_neg_numeral_simps [simp] =
huffman@47108:   neg_numeral_less_iff
huffman@47108:   neg_numeral_less_numeral not_numeral_less_neg_numeral
huffman@47108:   neg_numeral_less_zero not_zero_less_neg_numeral
huffman@47108:   neg_numeral_less_one not_one_less_neg_numeral
huffman@47108: 
huffman@47108: lemma abs_numeral [simp]: "abs (numeral n) = numeral n"
huffman@47108:   by simp
huffman@47108: 
huffman@47108: lemma abs_neg_numeral [simp]: "abs (neg_numeral n) = numeral n"
huffman@47108:   by (simp only: neg_numeral_def abs_minus_cancel abs_numeral)
huffman@47108: 
huffman@47108: end
huffman@47108: 
huffman@47108: subsubsection {*
huffman@47108:   Natural numbers
huffman@47108: *}
huffman@47108: 
huffman@47299: lemma Suc_1 [simp]: "Suc 1 = 2"
huffman@47299:   unfolding Suc_eq_plus1 by (rule one_add_one)
huffman@47299: 
huffman@47108: lemma Suc_numeral [simp]: "Suc (numeral n) = numeral (n + One)"
huffman@47299:   unfolding Suc_eq_plus1 by (rule numeral_plus_one)
huffman@47108: 
huffman@47209: definition pred_numeral :: "num \<Rightarrow> nat"
huffman@47209:   where [code del]: "pred_numeral k = numeral k - 1"
huffman@47209: 
huffman@47209: lemma numeral_eq_Suc: "numeral k = Suc (pred_numeral k)"
huffman@47209:   unfolding pred_numeral_def by simp
huffman@47209: 
huffman@47220: lemma eval_nat_numeral:
huffman@47108:   "numeral One = Suc 0"
huffman@47108:   "numeral (Bit0 n) = Suc (numeral (BitM n))"
huffman@47108:   "numeral (Bit1 n) = Suc (numeral (Bit0 n))"
huffman@47108:   by (simp_all add: numeral.simps BitM_plus_one)
huffman@47108: 
huffman@47209: lemma pred_numeral_simps [simp]:
huffman@47300:   "pred_numeral One = 0"
huffman@47300:   "pred_numeral (Bit0 k) = numeral (BitM k)"
huffman@47300:   "pred_numeral (Bit1 k) = numeral (Bit0 k)"
huffman@47220:   unfolding pred_numeral_def eval_nat_numeral
huffman@47209:   by (simp_all only: diff_Suc_Suc diff_0)
huffman@47209: 
huffman@47192: lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
huffman@47220:   by (simp add: eval_nat_numeral)
huffman@47192: 
huffman@47192: lemma numeral_3_eq_3: "3 = Suc (Suc (Suc 0))"
huffman@47220:   by (simp add: eval_nat_numeral)
huffman@47192: 
huffman@47207: lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
huffman@47207:   by (simp only: numeral_One One_nat_def)
huffman@47207: 
huffman@47207: lemma Suc_nat_number_of_add:
huffman@47300:   "Suc (numeral v + n) = numeral (v + One) + n"
huffman@47207:   by simp
huffman@47207: 
huffman@47207: (*Maps #n to n for n = 1, 2*)
huffman@47207: lemmas numerals = numeral_One [where 'a=nat] numeral_2_eq_2
huffman@47207: 
huffman@47209: text {* Comparisons involving @{term Suc}. *}
huffman@47209: 
huffman@47209: lemma eq_numeral_Suc [simp]: "numeral k = Suc n \<longleftrightarrow> pred_numeral k = n"
huffman@47209:   by (simp add: numeral_eq_Suc)
huffman@47209: 
huffman@47209: lemma Suc_eq_numeral [simp]: "Suc n = numeral k \<longleftrightarrow> n = pred_numeral k"
huffman@47209:   by (simp add: numeral_eq_Suc)
huffman@47209: 
huffman@47209: lemma less_numeral_Suc [simp]: "numeral k < Suc n \<longleftrightarrow> pred_numeral k < n"
huffman@47209:   by (simp add: numeral_eq_Suc)
huffman@47209: 
huffman@47209: lemma less_Suc_numeral [simp]: "Suc n < numeral k \<longleftrightarrow> n < pred_numeral k"
huffman@47209:   by (simp add: numeral_eq_Suc)
huffman@47209: 
huffman@47209: lemma le_numeral_Suc [simp]: "numeral k \<le> Suc n \<longleftrightarrow> pred_numeral k \<le> n"
huffman@47209:   by (simp add: numeral_eq_Suc)
huffman@47209: 
huffman@47209: lemma le_Suc_numeral [simp]: "Suc n \<le> numeral k \<longleftrightarrow> n \<le> pred_numeral k"
huffman@47209:   by (simp add: numeral_eq_Suc)
huffman@47209: 
huffman@47218: lemma diff_Suc_numeral [simp]: "Suc n - numeral k = n - pred_numeral k"
huffman@47218:   by (simp add: numeral_eq_Suc)
huffman@47218: 
huffman@47218: lemma diff_numeral_Suc [simp]: "numeral k - Suc n = pred_numeral k - n"
huffman@47218:   by (simp add: numeral_eq_Suc)
huffman@47218: 
huffman@47209: lemma max_Suc_numeral [simp]:
huffman@47209:   "max (Suc n) (numeral k) = Suc (max n (pred_numeral k))"
huffman@47209:   by (simp add: numeral_eq_Suc)
huffman@47209: 
huffman@47209: lemma max_numeral_Suc [simp]:
huffman@47209:   "max (numeral k) (Suc n) = Suc (max (pred_numeral k) n)"
huffman@47209:   by (simp add: numeral_eq_Suc)
huffman@47209: 
huffman@47209: lemma min_Suc_numeral [simp]:
huffman@47209:   "min (Suc n) (numeral k) = Suc (min n (pred_numeral k))"
huffman@47209:   by (simp add: numeral_eq_Suc)
huffman@47209: 
huffman@47209: lemma min_numeral_Suc [simp]:
huffman@47209:   "min (numeral k) (Suc n) = Suc (min (pred_numeral k) n)"
huffman@47209:   by (simp add: numeral_eq_Suc)
huffman@47209: 
huffman@47216: text {* For @{term nat_case} and @{term nat_rec}. *}
huffman@47216: 
huffman@47216: lemma nat_case_numeral [simp]:
huffman@47216:   "nat_case a f (numeral v) = (let pv = pred_numeral v in f pv)"
huffman@47216:   by (simp add: numeral_eq_Suc)
huffman@47216: 
huffman@47216: lemma nat_case_add_eq_if [simp]:
huffman@47216:   "nat_case a f ((numeral v) + n) = (let pv = pred_numeral v in f (pv + n))"
huffman@47216:   by (simp add: numeral_eq_Suc)
huffman@47216: 
huffman@47216: lemma nat_rec_numeral [simp]:
huffman@47216:   "nat_rec a f (numeral v) =
huffman@47216:     (let pv = pred_numeral v in f pv (nat_rec a f pv))"
huffman@47216:   by (simp add: numeral_eq_Suc Let_def)
huffman@47216: 
huffman@47216: lemma nat_rec_add_eq_if [simp]:
huffman@47216:   "nat_rec a f (numeral v + n) =
huffman@47216:     (let pv = pred_numeral v in f (pv + n) (nat_rec a f (pv + n)))"
huffman@47216:   by (simp add: numeral_eq_Suc Let_def)
huffman@47216: 
huffman@47255: text {* Case analysis on @{term "n < 2"} *}
huffman@47255: 
huffman@47255: lemma less_2_cases: "n < 2 \<Longrightarrow> n = 0 \<or> n = Suc 0"
huffman@47255:   by (auto simp add: numeral_2_eq_2)
huffman@47255: 
huffman@47255: text {* Removal of Small Numerals: 0, 1 and (in additive positions) 2 *}
huffman@47255: text {* bh: Are these rules really a good idea? *}
huffman@47255: 
huffman@47255: lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
huffman@47255:   by simp
huffman@47255: 
huffman@47255: lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
huffman@47255:   by simp
huffman@47255: 
huffman@47255: text {* Can be used to eliminate long strings of Sucs, but not by default. *}
huffman@47255: 
huffman@47255: lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
huffman@47255:   by simp
huffman@47255: 
huffman@47255: lemmas nat_1_add_1 = one_add_one [where 'a=nat] (* legacy *)
huffman@47255: 
huffman@47108: 
huffman@47108: subsection {* Numeral equations as default simplification rules *}
huffman@47108: 
huffman@47108: declare (in numeral) numeral_One [simp]
huffman@47108: declare (in numeral) numeral_plus_numeral [simp]
huffman@47108: declare (in numeral) add_numeral_special [simp]
huffman@47108: declare (in neg_numeral) add_neg_numeral_simps [simp]
huffman@47108: declare (in neg_numeral) add_neg_numeral_special [simp]
huffman@47108: declare (in neg_numeral) diff_numeral_simps [simp]
huffman@47108: declare (in neg_numeral) diff_numeral_special [simp]
huffman@47108: declare (in semiring_numeral) numeral_times_numeral [simp]
huffman@47108: declare (in ring_1) mult_neg_numeral_simps [simp]
huffman@47108: 
huffman@47108: subsection {* Setting up simprocs *}
huffman@47108: 
huffman@47108: lemma mult_numeral_1: "Numeral1 * a = (a::'a::semiring_numeral)"
huffman@47108:   by simp
huffman@47108: 
huffman@47108: lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::semiring_numeral)"
huffman@47108:   by simp
huffman@47108: 
huffman@47108: lemma divide_numeral_1: "a / Numeral1 = (a::'a::field)"
huffman@47108:   by simp
huffman@47108: 
huffman@47108: lemma inverse_numeral_1:
huffman@47108:   "inverse Numeral1 = (Numeral1::'a::division_ring)"
huffman@47108:   by simp
huffman@47108: 
huffman@47211: text{*Theorem lists for the cancellation simprocs. The use of a binary
huffman@47108: numeral for 1 reduces the number of special cases.*}
huffman@47108: 
huffman@47108: lemmas mult_1s =
huffman@47108:   mult_numeral_1 mult_numeral_1_right 
huffman@47108:   mult_minus1 mult_minus1_right
huffman@47108: 
huffman@47226: setup {*
huffman@47226:   Reorient_Proc.add
huffman@47226:     (fn Const (@{const_name numeral}, _) $ _ => true
huffman@47226:     | Const (@{const_name neg_numeral}, _) $ _ => true
huffman@47226:     | _ => false)
huffman@47226: *}
huffman@47226: 
huffman@47226: simproc_setup reorient_numeral
huffman@47226:   ("numeral w = x" | "neg_numeral w = y") = Reorient_Proc.proc
huffman@47226: 
huffman@47108: 
huffman@47108: subsubsection {* Simplification of arithmetic operations on integer constants. *}
huffman@47108: 
huffman@47108: lemmas arith_special = (* already declared simp above *)
huffman@47108:   add_numeral_special add_neg_numeral_special
huffman@47108:   diff_numeral_special minus_one
huffman@47108: 
huffman@47108: (* rules already in simpset *)
huffman@47108: lemmas arith_extra_simps =
huffman@47108:   numeral_plus_numeral add_neg_numeral_simps add_0_left add_0_right
huffman@47108:   minus_numeral minus_neg_numeral minus_zero minus_one
huffman@47108:   diff_numeral_simps diff_0 diff_0_right
huffman@47108:   numeral_times_numeral mult_neg_numeral_simps
huffman@47108:   mult_zero_left mult_zero_right
huffman@47108:   abs_numeral abs_neg_numeral
huffman@47108: 
huffman@47108: text {*
huffman@47108:   For making a minimal simpset, one must include these default simprules.
huffman@47108:   Also include @{text simp_thms}.
huffman@47108: *}
huffman@47108: 
huffman@47108: lemmas arith_simps =
huffman@47108:   add_num_simps mult_num_simps sub_num_simps
huffman@47108:   BitM.simps dbl_simps dbl_inc_simps dbl_dec_simps
huffman@47108:   abs_zero abs_one arith_extra_simps
huffman@47108: 
huffman@47108: text {* Simplification of relational operations *}
huffman@47108: 
huffman@47108: lemmas eq_numeral_extra =
huffman@47108:   zero_neq_one one_neq_zero
huffman@47108: 
huffman@47108: lemmas rel_simps =
huffman@47108:   le_num_simps less_num_simps eq_num_simps
huffman@47108:   le_numeral_simps le_neg_numeral_simps le_numeral_extra
huffman@47108:   less_numeral_simps less_neg_numeral_simps less_numeral_extra
huffman@47108:   eq_numeral_simps eq_neg_numeral_simps eq_numeral_extra
huffman@47108: 
huffman@47108: 
huffman@47108: subsubsection {* Simplification of arithmetic when nested to the right. *}
huffman@47108: 
huffman@47108: lemma add_numeral_left [simp]:
huffman@47108:   "numeral v + (numeral w + z) = (numeral(v + w) + z)"
huffman@47108:   by (simp_all add: add_assoc [symmetric])
huffman@47108: 
huffman@47108: lemma add_neg_numeral_left [simp]:
huffman@47108:   "numeral v + (neg_numeral w + y) = (sub v w + y)"
huffman@47108:   "neg_numeral v + (numeral w + y) = (sub w v + y)"
huffman@47108:   "neg_numeral v + (neg_numeral w + y) = (neg_numeral(v + w) + y)"
huffman@47108:   by (simp_all add: add_assoc [symmetric])
huffman@47108: 
huffman@47108: lemma mult_numeral_left [simp]:
huffman@47108:   "numeral v * (numeral w * z) = (numeral(v * w) * z :: 'a::semiring_numeral)"
huffman@47108:   "neg_numeral v * (numeral w * y) = (neg_numeral(v * w) * y :: 'b::ring_1)"
huffman@47108:   "numeral v * (neg_numeral w * y) = (neg_numeral(v * w) * y :: 'b::ring_1)"
huffman@47108:   "neg_numeral v * (neg_numeral w * y) = (numeral(v * w) * y :: 'b::ring_1)"
huffman@47108:   by (simp_all add: mult_assoc [symmetric])
huffman@47108: 
huffman@47108: hide_const (open) One Bit0 Bit1 BitM inc pow sqr sub dbl dbl_inc dbl_dec
huffman@47108: 
haftmann@51143: 
huffman@47108: subsection {* code module namespace *}
huffman@47108: 
haftmann@52435: code_identifier
haftmann@52435:   code_module Num \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
huffman@47108: 
huffman@47108: end
haftmann@50817: 
haftmann@51143: