src/HOL/Library/Abstract_Rat.thy
 author Christian Sternagel Wed Aug 29 12:23:14 2012 +0900 (2012-08-29) changeset 49083 01081bca31b6 parent 47162 9d7d919b9fd8 child 50282 fe4d4bb9f4c2 permissions -rw-r--r--
dropped ord and bot instance for list prefixes (use locale interpretation instead, which allows users to decide what order to use on lists)
1 (*  Title:      HOL/Library/Abstract_Rat.thy
2     Author:     Amine Chaieb
3 *)
5 header {* Abstract rational numbers *}
7 theory Abstract_Rat
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" where
20   "isnormNum = (\<lambda>(a,b). (if a = 0 then b = 0 else b > 0 \<and> gcd a b = 1))"
22 definition normNum :: "Num \<Rightarrow> Num" where
23   "normNum = (\<lambda>(a,b).
24     (if a=0 \<or> b = 0 then (0,0) else
25       (let g = gcd a b
26        in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))"
28 declare gcd_dvd1_int[presburger] gcd_dvd2_int[presburger]
30 lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"
31 proof -
32   obtain a b where x: "x = (a, b)" by (cases x)
33   { assume "a=0 \<or> b = 0" hence ?thesis by (simp add: x normNum_def isnormNum_def) }
34   moreover
35   { assume anz: "a \<noteq> 0" and bnz: "b \<noteq> 0"
36     let ?g = "gcd a b"
37     let ?a' = "a div ?g"
38     let ?b' = "b div ?g"
39     let ?g' = "gcd ?a' ?b'"
40     from anz bnz have "?g \<noteq> 0" by simp  with gcd_ge_0_int[of a b]
41     have gpos: "?g > 0" by arith
42     have gdvd: "?g dvd a" "?g dvd b" by arith+
43     from dvd_mult_div_cancel[OF gdvd(1)] dvd_mult_div_cancel[OF gdvd(2)] anz bnz
44     have nz': "?a' \<noteq> 0" "?b' \<noteq> 0" by - (rule notI, simp)+
45     from anz bnz have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith
46     from div_gcd_coprime_int[OF stupid] have gp1: "?g' = 1" .
47     from bnz have "b < 0 \<or> b > 0" by arith
48     moreover
49     { assume b: "b > 0"
50       from b have "?b' \<ge> 0"
51         by (presburger add: pos_imp_zdiv_nonneg_iff[OF gpos])
52       with nz' have b': "?b' > 0" by arith
53       from b b' anz bnz nz' gp1 have ?thesis
54         by (simp add: x isnormNum_def normNum_def Let_def split_def) }
55     moreover {
56       assume b: "b < 0"
57       { assume b': "?b' \<ge> 0"
58         from gpos have th: "?g \<ge> 0" by arith
59         from mult_nonneg_nonneg[OF th b'] dvd_mult_div_cancel[OF gdvd(2)]
60         have False using b by arith }
61       hence b': "?b' < 0" by (presburger add: linorder_not_le[symmetric])
62       from anz bnz nz' b b' gp1 have ?thesis
63         by (simp add: x isnormNum_def normNum_def Let_def split_def) }
64     ultimately have ?thesis by blast
65   }
66   ultimately show ?thesis by blast
67 qed
69 text {* Arithmetic over Num *}
71 definition Nadd :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "+\<^sub>N" 60) where
72   "Nadd = (\<lambda>(a,b) (a',b'). if a = 0 \<or> b = 0 then normNum(a',b')
73     else if a'=0 \<or> b' = 0 then normNum(a,b)
74     else normNum(a*b' + b*a', b*b'))"
76 definition Nmul :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "*\<^sub>N" 60) where
77   "Nmul = (\<lambda>(a,b) (a',b'). let g = gcd (a*a') (b*b')
78     in (a*a' div g, b*b' div g))"
80 definition Nneg :: "Num \<Rightarrow> Num" ("~\<^sub>N")
81   where "Nneg \<equiv> (\<lambda>(a,b). (-a,b))"
83 definition Nsub :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "-\<^sub>N" 60)
84   where "Nsub = (\<lambda>a b. a +\<^sub>N ~\<^sub>N b)"
86 definition Ninv :: "Num \<Rightarrow> Num"
87   where "Ninv = (\<lambda>(a,b). if a < 0 then (-b, \<bar>a\<bar>) else (b,a))"
89 definition Ndiv :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "\<div>\<^sub>N" 60)
90   where "Ndiv = (\<lambda>a b. a *\<^sub>N Ninv b)"
92 lemma Nneg_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (~\<^sub>N x)"
93   by (simp add: isnormNum_def Nneg_def split_def)
95 lemma Nadd_normN[simp]: "isnormNum (x +\<^sub>N y)"
98 lemma Nsub_normN[simp]: "\<lbrakk> isnormNum y\<rbrakk> \<Longrightarrow> isnormNum (x -\<^sub>N y)"
99   by (simp add: Nsub_def split_def)
101 lemma Nmul_normN[simp]:
102   assumes xn: "isnormNum x" and yn: "isnormNum y"
103   shows "isnormNum (x *\<^sub>N y)"
104 proof -
105   obtain a b where x: "x = (a, b)" by (cases x)
106   obtain a' b' where y: "y = (a', b')" by (cases y)
107   { assume "a = 0"
108     hence ?thesis using xn x y
109       by (simp add: isnormNum_def Let_def Nmul_def split_def) }
110   moreover
111   { assume "a' = 0"
112     hence ?thesis using yn x y
113       by (simp add: isnormNum_def Let_def Nmul_def split_def) }
114   moreover
115   { assume a: "a \<noteq>0" and a': "a'\<noteq>0"
116     hence bp: "b > 0" "b' > 0" using xn yn x y by (simp_all add: isnormNum_def)
117     from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a * a', b * b')"
118       using x y a a' bp by (simp add: Nmul_def Let_def split_def normNum_def)
119     hence ?thesis by simp }
120   ultimately show ?thesis by blast
121 qed
123 lemma Ninv_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (Ninv x)"
124   by (simp add: Ninv_def isnormNum_def split_def)
125     (cases "fst x = 0", auto simp add: gcd_commute_int)
127 lemma isnormNum_int[simp]:
128   "isnormNum 0\<^sub>N" "isnormNum ((1::int)\<^sub>N)" "i \<noteq> 0 \<Longrightarrow> isnormNum (i\<^sub>N)"
129   by (simp_all add: isnormNum_def)
132 text {* Relations over Num *}
134 definition Nlt0:: "Num \<Rightarrow> bool"  ("0>\<^sub>N")
135   where "Nlt0 = (\<lambda>(a,b). a < 0)"
137 definition Nle0:: "Num \<Rightarrow> bool"  ("0\<ge>\<^sub>N")
138   where "Nle0 = (\<lambda>(a,b). a \<le> 0)"
140 definition Ngt0:: "Num \<Rightarrow> bool"  ("0<\<^sub>N")
141   where "Ngt0 = (\<lambda>(a,b). a > 0)"
143 definition Nge0:: "Num \<Rightarrow> bool"  ("0\<le>\<^sub>N")
144   where "Nge0 = (\<lambda>(a,b). a \<ge> 0)"
146 definition Nlt :: "Num \<Rightarrow> Num \<Rightarrow> bool"  (infix "<\<^sub>N" 55)
147   where "Nlt = (\<lambda>a b. 0>\<^sub>N (a -\<^sub>N b))"
149 definition Nle :: "Num \<Rightarrow> Num \<Rightarrow> bool"  (infix "\<le>\<^sub>N" 55)
150   where "Nle = (\<lambda>a b. 0\<ge>\<^sub>N (a -\<^sub>N b))"
152 definition "INum = (\<lambda>(a,b). of_int a / of_int b)"
154 lemma INum_int [simp]: "INum (i\<^sub>N) = ((of_int i) ::'a::field)" "INum 0\<^sub>N = (0::'a::field)"
155   by (simp_all add: INum_def)
157 lemma isnormNum_unique[simp]:
158   assumes na: "isnormNum x" and nb: "isnormNum y"
159   shows "((INum x ::'a::{field_char_0, field_inverse_zero}) = INum y) = (x = y)" (is "?lhs = ?rhs")
160 proof
161   obtain a b where x: "x = (a, b)" by (cases x)
162   obtain a' b' where y: "y = (a', b')" by (cases y)
163   assume H: ?lhs
164   { assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0"
165     hence ?rhs using na nb H
166       by (simp add: x y INum_def split_def isnormNum_def split: split_if_asm) }
167   moreover
168   { assume az: "a \<noteq> 0" and bz: "b \<noteq> 0" and a'z: "a'\<noteq>0" and b'z: "b'\<noteq>0"
169     from az bz a'z b'z na nb have pos: "b > 0" "b' > 0" by (simp_all add: x y isnormNum_def)
170     from H bz b'z have eq: "a * b' = a'*b"
171       by (simp add: x y INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
172     from az a'z na nb have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"
173       by (simp_all add: x y isnormNum_def add: gcd_commute_int)
174     from eq have raw_dvd: "a dvd a' * b" "b dvd b' * a" "a' dvd a * b'" "b' dvd b * a'"
175       apply -
176       apply algebra
177       apply algebra
178       apply simp
179       apply algebra
180       done
181     from zdvd_antisym_abs[OF coprime_dvd_mult_int[OF gcd1(2) raw_dvd(2)]
182         coprime_dvd_mult_int[OF gcd1(4) raw_dvd(4)]]
183       have eq1: "b = b'" using pos by arith
184       with eq have "a = a'" using pos by simp
185       with eq1 have ?rhs by (simp add: x y) }
186   ultimately show ?rhs by blast
187 next
188   assume ?rhs thus ?lhs by simp
189 qed
192 lemma isnormNum0[simp]:
193     "isnormNum x \<Longrightarrow> (INum x = (0::'a::{field_char_0, field_inverse_zero})) = (x = 0\<^sub>N)"
194   unfolding INum_int(2)[symmetric]
195   by (rule isnormNum_unique) simp_all
197 lemma of_int_div_aux: "d ~= 0 ==> ((of_int x)::'a::field_char_0) / (of_int d) =
198     of_int (x div d) + (of_int (x mod d)) / ((of_int d)::'a)"
199 proof -
200   assume "d ~= 0"
201   let ?t = "of_int (x div d) * ((of_int d)::'a) + of_int(x mod d)"
202   let ?f = "\<lambda>x. x / of_int d"
203   have "x = (x div d) * d + x mod d"
204     by auto
205   then have eq: "of_int x = ?t"
206     by (simp only: of_int_mult[symmetric] of_int_add [symmetric])
207   then have "of_int x / of_int d = ?t / of_int d"
208     using cong[OF refl[of ?f] eq] by simp
209   then show ?thesis by (simp add: add_divide_distrib algebra_simps `d ~= 0`)
210 qed
212 lemma of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
213     (of_int(n div d)::'a::field_char_0) = of_int n / of_int d"
214   apply (frule of_int_div_aux [of d n, where ?'a = 'a])
215   apply simp
216   apply (simp add: dvd_eq_mod_eq_0)
217   done
220 lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0, field_inverse_zero})"
221 proof -
222   obtain a b where x: "x = (a, b)" by (cases x)
223   { assume "a = 0 \<or> b = 0"
224     hence ?thesis by (simp add: x INum_def normNum_def split_def Let_def) }
225   moreover
226   { assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
227     let ?g = "gcd a b"
228     from a b have g: "?g \<noteq> 0"by simp
229     from of_int_div[OF g, where ?'a = 'a]
230     have ?thesis by (auto simp add: x INum_def normNum_def split_def Let_def) }
231   ultimately show ?thesis by blast
232 qed
234 lemma INum_normNum_iff:
235   "(INum x ::'a::{field_char_0, field_inverse_zero}) = INum y \<longleftrightarrow> normNum x = normNum y"
236   (is "?lhs = ?rhs")
237 proof -
238   have "normNum x = normNum y \<longleftrightarrow> (INum (normNum x) :: 'a) = INum (normNum y)"
239     by (simp del: normNum)
240   also have "\<dots> = ?lhs" by simp
241   finally show ?thesis by simp
242 qed
244 lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: {field_char_0, field_inverse_zero})"
245 proof -
246   let ?z = "0:: 'a"
247   obtain a b where x: "x = (a, b)" by (cases x)
248   obtain a' b' where y: "y = (a', b')" by (cases y)
249   { assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0"
250     hence ?thesis
251       apply (cases "a=0", simp_all add: x y Nadd_def)
252       apply (cases "b= 0", simp_all add: INum_def)
253        apply (cases "a'= 0", simp_all)
254        apply (cases "b'= 0", simp_all)
255        done }
256   moreover
257   { assume aa': "a \<noteq> 0" "a'\<noteq> 0" and bb': "b \<noteq> 0" "b' \<noteq> 0"
258     { assume z: "a * b' + b * a' = 0"
259       hence "of_int (a*b' + b*a') / (of_int b* of_int b') = ?z" by simp
260       hence "of_int b' * of_int a / (of_int b * of_int b') +
261           of_int b * of_int a' / (of_int b * of_int b') = ?z"
263       hence th: "of_int a / of_int b + of_int a' / of_int b' = ?z" using bb' aa'
264         by simp
265       from z aa' bb' have ?thesis
266         by (simp add: x y th Nadd_def normNum_def INum_def split_def) }
267     moreover {
268       assume z: "a * b' + b * a' \<noteq> 0"
269       let ?g = "gcd (a * b' + b * a') (b*b')"
270       have gz: "?g \<noteq> 0" using z by simp
271       have ?thesis using aa' bb' z gz
272         of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a * b' + b * a'" and y="b*b'"]]
273         of_int_div[where ?'a = 'a, OF gz gcd_dvd2_int[where x="a * b' + b * a'" and y="b*b'"]]
274         by (simp add: x y Nadd_def INum_def normNum_def Let_def add_divide_distrib) }
275     ultimately have ?thesis using aa' bb'
276       by (simp add: x y Nadd_def INum_def normNum_def Let_def) }
277   ultimately show ?thesis by blast
278 qed
280 lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a :: {field_char_0, field_inverse_zero})"
281 proof -
282   let ?z = "0::'a"
283   obtain a b where x: "x = (a, b)" by (cases x)
284   obtain a' b' where y: "y = (a', b')" by (cases y)
285   { assume "a=0 \<or> a'= 0 \<or> b = 0 \<or> b' = 0"
286     hence ?thesis
287       apply (cases "a=0", simp_all add: x y Nmul_def INum_def Let_def)
288       apply (cases "b=0", simp_all)
289       apply (cases "a'=0", simp_all)
290       done }
291   moreover
292   { assume z: "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
293     let ?g="gcd (a*a') (b*b')"
294     have gz: "?g \<noteq> 0" using z by simp
295     from z of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a*a'" and y="b*b'"]]
296       of_int_div[where ?'a = 'a , OF gz gcd_dvd2_int[where x="a*a'" and y="b*b'"]]
297     have ?thesis by (simp add: Nmul_def x y Let_def INum_def) }
298   ultimately show ?thesis by blast
299 qed
301 lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x ::'a:: field)"
302   by (simp add: Nneg_def split_def INum_def)
304 lemma Nsub[simp]: "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {field_char_0, field_inverse_zero})"
305   by (simp add: Nsub_def split_def)
307 lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: field_inverse_zero) / (INum x)"
308   by (simp add: Ninv_def INum_def split_def)
310 lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y ::'a :: {field_char_0, field_inverse_zero})"
311   by (simp add: Ndiv_def)
313 lemma Nlt0_iff[simp]:
314   assumes nx: "isnormNum x"
315   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})< 0) = 0>\<^sub>N x"
316 proof -
317   obtain a b where x: "x = (a, b)" by (cases x)
318   { assume "a = 0" hence ?thesis by (simp add: x Nlt0_def INum_def) }
319   moreover
320   { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0"
321       using nx by (simp add: x isnormNum_def)
322     from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]
323     have ?thesis by (simp add: x Nlt0_def INum_def) }
324   ultimately show ?thesis by blast
325 qed
327 lemma Nle0_iff[simp]:
328   assumes nx: "isnormNum x"
329   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<le> 0) = 0\<ge>\<^sub>N x"
330 proof -
331   obtain a b where x: "x = (a, b)" by (cases x)
332   { assume "a = 0" hence ?thesis by (simp add: x Nle0_def INum_def) }
333   moreover
334   { assume a: "a \<noteq> 0" hence b: "(of_int b :: 'a) > 0"
335       using nx by (simp add: x isnormNum_def)
336     from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]
337     have ?thesis by (simp add: x Nle0_def INum_def) }
338   ultimately show ?thesis by blast
339 qed
341 lemma Ngt0_iff[simp]:
342   assumes nx: "isnormNum x"
343   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})> 0) = 0<\<^sub>N x"
344 proof -
345   obtain a b where x: "x = (a, b)" by (cases x)
346   { assume "a = 0" hence ?thesis by (simp add: x Ngt0_def INum_def) }
347   moreover
348   { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx
349       by (simp add: x isnormNum_def)
350     from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]
351     have ?thesis by (simp add: x Ngt0_def INum_def) }
352   ultimately show ?thesis by blast
353 qed
355 lemma Nge0_iff[simp]:
356   assumes nx: "isnormNum x"
357   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<ge> 0) = 0\<le>\<^sub>N x"
358 proof -
359   obtain a b where x: "x = (a, b)" by (cases x)
360   { assume "a = 0" hence ?thesis by (simp add: x Nge0_def INum_def) }
361   moreover
362   { assume "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx
363       by (simp add: x isnormNum_def)
364     from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]
365     have ?thesis by (simp add: x Nge0_def INum_def) }
366   ultimately show ?thesis by blast
367 qed
369 lemma Nlt_iff[simp]:
370   assumes nx: "isnormNum x" and ny: "isnormNum y"
371   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) < INum y) = (x <\<^sub>N y)"
372 proof -
373   let ?z = "0::'a"
374   have "((INum x ::'a) < INum y) = (INum (x -\<^sub>N y) < ?z)"
375     using nx ny by simp
376   also have "\<dots> = (0>\<^sub>N (x -\<^sub>N y))"
377     using Nlt0_iff[OF Nsub_normN[OF ny]] by simp
378   finally show ?thesis by (simp add: Nlt_def)
379 qed
381 lemma Nle_iff[simp]:
382   assumes nx: "isnormNum x" and ny: "isnormNum y"
383   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})\<le> INum y) = (x \<le>\<^sub>N y)"
384 proof -
385   have "((INum x ::'a) \<le> INum y) = (INum (x -\<^sub>N y) \<le> (0::'a))"
386     using nx ny by simp
387   also have "\<dots> = (0\<ge>\<^sub>N (x -\<^sub>N y))"
388     using Nle0_iff[OF Nsub_normN[OF ny]] by simp
389   finally show ?thesis by (simp add: Nle_def)
390 qed
393   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
394   shows "x +\<^sub>N y = y +\<^sub>N x"
395 proof -
396   have n: "isnormNum (x +\<^sub>N y)" "isnormNum (y +\<^sub>N x)" by simp_all
397   have "(INum (x +\<^sub>N y)::'a) = INum (y +\<^sub>N x)" by simp
398   with isnormNum_unique[OF n] show ?thesis by simp
399 qed
401 lemma [simp]:
402   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
403   shows "(0, b) +\<^sub>N y = normNum y"
404     and "(a, 0) +\<^sub>N y = normNum y"
405     and "x +\<^sub>N (0, b) = normNum x"
406     and "x +\<^sub>N (a, 0) = normNum x"
411   done
413 lemma normNum_nilpotent_aux[simp]:
414   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
415   assumes nx: "isnormNum x"
416   shows "normNum x = x"
417 proof -
418   let ?a = "normNum x"
419   have n: "isnormNum ?a" by simp
420   have th: "INum ?a = (INum x ::'a)" by simp
421   with isnormNum_unique[OF n nx] show ?thesis by simp
422 qed
424 lemma normNum_nilpotent[simp]:
425   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
426   shows "normNum (normNum x) = normNum x"
427   by simp
429 lemma normNum0[simp]: "normNum (0,b) = 0\<^sub>N" "normNum (a,0) = 0\<^sub>N"
430   by (simp_all add: normNum_def)
433   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
434   shows "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp
437   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
438   shows "normNum x +\<^sub>N y = x +\<^sub>N y"
439 proof -
440   have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)" by simp_all
441   have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a)" by simp
442   also have "\<dots> = INum (x +\<^sub>N y)" by simp
443   finally show ?thesis using isnormNum_unique[OF n] by simp
444 qed
447   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
448   shows "x +\<^sub>N normNum y = x +\<^sub>N y"
449 proof -
450   have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)" by simp_all
451   have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a)" by simp
452   also have "\<dots> = INum (x +\<^sub>N y)" by simp
453   finally show ?thesis using isnormNum_unique[OF n] by simp
454 qed
457   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
458   shows "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)"
459 proof -
460   have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))" by simp_all
461   have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
462   with isnormNum_unique[OF n] show ?thesis by simp
463 qed
465 lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x"
466   by (simp add: Nmul_def split_def Let_def gcd_commute_int mult_commute)
468 lemma Nmul_assoc:
469   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
470   assumes nx: "isnormNum x" and ny: "isnormNum y" and nz: "isnormNum z"
471   shows "x *\<^sub>N y *\<^sub>N z = x *\<^sub>N (y *\<^sub>N z)"
472 proof -
473   from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))"
474     by simp_all
475   have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
476   with isnormNum_unique[OF n] show ?thesis by simp
477 qed
479 lemma Nsub0:
480   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
481   assumes x: "isnormNum x" and y: "isnormNum y"
482   shows "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = y"
483 proof -
484   fix h :: 'a
485   from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"]
486   have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)) " by simp
487   also have "\<dots> = (INum x = (INum y :: 'a))" by simp
488   also have "\<dots> = (x = y)" using x y by simp
489   finally show ?thesis .
490 qed
492 lemma Nmul0[simp]: "c *\<^sub>N 0\<^sub>N = 0\<^sub>N" " 0\<^sub>N *\<^sub>N c = 0\<^sub>N"
493   by (simp_all add: Nmul_def Let_def split_def)
495 lemma Nmul_eq0[simp]:
496   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
497   assumes nx: "isnormNum x" and ny: "isnormNum y"
498   shows "x*\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = 0\<^sub>N \<or> y = 0\<^sub>N"
499 proof -
500   fix h :: 'a
501   obtain a b where x: "x = (a, b)" by (cases x)
502   obtain a' b' where y: "y = (a', b')" by (cases y)
503   have n0: "isnormNum 0\<^sub>N" by simp
504   show ?thesis using nx ny
505     apply (simp only: isnormNum_unique[where ?'a = 'a, OF  Nmul_normN[OF nx ny] n0, symmetric]
506       Nmul[where ?'a = 'a])
507     apply (simp add: x y INum_def split_def isnormNum_def split: split_if_asm)
508     done
509 qed
511 lemma Nneg_Nneg[simp]: "~\<^sub>N (~\<^sub>N c) = c"
512   by (simp add: Nneg_def split_def)
514 lemma Nmul1[simp]:
515     "isnormNum c \<Longrightarrow> 1\<^sub>N *\<^sub>N c = c"
516     "isnormNum c \<Longrightarrow> c *\<^sub>N (1\<^sub>N) = c"
517   apply (simp_all add: Nmul_def Let_def split_def isnormNum_def)
518   apply (cases "fst c = 0", simp_all, cases c, simp_all)+
519   done
521 end