src/HOL/Integ/NatBin.thy
author nipkow
Thu May 24 07:27:44 2007 +0200 (2007-05-24)
changeset 23085 fd30d75a6614
parent 23051 e98ed26577a2
child 23095 45f10b70e891
permissions -rw-r--r--
Introduced new classes monoid_add and group_add
     1 (*  Title:      HOL/NatBin.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1999  University of Cambridge
     5 *)
     6 
     7 header {* Binary arithmetic for the natural numbers *}
     8 
     9 theory NatBin
    10 imports IntDiv
    11 begin
    12 
    13 text {*
    14   Arithmetic for naturals is reduced to that for the non-negative integers.
    15 *}
    16 
    17 instance nat :: number
    18   nat_number_of_def [code inline]: "number_of v == nat (number_of (v\<Colon>int))" ..
    19 
    20 abbreviation (xsymbols)
    21   square :: "'a::power => 'a"  ("(_\<twosuperior>)" [1000] 999) where
    22   "x\<twosuperior> == x^2"
    23 
    24 notation (latex output)
    25   square  ("(_\<twosuperior>)" [1000] 999)
    26 
    27 notation (HTML output)
    28   square  ("(_\<twosuperior>)" [1000] 999)
    29 
    30 
    31 subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}
    32 
    33 declare nat_0 [simp] nat_1 [simp]
    34 
    35 lemma nat_number_of [simp]: "nat (number_of w) = number_of w"
    36 by (simp add: nat_number_of_def)
    37 
    38 lemma nat_numeral_0_eq_0 [simp]: "Numeral0 = (0::nat)"
    39 by (simp add: nat_number_of_def)
    40 
    41 lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
    42 by (simp add: nat_1 nat_number_of_def)
    43 
    44 lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
    45 by (simp add: nat_numeral_1_eq_1)
    46 
    47 lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
    48 apply (unfold nat_number_of_def)
    49 apply (rule nat_2)
    50 done
    51 
    52 
    53 text{*Distributive laws for type @{text nat}.  The others are in theory
    54    @{text IntArith}, but these require div and mod to be defined for type
    55    "int".  They also need some of the lemmas proved above.*}
    56 
    57 lemma nat_div_distrib: "(0::int) <= z ==> nat (z div z') = nat z div nat z'"
    58 apply (case_tac "0 <= z'")
    59 apply (auto simp add: div_nonneg_neg_le0 DIVISION_BY_ZERO_DIV)
    60 apply (case_tac "z' = 0", simp add: DIVISION_BY_ZERO)
    61 apply (auto elim!: nonneg_eq_int)
    62 apply (rename_tac m m')
    63 apply (subgoal_tac "0 <= int m div int m'")
    64  prefer 2 apply (simp add: nat_numeral_0_eq_0 pos_imp_zdiv_nonneg_iff) 
    65 apply (rule inj_int [THEN injD], simp)
    66 apply (rule_tac r = "int (m mod m') " in quorem_div)
    67  prefer 2 apply force
    68 apply (simp add: nat_less_iff [symmetric] quorem_def nat_numeral_0_eq_0 zadd_int 
    69                  zmult_int)
    70 done
    71 
    72 (*Fails if z'<0: the LHS collapses to (nat z) but the RHS doesn't*)
    73 lemma nat_mod_distrib:
    74      "[| (0::int) <= z;  0 <= z' |] ==> nat (z mod z') = nat z mod nat z'"
    75 apply (case_tac "z' = 0", simp add: DIVISION_BY_ZERO)
    76 apply (auto elim!: nonneg_eq_int)
    77 apply (rename_tac m m')
    78 apply (subgoal_tac "0 <= int m mod int m'")
    79  prefer 2 apply (simp add: nat_less_iff nat_numeral_0_eq_0 pos_mod_sign) 
    80 apply (rule inj_int [THEN injD], simp)
    81 apply (rule_tac q = "int (m div m') " in quorem_mod)
    82  prefer 2 apply force
    83 apply (simp add: nat_less_iff [symmetric] quorem_def nat_numeral_0_eq_0 zadd_int zmult_int)
    84 done
    85 
    86 text{*Suggested by Matthias Daum*}
    87 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
    88 apply (subgoal_tac "nat x div nat k < nat x")
    89  apply (simp (asm_lr) add: nat_div_distrib [symmetric])
    90 apply (rule Divides.div_less_dividend, simp_all) 
    91 done
    92 
    93 subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
    94 
    95 (*"neg" is used in rewrite rules for binary comparisons*)
    96 lemma int_nat_number_of [simp]:
    97      "int (number_of v :: nat) =  
    98          (if neg (number_of v :: int) then 0  
    99           else (number_of v :: int))"
   100 by (simp del: nat_number_of
   101 	 add: neg_nat nat_number_of_def not_neg_nat add_assoc)
   102 
   103 
   104 subsubsection{*Successor *}
   105 
   106 lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
   107 apply (rule sym)
   108 apply (simp add: nat_eq_iff int_Suc)
   109 done
   110 
   111 lemma Suc_nat_number_of_add:
   112      "Suc (number_of v + n) =  
   113         (if neg (number_of v :: int) then 1+n else number_of (Numeral.succ v) + n)" 
   114 by (simp del: nat_number_of 
   115          add: nat_number_of_def neg_nat
   116               Suc_nat_eq_nat_zadd1 number_of_succ) 
   117 
   118 lemma Suc_nat_number_of [simp]:
   119      "Suc (number_of v) =  
   120         (if neg (number_of v :: int) then 1 else number_of (Numeral.succ v))"
   121 apply (cut_tac n = 0 in Suc_nat_number_of_add)
   122 apply (simp cong del: if_weak_cong)
   123 done
   124 
   125 
   126 subsubsection{*Addition *}
   127 
   128 (*"neg" is used in rewrite rules for binary comparisons*)
   129 lemma add_nat_number_of [simp]:
   130      "(number_of v :: nat) + number_of v' =  
   131          (if neg (number_of v :: int) then number_of v'  
   132           else if neg (number_of v' :: int) then number_of v  
   133           else number_of (v + v'))"
   134 by (force dest!: neg_nat
   135           simp del: nat_number_of
   136           simp add: nat_number_of_def nat_add_distrib [symmetric]) 
   137 
   138 
   139 subsubsection{*Subtraction *}
   140 
   141 lemma diff_nat_eq_if:
   142      "nat z - nat z' =  
   143         (if neg z' then nat z   
   144          else let d = z-z' in     
   145               if neg d then 0 else nat d)"
   146 apply (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
   147 done
   148 
   149 lemma diff_nat_number_of [simp]: 
   150      "(number_of v :: nat) - number_of v' =  
   151         (if neg (number_of v' :: int) then number_of v  
   152          else let d = number_of (v + uminus v') in     
   153               if neg d then 0 else nat d)"
   154 by (simp del: nat_number_of add: diff_nat_eq_if nat_number_of_def) 
   155 
   156 
   157 
   158 subsubsection{*Multiplication *}
   159 
   160 lemma mult_nat_number_of [simp]:
   161      "(number_of v :: nat) * number_of v' =  
   162        (if neg (number_of v :: int) then 0 else number_of (v * v'))"
   163 by (force dest!: neg_nat
   164           simp del: nat_number_of
   165           simp add: nat_number_of_def nat_mult_distrib [symmetric]) 
   166 
   167 
   168 
   169 subsubsection{*Quotient *}
   170 
   171 lemma div_nat_number_of [simp]:
   172      "(number_of v :: nat)  div  number_of v' =  
   173           (if neg (number_of v :: int) then 0  
   174            else nat (number_of v div number_of v'))"
   175 by (force dest!: neg_nat
   176           simp del: nat_number_of
   177           simp add: nat_number_of_def nat_div_distrib [symmetric]) 
   178 
   179 lemma one_div_nat_number_of [simp]:
   180      "(Suc 0)  div  number_of v' = (nat (1 div number_of v'))" 
   181 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
   182 
   183 
   184 subsubsection{*Remainder *}
   185 
   186 lemma mod_nat_number_of [simp]:
   187      "(number_of v :: nat)  mod  number_of v' =  
   188         (if neg (number_of v :: int) then 0  
   189          else if neg (number_of v' :: int) then number_of v  
   190          else nat (number_of v mod number_of v'))"
   191 by (force dest!: neg_nat
   192           simp del: nat_number_of
   193           simp add: nat_number_of_def nat_mod_distrib [symmetric]) 
   194 
   195 lemma one_mod_nat_number_of [simp]:
   196      "(Suc 0)  mod  number_of v' =  
   197         (if neg (number_of v' :: int) then Suc 0
   198          else nat (1 mod number_of v'))"
   199 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
   200 
   201 
   202 subsubsection{* Divisibility *}
   203 
   204 lemmas dvd_eq_mod_eq_0_number_of =
   205   dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
   206 
   207 declare dvd_eq_mod_eq_0_number_of [simp]
   208 
   209 ML
   210 {*
   211 val nat_number_of_def = thm"nat_number_of_def";
   212 
   213 val nat_number_of = thm"nat_number_of";
   214 val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0";
   215 val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1";
   216 val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0";
   217 val numeral_2_eq_2 = thm"numeral_2_eq_2";
   218 val nat_div_distrib = thm"nat_div_distrib";
   219 val nat_mod_distrib = thm"nat_mod_distrib";
   220 val int_nat_number_of = thm"int_nat_number_of";
   221 val Suc_nat_eq_nat_zadd1 = thm"Suc_nat_eq_nat_zadd1";
   222 val Suc_nat_number_of_add = thm"Suc_nat_number_of_add";
   223 val Suc_nat_number_of = thm"Suc_nat_number_of";
   224 val add_nat_number_of = thm"add_nat_number_of";
   225 val diff_nat_eq_if = thm"diff_nat_eq_if";
   226 val diff_nat_number_of = thm"diff_nat_number_of";
   227 val mult_nat_number_of = thm"mult_nat_number_of";
   228 val div_nat_number_of = thm"div_nat_number_of";
   229 val mod_nat_number_of = thm"mod_nat_number_of";
   230 *}
   231 
   232 
   233 subsection{*Comparisons*}
   234 
   235 subsubsection{*Equals (=) *}
   236 
   237 lemma eq_nat_nat_iff:
   238      "[| (0::int) <= z;  0 <= z' |] ==> (nat z = nat z') = (z=z')"
   239 by (auto elim!: nonneg_eq_int)
   240 
   241 (*"neg" is used in rewrite rules for binary comparisons*)
   242 lemma eq_nat_number_of [simp]:
   243      "((number_of v :: nat) = number_of v') =  
   244       (if neg (number_of v :: int) then (iszero (number_of v' :: int) | neg (number_of v' :: int))  
   245        else if neg (number_of v' :: int) then iszero (number_of v :: int)  
   246        else iszero (number_of (v + uminus v') :: int))"
   247 apply (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def
   248                   eq_nat_nat_iff eq_number_of_eq nat_0 iszero_def
   249             split add: split_if cong add: imp_cong)
   250 apply (simp only: nat_eq_iff nat_eq_iff2)
   251 apply (simp add: not_neg_eq_ge_0 [symmetric])
   252 done
   253 
   254 
   255 subsubsection{*Less-than (<) *}
   256 
   257 (*"neg" is used in rewrite rules for binary comparisons*)
   258 lemma less_nat_number_of [simp]:
   259      "((number_of v :: nat) < number_of v') =  
   260          (if neg (number_of v :: int) then neg (number_of (uminus v') :: int)  
   261           else neg (number_of (v + uminus v') :: int))"
   262 by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def
   263                 nat_less_eq_zless less_number_of_eq_neg zless_nat_eq_int_zless
   264          cong add: imp_cong, simp add: Pls_def)
   265 
   266 
   267 (*Maps #n to n for n = 0, 1, 2*)
   268 lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
   269 
   270 
   271 subsection{*Powers with Numeric Exponents*}
   272 
   273 text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}.
   274 We cannot prove general results about the numeral @{term "-1"}, so we have to
   275 use @{term "- 1"} instead.*}
   276 
   277 lemma power2_eq_square: "(a::'a::{comm_semiring_1_cancel,recpower})\<twosuperior> = a * a"
   278   by (simp add: numeral_2_eq_2 Power.power_Suc)
   279 
   280 lemma zero_power2 [simp]: "(0::'a::{comm_semiring_1_cancel,recpower})\<twosuperior> = 0"
   281   by (simp add: power2_eq_square)
   282 
   283 lemma one_power2 [simp]: "(1::'a::{comm_semiring_1_cancel,recpower})\<twosuperior> = 1"
   284   by (simp add: power2_eq_square)
   285 
   286 lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x"
   287   apply (subgoal_tac "3 = Suc (Suc (Suc 0))")
   288   apply (erule ssubst)
   289   apply (simp add: power_Suc mult_ac)
   290   apply (unfold nat_number_of_def)
   291   apply (subst nat_eq_iff)
   292   apply simp
   293 done
   294 
   295 text{*Squares of literal numerals will be evaluated.*}
   296 lemmas power2_eq_square_number_of =
   297     power2_eq_square [of "number_of w", standard]
   298 declare power2_eq_square_number_of [simp]
   299 
   300 
   301 lemma zero_le_power2: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})"
   302   by (simp add: power2_eq_square zero_le_square)
   303 
   304 lemma zero_less_power2:
   305      "(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))"
   306   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
   307 
   308 lemma power2_less_0:
   309   fixes a :: "'a::{ordered_idom,recpower}"
   310   shows "~ (a\<twosuperior> < 0)"
   311 by (force simp add: power2_eq_square mult_less_0_iff) 
   312 
   313 lemma zero_eq_power2:
   314      "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))"
   315   by (force simp add: power2_eq_square mult_eq_0_iff)
   316 
   317 lemma abs_power2:
   318      "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})"
   319   by (simp add: power2_eq_square abs_mult abs_mult_self)
   320 
   321 lemma power2_abs:
   322      "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})"
   323   by (simp add: power2_eq_square abs_mult_self)
   324 
   325 lemma power2_minus:
   326      "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})"
   327   by (simp add: power2_eq_square)
   328 
   329 lemma power2_le_imp_le:
   330   fixes x y :: "'a::{ordered_semidom,recpower}"
   331   shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y"
   332 unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
   333 
   334 lemma power2_less_imp_less:
   335   fixes x y :: "'a::{ordered_semidom,recpower}"
   336   shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y"
   337 by (rule power_less_imp_less_base)
   338 
   339 lemma power2_eq_imp_eq:
   340   fixes x y :: "'a::{ordered_semidom,recpower}"
   341   shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y"
   342 unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp)
   343 
   344 lemma power_minus1_even: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"
   345 apply (induct "n")
   346 apply (auto simp add: power_Suc power_add power2_minus)
   347 done
   348 
   349 lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
   350 by (subst mult_commute) (simp add: power_mult)
   351 
   352 lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
   353 by (simp add: power_even_eq) 
   354 
   355 lemma power_minus_even [simp]:
   356      "(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)"
   357 by (simp add: power_minus1_even power_minus [of a]) 
   358 
   359 lemma zero_le_even_power':
   360      "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)"
   361 proof (induct "n")
   362   case 0
   363     show ?case by (simp add: zero_le_one)
   364 next
   365   case (Suc n)
   366     have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
   367       by (simp add: mult_ac power_add power2_eq_square)
   368     thus ?case
   369       by (simp add: prems zero_le_square zero_le_mult_iff)
   370 qed
   371 
   372 lemma odd_power_less_zero:
   373      "(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0"
   374 proof (induct "n")
   375   case 0
   376     show ?case by (simp add: Power.power_Suc)
   377 next
   378   case (Suc n)
   379     have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)" 
   380       by (simp add: mult_ac power_add power2_eq_square Power.power_Suc)
   381     thus ?case
   382       by (simp add: prems mult_less_0_iff mult_neg_neg)
   383 qed
   384 
   385 lemma odd_0_le_power_imp_0_le:
   386      "0 \<le> a  ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})"
   387 apply (insert odd_power_less_zero [of a n]) 
   388 apply (force simp add: linorder_not_less [symmetric]) 
   389 done
   390 
   391 text{*Simprules for comparisons where common factors can be cancelled.*}
   392 lemmas zero_compare_simps =
   393     add_strict_increasing add_strict_increasing2 add_increasing
   394     zero_le_mult_iff zero_le_divide_iff 
   395     zero_less_mult_iff zero_less_divide_iff 
   396     mult_le_0_iff divide_le_0_iff 
   397     mult_less_0_iff divide_less_0_iff 
   398     zero_le_power2 power2_less_0
   399 
   400 subsubsection{*Nat *}
   401 
   402 lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
   403 by (simp add: numerals)
   404 
   405 (*Expresses a natural number constant as the Suc of another one.
   406   NOT suitable for rewriting because n recurs in the condition.*)
   407 lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
   408 
   409 subsubsection{*Arith *}
   410 
   411 lemma Suc_eq_add_numeral_1: "Suc n = n + 1"
   412 by (simp add: numerals)
   413 
   414 lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"
   415 by (simp add: numerals)
   416 
   417 (* These two can be useful when m = number_of... *)
   418 
   419 lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
   420 apply (case_tac "m")
   421 apply (simp_all add: numerals)
   422 done
   423 
   424 lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
   425 apply (case_tac "m")
   426 apply (simp_all add: numerals)
   427 done
   428 
   429 lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
   430 apply (case_tac "m")
   431 apply (simp_all add: numerals)
   432 done
   433 
   434 
   435 subsection{*Comparisons involving (0::nat) *}
   436 
   437 text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
   438 
   439 lemma eq_number_of_0 [simp]:
   440      "(number_of v = (0::nat)) =  
   441       (if neg (number_of v :: int) then True else iszero (number_of v :: int))"
   442 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0)
   443 
   444 lemma eq_0_number_of [simp]:
   445      "((0::nat) = number_of v) =  
   446       (if neg (number_of v :: int) then True else iszero (number_of v :: int))"
   447 by (rule trans [OF eq_sym_conv eq_number_of_0])
   448 
   449 lemma less_0_number_of [simp]:
   450      "((0::nat) < number_of v) = neg (number_of (uminus v) :: int)"
   451 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] Pls_def)
   452 
   453 
   454 lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
   455 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0)
   456 
   457 
   458 
   459 subsection{*Comparisons involving  @{term Suc} *}
   460 
   461 lemma eq_number_of_Suc [simp]:
   462      "(number_of v = Suc n) =  
   463         (let pv = number_of (Numeral.pred v) in  
   464          if neg pv then False else nat pv = n)"
   465 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
   466                   number_of_pred nat_number_of_def 
   467             split add: split_if)
   468 apply (rule_tac x = "number_of v" in spec)
   469 apply (auto simp add: nat_eq_iff)
   470 done
   471 
   472 lemma Suc_eq_number_of [simp]:
   473      "(Suc n = number_of v) =  
   474         (let pv = number_of (Numeral.pred v) in  
   475          if neg pv then False else nat pv = n)"
   476 by (rule trans [OF eq_sym_conv eq_number_of_Suc])
   477 
   478 lemma less_number_of_Suc [simp]:
   479      "(number_of v < Suc n) =  
   480         (let pv = number_of (Numeral.pred v) in  
   481          if neg pv then True else nat pv < n)"
   482 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
   483                   number_of_pred nat_number_of_def  
   484             split add: split_if)
   485 apply (rule_tac x = "number_of v" in spec)
   486 apply (auto simp add: nat_less_iff)
   487 done
   488 
   489 lemma less_Suc_number_of [simp]:
   490      "(Suc n < number_of v) =  
   491         (let pv = number_of (Numeral.pred v) in  
   492          if neg pv then False else n < nat pv)"
   493 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 
   494                   number_of_pred nat_number_of_def
   495             split add: split_if)
   496 apply (rule_tac x = "number_of v" in spec)
   497 apply (auto simp add: zless_nat_eq_int_zless)
   498 done
   499 
   500 lemma le_number_of_Suc [simp]:
   501      "(number_of v <= Suc n) =  
   502         (let pv = number_of (Numeral.pred v) in  
   503          if neg pv then True else nat pv <= n)"
   504 by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])
   505 
   506 lemma le_Suc_number_of [simp]:
   507      "(Suc n <= number_of v) =  
   508         (let pv = number_of (Numeral.pred v) in  
   509          if neg pv then False else n <= nat pv)"
   510 by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])
   511 
   512 
   513 (* Push int(.) inwards: *)
   514 declare zadd_int [symmetric, simp]
   515 
   516 lemma lemma1: "(m+m = n+n) = (m = (n::int))"
   517 by auto
   518 
   519 lemma lemma2: "m+m ~= (1::int) + (n + n)"
   520 apply auto
   521 apply (drule_tac f = "%x. x mod 2" in arg_cong)
   522 apply (simp add: zmod_zadd1_eq)
   523 done
   524 
   525 lemma eq_number_of_BIT_BIT:
   526      "((number_of (v BIT x) ::int) = number_of (w BIT y)) =  
   527       (x=y & (((number_of v) ::int) = number_of w))"
   528 apply (simp only: number_of_BIT lemma1 lemma2 eq_commute
   529                OrderedGroup.add_left_cancel add_assoc OrderedGroup.add_0_left
   530             split add: bit.split)
   531 apply simp
   532 done
   533 
   534 lemma eq_number_of_BIT_Pls:
   535      "((number_of (v BIT x) ::int) = Numeral0) =  
   536       (x=bit.B0 & (((number_of v) ::int) = Numeral0))"
   537 apply (simp only: simp_thms  add: number_of_BIT number_of_Pls eq_commute
   538             split add: bit.split cong: imp_cong)
   539 apply (rule_tac x = "number_of v" in spec, safe)
   540 apply (simp_all (no_asm_use))
   541 apply (drule_tac f = "%x. x mod 2" in arg_cong)
   542 apply (simp add: zmod_zadd1_eq)
   543 done
   544 
   545 lemma eq_number_of_BIT_Min:
   546      "((number_of (v BIT x) ::int) = number_of Numeral.Min) =  
   547       (x=bit.B1 & (((number_of v) ::int) = number_of Numeral.Min))"
   548 apply (simp only: simp_thms  add: number_of_BIT number_of_Min eq_commute
   549             split add: bit.split cong: imp_cong)
   550 apply (rule_tac x = "number_of v" in spec, auto)
   551 apply (drule_tac f = "%x. x mod 2" in arg_cong, auto)
   552 done
   553 
   554 lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Numeral.Min"
   555 by auto
   556 
   557 
   558 
   559 subsection{*Max and Min Combined with @{term Suc} *}
   560 
   561 lemma max_number_of_Suc [simp]:
   562      "max (Suc n) (number_of v) =  
   563         (let pv = number_of (Numeral.pred v) in  
   564          if neg pv then Suc n else Suc(max n (nat pv)))"
   565 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
   566             split add: split_if nat.split)
   567 apply (rule_tac x = "number_of v" in spec) 
   568 apply auto
   569 done
   570  
   571 lemma max_Suc_number_of [simp]:
   572      "max (number_of v) (Suc n) =  
   573         (let pv = number_of (Numeral.pred v) in  
   574          if neg pv then Suc n else Suc(max (nat pv) n))"
   575 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
   576             split add: split_if nat.split)
   577 apply (rule_tac x = "number_of v" in spec) 
   578 apply auto
   579 done
   580  
   581 lemma min_number_of_Suc [simp]:
   582      "min (Suc n) (number_of v) =  
   583         (let pv = number_of (Numeral.pred v) in  
   584          if neg pv then 0 else Suc(min n (nat pv)))"
   585 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
   586             split add: split_if nat.split)
   587 apply (rule_tac x = "number_of v" in spec) 
   588 apply auto
   589 done
   590  
   591 lemma min_Suc_number_of [simp]:
   592      "min (number_of v) (Suc n) =  
   593         (let pv = number_of (Numeral.pred v) in  
   594          if neg pv then 0 else Suc(min (nat pv) n))"
   595 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 
   596             split add: split_if nat.split)
   597 apply (rule_tac x = "number_of v" in spec) 
   598 apply auto
   599 done
   600  
   601 subsection{*Literal arithmetic involving powers*}
   602 
   603 lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n"
   604 apply (induct "n")
   605 apply (simp_all (no_asm_simp) add: nat_mult_distrib)
   606 done
   607 
   608 lemma power_nat_number_of:
   609      "(number_of v :: nat) ^ n =  
   610        (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
   611 by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
   612          split add: split_if cong: imp_cong)
   613 
   614 
   615 lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
   616 declare power_nat_number_of_number_of [simp]
   617 
   618 
   619 
   620 text{*For the integers*}
   621 
   622 lemma zpower_number_of_even:
   623   "(z::int) ^ number_of (w BIT bit.B0) = (let w = z ^ (number_of w) in w * w)"
   624 unfolding Let_def nat_number_of_def number_of_BIT bit.cases
   625 apply (rule_tac x = "number_of w" in spec, clarify)
   626 apply (case_tac " (0::int) <= x")
   627 apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)
   628 done
   629 
   630 lemma zpower_number_of_odd:
   631   "(z::int) ^ number_of (w BIT bit.B1) = (if (0::int) <= number_of w                    
   632      then (let w = z ^ (number_of w) in z * w * w) else 1)"
   633 unfolding Let_def nat_number_of_def number_of_BIT bit.cases
   634 apply (rule_tac x = "number_of w" in spec, auto)
   635 apply (simp only: nat_add_distrib nat_mult_distrib)
   636 apply simp
   637 apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat)
   638 done
   639 
   640 lemmas zpower_number_of_even_number_of =
   641     zpower_number_of_even [of "number_of v", standard]
   642 declare zpower_number_of_even_number_of [simp]
   643 
   644 lemmas zpower_number_of_odd_number_of =
   645     zpower_number_of_odd [of "number_of v", standard]
   646 declare zpower_number_of_odd_number_of [simp]
   647 
   648 
   649 
   650 
   651 ML
   652 {*
   653 val numerals = thms"numerals";
   654 val numeral_ss = simpset() addsimps numerals;
   655 
   656 val nat_bin_arith_setup =
   657  Fast_Arith.map_data
   658    (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
   659      {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
   660       inj_thms = inj_thms,
   661       lessD = lessD, neqE = neqE,
   662       simpset = simpset addsimps [Suc_nat_number_of, int_nat_number_of,
   663                                   not_neg_number_of_Pls,
   664                                   neg_number_of_Min,neg_number_of_BIT]})
   665 *}
   666 
   667 setup nat_bin_arith_setup
   668 
   669 (* Enable arith to deal with div/mod k where k is a numeral: *)
   670 declare split_div[of _ _ "number_of k", standard, arith_split]
   671 declare split_mod[of _ _ "number_of k", standard, arith_split]
   672 
   673 lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
   674   by (simp add: number_of_Pls nat_number_of_def)
   675 
   676 lemma nat_number_of_Min: "number_of Numeral.Min = (0::nat)"
   677   apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
   678   done
   679 
   680 lemma nat_number_of_BIT_1:
   681   "number_of (w BIT bit.B1) =
   682     (if neg (number_of w :: int) then 0
   683      else let n = number_of w in Suc (n + n))"
   684   apply (simp only: nat_number_of_def Let_def split: split_if)
   685   apply (intro conjI impI)
   686    apply (simp add: neg_nat neg_number_of_BIT)
   687   apply (rule int_int_eq [THEN iffD1])
   688   apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms)
   689   apply (simp only: number_of_BIT zadd_assoc split: bit.split)
   690   apply simp
   691   done
   692 
   693 lemma nat_number_of_BIT_0:
   694     "number_of (w BIT bit.B0) = (let n::nat = number_of w in n + n)"
   695   apply (simp only: nat_number_of_def Let_def)
   696   apply (cases "neg (number_of w :: int)")
   697    apply (simp add: neg_nat neg_number_of_BIT)
   698   apply (rule int_int_eq [THEN iffD1])
   699   apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms)
   700   apply (simp only: number_of_BIT zadd_assoc)
   701   apply simp
   702   done
   703 
   704 lemmas nat_number =
   705   nat_number_of_Pls nat_number_of_Min
   706   nat_number_of_BIT_1 nat_number_of_BIT_0
   707 
   708 lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
   709   by (simp add: Let_def)
   710 
   711 lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})"
   712 by (simp add: power_mult); 
   713 
   714 lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})"
   715 by (simp add: power_mult power_Suc); 
   716 
   717 
   718 subsection{*Literal arithmetic and @{term of_nat}*}
   719 
   720 lemma of_nat_double:
   721      "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
   722 by (simp only: mult_2 nat_add_distrib of_nat_add) 
   723 
   724 lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
   725 by (simp only: nat_number_of_def)
   726 
   727 lemma of_nat_number_of_lemma:
   728      "of_nat (number_of v :: nat) =  
   729          (if 0 \<le> (number_of v :: int) 
   730           then (number_of v :: 'a :: number_ring)
   731           else 0)"
   732 by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);
   733 
   734 lemma of_nat_number_of_eq [simp]:
   735      "of_nat (number_of v :: nat) =  
   736          (if neg (number_of v :: int) then 0  
   737           else (number_of v :: 'a :: number_ring))"
   738 by (simp only: of_nat_number_of_lemma neg_def, simp) 
   739 
   740 
   741 subsection {*Lemmas for the Combination and Cancellation Simprocs*}
   742 
   743 lemma nat_number_of_add_left:
   744      "number_of v + (number_of v' + (k::nat)) =  
   745          (if neg (number_of v :: int) then number_of v' + k  
   746           else if neg (number_of v' :: int) then number_of v + k  
   747           else number_of (v + v') + k)"
   748 by simp
   749 
   750 lemma nat_number_of_mult_left:
   751      "number_of v * (number_of v' * (k::nat)) =  
   752          (if neg (number_of v :: int) then 0
   753           else number_of (v * v') * k)"
   754 by simp
   755 
   756 
   757 subsubsection{*For @{text combine_numerals}*}
   758 
   759 lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
   760 by (simp add: add_mult_distrib)
   761 
   762 
   763 subsubsection{*For @{text cancel_numerals}*}
   764 
   765 lemma nat_diff_add_eq1:
   766      "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
   767 by (simp split add: nat_diff_split add: add_mult_distrib)
   768 
   769 lemma nat_diff_add_eq2:
   770      "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
   771 by (simp split add: nat_diff_split add: add_mult_distrib)
   772 
   773 lemma nat_eq_add_iff1:
   774      "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
   775 by (auto split add: nat_diff_split simp add: add_mult_distrib)
   776 
   777 lemma nat_eq_add_iff2:
   778      "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
   779 by (auto split add: nat_diff_split simp add: add_mult_distrib)
   780 
   781 lemma nat_less_add_iff1:
   782      "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
   783 by (auto split add: nat_diff_split simp add: add_mult_distrib)
   784 
   785 lemma nat_less_add_iff2:
   786      "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
   787 by (auto split add: nat_diff_split simp add: add_mult_distrib)
   788 
   789 lemma nat_le_add_iff1:
   790      "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
   791 by (auto split add: nat_diff_split simp add: add_mult_distrib)
   792 
   793 lemma nat_le_add_iff2:
   794      "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
   795 by (auto split add: nat_diff_split simp add: add_mult_distrib)
   796 
   797 
   798 subsubsection{*For @{text cancel_numeral_factors} *}
   799 
   800 lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
   801 by auto
   802 
   803 lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
   804 by auto
   805 
   806 lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
   807 by auto
   808 
   809 lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
   810 by auto
   811 
   812 
   813 subsubsection{*For @{text cancel_factor} *}
   814 
   815 lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
   816 by auto
   817 
   818 lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
   819 by auto
   820 
   821 lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
   822 by auto
   823 
   824 lemma nat_mult_div_cancel_disj:
   825      "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
   826 by (simp add: nat_mult_div_cancel1)
   827 
   828 
   829 subsection {* legacy ML bindings *}
   830 
   831 ML
   832 {*
   833 val eq_nat_nat_iff = thm"eq_nat_nat_iff";
   834 val eq_nat_number_of = thm"eq_nat_number_of";
   835 val less_nat_number_of = thm"less_nat_number_of";
   836 val power2_eq_square = thm "power2_eq_square";
   837 val zero_le_power2 = thm "zero_le_power2";
   838 val zero_less_power2 = thm "zero_less_power2";
   839 val zero_eq_power2 = thm "zero_eq_power2";
   840 val abs_power2 = thm "abs_power2";
   841 val power2_abs = thm "power2_abs";
   842 val power2_minus = thm "power2_minus";
   843 val power_minus1_even = thm "power_minus1_even";
   844 val power_minus_even = thm "power_minus_even";
   845 val odd_power_less_zero = thm "odd_power_less_zero";
   846 val odd_0_le_power_imp_0_le = thm "odd_0_le_power_imp_0_le";
   847 
   848 val Suc_pred' = thm"Suc_pred'";
   849 val expand_Suc = thm"expand_Suc";
   850 val Suc_eq_add_numeral_1 = thm"Suc_eq_add_numeral_1";
   851 val Suc_eq_add_numeral_1_left = thm"Suc_eq_add_numeral_1_left";
   852 val add_eq_if = thm"add_eq_if";
   853 val mult_eq_if = thm"mult_eq_if";
   854 val power_eq_if = thm"power_eq_if";
   855 val eq_number_of_0 = thm"eq_number_of_0";
   856 val eq_0_number_of = thm"eq_0_number_of";
   857 val less_0_number_of = thm"less_0_number_of";
   858 val neg_imp_number_of_eq_0 = thm"neg_imp_number_of_eq_0";
   859 val eq_number_of_Suc = thm"eq_number_of_Suc";
   860 val Suc_eq_number_of = thm"Suc_eq_number_of";
   861 val less_number_of_Suc = thm"less_number_of_Suc";
   862 val less_Suc_number_of = thm"less_Suc_number_of";
   863 val le_number_of_Suc = thm"le_number_of_Suc";
   864 val le_Suc_number_of = thm"le_Suc_number_of";
   865 val eq_number_of_BIT_BIT = thm"eq_number_of_BIT_BIT";
   866 val eq_number_of_BIT_Pls = thm"eq_number_of_BIT_Pls";
   867 val eq_number_of_BIT_Min = thm"eq_number_of_BIT_Min";
   868 val eq_number_of_Pls_Min = thm"eq_number_of_Pls_Min";
   869 val of_nat_number_of_eq = thm"of_nat_number_of_eq";
   870 val nat_power_eq = thm"nat_power_eq";
   871 val power_nat_number_of = thm"power_nat_number_of";
   872 val zpower_number_of_even = thm"zpower_number_of_even";
   873 val zpower_number_of_odd = thm"zpower_number_of_odd";
   874 val nat_number_of_Pls = thm"nat_number_of_Pls";
   875 val nat_number_of_Min = thm"nat_number_of_Min";
   876 val Let_Suc = thm"Let_Suc";
   877 
   878 val nat_number = thms"nat_number";
   879 
   880 val nat_number_of_add_left = thm"nat_number_of_add_left";
   881 val nat_number_of_mult_left = thm"nat_number_of_mult_left";
   882 val left_add_mult_distrib = thm"left_add_mult_distrib";
   883 val nat_diff_add_eq1 = thm"nat_diff_add_eq1";
   884 val nat_diff_add_eq2 = thm"nat_diff_add_eq2";
   885 val nat_eq_add_iff1 = thm"nat_eq_add_iff1";
   886 val nat_eq_add_iff2 = thm"nat_eq_add_iff2";
   887 val nat_less_add_iff1 = thm"nat_less_add_iff1";
   888 val nat_less_add_iff2 = thm"nat_less_add_iff2";
   889 val nat_le_add_iff1 = thm"nat_le_add_iff1";
   890 val nat_le_add_iff2 = thm"nat_le_add_iff2";
   891 val nat_mult_le_cancel1 = thm"nat_mult_le_cancel1";
   892 val nat_mult_less_cancel1 = thm"nat_mult_less_cancel1";
   893 val nat_mult_eq_cancel1 = thm"nat_mult_eq_cancel1";
   894 val nat_mult_div_cancel1 = thm"nat_mult_div_cancel1";
   895 val nat_mult_le_cancel_disj = thm"nat_mult_le_cancel_disj";
   896 val nat_mult_less_cancel_disj = thm"nat_mult_less_cancel_disj";
   897 val nat_mult_eq_cancel_disj = thm"nat_mult_eq_cancel_disj";
   898 val nat_mult_div_cancel_disj = thm"nat_mult_div_cancel_disj";
   899 
   900 val power_minus_even = thm"power_minus_even";
   901 *}
   902 
   903 end