src/HOL/Decision_Procs/Rat_Pair.thy
 author nipkow Thu Jun 14 15:45:53 2018 +0200 (10 months ago) changeset 68442 477b3f7067c9 parent 67123 3fe40ff1b921 child 69597 ff784d5a5bfb permissions -rw-r--r--
tuned
1 (*  Title:      HOL/Decision_Procs/Rat_Pair.thy
2     Author:     Amine Chaieb
3 *)
5 section \<open>Rational numbers as pairs\<close>
7 theory Rat_Pair
8   imports Complex_Main
9 begin
11 type_synonym Num = "int \<times> int"
13 abbreviation Num0_syn :: Num  ("0\<^sub>N")
14   where "0\<^sub>N \<equiv> (0, 0)"
16 abbreviation Numi_syn :: "int \<Rightarrow> Num"  ("'((_)')\<^sub>N")
17   where "(i)\<^sub>N \<equiv> (i, 1)"
19 definition isnormNum :: "Num \<Rightarrow> bool"
20   where "isnormNum = (\<lambda>(a, b). if a = 0 then b = 0 else b > 0 \<and> gcd a b = 1)"
22 definition normNum :: "Num \<Rightarrow> Num"
23   where "normNum = (\<lambda>(a,b).
24     (if a = 0 \<or> b = 0 then (0, 0)
25      else
26       (let g = gcd a b
27        in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))"
29 declare gcd_dvd1[presburger] gcd_dvd2[presburger]
31 lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"
32 proof -
33   obtain a b where x: "x = (a, b)" by (cases x)
34   consider "a = 0 \<or> b = 0" | "a \<noteq> 0" "b \<noteq> 0"
35     by blast
36   then show ?thesis
37   proof cases
38     case 1
39     then show ?thesis
40       by (simp add: x normNum_def isnormNum_def)
41   next
42     case ab: 2
43     let ?g = "gcd a b"
44     let ?a' = "a div ?g"
45     let ?b' = "b div ?g"
46     let ?g' = "gcd ?a' ?b'"
47     from ab have "?g \<noteq> 0" by simp
48     with gcd_ge_0_int[of a b] have gpos: "?g > 0" by arith
49     have gdvd: "?g dvd a" "?g dvd b" by arith+
50     from dvd_mult_div_cancel[OF gdvd(1)] dvd_mult_div_cancel[OF gdvd(2)] ab
51     have nz': "?a' \<noteq> 0" "?b' \<noteq> 0" by - (rule notI, simp)+
52     from ab have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith
53     from div_gcd_coprime[OF stupid] have gp1: "?g' = 1"
54       by simp
55     from ab consider "b < 0" | "b > 0" by arith
56     then show ?thesis
57     proof cases
58       case b: 1
59       have False if b': "?b' \<ge> 0"
60       proof -
61         from gpos have th: "?g \<ge> 0" by arith
62         from mult_nonneg_nonneg[OF th b'] dvd_mult_div_cancel[OF gdvd(2)]
63         show ?thesis using b by arith
64       qed
65       then have b': "?b' < 0" by (presburger add: linorder_not_le[symmetric])
66       from ab(1) nz' b b' gp1 show ?thesis
67         by (simp add: x isnormNum_def normNum_def Let_def split_def)
68     next
69       case b: 2
70       then have "?b' \<ge> 0"
71         by (presburger add: pos_imp_zdiv_nonneg_iff[OF gpos])
72       with nz' have b': "?b' > 0" by arith
73       from b b' ab(1) nz' gp1 show ?thesis
74         by (simp add: x isnormNum_def normNum_def Let_def split_def)
75     qed
76   qed
77 qed
79 text \<open>Arithmetic over Num\<close>
81 definition Nadd :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "+\<^sub>N" 60)
82 where
83   "Nadd = (\<lambda>(a, b) (a', b').
84     if a = 0 \<or> b = 0 then normNum (a', b')
85     else if a' = 0 \<or> b' = 0 then normNum (a, b)
86     else normNum (a * b' + b * a', b * b'))"
88 definition Nmul :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "*\<^sub>N" 60)
89 where
90   "Nmul = (\<lambda>(a, b) (a', b').
91     let g = gcd (a * a') (b * b')
92     in (a * a' div g, b * b' div g))"
94 definition Nneg :: "Num \<Rightarrow> Num" ("~\<^sub>N")
95   where "Nneg = (\<lambda>(a, b). (- a, b))"
97 definition Nsub :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "-\<^sub>N" 60)
98   where "Nsub = (\<lambda>a b. a +\<^sub>N ~\<^sub>N b)"
100 definition Ninv :: "Num \<Rightarrow> Num"
101   where "Ninv = (\<lambda>(a, b). if a < 0 then (- b, \<bar>a\<bar>) else (b, a))"
103 definition Ndiv :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "\<div>\<^sub>N" 60)
104   where "Ndiv = (\<lambda>a b. a *\<^sub>N Ninv b)"
106 lemma Nneg_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (~\<^sub>N x)"
107   by (simp add: isnormNum_def Nneg_def split_def)
109 lemma Nadd_normN[simp]: "isnormNum (x +\<^sub>N y)"
112 lemma Nsub_normN[simp]: "isnormNum y \<Longrightarrow> isnormNum (x -\<^sub>N y)"
113   by (simp add: Nsub_def split_def)
115 lemma Nmul_normN[simp]:
116   assumes xn: "isnormNum x"
117     and yn: "isnormNum y"
118   shows "isnormNum (x *\<^sub>N y)"
119 proof -
120   obtain a b where x: "x = (a, b)" by (cases x)
121   obtain a' b' where y: "y = (a', b')" by (cases y)
122   consider "a = 0" | "a' = 0" | "a \<noteq> 0" "a' \<noteq> 0" by blast
123   then show ?thesis
124   proof cases
125     case 1
126     then show ?thesis
127       using xn x y by (simp add: isnormNum_def Let_def Nmul_def split_def)
128   next
129     case 2
130     then show ?thesis
131       using yn x y by (simp add: isnormNum_def Let_def Nmul_def split_def)
132   next
133     case aa': 3
134     then have bp: "b > 0" "b' > 0"
135       using xn yn x y by (simp_all add: isnormNum_def)
136     from bp have "x *\<^sub>N y = normNum (a * a', b * b')"
137       using x y aa' bp by (simp add: Nmul_def Let_def split_def normNum_def)
138     then show ?thesis by simp
139   qed
140 qed
142 lemma Ninv_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (Ninv x)"
143   apply (simp add: Ninv_def isnormNum_def split_def)
144   apply (cases "fst x = 0")
145   apply (auto simp add: gcd.commute)
146   done
148 lemma isnormNum_int[simp]: "isnormNum 0\<^sub>N" "isnormNum ((1::int)\<^sub>N)" "i \<noteq> 0 \<Longrightarrow> isnormNum (i)\<^sub>N"
152 text \<open>Relations over Num\<close>
154 definition Nlt0:: "Num \<Rightarrow> bool"  ("0>\<^sub>N")
155   where "Nlt0 = (\<lambda>(a, b). a < 0)"
157 definition Nle0:: "Num \<Rightarrow> bool"  ("0\<ge>\<^sub>N")
158   where "Nle0 = (\<lambda>(a, b). a \<le> 0)"
160 definition Ngt0:: "Num \<Rightarrow> bool"  ("0<\<^sub>N")
161   where "Ngt0 = (\<lambda>(a, b). a > 0)"
163 definition Nge0:: "Num \<Rightarrow> bool"  ("0\<le>\<^sub>N")
164   where "Nge0 = (\<lambda>(a, b). a \<ge> 0)"
166 definition Nlt :: "Num \<Rightarrow> Num \<Rightarrow> bool"  (infix "<\<^sub>N" 55)
167   where "Nlt = (\<lambda>a b. 0>\<^sub>N (a -\<^sub>N b))"
169 definition Nle :: "Num \<Rightarrow> Num \<Rightarrow> bool"  (infix "\<le>\<^sub>N" 55)
170   where "Nle = (\<lambda>a b. 0\<ge>\<^sub>N (a -\<^sub>N b))"
172 definition "INum = (\<lambda>(a, b). of_int a / of_int b)"
174 lemma INum_int [simp]: "INum (i)\<^sub>N = (of_int i ::'a::field)" "INum 0\<^sub>N = (0::'a::field)"
177 lemma isnormNum_unique[simp]:
178   assumes na: "isnormNum x"
179     and nb: "isnormNum y"
180   shows "(INum x ::'a::field_char_0) = INum y \<longleftrightarrow> x = y"
181   (is "?lhs = ?rhs")
182 proof
183   obtain a b where x: "x = (a, b)" by (cases x)
184   obtain a' b' where y: "y = (a', b')" by (cases y)
185   consider "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0" | "a \<noteq> 0" "b \<noteq> 0" "a' \<noteq> 0" "b' \<noteq> 0"
186     by blast
187   then show ?rhs if H: ?lhs
188   proof cases
189     case 1
190     then show ?thesis
191       using na nb H by (simp add: x y INum_def split_def isnormNum_def split: if_split_asm)
192   next
193     case 2
194     with na nb have pos: "b > 0" "b' > 0"
195       by (simp_all add: x y isnormNum_def)
196     from H \<open>b \<noteq> 0\<close> \<open>b' \<noteq> 0\<close> have eq: "a * b' = a' * b"
197       by (simp add: x y INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
198     from \<open>a \<noteq> 0\<close> \<open>a' \<noteq> 0\<close> na nb
199     have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"
201     then have "coprime a b" "coprime b a" "coprime a' b'" "coprime b' a'"
203     from eq have raw_dvd: "a dvd a' * b" "b dvd b' * a" "a' dvd a * b'" "b' dvd b * a'"
204       apply -
205       apply algebra
206       apply algebra
207       apply simp
208       apply algebra
209       done
210     then have eq1: "b = b'"
211       using pos \<open>coprime b a\<close> \<open>coprime b' a'\<close>
212       by (auto simp add: coprime_dvd_mult_left_iff intro: associated_eqI)
213     with eq have "a = a'" using pos by simp
214     with \<open>b = b'\<close> show ?thesis by (simp add: x y)
215   qed
216   show ?lhs if ?rhs
217     using that by simp
218 qed
220 lemma isnormNum0[simp]: "isnormNum x \<Longrightarrow> INum x = (0::'a::field_char_0) \<longleftrightarrow> x = 0\<^sub>N"
221   unfolding INum_int(2)[symmetric]
222   by (rule isnormNum_unique) simp_all
224 lemma of_int_div_aux:
225   assumes "d \<noteq> 0"
226   shows "(of_int x ::'a::field_char_0) / of_int d =
227     of_int (x div d) + (of_int (x mod d)) / of_int d"
228 proof -
229   let ?t = "of_int (x div d) * (of_int d ::'a) + of_int (x mod d)"
230   let ?f = "\<lambda>x. x / of_int d"
231   have "x = (x div d) * d + x mod d"
232     by auto
233   then have eq: "of_int x = ?t"
234     by (simp only: of_int_mult[symmetric] of_int_add [symmetric])
235   then have "of_int x / of_int d = ?t / of_int d"
236     using cong[OF refl[of ?f] eq] by simp
237   then show ?thesis
239 qed
241 lemma of_int_div:
242   fixes d :: int
243   assumes "d \<noteq> 0" "d dvd n"
244   shows "(of_int (n div d) ::'a::field_char_0) = of_int n / of_int d"
245   using assms of_int_div_aux [of d n, where ?'a = 'a] by simp
247 lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::field_char_0)"
248 proof -
249   obtain a b where x: "x = (a, b)" by (cases x)
250   consider "a = 0 \<or> b = 0" | "a \<noteq> 0" "b \<noteq> 0" by blast
251   then show ?thesis
252   proof cases
253     case 1
254     then show ?thesis
255       by (simp add: x INum_def normNum_def split_def Let_def)
256   next
257     case ab: 2
258     let ?g = "gcd a b"
259     from ab have g: "?g \<noteq> 0"by simp
260     from of_int_div[OF g, where ?'a = 'a] show ?thesis
261       by (auto simp add: x INum_def normNum_def split_def Let_def)
262   qed
263 qed
265 lemma INum_normNum_iff: "(INum x ::'a::field_char_0) = INum y \<longleftrightarrow> normNum x = normNum y"
266   (is "?lhs \<longleftrightarrow> _")
267 proof -
268   have "normNum x = normNum y \<longleftrightarrow> (INum (normNum x) :: 'a) = INum (normNum y)"
269     by (simp del: normNum)
270   also have "\<dots> = ?lhs" by simp
271   finally show ?thesis by simp
272 qed
274 lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: field_char_0)"
275 proof -
276   let ?z = "0::'a"
277   obtain a b where x: "x = (a, b)" by (cases x)
278   obtain a' b' where y: "y = (a', b')" by (cases y)
279   consider "a = 0 \<or> a'= 0 \<or> b =0 \<or> b' = 0" | "a \<noteq> 0" "a'\<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
280     by blast
281   then show ?thesis
282   proof cases
283     case 1
284     then show ?thesis
285       apply (cases "a = 0")
287       apply (cases "b = 0")
289       apply (cases "a'= 0")
290       apply simp_all
291       apply (cases "b'= 0")
292       apply simp_all
293       done
294   next
295     case neq: 2
296     show ?thesis
297     proof (cases "a * b' + b * a' = 0")
298       case True
299       then have "of_int (a * b' + b * a') / (of_int b * of_int b') = ?z"
300         by simp
301       then have "of_int b' * of_int a / (of_int b * of_int b') +
302           of_int b * of_int a' / (of_int b * of_int b') = ?z"
304       then have th: "of_int a / of_int b + of_int a' / of_int b' = ?z"
305         using neq by simp
306       from True neq show ?thesis
308     next
309       case False
310       let ?g = "gcd (a * b' + b * a') (b * b')"
311       have gz: "?g \<noteq> 0"
312         using False by simp
313       show ?thesis
314         using neq False gz
315           of_int_div [where ?'a = 'a, OF gz gcd_dvd1 [of "a * b' + b * a'" "b * b'"]]
316           of_int_div [where ?'a = 'a, OF gz gcd_dvd2 [of "a * b' + b * a'" "b * b'"]]
318     qed
319   qed
320 qed
322 lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a::field_char_0)"
323 proof -
324   let ?z = "0::'a"
325   obtain a b where x: "x = (a, b)" by (cases x)
326   obtain a' b' where y: "y = (a', b')" by (cases y)
327   consider "a = 0 \<or> a' = 0 \<or> b = 0 \<or> b' = 0" | "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
328     by blast
329   then show ?thesis
330   proof cases
331     case 1
332     then show ?thesis
333       by (auto simp add: x y Nmul_def INum_def)
334   next
335     case neq: 2
336     let ?g = "gcd (a * a') (b * b')"
337     have gz: "?g \<noteq> 0"
338       using neq by simp
339     from neq of_int_div [where ?'a = 'a, OF gz gcd_dvd1 [of "a * a'" "b * b'"]]
340       of_int_div [where ?'a = 'a , OF gz gcd_dvd2 [of "a * a'" "b * b'"]]
341     show ?thesis
342       by (simp add: Nmul_def x y Let_def INum_def)
343   qed
344 qed
346 lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x :: 'a::field)"
347   by (simp add: Nneg_def split_def INum_def)
349 lemma Nsub[simp]: "INum (x -\<^sub>N y) = INum x - (INum y:: 'a::field_char_0)"
350   by (simp add: Nsub_def split_def)
352 lemma Ninv[simp]: "INum (Ninv x) = (1 :: 'a::field) / INum x"
353   by (simp add: Ninv_def INum_def split_def)
355 lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y :: 'a::field_char_0)"
358 lemma Nlt0_iff[simp]:
359   assumes nx: "isnormNum x"
360   shows "((INum x :: 'a::{field_char_0,linordered_field}) < 0) = 0>\<^sub>N x"
361 proof -
362   obtain a b where x: "x = (a, b)" by (cases x)
363   show ?thesis
364   proof (cases "a = 0")
365     case True
366     then show ?thesis
367       by (simp add: x Nlt0_def INum_def)
368   next
369     case False
370     then have b: "(of_int b::'a) > 0"
371       using nx by (simp add: x isnormNum_def)
372     from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]
373     show ?thesis
374       by (simp add: x Nlt0_def INum_def)
375   qed
376 qed
378 lemma Nle0_iff[simp]:
379   assumes nx: "isnormNum x"
380   shows "((INum x :: 'a::{field_char_0,linordered_field}) \<le> 0) = 0\<ge>\<^sub>N x"
381 proof -
382   obtain a b where x: "x = (a, b)" by (cases x)
383   show ?thesis
384   proof (cases "a = 0")
385     case True
386     then show ?thesis
387       by (simp add: x Nle0_def INum_def)
388   next
389     case False
390     then have b: "(of_int b :: 'a) > 0"
391       using nx by (simp add: x isnormNum_def)
392     from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]
393     show ?thesis
394       by (simp add: x Nle0_def INum_def)
395   qed
396 qed
398 lemma Ngt0_iff[simp]:
399   assumes nx: "isnormNum x"
400   shows "((INum x :: 'a::{field_char_0,linordered_field}) > 0) = 0<\<^sub>N x"
401 proof -
402   obtain a b where x: "x = (a, b)" by (cases x)
403   show ?thesis
404   proof (cases "a = 0")
405     case True
406     then show ?thesis
407       by (simp add: x Ngt0_def INum_def)
408   next
409     case False
410     then have b: "(of_int b::'a) > 0"
411       using nx by (simp add: x isnormNum_def)
412     from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]
413     show ?thesis
414       by (simp add: x Ngt0_def INum_def)
415   qed
416 qed
418 lemma Nge0_iff[simp]:
419   assumes nx: "isnormNum x"
420   shows "(INum x :: 'a::{field_char_0,linordered_field}) \<ge> 0 \<longleftrightarrow> 0\<le>\<^sub>N x"
421 proof -
422   obtain a b where x: "x = (a, b)" by (cases x)
423   show ?thesis
424   proof (cases "a = 0")
425     case True
426     then show ?thesis
427       by (simp add: x Nge0_def INum_def)
428   next
429     case False
430     then have b: "(of_int b::'a) > 0"
431       using nx by (simp add: x isnormNum_def)
432     from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]
433     show ?thesis
434       by (simp add: x Nge0_def INum_def)
435   qed
436 qed
438 lemma Nlt_iff[simp]:
439   assumes nx: "isnormNum x"
440     and ny: "isnormNum y"
441   shows "((INum x :: 'a::{field_char_0,linordered_field}) < INum y) \<longleftrightarrow> x <\<^sub>N y"
442 proof -
443   let ?z = "0::'a"
444   have "((INum x ::'a) < INum y) \<longleftrightarrow> INum (x -\<^sub>N y) < ?z"
445     using nx ny by simp
446   also have "\<dots> \<longleftrightarrow> 0>\<^sub>N (x -\<^sub>N y)"
447     using Nlt0_iff[OF Nsub_normN[OF ny]] by simp
448   finally show ?thesis
450 qed
452 lemma Nle_iff[simp]:
453   assumes nx: "isnormNum x"
454     and ny: "isnormNum y"
455   shows "((INum x :: 'a::{field_char_0,linordered_field}) \<le> INum y) \<longleftrightarrow> x \<le>\<^sub>N y"
456 proof -
457   have "((INum x ::'a) \<le> INum y) \<longleftrightarrow> INum (x -\<^sub>N y) \<le> (0::'a)"
458     using nx ny by simp
459   also have "\<dots> \<longleftrightarrow> 0\<ge>\<^sub>N (x -\<^sub>N y)"
460     using Nle0_iff[OF Nsub_normN[OF ny]] by simp
461   finally show ?thesis
463 qed
466   assumes "SORT_CONSTRAINT('a::field_char_0)"
467   shows "x +\<^sub>N y = y +\<^sub>N x"
468 proof -
469   have n: "isnormNum (x +\<^sub>N y)" "isnormNum (y +\<^sub>N x)"
470     by simp_all
471   have "(INum (x +\<^sub>N y)::'a) = INum (y +\<^sub>N x)"
472     by simp
473   with isnormNum_unique[OF n] show ?thesis
474     by simp
475 qed
477 lemma [simp]:
478   assumes "SORT_CONSTRAINT('a::field_char_0)"
479   shows "(0, b) +\<^sub>N y = normNum y"
480     and "(a, 0) +\<^sub>N y = normNum y"
481     and "x +\<^sub>N (0, b) = normNum x"
482     and "x +\<^sub>N (a, 0) = normNum x"
489   done
491 lemma normNum_nilpotent_aux[simp]:
492   assumes "SORT_CONSTRAINT('a::field_char_0)"
493   assumes nx: "isnormNum x"
494   shows "normNum x = x"
495 proof -
496   let ?a = "normNum x"
497   have n: "isnormNum ?a" by simp
498   have th: "INum ?a = (INum x ::'a)" by simp
499   with isnormNum_unique[OF n nx] show ?thesis by simp
500 qed
502 lemma normNum_nilpotent[simp]:
503   assumes "SORT_CONSTRAINT('a::field_char_0)"
504   shows "normNum (normNum x) = normNum x"
505   by simp
507 lemma normNum0[simp]: "normNum (0, b) = 0\<^sub>N" "normNum (a, 0) = 0\<^sub>N"
511   assumes "SORT_CONSTRAINT('a::field_char_0)"
512   shows "normNum (x +\<^sub>N y) = x +\<^sub>N y"
513   by simp
516   assumes "SORT_CONSTRAINT('a::field_char_0)"
517   shows "normNum x +\<^sub>N y = x +\<^sub>N y"
518 proof -
519   have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)"
520     by simp_all
521   have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a)"
522     by simp
523   also have "\<dots> = INum (x +\<^sub>N y)"
524     by simp
525   finally show ?thesis
526     using isnormNum_unique[OF n] by simp
527 qed
530   assumes "SORT_CONSTRAINT('a::field_char_0)"
531   shows "x +\<^sub>N normNum y = x +\<^sub>N y"
532 proof -
533   have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)"
534     by simp_all
535   have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a)"
536     by simp
537   also have "\<dots> = INum (x +\<^sub>N y)"
538     by simp
539   finally show ?thesis
540     using isnormNum_unique[OF n] by simp
541 qed
544   assumes "SORT_CONSTRAINT('a::field_char_0)"
545   shows "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)"
546 proof -
547   have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))"
548     by simp_all
549   have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)"
550     by simp
551   with isnormNum_unique[OF n] show ?thesis
552     by simp
553 qed
555 lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x"
556   by (simp add: Nmul_def split_def Let_def gcd.commute mult.commute)
558 lemma Nmul_assoc:
559   assumes "SORT_CONSTRAINT('a::field_char_0)"
560   assumes nx: "isnormNum x"
561     and ny: "isnormNum y"
562     and nz: "isnormNum z"
563   shows "x *\<^sub>N y *\<^sub>N z = x *\<^sub>N (y *\<^sub>N z)"
564 proof -
565   from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))"
566     by simp_all
567   have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)"
568     by simp
569   with isnormNum_unique[OF n] show ?thesis
570     by simp
571 qed
573 lemma Nsub0:
574   assumes "SORT_CONSTRAINT('a::field_char_0)"
575   assumes x: "isnormNum x"
576     and y: "isnormNum y"
577   shows "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = y"
578 proof -
579   from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"]
580   have "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)"
581     by simp
582   also have "\<dots> \<longleftrightarrow> INum x = (INum y :: 'a)"
583     by simp
584   also have "\<dots> \<longleftrightarrow> x = y"
585     using x y by simp
586   finally show ?thesis .
587 qed
589 lemma Nmul0[simp]: "c *\<^sub>N 0\<^sub>N = 0\<^sub>N" " 0\<^sub>N *\<^sub>N c = 0\<^sub>N"
590   by (simp_all add: Nmul_def Let_def split_def)
592 lemma Nmul_eq0[simp]:
593   assumes "SORT_CONSTRAINT('a::field_char_0)"
594   assumes nx: "isnormNum x"
595     and ny: "isnormNum y"
596   shows "x*\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = 0\<^sub>N \<or> y = 0\<^sub>N"
597 proof -
598   obtain a b where x: "x = (a, b)" by (cases x)
599   obtain a' b' where y: "y = (a', b')" by (cases y)
600   have n0: "isnormNum 0\<^sub>N" by simp
601   show ?thesis using nx ny
602     apply (simp only: isnormNum_unique[where ?'a = 'a, OF  Nmul_normN[OF nx ny] n0, symmetric]
603       Nmul[where ?'a = 'a])
604     apply (simp add: x y INum_def split_def isnormNum_def split: if_split_asm)
605     done
606 qed
608 lemma Nneg_Nneg[simp]: "~\<^sub>N (~\<^sub>N c) = c"
609   by (simp add: Nneg_def split_def)
611 lemma Nmul1[simp]: "isnormNum c \<Longrightarrow> (1)\<^sub>N *\<^sub>N c = c" "isnormNum c \<Longrightarrow> c *\<^sub>N (1)\<^sub>N = c"
612   apply (simp_all add: Nmul_def Let_def split_def isnormNum_def)
613   apply (cases "fst c = 0", simp_all, cases c, simp_all)+
614   done
616 end