author  nipkow 
Fri, 24 Nov 2000 16:49:27 +0100  
changeset 10519  ade64af4c57c 
parent 10198  2b255b772585 
child 10658  b9d43a2add79 
permissions  rwrr 
9508
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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1 
(* Title: EulerFermat.ML 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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2 
ID: $Id$ 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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3 
Author: Thomas M. Rasmussen 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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4 
Copyright 2000 University of Cambridge 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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5 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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6 
Fermat's Little Theorem extended to Euler's Totient function. 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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7 
More abstract approach than BoyerMoore (which seems necessary 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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8 
to achieve the extended version) 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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9 
*) 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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10 

9943  11 
(*LCP: not sure why this lemma is needed now*) 
12 
Goal "(abs z = (#1::int)) = (z = #1  z = #1)"; 

13 
by (auto_tac (claset(), simpset() addsimps [zabs_def])); 

14 
qed "abs_eq_1_iff"; 

15 
AddIffs [abs_eq_1_iff]; 

16 

9508
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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17 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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18 
(*** norRRset ***) 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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19 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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20 
Addsimps [RsetR.empty]; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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21 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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22 
val [BnorRset_eq] = BnorRset.simps; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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23 
Delsimps BnorRset.simps; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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24 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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25 
val [prem1,prem2] = 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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26 
Goal "[ !! a m. P {} a m; \ 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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27 
\ (!!a m. [ #0 < (a::int); P (BnorRset(a#1,m::int)) (a#1) m ] \ 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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28 
\ ==> P (BnorRset(a,m)) a m) ] \ 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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29 
\ ==> P (BnorRset(u,v)) u v"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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30 
by (rtac BnorRset.induct 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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31 
by Safe_tac; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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32 
by (case_tac "#0<a" 2); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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33 
by (rtac prem2 2); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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34 
by (ALLGOALS Asm_simp_tac); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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35 
by (ALLGOALS (asm_simp_tac (simpset() addsimps [BnorRset_eq,prem1]))); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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36 
qed "BnorRset_induct"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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37 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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38 
Goal "b:BnorRset(a,m) > b<=a"; 
9747  39 
by (induct_thm_tac BnorRset_induct "a m" 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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40 
by (stac BnorRset_eq 2); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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41 
by (rewtac Let_def); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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42 
by Auto_tac; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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43 
qed_spec_mp "Bnor_mem_zle"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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44 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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45 
Goal "a<b ==> b~:BnorRset(a,m)"; 
10198  46 
by (auto_tac (claset() addDs [Bnor_mem_zle], simpset())); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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47 
qed "Bnor_mem_zle_swap"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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48 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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49 
Goal "b:BnorRset(a,m) > #0<b"; 
9747  50 
by (induct_thm_tac BnorRset_induct "a m" 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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51 
by (stac BnorRset_eq 2); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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52 
by (rewtac Let_def); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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53 
by Auto_tac; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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54 
qed_spec_mp "Bnor_mem_zg"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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55 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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56 
Goal "zgcd(b,m) = #1 > #0<b > b<=a > b:BnorRset(a,m)"; 
9747  57 
by (induct_thm_tac BnorRset.induct "a m" 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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58 
by Auto_tac; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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59 
by (case_tac "a=b" 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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60 
by (asm_full_simp_tac (simpset() addsimps [zle_neq_implies_zless]) 2); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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61 
by (Asm_simp_tac 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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62 
by (ALLGOALS (stac BnorRset_eq)); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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63 
by (rewtac Let_def); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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64 
by Auto_tac; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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65 
qed_spec_mp "Bnor_mem_if"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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66 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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67 
Goal "a<m > BnorRset (a,m) : RsetR m"; 
9747  68 
by (induct_thm_tac BnorRset_induct "a m" 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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69 
by (Simp_tac 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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70 
by (stac BnorRset_eq 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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71 
by (rewtac Let_def); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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72 
by Auto_tac; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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73 
by (rtac RsetR.insert 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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74 
by (rtac allI 3); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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75 
by (rtac impI 3); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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76 
by (rtac zcong_not 3); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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77 
by (subgoal_tac "a' <= a#1" 6); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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78 
by (rtac Bnor_mem_zle 7); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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79 
by (rtac Bnor_mem_zg 5); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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80 
by Auto_tac; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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81 
qed_spec_mp "Bnor_in_RsetR"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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82 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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83 
Goal "finite (BnorRset (a,m))"; 
9747  84 
by (induct_thm_tac BnorRset_induct "a m" 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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85 
by (stac BnorRset_eq 2); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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86 
by (rewtac Let_def); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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parents:
diff
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87 
by Auto_tac; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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88 
qed "Bnor_fin"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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89 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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90 
Goal "a <= b  #1 ==> a < (b::int)"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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91 
by Auto_tac; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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92 
val lemma = result(); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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93 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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94 
Goalw [norRRset_def] 
9572  95 
"[ #1<m; zgcd(a,m) = #1 ] ==> (EX! b. [a = b](mod m) & b:(norRRset m))"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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96 
by (cut_inst_tac [("a","a"),("m","m")] zcong_zless_unique 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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97 
by Auto_tac; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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diff
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98 
by (res_inst_tac [("m","m")] zcong_zless_imp_eq 2); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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99 
by (auto_tac (claset() addIs [Bnor_mem_zle,Bnor_mem_zg,zcong_trans, 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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100 
zless_imp_zle,lemma], 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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101 
simpset() addsimps [zcong_sym])); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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102 
by (res_inst_tac [("x","b")] exI 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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diff
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103 
by Safe_tac; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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parents:
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104 
by (rtac Bnor_mem_if 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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105 
by (case_tac "b=#0" 2); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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diff
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106 
by (auto_tac (claset() addIs [zle_neq_implies_zless], simpset())); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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107 
by (SELECT_GOAL (rewtac zcong_def) 2); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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diff
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108 
by (subgoal_tac "zgcd(a,m) = m" 2); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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109 
by (stac (zdvd_iff_zgcd RS sym) 3); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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diff
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110 
by (rtac zgcd_zcong_zgcd 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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parents:
diff
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111 
by (ALLGOALS (asm_full_simp_tac (simpset() 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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diff
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112 
addsimps [zdvd_zminus_iff,zcong_sym]))); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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parents:
diff
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113 
qed "norR_mem_unique"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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parents:
diff
changeset

114 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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diff
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115 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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116 
(*** noXRRset ***) 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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parents:
diff
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117 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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118 
Goalw [is_RRset_def] 
9943  119 
"is_RRset A m ==> a:A > zgcd (a,m) = #1"; 
9508
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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120 
by (rtac RsetR.induct 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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parents:
diff
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121 
by Auto_tac; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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122 
qed_spec_mp "RRset_gcd"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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123 

9943  124 
Goal "[ A : RsetR m; #0<m; zgcd(x, m) = #1 ] ==> (%a. a*x)``A : RsetR m"; 
9508
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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parents:
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125 
by (etac RsetR.induct 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
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126 
by (ALLGOALS Simp_tac); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

127 
by (rtac RsetR.insert 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

128 
by Auto_tac; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
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129 
by (asm_full_simp_tac (simpset() addsimps [zcong_cancel]) 2); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
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130 
by (blast_tac (claset() addIs [zgcd_zgcd_zmult]) 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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parents:
diff
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131 
qed "RsetR_zmult_mono"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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parents:
diff
changeset

132 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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133 
Goalw [norRRset_def,noXRRset_def] 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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diff
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134 
"[ #0<m; zgcd(x,m) = #1 ] \ 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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parents:
diff
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135 
\ ==> card (noXRRset m x) = card (norRRset m)"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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parents:
diff
changeset

136 
by (rtac card_image 1); 
9634
61b57cc1cb5a
modified proofs: better rules for cancellation of common factors across comparisons
paulson
parents:
9572
diff
changeset

137 
by (auto_tac (claset(),simpset() addsimps [inj_on_def, Bnor_fin])); 
9508
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

138 
by (asm_full_simp_tac (simpset() addsimps [BnorRset_eq]) 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

139 
qed "card_nor_eq_noX"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

140 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

141 
Goalw [is_RRset_def,phi_def] 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

142 
"[ #0<m; zgcd(x,m) = #1 ] ==> is_RRset (noXRRset m x) m"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

143 
by (auto_tac (claset(),simpset() addsimps [card_nor_eq_noX])); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

144 
by (rewrite_goals_tac [noXRRset_def,norRRset_def]); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

145 
by (rtac RsetR_zmult_mono 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

146 
by (rtac Bnor_in_RsetR 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

147 
by (ALLGOALS Asm_simp_tac); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

148 
qed "noX_is_RRset"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

149 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

150 
Goal "[ #1<m; is_RRset A m; a:A ] \ 
9943  151 
\ ==> zcong a (@ b. [a = b](mod m) & b : norRRset m) m & \ 
152 
\ (@ b. [a = b](mod m) & b : norRRset m) : norRRset m"; 

10175  153 
by (rtac (norR_mem_unique RS ex1_implies_ex RS someI_ex) 1); 
9508
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

154 
by (rtac RRset_gcd 2); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

155 
by (ALLGOALS Asm_simp_tac); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

156 
val lemma_some = result(); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

157 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

158 
Goalw [RRset2norRR_def] 
9943  159 
"[ #1<m; is_RRset A m; a:A ] \ 
160 
\ ==> [a = RRset2norRR A m a] (mod m) & (RRset2norRR A m a):(norRRset m)"; 

9508
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

161 
by (Asm_simp_tac 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

162 
by (rtac lemma_some 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

163 
by (ALLGOALS Asm_simp_tac); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

164 
qed "RRset2norRR_correct"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

165 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

166 
bind_thm ("RRset2norRR_correct1", RRset2norRR_correct RS conjunct1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

167 
bind_thm ("RRset2norRR_correct2", RRset2norRR_correct RS conjunct2); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

168 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

169 
Goal "A : (RsetR m) ==> finite A"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

170 
by (etac RsetR.induct 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

171 
by Auto_tac; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

172 
qed "RsetR_fin"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

173 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

174 
Goalw [is_RRset_def] 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

175 
"[ #1<m; is_RRset A m; [a = b](mod m) ] ==> a:A > b:A > a = b"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

176 
by (rtac RsetR.induct 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

177 
by (auto_tac (claset(), simpset() addsimps [zcong_sym])); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

178 
qed_spec_mp "RRset_zcong_eq"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

179 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

180 
Goal "[ P (@ a. P a); Q (@ a. Q a); (@ a. P a) = (@ a. Q a) ] \ 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

181 
\ ==> (EX a. P a & Q a)"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

182 
by Auto_tac; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

183 
val lemma = result(); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

184 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

185 
Goalw [RRset2norRR_def,inj_on_def] 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

186 
"[ #1<m; is_RRset A m ] ==> inj_on (RRset2norRR A m) A"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

187 
by Auto_tac; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

188 
by (subgoal_tac "(EX b. ([x = b](mod m) & b : norRRset m) & \ 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

189 
\ ([y = b](mod m) & b : norRRset m))" 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

190 
by (rtac lemma 2); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

191 
by (rtac lemma_some 3); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

192 
by (rtac lemma_some 2); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

193 
by (rtac RRset_zcong_eq 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

194 
by Auto_tac; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

195 
by (res_inst_tac [("b","b")] zcong_trans 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

196 
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zcong_sym]))); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

197 
qed "RRset2norRR_inj"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

198 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

199 
Goal "[ #1<m; is_RRset A m ] ==> (RRset2norRR A m)``A = (norRRset m)"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

200 
by (rtac card_seteq 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

201 
by (stac card_image 3); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

202 
by (rtac RRset2norRR_inj 4); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

203 
by Auto_tac; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

204 
by (rtac RRset2norRR_correct2 2); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

205 
by Auto_tac; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

206 
by (rewrite_goals_tac [is_RRset_def,phi_def,norRRset_def]); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

207 
by (auto_tac (claset(),simpset() addsimps [RsetR_fin,Bnor_fin])); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

208 
qed "RRset2norRR_eq_norR"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

209 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

210 
Goalw [inj_on_def] "[ a ~: A ; inj f ] ==> (f a) ~: f``A"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

211 
by Auto_tac; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

212 
val lemma = result(); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

213 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

214 
Goal "x~=#0 ==> a<m > setprod ((%a. a*x) `` BnorRset(a,m)) = \ 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

215 
\ setprod (BnorRset(a,m)) * x^card(BnorRset(a,m))"; 
9747  216 
by (induct_thm_tac BnorRset_induct "a m" 1); 
9508
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

217 
by (stac BnorRset_eq 2); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

218 
by (rewtac Let_def); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

219 
by Auto_tac; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

220 
by (asm_simp_tac (simpset() addsimps [Bnor_fin,Bnor_mem_zle_swap]) 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

221 
by (stac setprod_insert 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

222 
by (rtac lemma 2); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

223 
by (rewtac inj_on_def); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

224 
by (ALLGOALS (asm_full_simp_tac (simpset() 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

225 
addsimps zmult_ac@[Bnor_fin,finite_imageI,Bnor_mem_zle_swap]))); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

226 
qed_spec_mp "Bnor_prod_power"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

227 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

228 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

229 
(*** Fermat ***) 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

230 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

231 
Goalw [zcongm_def] 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

232 
"(A,B) : bijR (zcongm m) ==> [setprod A = setprod B](mod m)"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

233 
by (etac bijR.induct 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

234 
by (subgoal_tac "a~:A & b~:B & finite A & finite B" 2); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

235 
by (auto_tac (claset() addIs [fin_bijRl,fin_bijRr,zcong_zmult], 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

236 
simpset())); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

237 
qed "bijzcong_zcong_prod"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

238 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

239 
Goalw [norRRset_def,phi_def] 
9943  240 
"a<m > zgcd (setprod (BnorRset (a,m)),m) = #1"; 
9747  241 
by (induct_thm_tac BnorRset_induct "a m" 1); 
9508
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

242 
by (stac BnorRset_eq 2); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

243 
by (rewtac Let_def); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

244 
by Auto_tac; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

245 
by (asm_simp_tac (simpset() addsimps [Bnor_fin,Bnor_mem_zle_swap]) 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

246 
by (blast_tac (claset() addIs [zgcd_zgcd_zmult]) 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

247 
qed_spec_mp "Bnor_prod_zgcd"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

248 

4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

249 
Goalw [norRRset_def,phi_def] 
9943  250 
"[ #0<m; zgcd(x,m) = #1 ] ==> [x^phi(m) = #1] (mod m)"; 
9508
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

251 
by (case_tac "x=#0" 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

252 
by (case_tac "m=#1" 2); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

253 
by (rtac iffD1 3); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

254 
by (res_inst_tac [("k","setprod (BnorRset (m#1,m))")] zcong_cancel2 3); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

255 
by (stac (Bnor_prod_power RS sym) 5); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

256 
by (rtac Bnor_prod_zgcd 4); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

257 
by (ALLGOALS Asm_full_simp_tac); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

258 
by (rtac bijzcong_zcong_prod 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

259 
by (fold_goals_tac [norRRset_def,noXRRset_def]); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

260 
by (stac (RRset2norRR_eq_norR RS sym) 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

261 
by (rtac inj_func_bijR 3); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

262 
by Auto_tac; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

263 
by (rewtac zcongm_def); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

264 
by (rtac RRset2norRR_correct1 3); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

265 
by (rtac RRset2norRR_inj 6); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

266 
by (auto_tac (claset() addIs [zle_neq_implies_zless], 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

267 
simpset() addsimps [noX_is_RRset])); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

268 
by (rewrite_goals_tac [noXRRset_def,norRRset_def]); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

269 
by (rtac finite_imageI 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

270 
by (rtac Bnor_fin 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

271 
qed "EulerFermatTheorem"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

272 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

273 
Goalw [zprime_def] 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

274 
"p:zprime ==> a<p > (ALL b. #0<b & b<=a > zgcd(b,p) = #1) \ 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

275 
\ > card (BnorRset(a, p)) = nat a"; 
9747  276 
by (induct_thm_tac BnorRset.induct "a p" 1); 
9508
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

277 
by (stac BnorRset_eq 1); 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

278 
by (rewtac Let_def); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

279 
by Auto_tac; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

280 
qed_spec_mp "Bnor_prime"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

281 

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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

282 
Goalw [phi_def,norRRset_def] 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

283 
"p:zprime ==> phi(p) = nat (p#1)"; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

284 
by (rtac Bnor_prime 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

285 
by Auto_tac; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

286 
by (etac zless_zprime_imp_zrelprime 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

287 
by (ALLGOALS Asm_simp_tac); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

288 
qed "phi_prime"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

289 

9943  290 
Goal "[ p:zprime; ~p dvd x ] ==> [x^(nat (p#1)) = #1] (mod p)"; 
9508
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

291 
by (stac (phi_prime RS sym) 1); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

292 
by (rtac EulerFermatTheorem 2); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

293 
by (etac zprime_imp_zrelprime 3); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

294 
by (rewtac zprime_def); 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

295 
by Auto_tac; 
4d01dbf6ded7
Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

296 
qed "Little_Fermat"; 
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Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff
changeset

297 