src/HOL/Num.thy
 author wenzelm Mon May 27 21:00:30 2013 +0200 (2013-05-27) changeset 52187 1f7b3aadec69 parent 52143 36ffe23b25f8 child 52210 0226035df99d permissions -rw-r--r--
tuned;
1 (*  Title:      HOL/Num.thy
2     Author:     Florian Haftmann
3     Author:     Brian Huffman
4 *)
6 header {* Binary Numerals *}
8 theory Num
9 imports Datatype
10 begin
12 subsection {* The @{text num} type *}
14 datatype num = One | Bit0 num | Bit1 num
16 text {* Increment function for type @{typ num} *}
18 primrec inc :: "num \<Rightarrow> num" where
19   "inc One = Bit0 One" |
20   "inc (Bit0 x) = Bit1 x" |
21   "inc (Bit1 x) = Bit0 (inc x)"
23 text {* Converting between type @{typ num} and type @{typ nat} *}
25 primrec nat_of_num :: "num \<Rightarrow> nat" where
26   "nat_of_num One = Suc 0" |
27   "nat_of_num (Bit0 x) = nat_of_num x + nat_of_num x" |
28   "nat_of_num (Bit1 x) = Suc (nat_of_num x + nat_of_num x)"
30 primrec num_of_nat :: "nat \<Rightarrow> num" where
31   "num_of_nat 0 = One" |
32   "num_of_nat (Suc n) = (if 0 < n then inc (num_of_nat n) else One)"
34 lemma nat_of_num_pos: "0 < nat_of_num x"
35   by (induct x) simp_all
37 lemma nat_of_num_neq_0: " nat_of_num x \<noteq> 0"
38   by (induct x) simp_all
40 lemma nat_of_num_inc: "nat_of_num (inc x) = Suc (nat_of_num x)"
41   by (induct x) simp_all
43 lemma num_of_nat_double:
44   "0 < n \<Longrightarrow> num_of_nat (n + n) = Bit0 (num_of_nat n)"
45   by (induct n) simp_all
47 text {*
48   Type @{typ num} is isomorphic to the strictly positive
49   natural numbers.
50 *}
52 lemma nat_of_num_inverse: "num_of_nat (nat_of_num x) = x"
53   by (induct x) (simp_all add: num_of_nat_double nat_of_num_pos)
55 lemma num_of_nat_inverse: "0 < n \<Longrightarrow> nat_of_num (num_of_nat n) = n"
56   by (induct n) (simp_all add: nat_of_num_inc)
58 lemma num_eq_iff: "x = y \<longleftrightarrow> nat_of_num x = nat_of_num y"
59   apply safe
60   apply (drule arg_cong [where f=num_of_nat])
61   apply (simp add: nat_of_num_inverse)
62   done
64 lemma num_induct [case_names One inc]:
65   fixes P :: "num \<Rightarrow> bool"
66   assumes One: "P One"
67     and inc: "\<And>x. P x \<Longrightarrow> P (inc x)"
68   shows "P x"
69 proof -
70   obtain n where n: "Suc n = nat_of_num x"
71     by (cases "nat_of_num x", simp_all add: nat_of_num_neq_0)
72   have "P (num_of_nat (Suc n))"
73   proof (induct n)
74     case 0 show ?case using One by simp
75   next
76     case (Suc n)
77     then have "P (inc (num_of_nat (Suc n)))" by (rule inc)
78     then show "P (num_of_nat (Suc (Suc n)))" by simp
79   qed
80   with n show "P x"
81     by (simp add: nat_of_num_inverse)
82 qed
84 text {*
85   From now on, there are two possible models for @{typ num}:
86   as positive naturals (rule @{text "num_induct"})
87   and as digit representation (rules @{text "num.induct"}, @{text "num.cases"}).
88 *}
91 subsection {* Numeral operations *}
93 instantiation num :: "{plus,times,linorder}"
94 begin
96 definition [code del]:
97   "m + n = num_of_nat (nat_of_num m + nat_of_num n)"
99 definition [code del]:
100   "m * n = num_of_nat (nat_of_num m * nat_of_num n)"
102 definition [code del]:
103   "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"
105 definition [code del]:
106   "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"
108 instance
109   by (default, auto simp add: less_num_def less_eq_num_def num_eq_iff)
111 end
113 lemma nat_of_num_add: "nat_of_num (x + y) = nat_of_num x + nat_of_num y"
114   unfolding plus_num_def
115   by (intro num_of_nat_inverse add_pos_pos nat_of_num_pos)
117 lemma nat_of_num_mult: "nat_of_num (x * y) = nat_of_num x * nat_of_num y"
118   unfolding times_num_def
119   by (intro num_of_nat_inverse mult_pos_pos nat_of_num_pos)
121 lemma add_num_simps [simp, code]:
122   "One + One = Bit0 One"
123   "One + Bit0 n = Bit1 n"
124   "One + Bit1 n = Bit0 (n + One)"
125   "Bit0 m + One = Bit1 m"
126   "Bit0 m + Bit0 n = Bit0 (m + n)"
127   "Bit0 m + Bit1 n = Bit1 (m + n)"
128   "Bit1 m + One = Bit0 (m + One)"
129   "Bit1 m + Bit0 n = Bit1 (m + n)"
130   "Bit1 m + Bit1 n = Bit0 (m + n + One)"
133 lemma mult_num_simps [simp, code]:
134   "m * One = m"
135   "One * n = n"
136   "Bit0 m * Bit0 n = Bit0 (Bit0 (m * n))"
137   "Bit0 m * Bit1 n = Bit0 (m * Bit1 n)"
138   "Bit1 m * Bit0 n = Bit0 (Bit1 m * n)"
139   "Bit1 m * Bit1 n = Bit1 (m + n + Bit0 (m * n))"
141     nat_of_num_mult distrib_right distrib_left)
143 lemma eq_num_simps:
144   "One = One \<longleftrightarrow> True"
145   "One = Bit0 n \<longleftrightarrow> False"
146   "One = Bit1 n \<longleftrightarrow> False"
147   "Bit0 m = One \<longleftrightarrow> False"
148   "Bit1 m = One \<longleftrightarrow> False"
149   "Bit0 m = Bit0 n \<longleftrightarrow> m = n"
150   "Bit0 m = Bit1 n \<longleftrightarrow> False"
151   "Bit1 m = Bit0 n \<longleftrightarrow> False"
152   "Bit1 m = Bit1 n \<longleftrightarrow> m = n"
153   by simp_all
155 lemma le_num_simps [simp, code]:
156   "One \<le> n \<longleftrightarrow> True"
157   "Bit0 m \<le> One \<longleftrightarrow> False"
158   "Bit1 m \<le> One \<longleftrightarrow> False"
159   "Bit0 m \<le> Bit0 n \<longleftrightarrow> m \<le> n"
160   "Bit0 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"
161   "Bit1 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"
162   "Bit1 m \<le> Bit0 n \<longleftrightarrow> m < n"
163   using nat_of_num_pos [of n] nat_of_num_pos [of m]
164   by (auto simp add: less_eq_num_def less_num_def)
166 lemma less_num_simps [simp, code]:
167   "m < One \<longleftrightarrow> False"
168   "One < Bit0 n \<longleftrightarrow> True"
169   "One < Bit1 n \<longleftrightarrow> True"
170   "Bit0 m < Bit0 n \<longleftrightarrow> m < n"
171   "Bit0 m < Bit1 n \<longleftrightarrow> m \<le> n"
172   "Bit1 m < Bit1 n \<longleftrightarrow> m < n"
173   "Bit1 m < Bit0 n \<longleftrightarrow> m < n"
174   using nat_of_num_pos [of n] nat_of_num_pos [of m]
175   by (auto simp add: less_eq_num_def less_num_def)
177 text {* Rules using @{text One} and @{text inc} as constructors *}
179 lemma add_One: "x + One = inc x"
180   by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
182 lemma add_One_commute: "One + n = n + One"
183   by (induct n) simp_all
185 lemma add_inc: "x + inc y = inc (x + y)"
186   by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
188 lemma mult_inc: "x * inc y = x * y + x"
189   by (simp add: num_eq_iff nat_of_num_mult nat_of_num_add nat_of_num_inc)
191 text {* The @{const num_of_nat} conversion *}
193 lemma num_of_nat_One:
194   "n \<le> 1 \<Longrightarrow> num_of_nat n = One"
195   by (cases n) simp_all
197 lemma num_of_nat_plus_distrib:
198   "0 < m \<Longrightarrow> 0 < n \<Longrightarrow> num_of_nat (m + n) = num_of_nat m + num_of_nat n"
201 text {* A double-and-decrement function *}
203 primrec BitM :: "num \<Rightarrow> num" where
204   "BitM One = One" |
205   "BitM (Bit0 n) = Bit1 (BitM n)" |
206   "BitM (Bit1 n) = Bit1 (Bit0 n)"
208 lemma BitM_plus_one: "BitM n + One = Bit0 n"
209   by (induct n) simp_all
211 lemma one_plus_BitM: "One + BitM n = Bit0 n"
212   unfolding add_One_commute BitM_plus_one ..
214 text {* Squaring and exponentiation *}
216 primrec sqr :: "num \<Rightarrow> num" where
217   "sqr One = One" |
218   "sqr (Bit0 n) = Bit0 (Bit0 (sqr n))" |
219   "sqr (Bit1 n) = Bit1 (Bit0 (sqr n + n))"
221 primrec pow :: "num \<Rightarrow> num \<Rightarrow> num" where
222   "pow x One = x" |
223   "pow x (Bit0 y) = sqr (pow x y)" |
224   "pow x (Bit1 y) = sqr (pow x y) * x"
226 lemma nat_of_num_sqr: "nat_of_num (sqr x) = nat_of_num x * nat_of_num x"
227   by (induct x, simp_all add: algebra_simps nat_of_num_add)
229 lemma sqr_conv_mult: "sqr x = x * x"
230   by (simp add: num_eq_iff nat_of_num_sqr nat_of_num_mult)
233 subsection {* Binary numerals *}
235 text {*
236   We embed binary representations into a generic algebraic
237   structure using @{text numeral}.
238 *}
240 class numeral = one + semigroup_add
241 begin
243 primrec numeral :: "num \<Rightarrow> 'a" where
244   numeral_One: "numeral One = 1" |
245   numeral_Bit0: "numeral (Bit0 n) = numeral n + numeral n" |
246   numeral_Bit1: "numeral (Bit1 n) = numeral n + numeral n + 1"
248 lemma numeral_code [code]:
249   "numeral One = 1"
250   "numeral (Bit0 n) = (let m = numeral n in m + m)"
251   "numeral (Bit1 n) = (let m = numeral n in m + m + 1)"
252   by (simp_all add: Let_def)
254 lemma one_plus_numeral_commute: "1 + numeral x = numeral x + 1"
255   apply (induct x)
256   apply simp
259   done
261 lemma numeral_inc: "numeral (inc x) = numeral x + 1"
262 proof (induct x)
263   case (Bit1 x)
264   have "numeral x + (1 + numeral x) + 1 = numeral x + (numeral x + 1) + 1"
265     by (simp only: one_plus_numeral_commute)
266   with Bit1 show ?case
268 qed simp_all
270 declare numeral.simps [simp del]
272 abbreviation "Numeral1 \<equiv> numeral One"
274 declare numeral_One [code_post]
276 end
278 text {* Negative numerals. *}
280 class neg_numeral = numeral + group_add
281 begin
283 definition neg_numeral :: "num \<Rightarrow> 'a" where
284   "neg_numeral k = - numeral k"
286 end
288 text {* Numeral syntax. *}
290 syntax
291   "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
293 parse_translation {*
294   let
295     fun num_of_int n =
296       if n > 0 then
297         (case IntInf.quotRem (n, 2) of
298           (0, 1) => Syntax.const @{const_name One}
299         | (n, 0) => Syntax.const @{const_name Bit0} \$ num_of_int n
300         | (n, 1) => Syntax.const @{const_name Bit1} \$ num_of_int n)
301       else raise Match
302     val pos = Syntax.const @{const_name numeral}
303     val neg = Syntax.const @{const_name neg_numeral}
304     val one = Syntax.const @{const_name Groups.one}
305     val zero = Syntax.const @{const_name Groups.zero}
306     fun numeral_tr [(c as Const (@{syntax_const "_constrain"}, _)) \$ t \$ u] =
307           c \$ numeral_tr [t] \$ u
308       | numeral_tr [Const (num, _)] =
309           let
310             val {value, ...} = Lexicon.read_xnum num;
311           in
312             if value = 0 then zero else
313             if value > 0
314             then pos \$ num_of_int value
315             else neg \$ num_of_int (~value)
316           end
317       | numeral_tr ts = raise TERM ("numeral_tr", ts);
318   in [("_Numeral", K numeral_tr)] end
319 *}
321 typed_print_translation {*
322   let
323     fun dest_num (Const (@{const_syntax Bit0}, _) \$ n) = 2 * dest_num n
324       | dest_num (Const (@{const_syntax Bit1}, _) \$ n) = 2 * dest_num n + 1
325       | dest_num (Const (@{const_syntax One}, _)) = 1;
326     fun num_tr' sign ctxt T [n] =
327       let
328         val k = dest_num n;
329         val t' =
330           Syntax.const @{syntax_const "_Numeral"} \$
331             Syntax.free (sign ^ string_of_int k);
332       in
333         (case T of
334           Type (@{type_name fun}, [_, T']) =>
335             if not (Printer.show_type_constraint ctxt) andalso can Term.dest_Type T' then t'
336             else Syntax.const @{syntax_const "_constrain"} \$ t' \$ Syntax_Phases.term_of_typ ctxt T'
337         | _ => if T = dummyT then t' else raise Match)
338       end;
339   in
340    [(@{const_syntax numeral}, num_tr' ""),
341     (@{const_syntax neg_numeral}, num_tr' "-")]
342   end
343 *}
345 ML_file "Tools/numeral.ML"
348 subsection {* Class-specific numeral rules *}
350 text {*
351   @{const numeral} is a morphism.
352 *}
354 subsubsection {* Structures with addition: class @{text numeral} *}
356 context numeral
357 begin
359 lemma numeral_add: "numeral (m + n) = numeral m + numeral n"
360   by (induct n rule: num_induct)
363 lemma numeral_plus_numeral: "numeral m + numeral n = numeral (m + n)"
364   by (rule numeral_add [symmetric])
366 lemma numeral_plus_one: "numeral n + 1 = numeral (n + One)"
367   using numeral_add [of n One] by (simp add: numeral_One)
369 lemma one_plus_numeral: "1 + numeral n = numeral (One + n)"
370   using numeral_add [of One n] by (simp add: numeral_One)
372 lemma one_add_one: "1 + 1 = 2"
373   using numeral_add [of One One] by (simp add: numeral_One)
375 lemmas add_numeral_special =
376   numeral_plus_one one_plus_numeral one_add_one
378 end
380 subsubsection {*
381   Structures with negation: class @{text neg_numeral}
382 *}
384 context neg_numeral
385 begin
387 text {* Numerals form an abelian subgroup. *}
389 inductive is_num :: "'a \<Rightarrow> bool" where
390   "is_num 1" |
391   "is_num x \<Longrightarrow> is_num (- x)" |
392   "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> is_num (x + y)"
394 lemma is_num_numeral: "is_num (numeral k)"
395   by (induct k, simp_all add: numeral.simps is_num.intros)
398   "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + y = y + x"
399   apply (induct x rule: is_num.induct)
400   apply (induct y rule: is_num.induct)
401   apply simp
402   apply (rule_tac a=x in add_left_imp_eq)
403   apply (rule_tac a=x in add_right_imp_eq)
406   apply (rule_tac a=x in add_left_imp_eq)
407   apply (rule_tac a=x in add_right_imp_eq)
410   done
413   "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + (y + z) = y + (x + z)"
414   by (simp only: add_assoc [symmetric] is_num_add_commute)
416 lemmas is_num_normalize =
418   is_num.intros is_num_numeral
421 definition dbl :: "'a \<Rightarrow> 'a" where "dbl x = x + x"
422 definition dbl_inc :: "'a \<Rightarrow> 'a" where "dbl_inc x = x + x + 1"
423 definition dbl_dec :: "'a \<Rightarrow> 'a" where "dbl_dec x = x + x - 1"
425 definition sub :: "num \<Rightarrow> num \<Rightarrow> 'a" where
426   "sub k l = numeral k - numeral l"
428 lemma numeral_BitM: "numeral (BitM n) = numeral (Bit0 n) - 1"
429   by (simp only: BitM_plus_one [symmetric] numeral_add numeral_One eq_diff_eq)
431 lemma dbl_simps [simp]:
432   "dbl (neg_numeral k) = neg_numeral (Bit0 k)"
433   "dbl 0 = 0"
434   "dbl 1 = 2"
435   "dbl (numeral k) = numeral (Bit0 k)"
436   unfolding dbl_def neg_numeral_def numeral.simps
439 lemma dbl_inc_simps [simp]:
440   "dbl_inc (neg_numeral k) = neg_numeral (BitM k)"
441   "dbl_inc 0 = 1"
442   "dbl_inc 1 = 3"
443   "dbl_inc (numeral k) = numeral (Bit1 k)"
444   unfolding dbl_inc_def neg_numeral_def numeral.simps numeral_BitM
445   by (simp_all add: is_num_normalize)
447 lemma dbl_dec_simps [simp]:
448   "dbl_dec (neg_numeral k) = neg_numeral (Bit1 k)"
449   "dbl_dec 0 = -1"
450   "dbl_dec 1 = 1"
451   "dbl_dec (numeral k) = numeral (BitM k)"
452   unfolding dbl_dec_def neg_numeral_def numeral.simps numeral_BitM
453   by (simp_all add: is_num_normalize)
455 lemma sub_num_simps [simp]:
456   "sub One One = 0"
457   "sub One (Bit0 l) = neg_numeral (BitM l)"
458   "sub One (Bit1 l) = neg_numeral (Bit0 l)"
459   "sub (Bit0 k) One = numeral (BitM k)"
460   "sub (Bit1 k) One = numeral (Bit0 k)"
461   "sub (Bit0 k) (Bit0 l) = dbl (sub k l)"
462   "sub (Bit0 k) (Bit1 l) = dbl_dec (sub k l)"
463   "sub (Bit1 k) (Bit0 l) = dbl_inc (sub k l)"
464   "sub (Bit1 k) (Bit1 l) = dbl (sub k l)"
465   unfolding dbl_def dbl_dec_def dbl_inc_def sub_def
466   unfolding neg_numeral_def numeral.simps numeral_BitM
467   by (simp_all add: is_num_normalize)
470   "numeral m + neg_numeral n = sub m n"
471   "neg_numeral m + numeral n = sub n m"
472   "neg_numeral m + neg_numeral n = neg_numeral (m + n)"
473   unfolding sub_def diff_minus neg_numeral_def numeral_add numeral.simps
474   by (simp_all add: is_num_normalize)
477   "1 + neg_numeral m = sub One m"
478   "neg_numeral m + 1 = sub One m"
479   unfolding sub_def diff_minus neg_numeral_def numeral_add numeral.simps
480   by (simp_all add: is_num_normalize)
482 lemma diff_numeral_simps:
483   "numeral m - numeral n = sub m n"
484   "numeral m - neg_numeral n = numeral (m + n)"
485   "neg_numeral m - numeral n = neg_numeral (m + n)"
486   "neg_numeral m - neg_numeral n = sub n m"
487   unfolding neg_numeral_def sub_def diff_minus numeral_add numeral.simps
488   by (simp_all add: is_num_normalize)
490 lemma diff_numeral_special:
491   "1 - numeral n = sub One n"
492   "1 - neg_numeral n = numeral (One + n)"
493   "numeral m - 1 = sub m One"
494   "neg_numeral m - 1 = neg_numeral (m + One)"
495   unfolding neg_numeral_def sub_def diff_minus numeral_add numeral.simps
496   by (simp_all add: is_num_normalize)
498 lemma minus_one: "- 1 = -1"
499   unfolding neg_numeral_def numeral.simps ..
501 lemma minus_numeral: "- numeral n = neg_numeral n"
502   unfolding neg_numeral_def ..
504 lemma minus_neg_numeral: "- neg_numeral n = numeral n"
505   unfolding neg_numeral_def by simp
507 lemmas minus_numeral_simps [simp] =
508   minus_one minus_numeral minus_neg_numeral
510 end
512 subsubsection {*
513   Structures with multiplication: class @{text semiring_numeral}
514 *}
516 class semiring_numeral = semiring + monoid_mult
517 begin
519 subclass numeral ..
521 lemma numeral_mult: "numeral (m * n) = numeral m * numeral n"
522   apply (induct n rule: num_induct)
523   apply (simp add: numeral_One)
524   apply (simp add: mult_inc numeral_inc numeral_add distrib_left)
525   done
527 lemma numeral_times_numeral: "numeral m * numeral n = numeral (m * n)"
528   by (rule numeral_mult [symmetric])
530 end
532 subsubsection {*
533   Structures with a zero: class @{text semiring_1}
534 *}
536 context semiring_1
537 begin
539 subclass semiring_numeral ..
541 lemma of_nat_numeral [simp]: "of_nat (numeral n) = numeral n"
542   by (induct n,
543     simp_all only: numeral.simps numeral_class.numeral.simps of_nat_add of_nat_1)
545 lemma mult_2: "2 * z = z + z"
546   unfolding one_add_one [symmetric] distrib_right by simp
548 lemma mult_2_right: "z * 2 = z + z"
549   unfolding one_add_one [symmetric] distrib_left by simp
551 end
553 lemma nat_of_num_numeral [code_abbrev]:
554   "nat_of_num = numeral"
555 proof
556   fix n
557   have "numeral n = nat_of_num n"
558     by (induct n) (simp_all add: numeral.simps)
559   then show "nat_of_num n = numeral n" by simp
560 qed
562 lemma nat_of_num_code [code]:
563   "nat_of_num One = 1"
564   "nat_of_num (Bit0 n) = (let m = nat_of_num n in m + m)"
565   "nat_of_num (Bit1 n) = (let m = nat_of_num n in Suc (m + m))"
566   by (simp_all add: Let_def)
568 subsubsection {*
569   Equality: class @{text semiring_char_0}
570 *}
572 context semiring_char_0
573 begin
575 lemma numeral_eq_iff: "numeral m = numeral n \<longleftrightarrow> m = n"
576   unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
577     of_nat_eq_iff num_eq_iff ..
579 lemma numeral_eq_one_iff: "numeral n = 1 \<longleftrightarrow> n = One"
580   by (rule numeral_eq_iff [of n One, unfolded numeral_One])
582 lemma one_eq_numeral_iff: "1 = numeral n \<longleftrightarrow> One = n"
583   by (rule numeral_eq_iff [of One n, unfolded numeral_One])
585 lemma numeral_neq_zero: "numeral n \<noteq> 0"
586   unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
587   by (simp add: nat_of_num_pos)
589 lemma zero_neq_numeral: "0 \<noteq> numeral n"
590   unfolding eq_commute [of 0] by (rule numeral_neq_zero)
592 lemmas eq_numeral_simps [simp] =
593   numeral_eq_iff
594   numeral_eq_one_iff
595   one_eq_numeral_iff
596   numeral_neq_zero
597   zero_neq_numeral
599 end
601 subsubsection {*
602   Comparisons: class @{text linordered_semidom}
603 *}
605 text {*  Could be perhaps more general than here. *}
607 context linordered_semidom
608 begin
610 lemma numeral_le_iff: "numeral m \<le> numeral n \<longleftrightarrow> m \<le> n"
611 proof -
612   have "of_nat (numeral m) \<le> of_nat (numeral n) \<longleftrightarrow> m \<le> n"
613     unfolding less_eq_num_def nat_of_num_numeral of_nat_le_iff ..
614   then show ?thesis by simp
615 qed
617 lemma one_le_numeral: "1 \<le> numeral n"
618 using numeral_le_iff [of One n] by (simp add: numeral_One)
620 lemma numeral_le_one_iff: "numeral n \<le> 1 \<longleftrightarrow> n \<le> One"
621 using numeral_le_iff [of n One] by (simp add: numeral_One)
623 lemma numeral_less_iff: "numeral m < numeral n \<longleftrightarrow> m < n"
624 proof -
625   have "of_nat (numeral m) < of_nat (numeral n) \<longleftrightarrow> m < n"
626     unfolding less_num_def nat_of_num_numeral of_nat_less_iff ..
627   then show ?thesis by simp
628 qed
630 lemma not_numeral_less_one: "\<not> numeral n < 1"
631   using numeral_less_iff [of n One] by (simp add: numeral_One)
633 lemma one_less_numeral_iff: "1 < numeral n \<longleftrightarrow> One < n"
634   using numeral_less_iff [of One n] by (simp add: numeral_One)
636 lemma zero_le_numeral: "0 \<le> numeral n"
637   by (induct n) (simp_all add: numeral.simps)
639 lemma zero_less_numeral: "0 < numeral n"
640   by (induct n) (simp_all add: numeral.simps add_pos_pos)
642 lemma not_numeral_le_zero: "\<not> numeral n \<le> 0"
643   by (simp add: not_le zero_less_numeral)
645 lemma not_numeral_less_zero: "\<not> numeral n < 0"
646   by (simp add: not_less zero_le_numeral)
648 lemmas le_numeral_extra =
649   zero_le_one not_one_le_zero
650   order_refl [of 0] order_refl [of 1]
652 lemmas less_numeral_extra =
653   zero_less_one not_one_less_zero
654   less_irrefl [of 0] less_irrefl [of 1]
656 lemmas le_numeral_simps [simp] =
657   numeral_le_iff
658   one_le_numeral
659   numeral_le_one_iff
660   zero_le_numeral
661   not_numeral_le_zero
663 lemmas less_numeral_simps [simp] =
664   numeral_less_iff
665   one_less_numeral_iff
666   not_numeral_less_one
667   zero_less_numeral
668   not_numeral_less_zero
670 end
672 subsubsection {*
673   Multiplication and negation: class @{text ring_1}
674 *}
676 context ring_1
677 begin
679 subclass neg_numeral ..
681 lemma mult_neg_numeral_simps:
682   "neg_numeral m * neg_numeral n = numeral (m * n)"
683   "neg_numeral m * numeral n = neg_numeral (m * n)"
684   "numeral m * neg_numeral n = neg_numeral (m * n)"
685   unfolding neg_numeral_def mult_minus_left mult_minus_right
686   by (simp_all only: minus_minus numeral_mult)
688 lemma mult_minus1 [simp]: "-1 * z = - z"
689   unfolding neg_numeral_def numeral.simps mult_minus_left by simp
691 lemma mult_minus1_right [simp]: "z * -1 = - z"
692   unfolding neg_numeral_def numeral.simps mult_minus_right by simp
694 end
696 subsubsection {*
697   Equality using @{text iszero} for rings with non-zero characteristic
698 *}
700 context ring_1
701 begin
703 definition iszero :: "'a \<Rightarrow> bool"
704   where "iszero z \<longleftrightarrow> z = 0"
706 lemma iszero_0 [simp]: "iszero 0"
707   by (simp add: iszero_def)
709 lemma not_iszero_1 [simp]: "\<not> iszero 1"
710   by (simp add: iszero_def)
712 lemma not_iszero_Numeral1: "\<not> iszero Numeral1"
713   by (simp add: numeral_One)
715 lemma iszero_neg_numeral [simp]:
716   "iszero (neg_numeral w) \<longleftrightarrow> iszero (numeral w)"
717   unfolding iszero_def neg_numeral_def
718   by (rule neg_equal_0_iff_equal)
720 lemma eq_iff_iszero_diff: "x = y \<longleftrightarrow> iszero (x - y)"
721   unfolding iszero_def by (rule eq_iff_diff_eq_0)
723 text {* The @{text "eq_numeral_iff_iszero"} lemmas are not declared
724 @{text "[simp]"} by default, because for rings of characteristic zero,
725 better simp rules are possible. For a type like integers mod @{text
726 "n"}, type-instantiated versions of these rules should be added to the
727 simplifier, along with a type-specific rule for deciding propositions
728 of the form @{text "iszero (numeral w)"}.
730 bh: Maybe it would not be so bad to just declare these as simp
731 rules anyway? I should test whether these rules take precedence over
732 the @{text "ring_char_0"} rules in the simplifier.
733 *}
735 lemma eq_numeral_iff_iszero:
736   "numeral x = numeral y \<longleftrightarrow> iszero (sub x y)"
737   "numeral x = neg_numeral y \<longleftrightarrow> iszero (numeral (x + y))"
738   "neg_numeral x = numeral y \<longleftrightarrow> iszero (numeral (x + y))"
739   "neg_numeral x = neg_numeral y \<longleftrightarrow> iszero (sub y x)"
740   "numeral x = 1 \<longleftrightarrow> iszero (sub x One)"
741   "1 = numeral y \<longleftrightarrow> iszero (sub One y)"
742   "neg_numeral x = 1 \<longleftrightarrow> iszero (numeral (x + One))"
743   "1 = neg_numeral y \<longleftrightarrow> iszero (numeral (One + y))"
744   "numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
745   "0 = numeral y \<longleftrightarrow> iszero (numeral y)"
746   "neg_numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
747   "0 = neg_numeral y \<longleftrightarrow> iszero (numeral y)"
748   unfolding eq_iff_iszero_diff diff_numeral_simps diff_numeral_special
749   by simp_all
751 end
753 subsubsection {*
754   Equality and negation: class @{text ring_char_0}
755 *}
757 class ring_char_0 = ring_1 + semiring_char_0
758 begin
760 lemma not_iszero_numeral [simp]: "\<not> iszero (numeral w)"
761   by (simp add: iszero_def)
763 lemma neg_numeral_eq_iff: "neg_numeral m = neg_numeral n \<longleftrightarrow> m = n"
764   by (simp only: neg_numeral_def neg_equal_iff_equal numeral_eq_iff)
766 lemma numeral_neq_neg_numeral: "numeral m \<noteq> neg_numeral n"
767   unfolding neg_numeral_def eq_neg_iff_add_eq_0
768   by (simp add: numeral_plus_numeral)
770 lemma neg_numeral_neq_numeral: "neg_numeral m \<noteq> numeral n"
771   by (rule numeral_neq_neg_numeral [symmetric])
773 lemma zero_neq_neg_numeral: "0 \<noteq> neg_numeral n"
774   unfolding neg_numeral_def neg_0_equal_iff_equal by simp
776 lemma neg_numeral_neq_zero: "neg_numeral n \<noteq> 0"
777   unfolding neg_numeral_def neg_equal_0_iff_equal by simp
779 lemma one_neq_neg_numeral: "1 \<noteq> neg_numeral n"
780   using numeral_neq_neg_numeral [of One n] by (simp add: numeral_One)
782 lemma neg_numeral_neq_one: "neg_numeral n \<noteq> 1"
783   using neg_numeral_neq_numeral [of n One] by (simp add: numeral_One)
785 lemmas eq_neg_numeral_simps [simp] =
786   neg_numeral_eq_iff
787   numeral_neq_neg_numeral neg_numeral_neq_numeral
788   one_neq_neg_numeral neg_numeral_neq_one
789   zero_neq_neg_numeral neg_numeral_neq_zero
791 end
793 subsubsection {*
794   Structures with negation and order: class @{text linordered_idom}
795 *}
797 context linordered_idom
798 begin
800 subclass ring_char_0 ..
802 lemma neg_numeral_le_iff: "neg_numeral m \<le> neg_numeral n \<longleftrightarrow> n \<le> m"
803   by (simp only: neg_numeral_def neg_le_iff_le numeral_le_iff)
805 lemma neg_numeral_less_iff: "neg_numeral m < neg_numeral n \<longleftrightarrow> n < m"
806   by (simp only: neg_numeral_def neg_less_iff_less numeral_less_iff)
808 lemma neg_numeral_less_zero: "neg_numeral n < 0"
809   by (simp only: neg_numeral_def neg_less_0_iff_less zero_less_numeral)
811 lemma neg_numeral_le_zero: "neg_numeral n \<le> 0"
812   by (simp only: neg_numeral_def neg_le_0_iff_le zero_le_numeral)
814 lemma not_zero_less_neg_numeral: "\<not> 0 < neg_numeral n"
815   by (simp only: not_less neg_numeral_le_zero)
817 lemma not_zero_le_neg_numeral: "\<not> 0 \<le> neg_numeral n"
818   by (simp only: not_le neg_numeral_less_zero)
820 lemma neg_numeral_less_numeral: "neg_numeral m < numeral n"
821   using neg_numeral_less_zero zero_less_numeral by (rule less_trans)
823 lemma neg_numeral_le_numeral: "neg_numeral m \<le> numeral n"
824   by (simp only: less_imp_le neg_numeral_less_numeral)
826 lemma not_numeral_less_neg_numeral: "\<not> numeral m < neg_numeral n"
827   by (simp only: not_less neg_numeral_le_numeral)
829 lemma not_numeral_le_neg_numeral: "\<not> numeral m \<le> neg_numeral n"
830   by (simp only: not_le neg_numeral_less_numeral)
832 lemma neg_numeral_less_one: "neg_numeral m < 1"
833   by (rule neg_numeral_less_numeral [of m One, unfolded numeral_One])
835 lemma neg_numeral_le_one: "neg_numeral m \<le> 1"
836   by (rule neg_numeral_le_numeral [of m One, unfolded numeral_One])
838 lemma not_one_less_neg_numeral: "\<not> 1 < neg_numeral m"
839   by (simp only: not_less neg_numeral_le_one)
841 lemma not_one_le_neg_numeral: "\<not> 1 \<le> neg_numeral m"
842   by (simp only: not_le neg_numeral_less_one)
844 lemma sub_non_negative:
845   "sub n m \<ge> 0 \<longleftrightarrow> n \<ge> m"
846   by (simp only: sub_def le_diff_eq) simp
848 lemma sub_positive:
849   "sub n m > 0 \<longleftrightarrow> n > m"
850   by (simp only: sub_def less_diff_eq) simp
852 lemma sub_non_positive:
853   "sub n m \<le> 0 \<longleftrightarrow> n \<le> m"
854   by (simp only: sub_def diff_le_eq) simp
856 lemma sub_negative:
857   "sub n m < 0 \<longleftrightarrow> n < m"
858   by (simp only: sub_def diff_less_eq) simp
860 lemmas le_neg_numeral_simps [simp] =
861   neg_numeral_le_iff
862   neg_numeral_le_numeral not_numeral_le_neg_numeral
863   neg_numeral_le_zero not_zero_le_neg_numeral
864   neg_numeral_le_one not_one_le_neg_numeral
866 lemmas less_neg_numeral_simps [simp] =
867   neg_numeral_less_iff
868   neg_numeral_less_numeral not_numeral_less_neg_numeral
869   neg_numeral_less_zero not_zero_less_neg_numeral
870   neg_numeral_less_one not_one_less_neg_numeral
872 lemma abs_numeral [simp]: "abs (numeral n) = numeral n"
873   by simp
875 lemma abs_neg_numeral [simp]: "abs (neg_numeral n) = numeral n"
876   by (simp only: neg_numeral_def abs_minus_cancel abs_numeral)
878 end
880 subsubsection {*
881   Natural numbers
882 *}
884 lemma Suc_1 [simp]: "Suc 1 = 2"
885   unfolding Suc_eq_plus1 by (rule one_add_one)
887 lemma Suc_numeral [simp]: "Suc (numeral n) = numeral (n + One)"
888   unfolding Suc_eq_plus1 by (rule numeral_plus_one)
890 definition pred_numeral :: "num \<Rightarrow> nat"
891   where [code del]: "pred_numeral k = numeral k - 1"
893 lemma numeral_eq_Suc: "numeral k = Suc (pred_numeral k)"
894   unfolding pred_numeral_def by simp
896 lemma eval_nat_numeral:
897   "numeral One = Suc 0"
898   "numeral (Bit0 n) = Suc (numeral (BitM n))"
899   "numeral (Bit1 n) = Suc (numeral (Bit0 n))"
900   by (simp_all add: numeral.simps BitM_plus_one)
902 lemma pred_numeral_simps [simp]:
903   "pred_numeral One = 0"
904   "pred_numeral (Bit0 k) = numeral (BitM k)"
905   "pred_numeral (Bit1 k) = numeral (Bit0 k)"
906   unfolding pred_numeral_def eval_nat_numeral
907   by (simp_all only: diff_Suc_Suc diff_0)
909 lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
910   by (simp add: eval_nat_numeral)
912 lemma numeral_3_eq_3: "3 = Suc (Suc (Suc 0))"
913   by (simp add: eval_nat_numeral)
915 lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
916   by (simp only: numeral_One One_nat_def)
919   "Suc (numeral v + n) = numeral (v + One) + n"
920   by simp
922 (*Maps #n to n for n = 1, 2*)
923 lemmas numerals = numeral_One [where 'a=nat] numeral_2_eq_2
925 text {* Comparisons involving @{term Suc}. *}
927 lemma eq_numeral_Suc [simp]: "numeral k = Suc n \<longleftrightarrow> pred_numeral k = n"
928   by (simp add: numeral_eq_Suc)
930 lemma Suc_eq_numeral [simp]: "Suc n = numeral k \<longleftrightarrow> n = pred_numeral k"
931   by (simp add: numeral_eq_Suc)
933 lemma less_numeral_Suc [simp]: "numeral k < Suc n \<longleftrightarrow> pred_numeral k < n"
934   by (simp add: numeral_eq_Suc)
936 lemma less_Suc_numeral [simp]: "Suc n < numeral k \<longleftrightarrow> n < pred_numeral k"
937   by (simp add: numeral_eq_Suc)
939 lemma le_numeral_Suc [simp]: "numeral k \<le> Suc n \<longleftrightarrow> pred_numeral k \<le> n"
940   by (simp add: numeral_eq_Suc)
942 lemma le_Suc_numeral [simp]: "Suc n \<le> numeral k \<longleftrightarrow> n \<le> pred_numeral k"
943   by (simp add: numeral_eq_Suc)
945 lemma diff_Suc_numeral [simp]: "Suc n - numeral k = n - pred_numeral k"
946   by (simp add: numeral_eq_Suc)
948 lemma diff_numeral_Suc [simp]: "numeral k - Suc n = pred_numeral k - n"
949   by (simp add: numeral_eq_Suc)
951 lemma max_Suc_numeral [simp]:
952   "max (Suc n) (numeral k) = Suc (max n (pred_numeral k))"
953   by (simp add: numeral_eq_Suc)
955 lemma max_numeral_Suc [simp]:
956   "max (numeral k) (Suc n) = Suc (max (pred_numeral k) n)"
957   by (simp add: numeral_eq_Suc)
959 lemma min_Suc_numeral [simp]:
960   "min (Suc n) (numeral k) = Suc (min n (pred_numeral k))"
961   by (simp add: numeral_eq_Suc)
963 lemma min_numeral_Suc [simp]:
964   "min (numeral k) (Suc n) = Suc (min (pred_numeral k) n)"
965   by (simp add: numeral_eq_Suc)
967 text {* For @{term nat_case} and @{term nat_rec}. *}
969 lemma nat_case_numeral [simp]:
970   "nat_case a f (numeral v) = (let pv = pred_numeral v in f pv)"
971   by (simp add: numeral_eq_Suc)
973 lemma nat_case_add_eq_if [simp]:
974   "nat_case a f ((numeral v) + n) = (let pv = pred_numeral v in f (pv + n))"
975   by (simp add: numeral_eq_Suc)
977 lemma nat_rec_numeral [simp]:
978   "nat_rec a f (numeral v) =
979     (let pv = pred_numeral v in f pv (nat_rec a f pv))"
980   by (simp add: numeral_eq_Suc Let_def)
982 lemma nat_rec_add_eq_if [simp]:
983   "nat_rec a f (numeral v + n) =
984     (let pv = pred_numeral v in f (pv + n) (nat_rec a f (pv + n)))"
985   by (simp add: numeral_eq_Suc Let_def)
987 text {* Case analysis on @{term "n < 2"} *}
989 lemma less_2_cases: "n < 2 \<Longrightarrow> n = 0 \<or> n = Suc 0"
990   by (auto simp add: numeral_2_eq_2)
992 text {* Removal of Small Numerals: 0, 1 and (in additive positions) 2 *}
993 text {* bh: Are these rules really a good idea? *}
995 lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
996   by simp
998 lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
999   by simp
1001 text {* Can be used to eliminate long strings of Sucs, but not by default. *}
1003 lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
1004   by simp
1006 lemmas nat_1_add_1 = one_add_one [where 'a=nat] (* legacy *)
1009 subsection {* Numeral equations as default simplification rules *}
1011 declare (in numeral) numeral_One [simp]
1012 declare (in numeral) numeral_plus_numeral [simp]
1013 declare (in numeral) add_numeral_special [simp]
1014 declare (in neg_numeral) add_neg_numeral_simps [simp]
1015 declare (in neg_numeral) add_neg_numeral_special [simp]
1016 declare (in neg_numeral) diff_numeral_simps [simp]
1017 declare (in neg_numeral) diff_numeral_special [simp]
1018 declare (in semiring_numeral) numeral_times_numeral [simp]
1019 declare (in ring_1) mult_neg_numeral_simps [simp]
1021 subsection {* Setting up simprocs *}
1023 lemma mult_numeral_1: "Numeral1 * a = (a::'a::semiring_numeral)"
1024   by simp
1026 lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::semiring_numeral)"
1027   by simp
1029 lemma divide_numeral_1: "a / Numeral1 = (a::'a::field)"
1030   by simp
1032 lemma inverse_numeral_1:
1033   "inverse Numeral1 = (Numeral1::'a::division_ring)"
1034   by simp
1036 text{*Theorem lists for the cancellation simprocs. The use of a binary
1037 numeral for 1 reduces the number of special cases.*}
1039 lemmas mult_1s =
1040   mult_numeral_1 mult_numeral_1_right
1041   mult_minus1 mult_minus1_right
1043 setup {*
1045     (fn Const (@{const_name numeral}, _) \$ _ => true
1046     | Const (@{const_name neg_numeral}, _) \$ _ => true
1047     | _ => false)
1048 *}
1050 simproc_setup reorient_numeral
1051   ("numeral w = x" | "neg_numeral w = y") = Reorient_Proc.proc
1054 subsubsection {* Simplification of arithmetic operations on integer constants. *}
1056 lemmas arith_special = (* already declared simp above *)
1058   diff_numeral_special minus_one
1060 (* rules already in simpset *)
1061 lemmas arith_extra_simps =
1063   minus_numeral minus_neg_numeral minus_zero minus_one
1064   diff_numeral_simps diff_0 diff_0_right
1065   numeral_times_numeral mult_neg_numeral_simps
1066   mult_zero_left mult_zero_right
1067   abs_numeral abs_neg_numeral
1069 text {*
1070   For making a minimal simpset, one must include these default simprules.
1071   Also include @{text simp_thms}.
1072 *}
1074 lemmas arith_simps =
1075   add_num_simps mult_num_simps sub_num_simps
1076   BitM.simps dbl_simps dbl_inc_simps dbl_dec_simps
1077   abs_zero abs_one arith_extra_simps
1079 text {* Simplification of relational operations *}
1081 lemmas eq_numeral_extra =
1082   zero_neq_one one_neq_zero
1084 lemmas rel_simps =
1085   le_num_simps less_num_simps eq_num_simps
1086   le_numeral_simps le_neg_numeral_simps le_numeral_extra
1087   less_numeral_simps less_neg_numeral_simps less_numeral_extra
1088   eq_numeral_simps eq_neg_numeral_simps eq_numeral_extra
1091 subsubsection {* Simplification of arithmetic when nested to the right. *}
1093 lemma add_numeral_left [simp]:
1094   "numeral v + (numeral w + z) = (numeral(v + w) + z)"
1097 lemma add_neg_numeral_left [simp]:
1098   "numeral v + (neg_numeral w + y) = (sub v w + y)"
1099   "neg_numeral v + (numeral w + y) = (sub w v + y)"
1100   "neg_numeral v + (neg_numeral w + y) = (neg_numeral(v + w) + y)"
1103 lemma mult_numeral_left [simp]:
1104   "numeral v * (numeral w * z) = (numeral(v * w) * z :: 'a::semiring_numeral)"
1105   "neg_numeral v * (numeral w * y) = (neg_numeral(v * w) * y :: 'b::ring_1)"
1106   "numeral v * (neg_numeral w * y) = (neg_numeral(v * w) * y :: 'b::ring_1)"
1107   "neg_numeral v * (neg_numeral w * y) = (numeral(v * w) * y :: 'b::ring_1)"
1108   by (simp_all add: mult_assoc [symmetric])
1110 hide_const (open) One Bit0 Bit1 BitM inc pow sqr sub dbl dbl_inc dbl_dec
1113 subsection {* code module namespace *}
1115 code_modulename SML
1116   Num Arith
1118 code_modulename OCaml
1119   Num Arith