src/HOL/Old_Number_Theory/Int2.thy
 author blanchet Thu Sep 11 18:54:36 2014 +0200 (2014-09-11) changeset 58306 117ba6cbe414 parent 57514 bdc2c6b40bf2 child 58889 5b7a9633cfa8 permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
1 (*  Title:      HOL/Old_Number_Theory/Int2.thy
3 *)
5 header {*Integers: Divisibility and Congruences*}
7 theory Int2
8 imports Finite2 WilsonRuss
9 begin
11 definition MultInv :: "int => int => int"
12   where "MultInv p x = x ^ nat (p - 2)"
15 subsection {* Useful lemmas about dvd and powers *}
17 lemma zpower_zdvd_prop1:
18   "0 < n \<Longrightarrow> p dvd y \<Longrightarrow> p dvd ((y::int) ^ n)"
19   by (induct n) (auto simp add: dvd_mult2 [of p y])
21 lemma zdvd_bounds: "n dvd m ==> m \<le> (0::int) | n \<le> m"
22 proof -
23   assume "n dvd m"
24   then have "~(0 < m & m < n)"
25     using zdvd_not_zless [of m n] by auto
26   then show ?thesis by auto
27 qed
29 lemma zprime_zdvd_zmult_better: "[| zprime p;  p dvd (m * n) |] ==>
30     (p dvd m) | (p dvd n)"
31   apply (cases "0 \<le> m")
33   apply (insert zprime_zdvd_zmult [of "-m" p n])
34   apply auto
35   done
37 lemma zpower_zdvd_prop2:
38     "zprime p \<Longrightarrow> p dvd ((y::int) ^ n) \<Longrightarrow> 0 < n \<Longrightarrow> p dvd y"
39   apply (induct n)
40    apply simp
41   apply (frule zprime_zdvd_zmult_better)
42    apply simp
43   apply (force simp del:dvd_mult)
44   done
46 lemma div_prop1:
47   assumes "0 < z" and "(x::int) < y * z"
48   shows "x div z < y"
49 proof -
50   from `0 < z` have modth: "x mod z \<ge> 0" by simp
51   have "(x div z) * z \<le> (x div z) * z" by simp
52   then have "(x div z) * z \<le> (x div z) * z + x mod z" using modth by arith
53   also have "\<dots> = x"
54     by (auto simp add: zmod_zdiv_equality [symmetric] ac_simps)
55   also note `x < y * z`
56   finally show ?thesis
57     apply (auto simp add: mult_less_cancel_right)
58     using assms apply arith
59     done
60 qed
62 lemma div_prop2:
63   assumes "0 < z" and "(x::int) < (y * z) + z"
64   shows "x div z \<le> y"
65 proof -
66   from assms have "x < (y + 1) * z" by (auto simp add: int_distrib)
67   then have "x div z < y + 1"
68     apply (rule_tac y = "y + 1" in div_prop1)
69     apply (auto simp add: `0 < z`)
70     done
71   then show ?thesis by auto
72 qed
74 lemma zdiv_leq_prop: assumes "0 < y" shows "y * (x div y) \<le> (x::int)"
75 proof-
76   from zmod_zdiv_equality have "x = y * (x div y) + x mod y" by auto
77   moreover have "0 \<le> x mod y" by (auto simp add: assms)
78   ultimately show ?thesis by arith
79 qed
82 subsection {* Useful properties of congruences *}
84 lemma zcong_eq_zdvd_prop: "[x = 0](mod p) = (p dvd x)"
85   by (auto simp add: zcong_def)
87 lemma zcong_id: "[m = 0] (mod m)"
88   by (auto simp add: zcong_def)
90 lemma zcong_shift: "[a = b] (mod m) ==> [a + c = b + c] (mod m)"
93 lemma zcong_zpower: "[x = y](mod m) ==> [x^z = y^z](mod m)"
94   by (induct z) (auto simp add: zcong_zmult)
96 lemma zcong_eq_trans: "[| [a = b](mod m); b = c; [c = d](mod m) |] ==>
97     [a = d](mod m)"
98   apply (erule zcong_trans)
99   apply simp
100   done
102 lemma aux1: "a - b = (c::int) ==> a = c + b"
103   by auto
105 lemma zcong_zmult_prop1: "[a = b](mod m) ==> ([c = a * d](mod m) =
106     [c = b * d] (mod m))"
107   apply (auto simp add: zcong_def dvd_def)
108   apply (rule_tac x = "ka + k * d" in exI)
109   apply (drule aux1)+
110   apply (auto simp add: int_distrib)
111   apply (rule_tac x = "ka - k * d" in exI)
112   apply (drule aux1)+
113   apply (auto simp add: int_distrib)
114   done
116 lemma zcong_zmult_prop2: "[a = b](mod m) ==>
117     ([c = d * a](mod m) = [c = d * b] (mod m))"
118   by (auto simp add: ac_simps zcong_zmult_prop1)
120 lemma zcong_zmult_prop3: "[| zprime p; ~[x = 0] (mod p);
121     ~[y = 0] (mod p) |] ==> ~[x * y = 0] (mod p)"
122   apply (auto simp add: zcong_def)
123   apply (drule zprime_zdvd_zmult_better, auto)
124   done
126 lemma zcong_less_eq: "[| 0 < x; 0 < y; 0 < m; [x = y] (mod m);
127     x < m; y < m |] ==> x = y"
128   by (metis zcong_not zcong_sym less_linear)
130 lemma zcong_neg_1_impl_ne_1:
131   assumes "2 < p" and "[x = -1] (mod p)"
132   shows "~([x = 1] (mod p))"
133 proof
134   assume "[x = 1] (mod p)"
135   with assms have "[1 = -1] (mod p)"
136     apply (auto simp add: zcong_sym)
137     apply (drule zcong_trans, auto)
138     done
139   then have "[1 + 1 = -1 + 1] (mod p)"
140     by (simp only: zcong_shift)
141   then have "[2 = 0] (mod p)"
142     by auto
143   then have "p dvd 2"
144     by (auto simp add: dvd_def zcong_def)
145   with `2 < p` show False
146     by (auto simp add: zdvd_not_zless)
147 qed
149 lemma zcong_zero_equiv_div: "[a = 0] (mod m) = (m dvd a)"
150   by (auto simp add: zcong_def)
152 lemma zcong_zprime_prod_zero: "[| zprime p; 0 < a |] ==>
153     [a * b = 0] (mod p) ==> [a = 0] (mod p) | [b = 0] (mod p)"
154   by (auto simp add: zcong_zero_equiv_div zprime_zdvd_zmult)
156 lemma zcong_zprime_prod_zero_contra: "[| zprime p; 0 < a |] ==>
157   ~[a = 0](mod p) & ~[b = 0](mod p) ==> ~[a * b = 0] (mod p)"
158   apply auto
159   apply (frule_tac a = a and b = b and p = p in zcong_zprime_prod_zero)
160   apply auto
161   done
163 lemma zcong_not_zero: "[| 0 < x; x < m |] ==> ~[x = 0] (mod m)"
164   by (auto simp add: zcong_zero_equiv_div zdvd_not_zless)
166 lemma zcong_zero: "[| 0 \<le> x; x < m; [x = 0](mod m) |] ==> x = 0"
167   apply (drule order_le_imp_less_or_eq, auto)
168   apply (frule_tac m = m in zcong_not_zero)
169   apply auto
170   done
172 lemma all_relprime_prod_relprime: "[| finite A; \<forall>x \<in> A. zgcd x y = 1 |]
173     ==> zgcd (setprod id A) y = 1"
174   by (induct set: finite) (auto simp add: zgcd_zgcd_zmult)
177 subsection {* Some properties of MultInv *}
179 lemma MultInv_prop1: "[| 2 < p; [x = y] (mod p) |] ==>
180     [(MultInv p x) = (MultInv p y)] (mod p)"
181   by (auto simp add: MultInv_def zcong_zpower)
183 lemma MultInv_prop2: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==>
184   [(x * (MultInv p x)) = 1] (mod p)"
185 proof (simp add: MultInv_def zcong_eq_zdvd_prop)
186   assume 1: "2 < p" and 2: "zprime p" and 3: "~ p dvd x"
187   have "x * x ^ nat (p - 2) = x ^ (nat (p - 2) + 1)"
188     by auto
189   also from 1 have "nat (p - 2) + 1 = nat (p - 2 + 1)"
191   also have "p - 2 + 1 = p - 1" by arith
192   finally have "[x * x ^ nat (p - 2) = x ^ nat (p - 1)] (mod p)"
193     by (rule ssubst, auto)
194   also from 2 3 have "[x ^ nat (p - 1) = 1] (mod p)"
195     by (auto simp add: Little_Fermat)
196   finally (zcong_trans) show "[x * x ^ nat (p - 2) = 1] (mod p)" .
197 qed
199 lemma MultInv_prop2a: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==>
200     [(MultInv p x) * x = 1] (mod p)"
201   by (auto simp add: MultInv_prop2 ac_simps)
203 lemma aux_1: "2 < p ==> ((nat p) - 2) = (nat (p - 2))"
206 lemma aux_2: "2 < p ==> 0 < nat (p - 2)"
207   by auto
209 lemma MultInv_prop3: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==>
210     ~([MultInv p x = 0](mod p))"
211   apply (auto simp add: MultInv_def zcong_eq_zdvd_prop aux_1)
212   apply (drule aux_2)
213   apply (drule zpower_zdvd_prop2, auto)
214   done
216 lemma aux__1: "[| 2 < p; zprime p; ~([x = 0](mod p))|] ==>
217     [(MultInv p (MultInv p x)) = (x * (MultInv p x) *
218       (MultInv p (MultInv p x)))] (mod p)"
219   apply (drule MultInv_prop2, auto)
220   apply (drule_tac k = "MultInv p (MultInv p x)" in zcong_scalar, auto)
221   apply (auto simp add: zcong_sym)
222   done
224 lemma aux__2: "[| 2 < p; zprime p; ~([x = 0](mod p))|] ==>
225     [(x * (MultInv p x) * (MultInv p (MultInv p x))) = x] (mod p)"
226   apply (frule MultInv_prop3, auto)
227   apply (insert MultInv_prop2 [of p "MultInv p x"], auto)
228   apply (drule MultInv_prop2, auto)
229   apply (drule_tac k = x in zcong_scalar2, auto)
230   apply (auto simp add: ac_simps)
231   done
233 lemma MultInv_prop4: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==>
234     [(MultInv p (MultInv p x)) = x] (mod p)"
235   apply (frule aux__1, auto)
236   apply (drule aux__2, auto)
237   apply (drule zcong_trans, auto)
238   done
240 lemma MultInv_prop5: "[| 2 < p; zprime p; ~([x = 0](mod p));
241     ~([y = 0](mod p)); [(MultInv p x) = (MultInv p y)] (mod p) |] ==>
242     [x = y] (mod p)"
243   apply (drule_tac a = "MultInv p x" and b = "MultInv p y" and
244     m = p and k = x in zcong_scalar)
245   apply (insert MultInv_prop2 [of p x], simp)
246   apply (auto simp only: zcong_sym [of "MultInv p x * x"])
247   apply (auto simp add: ac_simps)
248   apply (drule zcong_trans, auto)
249   apply (drule_tac a = "x * MultInv p y" and k = y in zcong_scalar, auto)
250   apply (insert MultInv_prop2a [of p y], auto simp add: ac_simps)
251   apply (insert zcong_zmult_prop2 [of "y * MultInv p y" 1 p y x])
252   apply (auto simp add: zcong_sym)
253   done
255 lemma MultInv_zcong_prop1: "[| 2 < p; [j = k] (mod p) |] ==>
256     [a * MultInv p j = a * MultInv p k] (mod p)"
257   by (drule MultInv_prop1, auto simp add: zcong_scalar2)
259 lemma aux___1: "[j = a * MultInv p k] (mod p) ==>
260     [j * k = a * MultInv p k * k] (mod p)"
261   by (auto simp add: zcong_scalar)
263 lemma aux___2: "[|2 < p; zprime p; ~([k = 0](mod p));
264     [j * k = a * MultInv p k * k] (mod p) |] ==> [j * k = a] (mod p)"
265   apply (insert MultInv_prop2a [of p k] zcong_zmult_prop2
266     [of "MultInv p k * k" 1 p "j * k" a])
267   apply (auto simp add: ac_simps)
268   done
270 lemma aux___3: "[j * k = a] (mod p) ==> [(MultInv p j) * j * k =
271      (MultInv p j) * a] (mod p)"
272   by (auto simp add: mult.assoc zcong_scalar2)
274 lemma aux___4: "[|2 < p; zprime p; ~([j = 0](mod p));
275     [(MultInv p j) * j * k = (MultInv p j) * a] (mod p) |]
276        ==> [k = a * (MultInv p j)] (mod p)"
277   apply (insert MultInv_prop2a [of p j] zcong_zmult_prop1
278     [of "MultInv p j * j" 1 p "MultInv p j * a" k])
279   apply (auto simp add: ac_simps zcong_sym)
280   done
282 lemma MultInv_zcong_prop2: "[| 2 < p; zprime p; ~([k = 0](mod p));
283     ~([j = 0](mod p)); [j = a * MultInv p k] (mod p) |] ==>
284     [k = a * MultInv p j] (mod p)"
285   apply (drule aux___1)
286   apply (frule aux___2, auto)
287   by (drule aux___3, drule aux___4, auto)
289 lemma MultInv_zcong_prop3: "[| 2 < p; zprime p; ~([a = 0](mod p));
290     ~([k = 0](mod p)); ~([j = 0](mod p));
291     [a * MultInv p j = a * MultInv p k] (mod p) |] ==>
292       [j = k] (mod p)"
293   apply (auto simp add: zcong_eq_zdvd_prop [of a p])
294   apply (frule zprime_imp_zrelprime, auto)
295   apply (insert zcong_cancel2 [of p a "MultInv p j" "MultInv p k"], auto)
296   apply (drule MultInv_prop5, auto)
297   done
299 end