author  haftmann 
Thu, 17 May 2007 19:49:16 +0200  
changeset 22993  838c66e760b5 
parent 22916  8caf6da610e2 
child 23017  00c0e4c42396 
permissions  rwrr 
3366  1 
(* Title: HOL/Divides.thy 
2 
ID: $Id$ 

3 
Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
3366
diff
changeset

4 
Copyright 1999 University of Cambridge 
18154  5 
*) 
3366  6 

18154  7 
header {* The division operators div, mod and the divides relation "dvd" *} 
3366  8 

15131  9 
theory Divides 
21408  10 
imports Datatype Power 
22993  11 
uses "~~/src/Provers/Arith/cancel_div_mod.ML" 
15131  12 
begin 
3366  13 

8902  14 
(*We use the same class for div and mod; 
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
3366
diff
changeset

15 
moreover, dvd is defined whenever multiplication is*) 
22473  16 
class div = type + 
21408  17 
fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" 
18 
fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" 

19 
begin 

20 

21 
notation 

22 
div (infixl "\<^loc>div" 70) 

23 

24 
notation 

25 
mod (infixl "\<^loc>mod" 70) 

26 

27 
end 

6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
3366
diff
changeset

28 

21408  29 
notation 
30 
div (infixl "div" 70) 

31 

32 
notation 

33 
mod (infixl "mod" 70) 

34 

22993  35 
instance nat :: Divides.div 
36 
div_def: "m div n == wfrec (pred_nat^+) 

37 
(%f j. if j<n  n=0 then 0 else Suc (f (jn))) m" 

22261
9e185f78e7d4
Adapted to changes in Transitive_Closure theory.
berghofe
parents:
21911
diff
changeset

38 
mod_def: "m mod n == wfrec (pred_nat^+) 
22993  39 
(%f j. if j<n  n=0 then j else f (jn)) m" .. 
21408  40 

41 
definition 

42 
(*The definition of dvd is polymorphic!*) 

43 
dvd :: "'a::times \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) where 

44 
dvd_def: "m dvd n \<longleftrightarrow> (\<exists>k. n = m*k)" 

6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
3366
diff
changeset

45 

22718  46 
definition 
47 
quorem :: "(nat*nat) * (nat*nat) => bool" where 

21408  48 
(*This definition helps prove the harder properties of div and mod. 
49 
It is copied from IntDiv.thy; should it be overloaded?*) 

22718  50 
"quorem = (%((a,b), (q,r)). 
21408  51 
a = b*q + r & 
52 
(if 0<b then 0\<le>r & r<b else b<r & r \<le>0))" 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

53 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

54 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

55 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

56 
subsection{*Initial Lemmas*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

57 

22718  58 
lemmas wf_less_trans = 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

59 
def_wfrec [THEN trans, OF eq_reflection wf_pred_nat [THEN wf_trancl], 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

60 
standard] 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

61 

22718  62 
lemma mod_eq: "(%m. m mod n) = 
22261
9e185f78e7d4
Adapted to changes in Transitive_Closure theory.
berghofe
parents:
21911
diff
changeset

63 
wfrec (pred_nat^+) (%f j. if j<n  n=0 then j else f (jn))" 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

64 
by (simp add: mod_def) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

65 

22718  66 
lemma div_eq: "(%m. m div n) = wfrec (pred_nat^+) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

67 
(%f j. if j<n  n=0 then 0 else Suc (f (jn)))" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

68 
by (simp add: div_def) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

69 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

70 

22718  71 
(** Aribtrary definitions for division by zero. Useful to simplify 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

72 
certain equations **) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

73 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

74 
lemma DIVISION_BY_ZERO_DIV [simp]: "a div 0 = (0::nat)" 
22718  75 
by (rule div_eq [THEN wf_less_trans], simp) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

76 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

77 
lemma DIVISION_BY_ZERO_MOD [simp]: "a mod 0 = (a::nat)" 
22718  78 
by (rule mod_eq [THEN wf_less_trans], simp) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

79 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

80 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

81 
subsection{*Remainder*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

82 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

83 
lemma mod_less [simp]: "m<n ==> m mod n = (m::nat)" 
22718  84 
by (rule mod_eq [THEN wf_less_trans]) simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

85 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

86 
lemma mod_geq: "~ m < (n::nat) ==> m mod n = (mn) mod n" 
22718  87 
apply (cases "n=0") 
88 
apply simp 

89 
apply (rule mod_eq [THEN wf_less_trans]) 

90 
apply (simp add: cut_apply less_eq) 

91 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

92 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

93 
(*Avoids the ugly ~m<n above*) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

94 
lemma le_mod_geq: "(n::nat) \<le> m ==> m mod n = (mn) mod n" 
22718  95 
by (simp add: mod_geq linorder_not_less) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

96 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

97 
lemma mod_if: "m mod (n::nat) = (if m<n then m else (mn) mod n)" 
22718  98 
by (simp add: mod_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

99 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

100 
lemma mod_1 [simp]: "m mod Suc 0 = 0" 
22718  101 
by (induct m) (simp_all add: mod_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

102 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

103 
lemma mod_self [simp]: "n mod n = (0::nat)" 
22718  104 
by (cases "n = 0") (simp_all add: mod_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

105 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

106 
lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)" 
22718  107 
apply (subgoal_tac "(n + m) mod n = (n+mn) mod n") 
108 
apply (simp add: add_commute) 

109 
apply (subst mod_geq [symmetric], simp_all) 

110 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

111 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

112 
lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)" 
22718  113 
by (simp add: add_commute mod_add_self2) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

114 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

115 
lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)" 
22718  116 
by (induct k) (simp_all add: add_left_commute [of _ n]) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

117 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

118 
lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)" 
22718  119 
by (simp add: mult_commute mod_mult_self1) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

120 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

121 
lemma mod_mult_distrib: "(m mod n) * (k::nat) = (m*k) mod (n*k)" 
22718  122 
apply (cases "n = 0", simp) 
123 
apply (cases "k = 0", simp) 

124 
apply (induct m rule: nat_less_induct) 

125 
apply (subst mod_if, simp) 

126 
apply (simp add: mod_geq diff_mult_distrib) 

127 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

128 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

129 
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)" 
22718  130 
by (simp add: mult_commute [of k] mod_mult_distrib) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

131 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

132 
lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)" 
22718  133 
apply (cases "n = 0", simp) 
134 
apply (induct m, simp) 

135 
apply (rename_tac k) 

136 
apply (cut_tac m = "k * n" and n = n in mod_add_self2) 

137 
apply (simp add: add_commute) 

138 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

139 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

140 
lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)" 
22718  141 
by (simp add: mult_commute mod_mult_self_is_0) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

142 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

143 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

144 
subsection{*Quotient*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

145 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

146 
lemma div_less [simp]: "m<n ==> m div n = (0::nat)" 
22718  147 
by (rule div_eq [THEN wf_less_trans], simp) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

148 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

149 
lemma div_geq: "[ 0<n; ~m<n ] ==> m div n = Suc((mn) div n)" 
22718  150 
apply (rule div_eq [THEN wf_less_trans]) 
151 
apply (simp add: cut_apply less_eq) 

152 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

153 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

154 
(*Avoids the ugly ~m<n above*) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

155 
lemma le_div_geq: "[ 0<n; n\<le>m ] ==> m div n = Suc((mn) div n)" 
22718  156 
by (simp add: div_geq linorder_not_less) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

157 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

158 
lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((mn) div n))" 
22718  159 
by (simp add: div_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

160 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

161 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

162 
(*Main Result about quotient and remainder.*) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

163 
lemma mod_div_equality: "(m div n)*n + m mod n = (m::nat)" 
22718  164 
apply (cases "n = 0", simp) 
165 
apply (induct m rule: nat_less_induct) 

166 
apply (subst mod_if) 

167 
apply (simp_all add: add_assoc div_geq add_diff_inverse) 

168 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

169 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

170 
lemma mod_div_equality2: "n * (m div n) + m mod n = (m::nat)" 
22718  171 
apply (cut_tac m = m and n = n in mod_div_equality) 
172 
apply (simp add: mult_commute) 

173 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

174 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

175 
subsection{*Simproc for Cancelling Div and Mod*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

176 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

177 
lemma div_mod_equality: "((m div n)*n + m mod n) + k = (m::nat) + k" 
22718  178 
by (simp add: mod_div_equality) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

179 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

180 
lemma div_mod_equality2: "(n*(m div n) + m mod n) + k = (m::nat) + k" 
22718  181 
by (simp add: mod_div_equality2) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

182 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

183 
ML 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

184 
{* 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

185 
structure CancelDivModData = 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

186 
struct 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

187 

22718  188 
val div_name = @{const_name Divides.div}; 
189 
val mod_name = @{const_name Divides.mod}; 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

190 
val mk_binop = HOLogic.mk_binop; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

191 
val mk_sum = NatArithUtils.mk_sum; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

192 
val dest_sum = NatArithUtils.dest_sum; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

193 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

194 
(*logic*) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

195 

22718  196 
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

197 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

198 
val trans = trans 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

199 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

200 
val prove_eq_sums = 
22718  201 
let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac} 
17609
5156b731ebc8
Provers/cancel_sums.ML: Simplifier.inherit_bounds;
wenzelm
parents:
17508
diff
changeset

202 
in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end; 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

203 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

204 
end; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

205 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

206 
structure CancelDivMod = CancelDivModFun(CancelDivModData); 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

207 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

208 
val cancel_div_mod_proc = NatArithUtils.prep_simproc 
20044
92cc2f4c7335
simprocs: no theory argument  use simpset context instead;
wenzelm
parents:
18702
diff
changeset

209 
("cancel_div_mod", ["(m::nat) + n"], K CancelDivMod.proc); 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

210 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

211 
Addsimprocs[cancel_div_mod_proc]; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

212 
*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

213 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

214 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

215 
(* a simple rearrangement of mod_div_equality: *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

216 
lemma mult_div_cancel: "(n::nat) * (m div n) = m  (m mod n)" 
22718  217 
by (cut_tac m = m and n = n in mod_div_equality2, arith) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

218 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

219 
lemma mod_less_divisor [simp]: "0<n ==> m mod n < (n::nat)" 
22718  220 
apply (induct m rule: nat_less_induct) 
221 
apply (rename_tac m) 

222 
apply (case_tac "m<n", simp) 

223 
txt{*case @{term "n \<le> m"}*} 

224 
apply (simp add: mod_geq) 

225 
done 

15439  226 

227 
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)" 

22718  228 
apply (drule mod_less_divisor [where m = m]) 
229 
apply simp 

230 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

231 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

232 
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)" 
22718  233 
by (cut_tac m = "m*n" and n = n in mod_div_equality, auto) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

234 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

235 
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)" 
22718  236 
by (simp add: mult_commute div_mult_self_is_m) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

237 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

238 
(*mod_mult_distrib2 above is the counterpart for remainder*) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

239 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

240 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

241 
subsection{*Proving facts about Quotient and Remainder*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

242 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

243 
lemma unique_quotient_lemma: 
22718  244 
"[ b*q' + r' \<le> b*q + r; x < b; r < b ] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

245 
==> q' \<le> (q::nat)" 
22718  246 
apply (rule leI) 
247 
apply (subst less_iff_Suc_add) 

248 
apply (auto simp add: add_mult_distrib2) 

249 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

250 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

251 
lemma unique_quotient: 
22718  252 
"[ quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); 0 < b ] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

253 
==> q = q'" 
22718  254 
apply (simp add: split_ifs quorem_def) 
255 
apply (blast intro: order_antisym 

256 
dest: order_eq_refl [THEN unique_quotient_lemma] sym) 

257 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

258 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

259 
lemma unique_remainder: 
22718  260 
"[ quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); 0 < b ] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

261 
==> r = r'" 
22718  262 
apply (subgoal_tac "q = q'") 
263 
prefer 2 apply (blast intro: unique_quotient) 

264 
apply (simp add: quorem_def) 

265 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

266 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

267 
lemma quorem_div_mod: "0 < b ==> quorem ((a, b), (a div b, a mod b))" 
22718  268 
unfolding quorem_def by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

269 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

270 
lemma quorem_div: "[ quorem((a,b),(q,r)); 0 < b ] ==> a div b = q" 
22718  271 
by (simp add: quorem_div_mod [THEN unique_quotient]) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

272 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

273 
lemma quorem_mod: "[ quorem((a,b),(q,r)); 0 < b ] ==> a mod b = r" 
22718  274 
by (simp add: quorem_div_mod [THEN unique_remainder]) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

275 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

276 
(** A dividend of zero **) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

277 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

278 
lemma div_0 [simp]: "0 div m = (0::nat)" 
22718  279 
by (cases "m = 0") simp_all 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

280 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

281 
lemma mod_0 [simp]: "0 mod m = (0::nat)" 
22718  282 
by (cases "m = 0") simp_all 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

283 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

284 
(** proving (a*b) div c = a * (b div c) + a * (b mod c) **) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

285 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

286 
lemma quorem_mult1_eq: 
22718  287 
"[ quorem((b,c),(q,r)); 0 < c ] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

288 
==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))" 
22718  289 
by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

290 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

291 
lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)" 
22718  292 
apply (cases "c = 0", simp) 
293 
apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div]) 

294 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

295 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

296 
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)" 
22718  297 
apply (cases "c = 0", simp) 
298 
apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod]) 

299 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

300 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

301 
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c" 
22718  302 
apply (rule trans) 
303 
apply (rule_tac s = "b*a mod c" in trans) 

304 
apply (rule_tac [2] mod_mult1_eq) 

305 
apply (simp_all add: mult_commute) 

306 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

307 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

308 
lemma mod_mult_distrib_mod: "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c" 
22718  309 
apply (rule mod_mult1_eq' [THEN trans]) 
310 
apply (rule mod_mult1_eq) 

311 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

312 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

313 
(** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

314 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

315 
lemma quorem_add1_eq: 
22718  316 
"[ quorem((a,c),(aq,ar)); quorem((b,c),(bq,br)); 0 < c ] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

317 
==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))" 
22718  318 
by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

319 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

320 
(*NOT suitable for rewriting: the RHS has an instance of the LHS*) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

321 
lemma div_add1_eq: 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

322 
"(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)" 
22718  323 
apply (cases "c = 0", simp) 
324 
apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod quorem_div_mod) 

325 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

326 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

327 
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c" 
22718  328 
apply (cases "c = 0", simp) 
329 
apply (blast intro: quorem_div_mod quorem_div_mod quorem_add1_eq [THEN quorem_mod]) 

330 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

331 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

332 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

333 
subsection{*Proving @{term "a div (b*c) = (a div b) div c"}*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

334 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

335 
(** first, a lemma to bound the remainder **) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

336 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

337 
lemma mod_lemma: "[ (0::nat) < c; r < b ] ==> b * (q mod c) + r < b * c" 
22718  338 
apply (cut_tac m = q and n = c in mod_less_divisor) 
339 
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto) 

340 
apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst) 

341 
apply (simp add: add_mult_distrib2) 

342 
done 

10559
d3fd54fc659b
many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents:
10214
diff
changeset

343 

22718  344 
lemma quorem_mult2_eq: "[ quorem ((a,b), (q,r)); 0 < b; 0 < c ] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

345 
==> quorem ((a, b*c), (q div c, b*(q mod c) + r))" 
22718  346 
by (auto simp add: mult_ac quorem_def add_mult_distrib2 [symmetric] mod_lemma) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

347 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

348 
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)" 
22718  349 
apply (cases "b = 0", simp) 
350 
apply (cases "c = 0", simp) 

351 
apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div]) 

352 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

353 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

354 
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)" 
22718  355 
apply (cases "b = 0", simp) 
356 
apply (cases "c = 0", simp) 

357 
apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod]) 

358 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

359 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

360 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

361 
subsection{*Cancellation of Common Factors in Division*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

362 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

363 
lemma div_mult_mult_lemma: 
22718  364 
"[ (0::nat) < b; 0 < c ] ==> (c*a) div (c*b) = a div b" 
365 
by (auto simp add: div_mult2_eq) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

366 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

367 
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b" 
22718  368 
apply (cases "b = 0") 
369 
apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma) 

370 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

371 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

372 
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b" 
22718  373 
apply (drule div_mult_mult1) 
374 
apply (auto simp add: mult_commute) 

375 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

376 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

377 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

378 
(*Distribution of Factors over Remainders: 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

379 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

380 
Could prove these as in Integ/IntDiv.ML, but we already have 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

381 
mod_mult_distrib and mod_mult_distrib2 above! 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

382 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

383 
Goal "(c*a) mod (c*b) = (c::nat) * (a mod b)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

384 
qed "mod_mult_mult1"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

385 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

386 
Goal "(a*c) mod (b*c) = (a mod b) * (c::nat)"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

387 
qed "mod_mult_mult2"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

388 
***) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

389 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

390 
subsection{*Further Facts about Quotient and Remainder*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

391 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

392 
lemma div_1 [simp]: "m div Suc 0 = m" 
22718  393 
by (induct m) (simp_all add: div_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

394 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

395 
lemma div_self [simp]: "0<n ==> n div n = (1::nat)" 
22718  396 
by (simp add: div_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

397 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

398 
lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)" 
22718  399 
apply (subgoal_tac "(n + m) div n = Suc ((n+mn) div n) ") 
400 
apply (simp add: add_commute) 

401 
apply (subst div_geq [symmetric], simp_all) 

402 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

403 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

404 
lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)" 
22718  405 
by (simp add: add_commute div_add_self2) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

406 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

407 
lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n" 
22718  408 
apply (subst div_add1_eq) 
409 
apply (subst div_mult1_eq, simp) 

410 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

411 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

412 
lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)" 
22718  413 
by (simp add: mult_commute div_mult_self1) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

414 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

415 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

416 
(* Monotonicity of div in first argument *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

417 
lemma div_le_mono [rule_format (no_asm)]: 
22718  418 
"\<forall>m::nat. m \<le> n > (m div k) \<le> (n div k)" 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

419 
apply (case_tac "k=0", simp) 
15251  420 
apply (induct "n" rule: nat_less_induct, clarify) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

421 
apply (case_tac "n<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

422 
(* 1 case n<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

423 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

424 
(* 2 case n >= k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

425 
apply (case_tac "m<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

426 
(* 2.1 case m<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

427 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

428 
(* 2.2 case m>=k *) 
15439  429 
apply (simp add: div_geq diff_le_mono) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

430 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

431 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

432 
(* Antimonotonicity of div in second argument *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

433 
lemma div_le_mono2: "!!m::nat. [ 0<m; m\<le>n ] ==> (k div n) \<le> (k div m)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

434 
apply (subgoal_tac "0<n") 
22718  435 
prefer 2 apply simp 
15251  436 
apply (induct_tac k rule: nat_less_induct) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

437 
apply (rename_tac "k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

438 
apply (case_tac "k<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

439 
apply (subgoal_tac "~ (k<m) ") 
22718  440 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

441 
apply (simp add: div_geq) 
15251  442 
apply (subgoal_tac "(kn) div n \<le> (km) div n") 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

443 
prefer 2 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

444 
apply (blast intro: div_le_mono diff_le_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

445 
apply (rule le_trans, simp) 
15439  446 
apply (simp) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

447 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

448 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

449 
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

450 
apply (case_tac "n=0", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

451 
apply (subgoal_tac "m div n \<le> m div 1", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

452 
apply (rule div_le_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

453 
apply (simp_all (no_asm_simp)) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

454 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

455 

22718  456 
(* Similar for "less than" *) 
17085  457 
lemma div_less_dividend [rule_format]: 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

458 
"!!n::nat. 1<n ==> 0 < m > m div n < m" 
15251  459 
apply (induct_tac m rule: nat_less_induct) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

460 
apply (rename_tac "m") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

461 
apply (case_tac "m<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

462 
apply (subgoal_tac "0<n") 
22718  463 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

464 
apply (simp add: div_geq) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

465 
apply (case_tac "n<m") 
15251  466 
apply (subgoal_tac "(mn) div n < (mn) ") 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

467 
apply (rule impI less_trans_Suc)+ 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

468 
apply assumption 
15439  469 
apply (simp_all) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

470 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

471 

17085  472 
declare div_less_dividend [simp] 
473 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

474 
text{*A fact for the mutilated chess board*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

475 
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

476 
apply (case_tac "n=0", simp) 
15251  477 
apply (induct "m" rule: nat_less_induct) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

478 
apply (case_tac "Suc (na) <n") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

479 
(* case Suc(na) < n *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

480 
apply (frule lessI [THEN less_trans], simp add: less_not_refl3) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

481 
(* case n \<le> Suc(na) *) 
16796  482 
apply (simp add: linorder_not_less le_Suc_eq mod_geq) 
15439  483 
apply (auto simp add: Suc_diff_le le_mod_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

484 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

485 

14437  486 
lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)" 
22718  487 
by (cases "n = 0") auto 
14437  488 

489 
lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)" 

22718  490 
by (cases "n = 0") auto 
14437  491 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

492 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

493 
subsection{*The Divides Relation*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

494 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

495 
lemma dvdI [intro?]: "n = m * k ==> m dvd n" 
22718  496 
unfolding dvd_def by blast 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

497 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

498 
lemma dvdE [elim?]: "!!P. [m dvd n; !!k. n = m*k ==> P] ==> P" 
22718  499 
unfolding dvd_def by blast 
13152  500 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

501 
lemma dvd_0_right [iff]: "m dvd (0::nat)" 
22718  502 
unfolding dvd_def by (blast intro: mult_0_right [symmetric]) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

503 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

504 
lemma dvd_0_left: "0 dvd m ==> m = (0::nat)" 
22718  505 
by (force simp add: dvd_def) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

506 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

507 
lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)" 
22718  508 
by (blast intro: dvd_0_left) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

509 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

510 
lemma dvd_1_left [iff]: "Suc 0 dvd k" 
22718  511 
unfolding dvd_def by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

512 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

513 
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)" 
22718  514 
by (simp add: dvd_def) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

515 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

516 
lemma dvd_refl [simp]: "m dvd (m::nat)" 
22718  517 
unfolding dvd_def by (blast intro: mult_1_right [symmetric]) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

518 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

519 
lemma dvd_trans [trans]: "[ m dvd n; n dvd p ] ==> m dvd (p::nat)" 
22718  520 
unfolding dvd_def by (blast intro: mult_assoc) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

521 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

522 
lemma dvd_anti_sym: "[ m dvd n; n dvd m ] ==> m = (n::nat)" 
22718  523 
unfolding dvd_def 
524 
by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

525 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

526 
lemma dvd_add: "[ k dvd m; k dvd n ] ==> k dvd (m+n :: nat)" 
22718  527 
unfolding dvd_def 
528 
by (blast intro: add_mult_distrib2 [symmetric]) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

529 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

530 
lemma dvd_diff: "[ k dvd m; k dvd n ] ==> k dvd (mn :: nat)" 
22718  531 
unfolding dvd_def 
532 
by (blast intro: diff_mult_distrib2 [symmetric]) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

533 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

534 
lemma dvd_diffD: "[ k dvd mn; k dvd n; n\<le>m ] ==> k dvd (m::nat)" 
22718  535 
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst]) 
536 
apply (blast intro: dvd_add) 

537 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

538 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

539 
lemma dvd_diffD1: "[ k dvd mn; k dvd m; n\<le>m ] ==> k dvd (n::nat)" 
22718  540 
by (drule_tac m = m in dvd_diff, auto) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

541 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

542 
lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)" 
22718  543 
unfolding dvd_def by (blast intro: mult_left_commute) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

544 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

545 
lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)" 
22718  546 
apply (subst mult_commute) 
547 
apply (erule dvd_mult) 

548 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

549 

17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16796
diff
changeset

550 
lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)" 
22718  551 
by (rule dvd_refl [THEN dvd_mult]) 
17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16796
diff
changeset

552 

fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16796
diff
changeset

553 
lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)" 
22718  554 
by (rule dvd_refl [THEN dvd_mult2]) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

555 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

556 
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))" 
22718  557 
apply (rule iffI) 
558 
apply (erule_tac [2] dvd_add) 

559 
apply (rule_tac [2] dvd_refl) 

560 
apply (subgoal_tac "n = (n+k) k") 

561 
prefer 2 apply simp 

562 
apply (erule ssubst) 

563 
apply (erule dvd_diff) 

564 
apply (rule dvd_refl) 

565 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

566 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

567 
lemma dvd_mod: "!!n::nat. [ f dvd m; f dvd n ] ==> f dvd m mod n" 
22718  568 
unfolding dvd_def 
569 
apply (case_tac "n = 0", auto) 

570 
apply (blast intro: mod_mult_distrib2 [symmetric]) 

571 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

572 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

573 
lemma dvd_mod_imp_dvd: "[ (k::nat) dvd m mod n; k dvd n ] ==> k dvd m" 
22718  574 
apply (subgoal_tac "k dvd (m div n) *n + m mod n") 
575 
apply (simp add: mod_div_equality) 

576 
apply (simp only: dvd_add dvd_mult) 

577 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

578 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

579 
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)" 
22718  580 
by (blast intro: dvd_mod_imp_dvd dvd_mod) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

581 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

582 
lemma dvd_mult_cancel: "!!k::nat. [ k*m dvd k*n; 0<k ] ==> m dvd n" 
22718  583 
unfolding dvd_def 
584 
apply (erule exE) 

585 
apply (simp add: mult_ac) 

586 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

587 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

588 
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))" 
22718  589 
apply auto 
590 
apply (subgoal_tac "m*n dvd m*1") 

591 
apply (drule dvd_mult_cancel, auto) 

592 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

593 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

594 
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))" 
22718  595 
apply (subst mult_commute) 
596 
apply (erule dvd_mult_cancel1) 

597 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

598 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

599 
lemma mult_dvd_mono: "[ i dvd m; j dvd n] ==> i*j dvd (m*n :: nat)" 
22718  600 
apply (unfold dvd_def, clarify) 
601 
apply (rule_tac x = "k*ka" in exI) 

602 
apply (simp add: mult_ac) 

603 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

604 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

605 
lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k" 
22718  606 
by (simp add: dvd_def mult_assoc, blast) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

607 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

608 
lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k" 
22718  609 
apply (unfold dvd_def, clarify) 
610 
apply (rule_tac x = "i*k" in exI) 

611 
apply (simp add: mult_ac) 

612 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

613 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

614 
lemma dvd_imp_le: "[ k dvd n; 0 < n ] ==> k \<le> (n::nat)" 
22718  615 
apply (unfold dvd_def, clarify) 
616 
apply (simp_all (no_asm_use) add: zero_less_mult_iff) 

617 
apply (erule conjE) 

618 
apply (rule le_trans) 

619 
apply (rule_tac [2] le_refl [THEN mult_le_mono]) 

620 
apply (erule_tac [2] Suc_leI, simp) 

621 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

622 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

623 
lemma dvd_eq_mod_eq_0: "!!k::nat. (k dvd n) = (n mod k = 0)" 
22718  624 
apply (unfold dvd_def) 
625 
apply (case_tac "k=0", simp, safe) 

626 
apply (simp add: mult_commute) 

627 
apply (rule_tac t = n and n1 = k in mod_div_equality [THEN subst]) 

628 
apply (subst mult_commute, simp) 

629 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

630 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

631 
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)" 
22718  632 
apply (subgoal_tac "m mod n = 0") 
633 
apply (simp add: mult_div_cancel) 

634 
apply (simp only: dvd_eq_mod_eq_0) 

635 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

636 

21408  637 
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n" 
22718  638 
apply (unfold dvd_def) 
639 
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst]) 

640 
apply (simp add: power_add) 

641 
done 

21408  642 

643 
lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat)  n=0)" 

22718  644 
by (induct n) auto 
21408  645 

646 
lemma power_le_dvd [rule_format]: "k^j dvd n > i\<le>j > k^i dvd (n::nat)" 

22718  647 
apply (induct j) 
648 
apply (simp_all add: le_Suc_eq) 

649 
apply (blast dest!: dvd_mult_right) 

650 
done 

21408  651 

652 
lemma power_dvd_imp_le: "[i^m dvd i^n; (1::nat) < i] ==> m \<le> n" 

22718  653 
apply (rule power_le_imp_le_exp, assumption) 
654 
apply (erule dvd_imp_le, simp) 

655 
done 

21408  656 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

657 
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)" 
22718  658 
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def) 
17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16796
diff
changeset

659 

22718  660 
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

661 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

662 
(*Loses information, namely we also have r<d provided d is nonzero*) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

663 
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d" 
22718  664 
apply (cut_tac m = m in mod_div_equality) 
665 
apply (simp only: add_ac) 

666 
apply (blast intro: sym) 

667 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

668 

14131  669 

13152  670 
lemma split_div: 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

671 
"P(n div k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

672 
((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

673 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

674 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

675 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

676 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

677 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

678 
assume "k = 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

679 
with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

680 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

681 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

682 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

683 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

684 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

685 
assume n: "n = k*i + j" and j: "j < k" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

686 
show "P i" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

687 
proof (cases) 
22718  688 
assume "i = 0" 
689 
with n j P show "P i" by simp 

13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

690 
next 
22718  691 
assume "i \<noteq> 0" 
692 
with not0 n j P show "P i" by(simp add:add_ac) 

13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

693 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

694 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

695 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

696 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

697 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

698 
show ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

699 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

700 
assume "k = 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

701 
with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

702 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

703 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

704 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

705 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  706 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

707 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

708 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

709 

13882  710 
lemma split_div_lemma: 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

711 
"0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))" 
13882  712 
apply (rule iffI) 
713 
apply (rule_tac a=m and r = "m  n * q" and r' = "m mod n" in unique_quotient) 

16733
236dfafbeb63
linear arithmetic now takes "&" in assumptions apart.
nipkow
parents:
15439
diff
changeset

714 
prefer 3; apply assumption 
20432
07ec57376051
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20380
diff
changeset

715 
apply (simp_all add: quorem_def) apply arith 
13882  716 
apply (rule conjI) 
717 
apply (rule_tac P="%x. n * (m div n) \<le> x" in 

718 
subst [OF mod_div_equality [of _ n]]) 

719 
apply (simp only: add: mult_ac) 

720 
apply (rule_tac P="%x. x < n + n * (m div n)" in 

721 
subst [OF mod_div_equality [of _ n]]) 

722 
apply (simp only: add: mult_ac add_ac) 

14208  723 
apply (rule add_less_mono1, simp) 
13882  724 
done 
725 

726 
theorem split_div': 

727 
"P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or> 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

728 
(\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))" 
13882  729 
apply (case_tac "0 < n") 
730 
apply (simp only: add: split_div_lemma) 

731 
apply (simp_all add: DIVISION_BY_ZERO_DIV) 

732 
done 

733 

13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

734 
lemma split_mod: 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

735 
"P(n mod k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

736 
((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

737 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

738 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

739 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

740 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

741 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

742 
assume "k = 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

743 
with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

744 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

745 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

746 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

747 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

748 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

749 
assume "n = k*i + j" "j < k" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

750 
thus "P j" using not0 P by(simp add:add_ac mult_ac) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

751 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

752 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

753 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

754 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

755 
show ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

756 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

757 
assume "k = 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

758 
with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

759 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

760 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

761 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

762 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  763 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

764 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

765 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

766 

13882  767 
theorem mod_div_equality': "(m::nat) mod n = m  (m div n) * n" 
768 
apply (rule_tac P="%x. m mod n = x  (m div n) * n" in 

769 
subst [OF mod_div_equality [of _ n]]) 

770 
apply arith 

771 
done 

772 

22800  773 
lemma div_mod_equality': 
774 
fixes m n :: nat 

775 
shows "m div n * n = m  m mod n" 

776 
proof  

777 
have "m mod n \<le> m mod n" .. 

778 
from div_mod_equality have 

779 
"m div n * n + m mod n  m mod n = m  m mod n" by simp 

780 
with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have 

781 
"m div n * n + (m mod n  m mod n) = m  m mod n" 

782 
by simp 

783 
then show ?thesis by simp 

784 
qed 

785 

786 

14640  787 
subsection {*An ``induction'' law for modulus arithmetic.*} 
788 

789 
lemma mod_induct_0: 

790 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

791 
and base: "P i" and i: "i<p" 

792 
shows "P 0" 

793 
proof (rule ccontr) 

794 
assume contra: "\<not>(P 0)" 

795 
from i have p: "0<p" by simp 

796 
have "\<forall>k. 0<k \<longrightarrow> \<not> P (pk)" (is "\<forall>k. ?A k") 

797 
proof 

798 
fix k 

799 
show "?A k" 

800 
proof (induct k) 

801 
show "?A 0" by simp  "by contradiction" 

802 
next 

803 
fix n 

804 
assume ih: "?A n" 

805 
show "?A (Suc n)" 

806 
proof (clarsimp) 

22718  807 
assume y: "P (p  Suc n)" 
808 
have n: "Suc n < p" 

809 
proof (rule ccontr) 

810 
assume "\<not>(Suc n < p)" 

811 
hence "p  Suc n = 0" 

812 
by simp 

813 
with y contra show "False" 

814 
by simp 

815 
qed 

816 
hence n2: "Suc (p  Suc n) = pn" by arith 

817 
from p have "p  Suc n < p" by arith 

818 
with y step have z: "P ((Suc (p  Suc n)) mod p)" 

819 
by blast 

820 
show "False" 

821 
proof (cases "n=0") 

822 
case True 

823 
with z n2 contra show ?thesis by simp 

824 
next 

825 
case False 

826 
with p have "pn < p" by arith 

827 
with z n2 False ih show ?thesis by simp 

828 
qed 

14640  829 
qed 
830 
qed 

831 
qed 

832 
moreover 

833 
from i obtain k where "0<k \<and> i+k=p" 

834 
by (blast dest: less_imp_add_positive) 

835 
hence "0<k \<and> i=pk" by auto 

836 
moreover 

837 
note base 

838 
ultimately 

839 
show "False" by blast 

840 
qed 

841 

842 
lemma mod_induct: 

843 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

844 
and base: "P i" and i: "i<p" and j: "j<p" 

845 
shows "P j" 

846 
proof  

847 
have "\<forall>j<p. P j" 

848 
proof 

849 
fix j 

850 
show "j<p \<longrightarrow> P j" (is "?A j") 

851 
proof (induct j) 

852 
from step base i show "?A 0" 

22718  853 
by (auto elim: mod_induct_0) 
14640  854 
next 
855 
fix k 

856 
assume ih: "?A k" 

857 
show "?A (Suc k)" 

858 
proof 

22718  859 
assume suc: "Suc k < p" 
860 
hence k: "k<p" by simp 

861 
with ih have "P k" .. 

862 
with step k have "P (Suc k mod p)" 

863 
by blast 

864 
moreover 

865 
from suc have "Suc k mod p = Suc k" 

866 
by simp 

867 
ultimately 

868 
show "P (Suc k)" by simp 

14640  869 
qed 
870 
qed 

871 
qed 

872 
with j show ?thesis by blast 

873 
qed 

874 

875 

18202
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

876 
lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c" 
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

877 
apply (rule trans [symmetric]) 
22718  878 
apply (rule mod_add1_eq, simp) 
18202
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

879 
apply (rule mod_add1_eq [symmetric]) 
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

880 
done 
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

881 

46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

882 
lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c" 
22718  883 
apply (rule trans [symmetric]) 
884 
apply (rule mod_add1_eq, simp) 

885 
apply (rule mod_add1_eq [symmetric]) 

886 
done 

18202
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

887 

22800  888 
lemma mod_div_decomp: 
889 
fixes n k :: nat 

890 
obtains m q where "m = n div k" and "q = n mod k" 

891 
and "n = m * k + q" 

892 
proof  

893 
from mod_div_equality have "n = n div k * k + n mod k" by auto 

894 
moreover have "n div k = n div k" .. 

895 
moreover have "n mod k = n mod k" .. 

896 
note that ultimately show thesis by blast 

897 
qed 

898 

20589  899 

22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22718
diff
changeset

900 
subsection {* Code generation for div, mod and dvd on nat *} 
20589  901 

22845  902 
definition [code func del]: 
20589  903 
"divmod (m\<Colon>nat) n = (m div n, m mod n)" 
904 

22718  905 
lemma divmod_zero [code]: "divmod m 0 = (0, m)" 
20589  906 
unfolding divmod_def by simp 
907 

908 
lemma divmod_succ [code]: 

909 
"divmod m (Suc k) = (if m < Suc k then (0, m) else 

910 
let 

911 
(p, q) = divmod (m  Suc k) (Suc k) 

22718  912 
in (Suc p, q))" 
20589  913 
unfolding divmod_def Let_def split_def 
914 
by (auto intro: div_geq mod_geq) 

915 

22718  916 
lemma div_divmod [code]: "m div n = fst (divmod m n)" 
20589  917 
unfolding divmod_def by simp 
918 

22718  919 
lemma mod_divmod [code]: "m mod n = snd (divmod m n)" 
20589  920 
unfolding divmod_def by simp 
921 

22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22718
diff
changeset

922 
definition 
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22718
diff
changeset

923 
dvd_nat :: "nat \<Rightarrow> nat \<Rightarrow> bool" 
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22718
diff
changeset

924 
where 
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22718
diff
changeset

925 
"dvd_nat m n \<longleftrightarrow> n mod m = (0 \<Colon> nat)" 
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22718
diff
changeset

926 

5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22718
diff
changeset

927 
lemma [code inline]: 
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22718
diff
changeset

928 
"op dvd = dvd_nat" 
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22718
diff
changeset

929 
by (auto simp add: dvd_nat_def dvd_eq_mod_eq_0 expand_fun_eq) 
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22718
diff
changeset

930 

21191  931 
code_modulename SML 
932 
Divides Integer 

20640  933 

21911
e29bcab0c81c
added OCaml code generation (without dictionaries)
haftmann
parents:
21408
diff
changeset

934 
code_modulename OCaml 
e29bcab0c81c
added OCaml code generation (without dictionaries)
haftmann
parents:
21408
diff
changeset

935 
Divides Integer 
e29bcab0c81c
added OCaml code generation (without dictionaries)
haftmann
parents:
21408
diff
changeset

936 

22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22718
diff
changeset

937 
hide (open) const divmod dvd_nat 
20589  938 

939 
subsection {* Legacy bindings *} 

940 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

941 
ML 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

942 
{* 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

943 
val div_def = thm "div_def" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

944 
val mod_def = thm "mod_def" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

945 
val dvd_def = thm "dvd_def" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

946 
val quorem_def = thm "quorem_def" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

947 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

948 
val wf_less_trans = thm "wf_less_trans"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

949 
val mod_eq = thm "mod_eq"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

950 
val div_eq = thm "div_eq"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

951 
val DIVISION_BY_ZERO_DIV = thm "DIVISION_BY_ZERO_DIV"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

952 
val DIVISION_BY_ZERO_MOD = thm "DIVISION_BY_ZERO_MOD"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

953 
val mod_less = thm "mod_less"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

954 
val mod_geq = thm "mod_geq"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

955 
val le_mod_geq = thm "le_mod_geq"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

956 
val mod_if = thm "mod_if"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

957 
val mod_1 = thm "mod_1"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

958 
val mod_self = thm "mod_self"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

959 
val mod_add_self2 = thm "mod_add_self2"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

960 
val mod_add_self1 = thm "mod_add_self1"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

961 
val mod_mult_self1 = thm "mod_mult_self1"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

962 
val mod_mult_self2 = thm "mod_mult_self2"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

963 
val mod_mult_distrib = thm "mod_mult_distrib"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

964 
val mod_mult_distrib2 = thm "mod_mult_distrib2"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

965 
val mod_mult_self_is_0 = thm "mod_mult_self_is_0"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

966 
val mod_mult_self1_is_0 = thm "mod_mult_self1_is_0"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

967 
val div_less = thm "div_less"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

968 
val div_geq = thm "div_geq"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

969 
val le_div_geq = thm "le_div_geq"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

970 
val div_if = thm "div_if"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

971 
val mod_div_equality = thm "mod_div_equality"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

972 
val mod_div_equality2 = thm "mod_div_equality2"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

973 
val div_mod_equality = thm "div_mod_equality"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

974 
val div_mod_equality2 = thm "div_mod_equality2"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

975 
val mult_div_cancel = thm "mult_div_cancel"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

976 
val mod_less_divisor = thm "mod_less_divisor"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

977 
val div_mult_self_is_m = thm "div_mult_self_is_m"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

978 
val div_mult_self1_is_m = thm "div_mult_self1_is_m"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

979 
val unique_quotient_lemma = thm "unique_quotient_lemma"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

980 
val unique_quotient = thm "unique_quotient"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

981 
val unique_remainder = thm "unique_remainder"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

982 
val div_0 = thm "div_0"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

983 
val mod_0 = thm "mod_0"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

984 
val div_mult1_eq = thm "div_mult1_eq"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

985 
val mod_mult1_eq = thm "mod_mult1_eq"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

986 
val mod_mult1_eq' = thm "mod_mult1_eq'"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

987 
val mod_mult_distrib_mod = thm "mod_mult_distrib_mod"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

988 
val div_add1_eq = thm "div_add1_eq"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

989 
val mod_add1_eq = thm "mod_add1_eq"; 
18202
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

990 
val mod_add_left_eq = thm "mod_add_left_eq"; 
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

991 
val mod_add_right_eq = thm "mod_add_right_eq"; 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

992 
val mod_lemma = thm "mod_lemma"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

993 
val div_mult2_eq = thm "div_mult2_eq"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

994 
val mod_mult2_eq = thm "mod_mult2_eq"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

995 
val div_mult_mult_lemma = thm "div_mult_mult_lemma"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

996 
val div_mult_mult1 = thm "div_mult_mult1"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

997 
val div_mult_mult2 = thm "div_mult_mult2"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

998 
val div_1 = thm "div_1"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

999 
val div_self = thm "div_self"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1000 
val div_add_self2 = thm "div_add_self2"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1001 
val div_add_self1 = thm "div_add_self1"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1002 
val div_mult_self1 = thm "div_mult_self1"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1003 
val div_mult_self2 = thm "div_mult_self2"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1004 
val div_le_mono = thm "div_le_mono"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1005 
val div_le_mono2 = thm "div_le_mono2"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1006 
val div_le_dividend = thm "div_le_dividend"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1007 
val div_less_dividend = thm "div_less_dividend"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1008 
val mod_Suc = thm "mod_Suc"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1009 
val dvdI = thm "dvdI"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1010 
val dvdE = thm "dvdE"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1011 
val dvd_0_right = thm "dvd_0_right"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1012 
val dvd_0_left = thm "dvd_0_left"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1013 
val dvd_0_left_iff = thm "dvd_0_left_iff"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1014 
val dvd_1_left = thm "dvd_1_left"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1015 
val dvd_1_iff_1 = thm "dvd_1_iff_1"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1016 
val dvd_refl = thm "dvd_refl"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1017 
val dvd_trans = thm "dvd_trans"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1018 
val dvd_anti_sym = thm "dvd_anti_sym"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1019 
val dvd_add = thm "dvd_add"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1020 
val dvd_diff = thm "dvd_diff"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1021 
val dvd_diffD = thm "dvd_diffD"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1022 
val dvd_diffD1 = thm "dvd_diffD1"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1023 
val dvd_mult = thm "dvd_mult"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1024 
val dvd_mult2 = thm "dvd_mult2"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1025 
val dvd_reduce = thm "dvd_reduce"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1026 
val dvd_mod = thm "dvd_mod"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1027 
val dvd_mod_imp_dvd = thm "dvd_mod_imp_dvd"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1028 
val dvd_mod_iff = thm "dvd_mod_iff"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1029 
val dvd_mult_cancel = thm "dvd_mult_cancel"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1030 
val dvd_mult_cancel1 = thm "dvd_mult_cancel1"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1031 
val dvd_mult_cancel2 = thm "dvd_mult_cancel2"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1032 
val mult_dvd_mono = thm "mult_dvd_mono"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1033 
val dvd_mult_left = thm "dvd_mult_left"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1034 
val dvd_mult_right = thm "dvd_mult_right"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1035 
val dvd_imp_le = thm "dvd_imp_le"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1036 
val dvd_eq_mod_eq_0 = thm "dvd_eq_mod_eq_0"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1037 
val dvd_mult_div_cancel = thm "dvd_mult_div_cancel"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1038 
val mod_eq_0_iff = thm "mod_eq_0_iff"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1039 
val mod_eqD = thm "mod_eqD"; 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1040 
*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1041 

13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1042 
(* 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1043 
lemma split_div: 
13152  1044 
assumes m: "m \<noteq> 0" 
1045 
shows "P(n div m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P i)" 

1046 
(is "?P = ?Q") 

1047 
proof 

1048 
assume P: ?P 

1049 
show ?Q 

1050 
proof (intro allI impI) 

1051 
fix i j 

1052 
assume n: "n = m*i + j" and j: "j < m" 

1053 
show "P i" 

1054 
proof (cases) 

1055 
assume "i = 0" 

1056 
with n j P show "P i" by simp 

1057 
next 

1058 
assume "i \<noteq> 0" 

1059 
with n j P show "P i" by (simp add:add_ac div_mult_self1) 

1060 
qed 

1061 
qed 

1062 
next 

1063 
assume Q: ?Q 

1064 
from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"] 

13517  1065 
show ?P by simp 
13152  1066 
qed 
1067 

1068 
lemma split_mod: 

1069 
assumes m: "m \<noteq> 0" 

1070 
shows "P(n mod m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P j)" 

1071 
(is "?P = ?Q") 

1072 
proof 

1073 
assume P: ?P 

1074 
show ?Q 

1075 
proof (intro allI impI) 

1076 
fix i j 

1077 
assume "n = m*i + j" "j < m" 

1078 
thus "P j" using m P by(simp add:add_ac mult_ac) 

1079 
qed 

1080 
next 

1081 
assume Q: ?Q 

1082 
from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"] 

13517  1083 
show ?P by simp 
13152  1084 
qed 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1085 
*) 
3366  1086 
end 