src/HOL/Nat_Numeral.thy
 author haftmann Wed Apr 22 19:09:21 2009 +0200 (2009-04-22) changeset 30960 fec1a04b7220 parent 30925 c38cbc0ac8d1 child 31002 bc4117fe72ab permissions -rw-r--r--
power operation defined generic
1 (*  Title:      HOL/Nat_Numeral.thy
2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
3     Copyright   1999  University of Cambridge
4 *)
6 header {* Binary numerals for the natural numbers *}
8 theory Nat_Numeral
9 imports IntDiv
10 uses ("Tools/nat_simprocs.ML")
11 begin
13 text {*
14   Arithmetic for naturals is reduced to that for the non-negative integers.
15 *}
17 instantiation nat :: number
18 begin
20 definition
21   nat_number_of_def [code inline, code del]: "number_of v = nat (number_of v)"
23 instance ..
25 end
27 lemma [code post]:
28   "nat (number_of v) = number_of v"
29   unfolding nat_number_of_def ..
31 context recpower
32 begin
34 abbreviation (xsymbols)
35   power2 :: "'a \<Rightarrow> 'a"  ("(_\<twosuperior>)" [1000] 999) where
36   "x\<twosuperior> \<equiv> x ^ 2"
38 notation (latex output)
39   power2  ("(_\<twosuperior>)" [1000] 999)
41 notation (HTML output)
42   power2  ("(_\<twosuperior>)" [1000] 999)
44 end
47 subsection {* Predicate for negative binary numbers *}
49 definition neg  :: "int \<Rightarrow> bool" where
50   "neg Z \<longleftrightarrow> Z < 0"
52 lemma not_neg_int [simp]: "~ neg (of_nat n)"
55 lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))"
56 by (simp add: neg_def neg_less_0_iff_less del: of_nat_Suc)
58 lemmas neg_eq_less_0 = neg_def
60 lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
61 by (simp add: neg_def linorder_not_less)
63 text{*To simplify inequalities when Numeral1 can get simplified to 1*}
65 lemma not_neg_0: "~ neg 0"
66 by (simp add: One_int_def neg_def)
68 lemma not_neg_1: "~ neg 1"
69 by (simp add: neg_def linorder_not_less zero_le_one)
71 lemma neg_nat: "neg z ==> nat z = 0"
72 by (simp add: neg_def order_less_imp_le)
74 lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z"
75 by (simp add: linorder_not_less neg_def)
77 text {*
78   If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
79   @{term Numeral0} IS @{term "number_of Pls"}
80 *}
82 lemma not_neg_number_of_Pls: "~ neg (number_of Int.Pls)"
85 lemma neg_number_of_Min: "neg (number_of Int.Min)"
88 lemma neg_number_of_Bit0:
89   "neg (number_of (Int.Bit0 w)) = neg (number_of w)"
92 lemma neg_number_of_Bit1:
93   "neg (number_of (Int.Bit1 w)) = neg (number_of w)"
96 lemmas neg_simps [simp] =
97   not_neg_0 not_neg_1
98   not_neg_number_of_Pls neg_number_of_Min
99   neg_number_of_Bit0 neg_number_of_Bit1
102 subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}
104 declare nat_0 [simp] nat_1 [simp]
106 lemma nat_number_of [simp]: "nat (number_of w) = number_of w"
109 lemma nat_numeral_0_eq_0 [simp]: "Numeral0 = (0::nat)"
112 lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
113 by (simp add: nat_1 nat_number_of_def)
115 lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
118 lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
119 apply (unfold nat_number_of_def)
120 apply (rule nat_2)
121 done
124 subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
126 lemma int_nat_number_of [simp]:
127      "int (number_of v) =
128          (if neg (number_of v :: int) then 0
129           else (number_of v :: int))"
130   unfolding nat_number_of_def number_of_is_id neg_def
131   by simp
134 subsubsection{*Successor *}
136 lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
137 apply (rule sym)
138 apply (simp add: nat_eq_iff int_Suc)
139 done
142      "Suc (number_of v + n) =
143         (if neg (number_of v :: int) then 1+n else number_of (Int.succ v) + n)"
144   unfolding nat_number_of_def number_of_is_id neg_def numeral_simps
147 lemma Suc_nat_number_of [simp]:
148      "Suc (number_of v) =
149         (if neg (number_of v :: int) then 1 else number_of (Int.succ v))"
150 apply (cut_tac n = 0 in Suc_nat_number_of_add)
151 apply (simp cong del: if_weak_cong)
152 done
158      "(number_of v :: nat) + number_of v' =
159          (if v < Int.Pls then number_of v'
160           else if v' < Int.Pls then number_of v
161           else number_of (v + v'))"
162   unfolding nat_number_of_def number_of_is_id numeral_simps
166   "number_of v + (1::nat) =
167     (if v < Int.Pls then 1 else number_of (Int.succ v))"
168   unfolding nat_number_of_def number_of_is_id numeral_simps
172   "(1::nat) + number_of v =
173     (if v < Int.Pls then 1 else number_of (Int.succ v))"
174   unfolding nat_number_of_def number_of_is_id numeral_simps
177 lemma nat_1_add_1 [simp]: "1 + 1 = (2::nat)"
178   by (rule int_int_eq [THEN iffD1]) simp
181 subsubsection{*Subtraction *}
183 lemma diff_nat_eq_if:
184      "nat z - nat z' =
185         (if neg z' then nat z
186          else let d = z-z' in
187               if neg d then 0 else nat d)"
188 by (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
191 lemma diff_nat_number_of [simp]:
192      "(number_of v :: nat) - number_of v' =
193         (if v' < Int.Pls then number_of v
194          else let d = number_of (v + uminus v') in
195               if neg d then 0 else nat d)"
196   unfolding nat_number_of_def number_of_is_id numeral_simps neg_def
197   by auto
199 lemma nat_number_of_diff_1 [simp]:
200   "number_of v - (1::nat) =
201     (if v \<le> Int.Pls then 0 else number_of (Int.pred v))"
202   unfolding nat_number_of_def number_of_is_id numeral_simps
203   by auto
206 subsubsection{*Multiplication *}
208 lemma mult_nat_number_of [simp]:
209      "(number_of v :: nat) * number_of v' =
210        (if v < Int.Pls then 0 else number_of (v * v'))"
211   unfolding nat_number_of_def number_of_is_id numeral_simps
215 subsubsection{*Quotient *}
217 lemma div_nat_number_of [simp]:
218      "(number_of v :: nat)  div  number_of v' =
219           (if neg (number_of v :: int) then 0
220            else nat (number_of v div number_of v'))"
221   unfolding nat_number_of_def number_of_is_id neg_def
224 lemma one_div_nat_number_of [simp]:
225      "Suc 0 div number_of v' = nat (1 div number_of v')"
226 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
229 subsubsection{*Remainder *}
231 lemma mod_nat_number_of [simp]:
232      "(number_of v :: nat)  mod  number_of v' =
233         (if neg (number_of v :: int) then 0
234          else if neg (number_of v' :: int) then number_of v
235          else nat (number_of v mod number_of v'))"
236   unfolding nat_number_of_def number_of_is_id neg_def
239 lemma one_mod_nat_number_of [simp]:
240      "Suc 0 mod number_of v' =
241         (if neg (number_of v' :: int) then Suc 0
242          else nat (1 mod number_of v'))"
243 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
246 subsubsection{* Divisibility *}
248 lemmas dvd_eq_mod_eq_0_number_of =
249   dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
251 declare dvd_eq_mod_eq_0_number_of [simp]
253 ML
254 {*
255 val nat_number_of_def = thm"nat_number_of_def";
257 val nat_number_of = thm"nat_number_of";
258 val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0";
259 val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1";
260 val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0";
261 val numeral_2_eq_2 = thm"numeral_2_eq_2";
262 val nat_div_distrib = thm"nat_div_distrib";
263 val nat_mod_distrib = thm"nat_mod_distrib";
264 val int_nat_number_of = thm"int_nat_number_of";
267 val Suc_nat_number_of = thm"Suc_nat_number_of";
269 val diff_nat_eq_if = thm"diff_nat_eq_if";
270 val diff_nat_number_of = thm"diff_nat_number_of";
271 val mult_nat_number_of = thm"mult_nat_number_of";
272 val div_nat_number_of = thm"div_nat_number_of";
273 val mod_nat_number_of = thm"mod_nat_number_of";
274 *}
277 subsection{*Comparisons*}
279 subsubsection{*Equals (=) *}
281 lemma eq_nat_nat_iff:
282      "[| (0::int) <= z;  0 <= z' |] ==> (nat z = nat z') = (z=z')"
283 by (auto elim!: nonneg_eq_int)
285 lemma eq_nat_number_of [simp]:
286      "((number_of v :: nat) = number_of v') =
287       (if neg (number_of v :: int) then (number_of v' :: int) \<le> 0
288        else if neg (number_of v' :: int) then (number_of v :: int) = 0
289        else v = v')"
290   unfolding nat_number_of_def number_of_is_id neg_def
291   by auto
294 subsubsection{*Less-than (<) *}
296 lemma less_nat_number_of [simp]:
297   "(number_of v :: nat) < number_of v' \<longleftrightarrow>
298     (if v < v' then Int.Pls < v' else False)"
299   unfolding nat_number_of_def number_of_is_id numeral_simps
300   by auto
303 subsubsection{*Less-than-or-equal *}
305 lemma le_nat_number_of [simp]:
306   "(number_of v :: nat) \<le> number_of v' \<longleftrightarrow>
307     (if v \<le> v' then True else v \<le> Int.Pls)"
308   unfolding nat_number_of_def number_of_is_id numeral_simps
309   by auto
311 (*Maps #n to n for n = 0, 1, 2*)
312 lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
315 subsection{*Powers with Numeric Exponents*}
317 text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}.
318 We cannot prove general results about the numeral @{term "-1"}, so we have to
319 use @{term "- 1"} instead.*}
321 lemma power2_eq_square: "(a::'a::recpower)\<twosuperior> = a * a"
322   by (simp add: numeral_2_eq_2 Power.power_Suc)
324 lemma zero_power2 [simp]: "(0::'a::{semiring_1,recpower})\<twosuperior> = 0"
327 lemma one_power2 [simp]: "(1::'a::{semiring_1,recpower})\<twosuperior> = 1"
330 lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x"
331   apply (subgoal_tac "3 = Suc (Suc (Suc 0))")
332   apply (erule ssubst)
333   apply (simp add: power_Suc mult_ac)
334   apply (unfold nat_number_of_def)
335   apply (subst nat_eq_iff)
336   apply simp
337 done
339 text{*Squares of literal numerals will be evaluated.*}
340 lemmas power2_eq_square_number_of =
341     power2_eq_square [of "number_of w", standard]
342 declare power2_eq_square_number_of [simp]
345 lemma zero_le_power2[simp]: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})"
348 lemma zero_less_power2[simp]:
349      "(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))"
350   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
352 lemma power2_less_0[simp]:
353   fixes a :: "'a::{ordered_idom,recpower}"
354   shows "~ (a\<twosuperior> < 0)"
355 by (force simp add: power2_eq_square mult_less_0_iff)
357 lemma zero_eq_power2[simp]:
358      "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))"
359   by (force simp add: power2_eq_square mult_eq_0_iff)
361 lemma abs_power2[simp]:
362      "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})"
363   by (simp add: power2_eq_square abs_mult abs_mult_self)
365 lemma power2_abs[simp]:
366      "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})"
367   by (simp add: power2_eq_square abs_mult_self)
369 lemma power2_minus[simp]:
370      "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})"
373 lemma power2_le_imp_le:
374   fixes x y :: "'a::{ordered_semidom,recpower}"
375   shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y"
376 unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
378 lemma power2_less_imp_less:
379   fixes x y :: "'a::{ordered_semidom,recpower}"
380   shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y"
381 by (rule power_less_imp_less_base)
383 lemma power2_eq_imp_eq:
384   fixes x y :: "'a::{ordered_semidom,recpower}"
385   shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y"
386 unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp)
388 lemma power_minus1_even[simp]: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"
389 proof (induct n)
390   case 0 show ?case by simp
391 next
392   case (Suc n) then show ?case by (simp add: power_Suc power_add)
393 qed
395 lemma power_minus1_odd: "(- 1) ^ Suc(2*n) = -(1::'a::{comm_ring_1,recpower})"
398 lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
399 by (subst mult_commute) (simp add: power_mult)
401 lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
404 lemma power_minus_even [simp]:
405   "(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)"
406   by (simp add: power_minus [of a])
408 lemma zero_le_even_power'[simp]:
409      "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)"
410 proof (induct "n")
411   case 0
412     show ?case by (simp add: zero_le_one)
413 next
414   case (Suc n)
415     have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
417     thus ?case
418       by (simp add: prems zero_le_mult_iff)
419 qed
421 lemma odd_power_less_zero:
422      "(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0"
423 proof (induct "n")
424   case 0
425   then show ?case by simp
426 next
427   case (Suc n)
428   have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
430   thus ?case
431     by (simp del: power_Suc add: prems mult_less_0_iff mult_neg_neg)
432 qed
434 lemma odd_0_le_power_imp_0_le:
435      "0 \<le> a  ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})"
436 apply (insert odd_power_less_zero [of a n])
437 apply (force simp add: linorder_not_less [symmetric])
438 done
440 text{*Simprules for comparisons where common factors can be cancelled.*}
441 lemmas zero_compare_simps =
443     zero_le_mult_iff zero_le_divide_iff
444     zero_less_mult_iff zero_less_divide_iff
445     mult_le_0_iff divide_le_0_iff
446     mult_less_0_iff divide_less_0_iff
447     zero_le_power2 power2_less_0
449 subsubsection{*Nat *}
451 lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
454 (*Expresses a natural number constant as the Suc of another one.
455   NOT suitable for rewriting because n recurs in the condition.*)
456 lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
458 subsubsection{*Arith *}
460 lemma Suc_eq_add_numeral_1: "Suc n = n + 1"
463 lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"
466 (* These two can be useful when m = number_of... *)
468 lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
469   unfolding One_nat_def by (cases m) simp_all
471 lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
472   unfolding One_nat_def by (cases m) simp_all
474 lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
475   unfolding One_nat_def by (cases m) simp_all
478 subsection{*Comparisons involving (0::nat) *}
480 text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
482 lemma eq_number_of_0 [simp]:
483   "number_of v = (0::nat) \<longleftrightarrow> v \<le> Int.Pls"
484   unfolding nat_number_of_def number_of_is_id numeral_simps
485   by auto
487 lemma eq_0_number_of [simp]:
488   "(0::nat) = number_of v \<longleftrightarrow> v \<le> Int.Pls"
489 by (rule trans [OF eq_sym_conv eq_number_of_0])
491 lemma less_0_number_of [simp]:
492    "(0::nat) < number_of v \<longleftrightarrow> Int.Pls < v"
493   unfolding nat_number_of_def number_of_is_id numeral_simps
494   by simp
496 lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
497 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])
501 subsection{*Comparisons involving  @{term Suc} *}
503 lemma eq_number_of_Suc [simp]:
504      "(number_of v = Suc n) =
505         (let pv = number_of (Int.pred v) in
506          if neg pv then False else nat pv = n)"
507 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
508                   number_of_pred nat_number_of_def
510 apply (rule_tac x = "number_of v" in spec)
511 apply (auto simp add: nat_eq_iff)
512 done
514 lemma Suc_eq_number_of [simp]:
515      "(Suc n = number_of v) =
516         (let pv = number_of (Int.pred v) in
517          if neg pv then False else nat pv = n)"
518 by (rule trans [OF eq_sym_conv eq_number_of_Suc])
520 lemma less_number_of_Suc [simp]:
521      "(number_of v < Suc n) =
522         (let pv = number_of (Int.pred v) in
523          if neg pv then True else nat pv < n)"
524 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
525                   number_of_pred nat_number_of_def
527 apply (rule_tac x = "number_of v" in spec)
528 apply (auto simp add: nat_less_iff)
529 done
531 lemma less_Suc_number_of [simp]:
532      "(Suc n < number_of v) =
533         (let pv = number_of (Int.pred v) in
534          if neg pv then False else n < nat pv)"
535 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
536                   number_of_pred nat_number_of_def
538 apply (rule_tac x = "number_of v" in spec)
539 apply (auto simp add: zless_nat_eq_int_zless)
540 done
542 lemma le_number_of_Suc [simp]:
543      "(number_of v <= Suc n) =
544         (let pv = number_of (Int.pred v) in
545          if neg pv then True else nat pv <= n)"
546 by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])
548 lemma le_Suc_number_of [simp]:
549      "(Suc n <= number_of v) =
550         (let pv = number_of (Int.pred v) in
551          if neg pv then False else n <= nat pv)"
552 by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])
555 lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Int.Min"
556 by auto
560 subsection{*Max and Min Combined with @{term Suc} *}
562 lemma max_number_of_Suc [simp]:
563      "max (Suc n) (number_of v) =
564         (let pv = number_of (Int.pred v) in
565          if neg pv then Suc n else Suc(max n (nat pv)))"
566 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
568 apply (rule_tac x = "number_of v" in spec)
569 apply auto
570 done
572 lemma max_Suc_number_of [simp]:
573      "max (number_of v) (Suc n) =
574         (let pv = number_of (Int.pred v) in
575          if neg pv then Suc n else Suc(max (nat pv) n))"
576 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
578 apply (rule_tac x = "number_of v" in spec)
579 apply auto
580 done
582 lemma min_number_of_Suc [simp]:
583      "min (Suc n) (number_of v) =
584         (let pv = number_of (Int.pred v) in
585          if neg pv then 0 else Suc(min n (nat pv)))"
586 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
588 apply (rule_tac x = "number_of v" in spec)
589 apply auto
590 done
592 lemma min_Suc_number_of [simp]:
593      "min (number_of v) (Suc n) =
594         (let pv = number_of (Int.pred v) in
595          if neg pv then 0 else Suc(min (nat pv) n))"
596 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
598 apply (rule_tac x = "number_of v" in spec)
599 apply auto
600 done
602 subsection{*Literal arithmetic involving powers*}
604 lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n"
605 apply (induct "n")
606 apply (simp_all (no_asm_simp) add: nat_mult_distrib)
607 done
609 lemma power_nat_number_of:
610      "(number_of v :: nat) ^ n =
611        (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
612 by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
613          split add: split_if cong: imp_cong)
616 lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
617 declare power_nat_number_of_number_of [simp]
621 text{*For arbitrary rings*}
623 lemma power_number_of_even:
624   fixes z :: "'a::{number_ring,recpower}"
625   shows "z ^ number_of (Int.Bit0 w) = (let w = z ^ (number_of w) in w * w)"
626 unfolding Let_def nat_number_of_def number_of_Bit0
627 apply (rule_tac x = "number_of w" in spec, clarify)
628 apply (case_tac " (0::int) <= x")
629 apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)
630 done
632 lemma power_number_of_odd:
633   fixes z :: "'a::{number_ring,recpower}"
634   shows "z ^ number_of (Int.Bit1 w) = (if (0::int) <= number_of w
635      then (let w = z ^ (number_of w) in z * w * w) else 1)"
636 unfolding Let_def nat_number_of_def number_of_Bit1
637 apply (rule_tac x = "number_of w" in spec, auto)
638 apply (simp only: nat_add_distrib nat_mult_distrib)
639 apply simp
641 done
643 lemmas zpower_number_of_even = power_number_of_even [where 'a=int]
644 lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]
646 lemmas power_number_of_even_number_of [simp] =
647     power_number_of_even [of "number_of v", standard]
649 lemmas power_number_of_odd_number_of [simp] =
650     power_number_of_odd [of "number_of v", standard]
654 ML
655 {*
656 val numeral_ss = @{simpset} addsimps @{thms numerals};
658 val nat_bin_arith_setup =
659  Lin_Arith.map_data
660    (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
662       inj_thms = inj_thms,
663       lessD = lessD, neqE = neqE,
664       simpset = simpset addsimps @{thms neg_simps} @
665         [@{thm Suc_nat_number_of}, @{thm int_nat_number_of}]})
666 *}
668 declaration {* K nat_bin_arith_setup *}
670 (* Enable arith to deal with div/mod k where k is a numeral: *)
671 declare split_div[of _ _ "number_of k", standard, arith_split]
672 declare split_mod[of _ _ "number_of k", standard, arith_split]
674 lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
675   by (simp add: number_of_Pls nat_number_of_def)
677 lemma nat_number_of_Min: "number_of Int.Min = (0::nat)"
678   apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
679   done
681 lemma nat_number_of_Bit0:
682     "number_of (Int.Bit0 w) = (let n::nat = number_of w in n + n)"
683   unfolding nat_number_of_def number_of_is_id numeral_simps Let_def
684   by auto
686 lemma nat_number_of_Bit1:
687   "number_of (Int.Bit1 w) =
688     (if neg (number_of w :: int) then 0
689      else let n = number_of w in Suc (n + n))"
690   unfolding nat_number_of_def number_of_is_id numeral_simps neg_def Let_def
691   by auto
693 lemmas nat_number =
694   nat_number_of_Pls nat_number_of_Min
695   nat_number_of_Bit0 nat_number_of_Bit1
697 lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
700 lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})"
701 by (simp add: power_mult power_Suc);
703 lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})"
704 by (simp add: power_mult power_Suc);
707 subsection{*Literal arithmetic and @{term of_nat}*}
709 lemma of_nat_double:
710      "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
713 lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
714 by (simp only: nat_number_of_def)
716 lemma of_nat_number_of_lemma:
717      "of_nat (number_of v :: nat) =
718          (if 0 \<le> (number_of v :: int)
719           then (number_of v :: 'a :: number_ring)
720           else 0)"
721 by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);
723 lemma of_nat_number_of_eq [simp]:
724      "of_nat (number_of v :: nat) =
725          (if neg (number_of v :: int) then 0
726           else (number_of v :: 'a :: number_ring))"
727 by (simp only: of_nat_number_of_lemma neg_def, simp)
730 subsection {*Lemmas for the Combination and Cancellation Simprocs*}
733      "number_of v + (number_of v' + (k::nat)) =
734          (if neg (number_of v :: int) then number_of v' + k
735           else if neg (number_of v' :: int) then number_of v + k
736           else number_of (v + v') + k)"
737   unfolding nat_number_of_def number_of_is_id neg_def
738   by auto
740 lemma nat_number_of_mult_left:
741      "number_of v * (number_of v' * (k::nat)) =
742          (if v < Int.Pls then 0
743           else number_of (v * v') * k)"
744 by simp
747 subsubsection{*For @{text combine_numerals}*}
749 lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
753 subsubsection{*For @{text cancel_numerals}*}
756      "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
760      "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
764      "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
768      "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
772      "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
776      "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
780      "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
784      "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
788 subsubsection{*For @{text cancel_numeral_factors} *}
790 lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
791 by auto
793 lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
794 by auto
796 lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
797 by auto
799 lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
800 by auto
802 lemma nat_mult_dvd_cancel_disj[simp]:
803   "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
804 by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
806 lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
807 by(auto)
810 subsubsection{*For @{text cancel_factor} *}
812 lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
813 by auto
815 lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
816 by auto
818 lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
819 by auto
821 lemma nat_mult_div_cancel_disj[simp]:
822      "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
826 subsection {* Simprocs for the Naturals *}
828 use "Tools/nat_simprocs.ML"
829 declaration {* K nat_simprocs_setup *}
831 subsubsection{*For simplifying @{term "Suc m - K"} and  @{term "K - Suc m"}*}
833 text{*Where K above is a literal*}
835 lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
838 text {*Now just instantiating @{text n} to @{text "number_of v"} does
839   the right simplification, but with some redundant inequality
840   tests.*}
841 lemma neg_number_of_pred_iff_0:
842   "neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))"
843 apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ")
844 apply (simp only: less_Suc_eq_le le_0_eq)
845 apply (subst less_number_of_Suc, simp)
846 done
848 text{*No longer required as a simprule because of the @{text inverse_fold}
849    simproc*}
850 lemma Suc_diff_number_of:
851      "Int.Pls < v ==>
852       Suc m - (number_of v) = m - (number_of (Int.pred v))"
853 apply (subst Suc_diff_eq_diff_pred)
854 apply simp
855 apply (simp del: nat_numeral_1_eq_1)
856 apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
857                         neg_number_of_pred_iff_0)
858 done
860 lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
864 subsubsection{*For @{term nat_case} and @{term nat_rec}*}
866 lemma nat_case_number_of [simp]:
867      "nat_case a f (number_of v) =
868         (let pv = number_of (Int.pred v) in
869          if neg pv then a else f (nat pv))"
873      "nat_case a f ((number_of v) + n) =
874        (let pv = number_of (Int.pred v) in
875          if neg pv then nat_case a f n else f (nat pv + n))"
877 apply (simp split add: nat.split
878             del: nat_numeral_1_eq_1
880                  numeral_1_eq_Suc_0 [symmetric]
881                  neg_number_of_pred_iff_0)
882 done
884 lemma nat_rec_number_of [simp]:
885      "nat_rec a f (number_of v) =
886         (let pv = number_of (Int.pred v) in
887          if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
888 apply (case_tac " (number_of v) ::nat")
889 apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
890 apply (simp split add: split_if_asm)
891 done
894      "nat_rec a f (number_of v + n) =
895         (let pv = number_of (Int.pred v) in
896          if neg pv then nat_rec a f n
897                    else f (nat pv + n) (nat_rec a f (nat pv + n)))"
899 apply (simp split add: nat.split
900             del: nat_numeral_1_eq_1
902                  numeral_1_eq_Suc_0 [symmetric]
903                  neg_number_of_pred_iff_0)
904 done
907 subsubsection{*Various Other Lemmas*}
909 text {*Evens and Odds, for Mutilated Chess Board*}
911 text{*Lemmas for specialist use, NOT as default simprules*}
912 lemma nat_mult_2: "2 * z = (z+z::nat)"
913 proof -
914   have "2*z = (1 + 1)*z" by simp
915   also have "... = z+z" by (simp add: left_distrib)
916   finally show ?thesis .
917 qed
919 lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
920 by (subst mult_commute, rule nat_mult_2)
922 text{*Case analysis on @{term "n<2"}*}
923 lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
924 by arith
926 lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)"
927 by arith
929 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
930 by (simp add: nat_mult_2 [symmetric])
932 lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
933 apply (subgoal_tac "m mod 2 < 2")
934 apply (erule less_2_cases [THEN disjE])
935 apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
936 done
938 lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)"
939 apply (subgoal_tac "m mod 2 < 2")
940 apply (force simp del: mod_less_divisor, simp)
941 done
943 text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
945 lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
946 by simp
948 lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
949 by simp
951 text{*Can be used to eliminate long strings of Sucs, but not by default*}
952 lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
953 by simp
956 text{*These lemmas collapse some needless occurrences of Suc:
957     at least three Sucs, since two and fewer are rewritten back to Suc again!
958     We already have some rules to simplify operands smaller than 3.*}
960 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"