src/HOL/Nat_Numeral.thy
author haftmann
Wed Apr 15 15:30:39 2009 +0200 (2009-04-15)
changeset 30925 c38cbc0ac8d1
parent 30685 src/HOL/NatBin.thy@dd5fe091ff04
child 30960 fec1a04b7220
permissions -rw-r--r--
theory NatBin now named Nat_Numeral
     1 (*  Title:      HOL/Nat_Numeral.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Copyright   1999  University of Cambridge
     4 *)
     5 
     6 header {* Binary numerals for the natural numbers *}
     7 
     8 theory Nat_Numeral
     9 imports IntDiv
    10 uses ("Tools/nat_simprocs.ML")
    11 begin
    12 
    13 text {*
    14   Arithmetic for naturals is reduced to that for the non-negative integers.
    15 *}
    16 
    17 instantiation nat :: number
    18 begin
    19 
    20 definition
    21   nat_number_of_def [code inline, code del]: "number_of v = nat (number_of v)"
    22 
    23 instance ..
    24 
    25 end
    26 
    27 lemma [code post]:
    28   "nat (number_of v) = number_of v"
    29   unfolding nat_number_of_def ..
    30 
    31 abbreviation (xsymbols)
    32   power2 :: "'a::power => 'a"  ("(_\<twosuperior>)" [1000] 999) where
    33   "x\<twosuperior> == x^2"
    34 
    35 notation (latex output)
    36   power2  ("(_\<twosuperior>)" [1000] 999)
    37 
    38 notation (HTML output)
    39   power2  ("(_\<twosuperior>)" [1000] 999)
    40 
    41 
    42 subsection {* Predicate for negative binary numbers *}
    43 
    44 definition neg  :: "int \<Rightarrow> bool" where
    45   "neg Z \<longleftrightarrow> Z < 0"
    46 
    47 lemma not_neg_int [simp]: "~ neg (of_nat n)"
    48 by (simp add: neg_def)
    49 
    50 lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))"
    51 by (simp add: neg_def neg_less_0_iff_less del: of_nat_Suc)
    52 
    53 lemmas neg_eq_less_0 = neg_def
    54 
    55 lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
    56 by (simp add: neg_def linorder_not_less)
    57 
    58 text{*To simplify inequalities when Numeral1 can get simplified to 1*}
    59 
    60 lemma not_neg_0: "~ neg 0"
    61 by (simp add: One_int_def neg_def)
    62 
    63 lemma not_neg_1: "~ neg 1"
    64 by (simp add: neg_def linorder_not_less zero_le_one)
    65 
    66 lemma neg_nat: "neg z ==> nat z = 0"
    67 by (simp add: neg_def order_less_imp_le) 
    68 
    69 lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z"
    70 by (simp add: linorder_not_less neg_def)
    71 
    72 text {*
    73   If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
    74   @{term Numeral0} IS @{term "number_of Pls"}
    75 *}
    76 
    77 lemma not_neg_number_of_Pls: "~ neg (number_of Int.Pls)"
    78   by (simp add: neg_def)
    79 
    80 lemma neg_number_of_Min: "neg (number_of Int.Min)"
    81   by (simp add: neg_def)
    82 
    83 lemma neg_number_of_Bit0:
    84   "neg (number_of (Int.Bit0 w)) = neg (number_of w)"
    85   by (simp add: neg_def)
    86 
    87 lemma neg_number_of_Bit1:
    88   "neg (number_of (Int.Bit1 w)) = neg (number_of w)"
    89   by (simp add: neg_def)
    90 
    91 lemmas neg_simps [simp] =
    92   not_neg_0 not_neg_1
    93   not_neg_number_of_Pls neg_number_of_Min
    94   neg_number_of_Bit0 neg_number_of_Bit1
    95 
    96 
    97 subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}
    98 
    99 declare nat_0 [simp] nat_1 [simp]
   100 
   101 lemma nat_number_of [simp]: "nat (number_of w) = number_of w"
   102 by (simp add: nat_number_of_def)
   103 
   104 lemma nat_numeral_0_eq_0 [simp]: "Numeral0 = (0::nat)"
   105 by (simp add: nat_number_of_def)
   106 
   107 lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
   108 by (simp add: nat_1 nat_number_of_def)
   109 
   110 lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
   111 by (simp add: nat_numeral_1_eq_1)
   112 
   113 lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
   114 apply (unfold nat_number_of_def)
   115 apply (rule nat_2)
   116 done
   117 
   118 
   119 subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
   120 
   121 lemma int_nat_number_of [simp]:
   122      "int (number_of v) =  
   123          (if neg (number_of v :: int) then 0  
   124           else (number_of v :: int))"
   125   unfolding nat_number_of_def number_of_is_id neg_def
   126   by simp
   127 
   128 
   129 subsubsection{*Successor *}
   130 
   131 lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
   132 apply (rule sym)
   133 apply (simp add: nat_eq_iff int_Suc)
   134 done
   135 
   136 lemma Suc_nat_number_of_add:
   137      "Suc (number_of v + n) =  
   138         (if neg (number_of v :: int) then 1+n else number_of (Int.succ v) + n)"
   139   unfolding nat_number_of_def number_of_is_id neg_def numeral_simps
   140   by (simp add: Suc_nat_eq_nat_zadd1 add_ac)
   141 
   142 lemma Suc_nat_number_of [simp]:
   143      "Suc (number_of v) =  
   144         (if neg (number_of v :: int) then 1 else number_of (Int.succ v))"
   145 apply (cut_tac n = 0 in Suc_nat_number_of_add)
   146 apply (simp cong del: if_weak_cong)
   147 done
   148 
   149 
   150 subsubsection{*Addition *}
   151 
   152 lemma add_nat_number_of [simp]:
   153      "(number_of v :: nat) + number_of v' =  
   154          (if v < Int.Pls then number_of v'  
   155           else if v' < Int.Pls then number_of v  
   156           else number_of (v + v'))"
   157   unfolding nat_number_of_def number_of_is_id numeral_simps
   158   by (simp add: nat_add_distrib)
   159 
   160 lemma nat_number_of_add_1 [simp]:
   161   "number_of v + (1::nat) =
   162     (if v < Int.Pls then 1 else number_of (Int.succ v))"
   163   unfolding nat_number_of_def number_of_is_id numeral_simps
   164   by (simp add: nat_add_distrib)
   165 
   166 lemma nat_1_add_number_of [simp]:
   167   "(1::nat) + number_of v =
   168     (if v < Int.Pls then 1 else number_of (Int.succ v))"
   169   unfolding nat_number_of_def number_of_is_id numeral_simps
   170   by (simp add: nat_add_distrib)
   171 
   172 lemma nat_1_add_1 [simp]: "1 + 1 = (2::nat)"
   173   by (rule int_int_eq [THEN iffD1]) simp
   174 
   175 
   176 subsubsection{*Subtraction *}
   177 
   178 lemma diff_nat_eq_if:
   179      "nat z - nat z' =  
   180         (if neg z' then nat z   
   181          else let d = z-z' in     
   182               if neg d then 0 else nat d)"
   183 by (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
   184 
   185 
   186 lemma diff_nat_number_of [simp]: 
   187      "(number_of v :: nat) - number_of v' =  
   188         (if v' < Int.Pls then number_of v  
   189          else let d = number_of (v + uminus v') in     
   190               if neg d then 0 else nat d)"
   191   unfolding nat_number_of_def number_of_is_id numeral_simps neg_def
   192   by auto
   193 
   194 lemma nat_number_of_diff_1 [simp]:
   195   "number_of v - (1::nat) =
   196     (if v \<le> Int.Pls then 0 else number_of (Int.pred v))"
   197   unfolding nat_number_of_def number_of_is_id numeral_simps
   198   by auto
   199 
   200 
   201 subsubsection{*Multiplication *}
   202 
   203 lemma mult_nat_number_of [simp]:
   204      "(number_of v :: nat) * number_of v' =  
   205        (if v < Int.Pls then 0 else number_of (v * v'))"
   206   unfolding nat_number_of_def number_of_is_id numeral_simps
   207   by (simp add: nat_mult_distrib)
   208 
   209 
   210 subsubsection{*Quotient *}
   211 
   212 lemma div_nat_number_of [simp]:
   213      "(number_of v :: nat)  div  number_of v' =  
   214           (if neg (number_of v :: int) then 0  
   215            else nat (number_of v div number_of v'))"
   216   unfolding nat_number_of_def number_of_is_id neg_def
   217   by (simp add: nat_div_distrib)
   218 
   219 lemma one_div_nat_number_of [simp]:
   220      "Suc 0 div number_of v' = nat (1 div number_of v')" 
   221 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
   222 
   223 
   224 subsubsection{*Remainder *}
   225 
   226 lemma mod_nat_number_of [simp]:
   227      "(number_of v :: nat)  mod  number_of v' =  
   228         (if neg (number_of v :: int) then 0  
   229          else if neg (number_of v' :: int) then number_of v  
   230          else nat (number_of v mod number_of v'))"
   231   unfolding nat_number_of_def number_of_is_id neg_def
   232   by (simp add: nat_mod_distrib)
   233 
   234 lemma one_mod_nat_number_of [simp]:
   235      "Suc 0 mod number_of v' =  
   236         (if neg (number_of v' :: int) then Suc 0
   237          else nat (1 mod number_of v'))"
   238 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
   239 
   240 
   241 subsubsection{* Divisibility *}
   242 
   243 lemmas dvd_eq_mod_eq_0_number_of =
   244   dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
   245 
   246 declare dvd_eq_mod_eq_0_number_of [simp]
   247 
   248 ML
   249 {*
   250 val nat_number_of_def = thm"nat_number_of_def";
   251 
   252 val nat_number_of = thm"nat_number_of";
   253 val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0";
   254 val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1";
   255 val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0";
   256 val numeral_2_eq_2 = thm"numeral_2_eq_2";
   257 val nat_div_distrib = thm"nat_div_distrib";
   258 val nat_mod_distrib = thm"nat_mod_distrib";
   259 val int_nat_number_of = thm"int_nat_number_of";
   260 val Suc_nat_eq_nat_zadd1 = thm"Suc_nat_eq_nat_zadd1";
   261 val Suc_nat_number_of_add = thm"Suc_nat_number_of_add";
   262 val Suc_nat_number_of = thm"Suc_nat_number_of";
   263 val add_nat_number_of = thm"add_nat_number_of";
   264 val diff_nat_eq_if = thm"diff_nat_eq_if";
   265 val diff_nat_number_of = thm"diff_nat_number_of";
   266 val mult_nat_number_of = thm"mult_nat_number_of";
   267 val div_nat_number_of = thm"div_nat_number_of";
   268 val mod_nat_number_of = thm"mod_nat_number_of";
   269 *}
   270 
   271 
   272 subsection{*Comparisons*}
   273 
   274 subsubsection{*Equals (=) *}
   275 
   276 lemma eq_nat_nat_iff:
   277      "[| (0::int) <= z;  0 <= z' |] ==> (nat z = nat z') = (z=z')"
   278 by (auto elim!: nonneg_eq_int)
   279 
   280 lemma eq_nat_number_of [simp]:
   281      "((number_of v :: nat) = number_of v') =  
   282       (if neg (number_of v :: int) then (number_of v' :: int) \<le> 0
   283        else if neg (number_of v' :: int) then (number_of v :: int) = 0
   284        else v = v')"
   285   unfolding nat_number_of_def number_of_is_id neg_def
   286   by auto
   287 
   288 
   289 subsubsection{*Less-than (<) *}
   290 
   291 lemma less_nat_number_of [simp]:
   292   "(number_of v :: nat) < number_of v' \<longleftrightarrow>
   293     (if v < v' then Int.Pls < v' else False)"
   294   unfolding nat_number_of_def number_of_is_id numeral_simps
   295   by auto
   296 
   297 
   298 subsubsection{*Less-than-or-equal *}
   299 
   300 lemma le_nat_number_of [simp]:
   301   "(number_of v :: nat) \<le> number_of v' \<longleftrightarrow>
   302     (if v \<le> v' then True else v \<le> Int.Pls)"
   303   unfolding nat_number_of_def number_of_is_id numeral_simps
   304   by auto
   305 
   306 (*Maps #n to n for n = 0, 1, 2*)
   307 lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
   308 
   309 
   310 subsection{*Powers with Numeric Exponents*}
   311 
   312 text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}.
   313 We cannot prove general results about the numeral @{term "-1"}, so we have to
   314 use @{term "- 1"} instead.*}
   315 
   316 lemma power2_eq_square: "(a::'a::recpower)\<twosuperior> = a * a"
   317   by (simp add: numeral_2_eq_2 Power.power_Suc)
   318 
   319 lemma zero_power2 [simp]: "(0::'a::{semiring_1,recpower})\<twosuperior> = 0"
   320   by (simp add: power2_eq_square)
   321 
   322 lemma one_power2 [simp]: "(1::'a::{semiring_1,recpower})\<twosuperior> = 1"
   323   by (simp add: power2_eq_square)
   324 
   325 lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x"
   326   apply (subgoal_tac "3 = Suc (Suc (Suc 0))")
   327   apply (erule ssubst)
   328   apply (simp add: power_Suc mult_ac)
   329   apply (unfold nat_number_of_def)
   330   apply (subst nat_eq_iff)
   331   apply simp
   332 done
   333 
   334 text{*Squares of literal numerals will be evaluated.*}
   335 lemmas power2_eq_square_number_of =
   336     power2_eq_square [of "number_of w", standard]
   337 declare power2_eq_square_number_of [simp]
   338 
   339 
   340 lemma zero_le_power2[simp]: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})"
   341   by (simp add: power2_eq_square)
   342 
   343 lemma zero_less_power2[simp]:
   344      "(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))"
   345   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
   346 
   347 lemma power2_less_0[simp]:
   348   fixes a :: "'a::{ordered_idom,recpower}"
   349   shows "~ (a\<twosuperior> < 0)"
   350 by (force simp add: power2_eq_square mult_less_0_iff) 
   351 
   352 lemma zero_eq_power2[simp]:
   353      "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))"
   354   by (force simp add: power2_eq_square mult_eq_0_iff)
   355 
   356 lemma abs_power2[simp]:
   357      "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})"
   358   by (simp add: power2_eq_square abs_mult abs_mult_self)
   359 
   360 lemma power2_abs[simp]:
   361      "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})"
   362   by (simp add: power2_eq_square abs_mult_self)
   363 
   364 lemma power2_minus[simp]:
   365      "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})"
   366   by (simp add: power2_eq_square)
   367 
   368 lemma power2_le_imp_le:
   369   fixes x y :: "'a::{ordered_semidom,recpower}"
   370   shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y"
   371 unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
   372 
   373 lemma power2_less_imp_less:
   374   fixes x y :: "'a::{ordered_semidom,recpower}"
   375   shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y"
   376 by (rule power_less_imp_less_base)
   377 
   378 lemma power2_eq_imp_eq:
   379   fixes x y :: "'a::{ordered_semidom,recpower}"
   380   shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y"
   381 unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp)
   382 
   383 lemma power_minus1_even[simp]: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"
   384 proof (induct n)
   385   case 0 show ?case by simp
   386 next
   387   case (Suc n) then show ?case by (simp add: power_Suc power_add)
   388 qed
   389 
   390 lemma power_minus1_odd: "(- 1) ^ Suc(2*n) = -(1::'a::{comm_ring_1,recpower})"
   391   by (simp add: power_Suc) 
   392 
   393 lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
   394 by (subst mult_commute) (simp add: power_mult)
   395 
   396 lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
   397 by (simp add: power_even_eq) 
   398 
   399 lemma power_minus_even [simp]:
   400      "(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)"
   401 by (simp add: power_minus1_even power_minus [of a]) 
   402 
   403 lemma zero_le_even_power'[simp]:
   404      "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)"
   405 proof (induct "n")
   406   case 0
   407     show ?case by (simp add: zero_le_one)
   408 next
   409   case (Suc n)
   410     have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
   411       by (simp add: mult_ac power_add power2_eq_square)
   412     thus ?case
   413       by (simp add: prems zero_le_mult_iff)
   414 qed
   415 
   416 lemma odd_power_less_zero:
   417      "(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0"
   418 proof (induct "n")
   419   case 0
   420   then show ?case by simp
   421 next
   422   case (Suc n)
   423   have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
   424     by (simp add: mult_ac power_add power2_eq_square)
   425   thus ?case
   426     by (simp del: power_Suc add: prems mult_less_0_iff mult_neg_neg)
   427 qed
   428 
   429 lemma odd_0_le_power_imp_0_le:
   430      "0 \<le> a  ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})"
   431 apply (insert odd_power_less_zero [of a n]) 
   432 apply (force simp add: linorder_not_less [symmetric]) 
   433 done
   434 
   435 text{*Simprules for comparisons where common factors can be cancelled.*}
   436 lemmas zero_compare_simps =
   437     add_strict_increasing add_strict_increasing2 add_increasing
   438     zero_le_mult_iff zero_le_divide_iff 
   439     zero_less_mult_iff zero_less_divide_iff 
   440     mult_le_0_iff divide_le_0_iff 
   441     mult_less_0_iff divide_less_0_iff 
   442     zero_le_power2 power2_less_0
   443 
   444 subsubsection{*Nat *}
   445 
   446 lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
   447 by (simp add: numerals)
   448 
   449 (*Expresses a natural number constant as the Suc of another one.
   450   NOT suitable for rewriting because n recurs in the condition.*)
   451 lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
   452 
   453 subsubsection{*Arith *}
   454 
   455 lemma Suc_eq_add_numeral_1: "Suc n = n + 1"
   456 by (simp add: numerals)
   457 
   458 lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"
   459 by (simp add: numerals)
   460 
   461 (* These two can be useful when m = number_of... *)
   462 
   463 lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
   464   unfolding One_nat_def by (cases m) simp_all
   465 
   466 lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
   467   unfolding One_nat_def by (cases m) simp_all
   468 
   469 lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
   470   unfolding One_nat_def by (cases m) simp_all
   471 
   472 
   473 subsection{*Comparisons involving (0::nat) *}
   474 
   475 text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
   476 
   477 lemma eq_number_of_0 [simp]:
   478   "number_of v = (0::nat) \<longleftrightarrow> v \<le> Int.Pls"
   479   unfolding nat_number_of_def number_of_is_id numeral_simps
   480   by auto
   481 
   482 lemma eq_0_number_of [simp]:
   483   "(0::nat) = number_of v \<longleftrightarrow> v \<le> Int.Pls"
   484 by (rule trans [OF eq_sym_conv eq_number_of_0])
   485 
   486 lemma less_0_number_of [simp]:
   487    "(0::nat) < number_of v \<longleftrightarrow> Int.Pls < v"
   488   unfolding nat_number_of_def number_of_is_id numeral_simps
   489   by simp
   490 
   491 lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
   492 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])
   493 
   494 
   495 
   496 subsection{*Comparisons involving  @{term Suc} *}
   497 
   498 lemma eq_number_of_Suc [simp]:
   499      "(number_of v = Suc n) =  
   500         (let pv = number_of (Int.pred v) in  
   501          if neg pv then False else nat pv = n)"
   502 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
   503                   number_of_pred nat_number_of_def 
   504             split add: split_if)
   505 apply (rule_tac x = "number_of v" in spec)
   506 apply (auto simp add: nat_eq_iff)
   507 done
   508 
   509 lemma Suc_eq_number_of [simp]:
   510      "(Suc n = number_of v) =  
   511         (let pv = number_of (Int.pred v) in  
   512          if neg pv then False else nat pv = n)"
   513 by (rule trans [OF eq_sym_conv eq_number_of_Suc])
   514 
   515 lemma less_number_of_Suc [simp]:
   516      "(number_of v < Suc n) =  
   517         (let pv = number_of (Int.pred v) in  
   518          if neg pv then True else nat pv < n)"
   519 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
   520                   number_of_pred nat_number_of_def  
   521             split add: split_if)
   522 apply (rule_tac x = "number_of v" in spec)
   523 apply (auto simp add: nat_less_iff)
   524 done
   525 
   526 lemma less_Suc_number_of [simp]:
   527      "(Suc n < number_of v) =  
   528         (let pv = number_of (Int.pred v) in  
   529          if neg pv then False else n < nat pv)"
   530 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
   531                   number_of_pred nat_number_of_def
   532             split add: split_if)
   533 apply (rule_tac x = "number_of v" in spec)
   534 apply (auto simp add: zless_nat_eq_int_zless)
   535 done
   536 
   537 lemma le_number_of_Suc [simp]:
   538      "(number_of v <= Suc n) =  
   539         (let pv = number_of (Int.pred v) in  
   540          if neg pv then True else nat pv <= n)"
   541 by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])
   542 
   543 lemma le_Suc_number_of [simp]:
   544      "(Suc n <= number_of v) =  
   545         (let pv = number_of (Int.pred v) in  
   546          if neg pv then False else n <= nat pv)"
   547 by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])
   548 
   549 
   550 lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Int.Min"
   551 by auto
   552 
   553 
   554 
   555 subsection{*Max and Min Combined with @{term Suc} *}
   556 
   557 lemma max_number_of_Suc [simp]:
   558      "max (Suc n) (number_of v) =  
   559         (let pv = number_of (Int.pred v) in  
   560          if neg pv then Suc n else Suc(max n (nat pv)))"
   561 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
   562             split add: split_if nat.split)
   563 apply (rule_tac x = "number_of v" in spec) 
   564 apply auto
   565 done
   566  
   567 lemma max_Suc_number_of [simp]:
   568      "max (number_of v) (Suc n) =  
   569         (let pv = number_of (Int.pred v) in  
   570          if neg pv then Suc n else Suc(max (nat pv) n))"
   571 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
   572             split add: split_if nat.split)
   573 apply (rule_tac x = "number_of v" in spec) 
   574 apply auto
   575 done
   576  
   577 lemma min_number_of_Suc [simp]:
   578      "min (Suc n) (number_of v) =  
   579         (let pv = number_of (Int.pred v) in  
   580          if neg pv then 0 else Suc(min n (nat pv)))"
   581 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
   582             split add: split_if nat.split)
   583 apply (rule_tac x = "number_of v" in spec) 
   584 apply auto
   585 done
   586  
   587 lemma min_Suc_number_of [simp]:
   588      "min (number_of v) (Suc n) =  
   589         (let pv = number_of (Int.pred v) in  
   590          if neg pv then 0 else Suc(min (nat pv) n))"
   591 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
   592             split add: split_if nat.split)
   593 apply (rule_tac x = "number_of v" in spec) 
   594 apply auto
   595 done
   596  
   597 subsection{*Literal arithmetic involving powers*}
   598 
   599 lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n"
   600 apply (induct "n")
   601 apply (simp_all (no_asm_simp) add: nat_mult_distrib)
   602 done
   603 
   604 lemma power_nat_number_of:
   605      "(number_of v :: nat) ^ n =  
   606        (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
   607 by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
   608          split add: split_if cong: imp_cong)
   609 
   610 
   611 lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
   612 declare power_nat_number_of_number_of [simp]
   613 
   614 
   615 
   616 text{*For arbitrary rings*}
   617 
   618 lemma power_number_of_even:
   619   fixes z :: "'a::{number_ring,recpower}"
   620   shows "z ^ number_of (Int.Bit0 w) = (let w = z ^ (number_of w) in w * w)"
   621 unfolding Let_def nat_number_of_def number_of_Bit0
   622 apply (rule_tac x = "number_of w" in spec, clarify)
   623 apply (case_tac " (0::int) <= x")
   624 apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)
   625 done
   626 
   627 lemma power_number_of_odd:
   628   fixes z :: "'a::{number_ring,recpower}"
   629   shows "z ^ number_of (Int.Bit1 w) = (if (0::int) <= number_of w
   630      then (let w = z ^ (number_of w) in z * w * w) else 1)"
   631 unfolding Let_def nat_number_of_def number_of_Bit1
   632 apply (rule_tac x = "number_of w" in spec, auto)
   633 apply (simp only: nat_add_distrib nat_mult_distrib)
   634 apply simp
   635 apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat power_Suc)
   636 done
   637 
   638 lemmas zpower_number_of_even = power_number_of_even [where 'a=int]
   639 lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]
   640 
   641 lemmas power_number_of_even_number_of [simp] =
   642     power_number_of_even [of "number_of v", standard]
   643 
   644 lemmas power_number_of_odd_number_of [simp] =
   645     power_number_of_odd [of "number_of v", standard]
   646 
   647 
   648 
   649 ML
   650 {*
   651 val numeral_ss = @{simpset} addsimps @{thms numerals};
   652 
   653 val nat_bin_arith_setup =
   654  Lin_Arith.map_data
   655    (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
   656      {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
   657       inj_thms = inj_thms,
   658       lessD = lessD, neqE = neqE,
   659       simpset = simpset addsimps @{thms neg_simps} @
   660         [@{thm Suc_nat_number_of}, @{thm int_nat_number_of}]})
   661 *}
   662 
   663 declaration {* K nat_bin_arith_setup *}
   664 
   665 (* Enable arith to deal with div/mod k where k is a numeral: *)
   666 declare split_div[of _ _ "number_of k", standard, arith_split]
   667 declare split_mod[of _ _ "number_of k", standard, arith_split]
   668 
   669 lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
   670   by (simp add: number_of_Pls nat_number_of_def)
   671 
   672 lemma nat_number_of_Min: "number_of Int.Min = (0::nat)"
   673   apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
   674   done
   675 
   676 lemma nat_number_of_Bit0:
   677     "number_of (Int.Bit0 w) = (let n::nat = number_of w in n + n)"
   678   unfolding nat_number_of_def number_of_is_id numeral_simps Let_def
   679   by auto
   680 
   681 lemma nat_number_of_Bit1:
   682   "number_of (Int.Bit1 w) =
   683     (if neg (number_of w :: int) then 0
   684      else let n = number_of w in Suc (n + n))"
   685   unfolding nat_number_of_def number_of_is_id numeral_simps neg_def Let_def
   686   by auto
   687 
   688 lemmas nat_number =
   689   nat_number_of_Pls nat_number_of_Min
   690   nat_number_of_Bit0 nat_number_of_Bit1
   691 
   692 lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
   693   by (simp add: Let_def)
   694 
   695 lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})"
   696 by (simp add: power_mult power_Suc); 
   697 
   698 lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})"
   699 by (simp add: power_mult power_Suc); 
   700 
   701 
   702 subsection{*Literal arithmetic and @{term of_nat}*}
   703 
   704 lemma of_nat_double:
   705      "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
   706 by (simp only: mult_2 nat_add_distrib of_nat_add) 
   707 
   708 lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
   709 by (simp only: nat_number_of_def)
   710 
   711 lemma of_nat_number_of_lemma:
   712      "of_nat (number_of v :: nat) =  
   713          (if 0 \<le> (number_of v :: int) 
   714           then (number_of v :: 'a :: number_ring)
   715           else 0)"
   716 by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);
   717 
   718 lemma of_nat_number_of_eq [simp]:
   719      "of_nat (number_of v :: nat) =  
   720          (if neg (number_of v :: int) then 0  
   721           else (number_of v :: 'a :: number_ring))"
   722 by (simp only: of_nat_number_of_lemma neg_def, simp) 
   723 
   724 
   725 subsection {*Lemmas for the Combination and Cancellation Simprocs*}
   726 
   727 lemma nat_number_of_add_left:
   728      "number_of v + (number_of v' + (k::nat)) =  
   729          (if neg (number_of v :: int) then number_of v' + k  
   730           else if neg (number_of v' :: int) then number_of v + k  
   731           else number_of (v + v') + k)"
   732   unfolding nat_number_of_def number_of_is_id neg_def
   733   by auto
   734 
   735 lemma nat_number_of_mult_left:
   736      "number_of v * (number_of v' * (k::nat)) =  
   737          (if v < Int.Pls then 0
   738           else number_of (v * v') * k)"
   739 by simp
   740 
   741 
   742 subsubsection{*For @{text combine_numerals}*}
   743 
   744 lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
   745 by (simp add: add_mult_distrib)
   746 
   747 
   748 subsubsection{*For @{text cancel_numerals}*}
   749 
   750 lemma nat_diff_add_eq1:
   751      "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
   752 by (simp split add: nat_diff_split add: add_mult_distrib)
   753 
   754 lemma nat_diff_add_eq2:
   755      "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
   756 by (simp split add: nat_diff_split add: add_mult_distrib)
   757 
   758 lemma nat_eq_add_iff1:
   759      "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
   760 by (auto split add: nat_diff_split simp add: add_mult_distrib)
   761 
   762 lemma nat_eq_add_iff2:
   763      "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
   764 by (auto split add: nat_diff_split simp add: add_mult_distrib)
   765 
   766 lemma nat_less_add_iff1:
   767      "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
   768 by (auto split add: nat_diff_split simp add: add_mult_distrib)
   769 
   770 lemma nat_less_add_iff2:
   771      "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
   772 by (auto split add: nat_diff_split simp add: add_mult_distrib)
   773 
   774 lemma nat_le_add_iff1:
   775      "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
   776 by (auto split add: nat_diff_split simp add: add_mult_distrib)
   777 
   778 lemma nat_le_add_iff2:
   779      "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
   780 by (auto split add: nat_diff_split simp add: add_mult_distrib)
   781 
   782 
   783 subsubsection{*For @{text cancel_numeral_factors} *}
   784 
   785 lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
   786 by auto
   787 
   788 lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
   789 by auto
   790 
   791 lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
   792 by auto
   793 
   794 lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
   795 by auto
   796 
   797 lemma nat_mult_dvd_cancel_disj[simp]:
   798   "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
   799 by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
   800 
   801 lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
   802 by(auto)
   803 
   804 
   805 subsubsection{*For @{text cancel_factor} *}
   806 
   807 lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
   808 by auto
   809 
   810 lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
   811 by auto
   812 
   813 lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
   814 by auto
   815 
   816 lemma nat_mult_div_cancel_disj[simp]:
   817      "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
   818 by (simp add: nat_mult_div_cancel1)
   819 
   820 
   821 subsection {* Simprocs for the Naturals *}
   822 
   823 use "Tools/nat_simprocs.ML"
   824 declaration {* K nat_simprocs_setup *}
   825 
   826 subsubsection{*For simplifying @{term "Suc m - K"} and  @{term "K - Suc m"}*}
   827 
   828 text{*Where K above is a literal*}
   829 
   830 lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
   831 by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split)
   832 
   833 text {*Now just instantiating @{text n} to @{text "number_of v"} does
   834   the right simplification, but with some redundant inequality
   835   tests.*}
   836 lemma neg_number_of_pred_iff_0:
   837   "neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))"
   838 apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ")
   839 apply (simp only: less_Suc_eq_le le_0_eq)
   840 apply (subst less_number_of_Suc, simp)
   841 done
   842 
   843 text{*No longer required as a simprule because of the @{text inverse_fold}
   844    simproc*}
   845 lemma Suc_diff_number_of:
   846      "Int.Pls < v ==>
   847       Suc m - (number_of v) = m - (number_of (Int.pred v))"
   848 apply (subst Suc_diff_eq_diff_pred)
   849 apply simp
   850 apply (simp del: nat_numeral_1_eq_1)
   851 apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
   852                         neg_number_of_pred_iff_0)
   853 done
   854 
   855 lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
   856 by (simp add: numerals split add: nat_diff_split)
   857 
   858 
   859 subsubsection{*For @{term nat_case} and @{term nat_rec}*}
   860 
   861 lemma nat_case_number_of [simp]:
   862      "nat_case a f (number_of v) =
   863         (let pv = number_of (Int.pred v) in
   864          if neg pv then a else f (nat pv))"
   865 by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)
   866 
   867 lemma nat_case_add_eq_if [simp]:
   868      "nat_case a f ((number_of v) + n) =
   869        (let pv = number_of (Int.pred v) in
   870          if neg pv then nat_case a f n else f (nat pv + n))"
   871 apply (subst add_eq_if)
   872 apply (simp split add: nat.split
   873             del: nat_numeral_1_eq_1
   874             add: nat_numeral_1_eq_1 [symmetric]
   875                  numeral_1_eq_Suc_0 [symmetric]
   876                  neg_number_of_pred_iff_0)
   877 done
   878 
   879 lemma nat_rec_number_of [simp]:
   880      "nat_rec a f (number_of v) =
   881         (let pv = number_of (Int.pred v) in
   882          if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
   883 apply (case_tac " (number_of v) ::nat")
   884 apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
   885 apply (simp split add: split_if_asm)
   886 done
   887 
   888 lemma nat_rec_add_eq_if [simp]:
   889      "nat_rec a f (number_of v + n) =
   890         (let pv = number_of (Int.pred v) in
   891          if neg pv then nat_rec a f n
   892                    else f (nat pv + n) (nat_rec a f (nat pv + n)))"
   893 apply (subst add_eq_if)
   894 apply (simp split add: nat.split
   895             del: nat_numeral_1_eq_1
   896             add: nat_numeral_1_eq_1 [symmetric]
   897                  numeral_1_eq_Suc_0 [symmetric]
   898                  neg_number_of_pred_iff_0)
   899 done
   900 
   901 
   902 subsubsection{*Various Other Lemmas*}
   903 
   904 text {*Evens and Odds, for Mutilated Chess Board*}
   905 
   906 text{*Lemmas for specialist use, NOT as default simprules*}
   907 lemma nat_mult_2: "2 * z = (z+z::nat)"
   908 proof -
   909   have "2*z = (1 + 1)*z" by simp
   910   also have "... = z+z" by (simp add: left_distrib)
   911   finally show ?thesis .
   912 qed
   913 
   914 lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
   915 by (subst mult_commute, rule nat_mult_2)
   916 
   917 text{*Case analysis on @{term "n<2"}*}
   918 lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
   919 by arith
   920 
   921 lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)"
   922 by arith
   923 
   924 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
   925 by (simp add: nat_mult_2 [symmetric])
   926 
   927 lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
   928 apply (subgoal_tac "m mod 2 < 2")
   929 apply (erule less_2_cases [THEN disjE])
   930 apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
   931 done
   932 
   933 lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)"
   934 apply (subgoal_tac "m mod 2 < 2")
   935 apply (force simp del: mod_less_divisor, simp)
   936 done
   937 
   938 text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
   939 
   940 lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
   941 by simp
   942 
   943 lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
   944 by simp
   945 
   946 text{*Can be used to eliminate long strings of Sucs, but not by default*}
   947 lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
   948 by simp
   949 
   950 
   951 text{*These lemmas collapse some needless occurrences of Suc:
   952     at least three Sucs, since two and fewer are rewritten back to Suc again!
   953     We already have some rules to simplify operands smaller than 3.*}
   954 
   955 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
   956 by (simp add: Suc3_eq_add_3)
   957 
   958 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
   959 by (simp add: Suc3_eq_add_3)
   960 
   961 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
   962 by (simp add: Suc3_eq_add_3)
   963 
   964 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
   965 by (simp add: Suc3_eq_add_3)
   966 
   967 lemmas Suc_div_eq_add3_div_number_of =
   968     Suc_div_eq_add3_div [of _ "number_of v", standard]
   969 declare Suc_div_eq_add3_div_number_of [simp]
   970 
   971 lemmas Suc_mod_eq_add3_mod_number_of =
   972     Suc_mod_eq_add3_mod [of _ "number_of v", standard]
   973 declare Suc_mod_eq_add3_mod_number_of [simp]
   974 
   975 end