author | wenzelm |
Thu, 01 Jul 1999 21:19:45 +0200 | |
changeset 6874 | 747f656e04ec |
parent 6865 | 5577ffe4c2f1 |
child 7007 | b46ccfee8e59 |
permissions | -rw-r--r-- |
3366 | 1 |
(* Title: HOL/Divides.ML |
2 |
ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1993 University of Cambridge |
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The division operators div, mod and the divides relation "dvd" |
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*) |
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(** Less-then properties **) |
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val wf_less_trans = [eq_reflection, wf_pred_nat RS wf_trancl] MRS |
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def_wfrec RS trans; |
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||
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(*** Remainder ***) |
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||
5069 | 17 |
Goal "(%m. m mod n) = wfrec (trancl pred_nat) \ |
5415 | 18 |
\ (%f j. if j<n then j else f (j-n))"; |
4089 | 19 |
by (simp_tac (simpset() addsimps [mod_def]) 1); |
3366 | 20 |
qed "mod_eq"; |
21 |
||
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Goal "m<n ==> m mod n = (m::nat)"; |
3366 | 23 |
by (rtac (mod_eq RS wf_less_trans) 1); |
24 |
by (Asm_simp_tac 1); |
|
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qed "mod_less"; |
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||
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Goal "[| 0<n; ~m<n |] ==> m mod n = (m-n) mod n"; |
3366 | 28 |
by (rtac (mod_eq RS wf_less_trans) 1); |
4089 | 29 |
by (asm_simp_tac (simpset() addsimps [diff_less, cut_apply, less_eq]) 1); |
3366 | 30 |
qed "mod_geq"; |
31 |
||
5415 | 32 |
(*Avoids the ugly ~m<n above*) |
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Goal "[| 0<n; n<=m |] ==> m mod n = (m-n) mod n"; |
|
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by (asm_simp_tac (simpset() addsimps [mod_geq, not_less_iff_le]) 1); |
|
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qed "le_mod_geq"; |
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||
4774 | 37 |
(*NOT suitable for rewriting: loops*) |
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Goal "0<n ==> m mod n = (if m<n then m else (m-n) mod n)"; |
4774 | 39 |
by (asm_simp_tac (simpset() addsimps [mod_less, mod_geq]) 1); |
40 |
qed "mod_if"; |
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41 |
||
5069 | 42 |
Goal "m mod 1 = 0"; |
3366 | 43 |
by (induct_tac "m" 1); |
4089 | 44 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [mod_less, mod_geq]))); |
3366 | 45 |
qed "mod_1"; |
46 |
Addsimps [mod_1]; |
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||
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Goal "0<n ==> n mod n = 0"; |
4089 | 49 |
by (asm_simp_tac (simpset() addsimps [mod_less, mod_geq]) 1); |
3366 | 50 |
qed "mod_self"; |
51 |
||
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Goal "0<n ==> (m+n) mod n = m mod n"; |
3366 | 53 |
by (subgoal_tac "(n + m) mod n = (n+m-n) mod n" 1); |
54 |
by (stac (mod_geq RS sym) 2); |
|
4089 | 55 |
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [add_commute]))); |
4811 | 56 |
qed "mod_add_self2"; |
4810 | 57 |
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Goal "0<n ==> (n+m) mod n = m mod n"; |
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by (asm_simp_tac (simpset() addsimps [add_commute, mod_add_self2]) 1); |
60 |
qed "mod_add_self1"; |
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4810 | 61 |
|
5069 | 62 |
Goal "!!n. 0<n ==> (m + k*n) mod n = m mod n"; |
4810 | 63 |
by (induct_tac "k" 1); |
5537 | 64 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps add_ac @ [mod_add_self1]))); |
4811 | 65 |
qed "mod_mult_self1"; |
4810 | 66 |
|
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Goal "0<n ==> (m + n*k) mod n = m mod n"; |
4811 | 68 |
by (asm_simp_tac (simpset() addsimps [mult_commute, mod_mult_self1]) 1); |
69 |
qed "mod_mult_self2"; |
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4810 | 70 |
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4811 | 71 |
Addsimps [mod_mult_self1, mod_mult_self2]; |
3366 | 72 |
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Goal "[| 0<k; 0<n |] ==> (m mod n)*k = (m*k) mod (n*k)"; |
3366 | 74 |
by (res_inst_tac [("n","m")] less_induct 1); |
4774 | 75 |
by (stac mod_if 1); |
76 |
by (Asm_simp_tac 1); |
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by (asm_simp_tac (simpset() addsimps [mod_less, mod_geq, |
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diff_less, diff_mult_distrib]) 1); |
|
3366 | 79 |
qed "mod_mult_distrib"; |
80 |
||
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Goal "[| 0<k; 0<n |] ==> k*(m mod n) = (k*m) mod (k*n)"; |
3366 | 82 |
by (res_inst_tac [("n","m")] less_induct 1); |
4774 | 83 |
by (stac mod_if 1); |
84 |
by (Asm_simp_tac 1); |
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by (asm_simp_tac (simpset() addsimps [mod_less, mod_geq, |
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diff_less, diff_mult_distrib2]) 1); |
|
3366 | 87 |
qed "mod_mult_distrib2"; |
88 |
||
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Goal "0<n ==> m*n mod n = 0"; |
3366 | 90 |
by (induct_tac "m" 1); |
4089 | 91 |
by (asm_simp_tac (simpset() addsimps [mod_less]) 1); |
5183 | 92 |
by (dres_inst_tac [("m","na*n")] mod_add_self2 1); |
4089 | 93 |
by (asm_full_simp_tac (simpset() addsimps [add_commute]) 1); |
3366 | 94 |
qed "mod_mult_self_is_0"; |
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Addsimps [mod_mult_self_is_0]; |
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||
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(*** Quotient ***) |
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||
5069 | 99 |
Goal "(%m. m div n) = wfrec (trancl pred_nat) \ |
3366 | 100 |
\ (%f j. if j<n then 0 else Suc (f (j-n)))"; |
4089 | 101 |
by (simp_tac (simpset() addsimps [div_def]) 1); |
3366 | 102 |
qed "div_eq"; |
103 |
||
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Goal "m<n ==> m div n = 0"; |
3366 | 105 |
by (rtac (div_eq RS wf_less_trans) 1); |
106 |
by (Asm_simp_tac 1); |
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qed "div_less"; |
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||
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Goal "[| 0<n; ~m<n |] ==> m div n = Suc((m-n) div n)"; |
3366 | 110 |
by (rtac (div_eq RS wf_less_trans) 1); |
4089 | 111 |
by (asm_simp_tac (simpset() addsimps [diff_less, cut_apply, less_eq]) 1); |
3366 | 112 |
qed "div_geq"; |
113 |
||
5415 | 114 |
(*Avoids the ugly ~m<n above*) |
115 |
Goal "[| 0<n; n<=m |] ==> m div n = Suc((m-n) div n)"; |
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by (asm_simp_tac (simpset() addsimps [div_geq, not_less_iff_le]) 1); |
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qed "le_div_geq"; |
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118 |
||
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(*NOT suitable for rewriting: loops*) |
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Goal "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))"; |
4774 | 121 |
by (asm_simp_tac (simpset() addsimps [div_less, div_geq]) 1); |
122 |
qed "div_if"; |
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||
3366 | 124 |
(*Main Result about quotient and remainder.*) |
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Goal "0<n ==> (m div n)*n + m mod n = m"; |
3366 | 126 |
by (res_inst_tac [("n","m")] less_induct 1); |
4774 | 127 |
by (stac mod_if 1); |
128 |
by (ALLGOALS (asm_simp_tac |
|
5537 | 129 |
(simpset() addsimps [add_assoc, div_less, div_geq, |
130 |
add_diff_inverse, diff_less]))); |
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3366 | 131 |
qed "mod_div_equality"; |
132 |
||
4358 | 133 |
(* a simple rearrangement of mod_div_equality: *) |
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Goal "0<k ==> k*(m div k) = m - (m mod k)"; |
4423 | 135 |
by (dres_inst_tac [("m","m")] mod_div_equality 1); |
4358 | 136 |
by (EVERY1[etac subst, simp_tac (simpset() addsimps mult_ac), |
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K(IF_UNSOLVED no_tac)]); |
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qed "mult_div_cancel"; |
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||
5069 | 140 |
Goal "m div 1 = m"; |
3366 | 141 |
by (induct_tac "m" 1); |
4089 | 142 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [div_less, div_geq]))); |
3366 | 143 |
qed "div_1"; |
144 |
Addsimps [div_1]; |
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||
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Goal "0<n ==> n div n = 1"; |
4089 | 147 |
by (asm_simp_tac (simpset() addsimps [div_less, div_geq]) 1); |
3366 | 148 |
qed "div_self"; |
149 |
||
4811 | 150 |
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Goal "0<n ==> (m+n) div n = Suc (m div n)"; |
4811 | 152 |
by (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n)" 1); |
153 |
by (stac (div_geq RS sym) 2); |
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by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [add_commute]))); |
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qed "div_add_self2"; |
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||
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Goal "0<n ==> (n+m) div n = Suc (m div n)"; |
4811 | 158 |
by (asm_simp_tac (simpset() addsimps [add_commute, div_add_self2]) 1); |
159 |
qed "div_add_self1"; |
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||
5069 | 161 |
Goal "!!n. 0<n ==> (m + k*n) div n = k + m div n"; |
4811 | 162 |
by (induct_tac "k" 1); |
5537 | 163 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps add_ac @ [div_add_self1]))); |
4811 | 164 |
qed "div_mult_self1"; |
165 |
||
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166 |
Goal "0<n ==> (m + n*k) div n = k + m div n"; |
4811 | 167 |
by (asm_simp_tac (simpset() addsimps [mult_commute, div_mult_self1]) 1); |
168 |
qed "div_mult_self2"; |
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||
170 |
Addsimps [div_mult_self1, div_mult_self2]; |
|
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||
172 |
||
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|
173 |
(* Monotonicity of div in first argument *) |
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174 |
Goal "0<k ==> ALL m. m <= n --> (m div k) <= (n div k)"; |
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|
175 |
by (res_inst_tac [("n","n")] less_induct 1); |
3718 | 176 |
by (Clarify_tac 1); |
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177 |
by (case_tac "n<k" 1); |
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|
178 |
(* 1 case n<k *) |
4089 | 179 |
by (asm_simp_tac (simpset() addsimps [div_less]) 1); |
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|
180 |
(* 2 case n >= k *) |
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181 |
by (case_tac "m<k" 1); |
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|
182 |
(* 2.1 case m<k *) |
4089 | 183 |
by (asm_simp_tac (simpset() addsimps [div_less]) 1); |
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|
184 |
(* 2.2 case m>=k *) |
4089 | 185 |
by (asm_simp_tac (simpset() addsimps [div_geq, diff_less, diff_le_mono]) 1); |
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186 |
qed_spec_mp "div_le_mono"; |
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|
187 |
|
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|
188 |
|
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|
189 |
(* Antimonotonicity of div in second argument *) |
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190 |
Goal "[| 0<m; m<=n |] ==> (k div n) <= (k div m)"; |
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|
191 |
by (subgoal_tac "0<n" 1); |
6073 | 192 |
by (Asm_simp_tac 2); |
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|
193 |
by (res_inst_tac [("n","k")] less_induct 1); |
3496 | 194 |
by (rename_tac "k" 1); |
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195 |
by (case_tac "k<n" 1); |
4089 | 196 |
by (asm_simp_tac (simpset() addsimps [div_less]) 1); |
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|
197 |
by (subgoal_tac "~(k<m)" 1); |
6073 | 198 |
by (Asm_simp_tac 2); |
4089 | 199 |
by (asm_simp_tac (simpset() addsimps [div_geq]) 1); |
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|
200 |
by (subgoal_tac "(k-n) div n <= (k-m) div n" 1); |
5316 | 201 |
by (REPEAT (eresolve_tac [div_le_mono,diff_le_mono2] 2)); |
5318 | 202 |
by (rtac le_trans 1); |
5316 | 203 |
by (Asm_simp_tac 1); |
204 |
by (asm_simp_tac (simpset() addsimps [diff_less]) 1); |
|
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|
205 |
qed "div_le_mono2"; |
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|
206 |
|
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|
207 |
Goal "0<n ==> m div n <= m"; |
3484
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nipkow
parents:
3457
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|
208 |
by (subgoal_tac "m div n <= m div 1" 1); |
1e93eb09ebb9
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nipkow
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|
209 |
by (Asm_full_simp_tac 1); |
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nipkow
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|
210 |
by (rtac div_le_mono2 1); |
6073 | 211 |
by (ALLGOALS Asm_simp_tac); |
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|
212 |
qed "div_le_dividend"; |
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|
213 |
Addsimps [div_le_dividend]; |
1e93eb09ebb9
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nipkow
parents:
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|
214 |
|
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|
215 |
(* Similar for "less than" *) |
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|
216 |
Goal "1<n ==> (0 < m) --> (m div n < m)"; |
3484
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nipkow
parents:
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|
217 |
by (res_inst_tac [("n","m")] less_induct 1); |
3496 | 218 |
by (rename_tac "m" 1); |
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|
219 |
by (case_tac "m<n" 1); |
4089 | 220 |
by (asm_full_simp_tac (simpset() addsimps [div_less]) 1); |
3484
1e93eb09ebb9
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nipkow
parents:
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|
221 |
by (subgoal_tac "0<n" 1); |
6073 | 222 |
by (Asm_simp_tac 2); |
4089 | 223 |
by (asm_full_simp_tac (simpset() addsimps [div_geq]) 1); |
3484
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Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
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|
224 |
by (case_tac "n<m" 1); |
1e93eb09ebb9
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nipkow
parents:
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changeset
|
225 |
by (subgoal_tac "(m-n) div n < (m-n)" 1); |
1e93eb09ebb9
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nipkow
parents:
3457
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|
226 |
by (REPEAT (ares_tac [impI,less_trans_Suc] 1)); |
4089 | 227 |
by (asm_full_simp_tac (simpset() addsimps [diff_less]) 1); |
228 |
by (asm_full_simp_tac (simpset() addsimps [diff_less]) 1); |
|
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
229 |
(* case n=m *) |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
230 |
by (subgoal_tac "m=n" 1); |
6073 | 231 |
by (Asm_simp_tac 2); |
4089 | 232 |
by (asm_simp_tac (simpset() addsimps [div_less]) 1); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
233 |
qed_spec_mp "div_less_dividend"; |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
234 |
Addsimps [div_less_dividend]; |
3366 | 235 |
|
236 |
(*** Further facts about mod (mainly for the mutilated chess board ***) |
|
237 |
||
5278 | 238 |
Goal "0<n ==> Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"; |
3366 | 239 |
by (res_inst_tac [("n","m")] less_induct 1); |
240 |
by (excluded_middle_tac "Suc(na)<n" 1); |
|
241 |
(* case Suc(na) < n *) |
|
242 |
by (forward_tac [lessI RS less_trans] 2); |
|
5355 | 243 |
by (asm_simp_tac (simpset() addsimps [mod_less, less_not_refl3]) 2); |
3366 | 244 |
(* case n <= Suc(na) *) |
5415 | 245 |
by (asm_full_simp_tac (simpset() addsimps [not_less_iff_le, le_Suc_eq, |
246 |
mod_geq]) 1); |
|
247 |
by (etac disjE 1); |
|
248 |
by (asm_simp_tac (simpset() addsimps [mod_less]) 2); |
|
249 |
by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_diff_less, |
|
250 |
le_mod_geq]) 1); |
|
3366 | 251 |
qed "mod_Suc"; |
252 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
253 |
Goal "0<n ==> m mod n < n"; |
3366 | 254 |
by (res_inst_tac [("n","m")] less_induct 1); |
5498 | 255 |
by (case_tac "na<n" 1); |
256 |
(*case n le na*) |
|
257 |
by (asm_full_simp_tac (simpset() addsimps [mod_geq, diff_less]) 2); |
|
3366 | 258 |
(*case na<n*) |
5498 | 259 |
by (asm_simp_tac (simpset() addsimps [mod_less]) 1); |
3366 | 260 |
qed "mod_less_divisor"; |
261 |
||
262 |
||
263 |
(** Evens and Odds **) |
|
264 |
||
265 |
(*With less_zeroE, causes case analysis on b<2*) |
|
266 |
AddSEs [less_SucE]; |
|
267 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
268 |
Goal "b<2 ==> k mod 2 = b | k mod 2 = (if b=1 then 0 else 1)"; |
3366 | 269 |
by (subgoal_tac "k mod 2 < 2" 1); |
4089 | 270 |
by (asm_simp_tac (simpset() addsimps [mod_less_divisor]) 2); |
4686 | 271 |
by (Asm_simp_tac 1); |
4356 | 272 |
by Safe_tac; |
3366 | 273 |
qed "mod2_cases"; |
274 |
||
5069 | 275 |
Goal "Suc(Suc(m)) mod 2 = m mod 2"; |
3366 | 276 |
by (subgoal_tac "m mod 2 < 2" 1); |
4089 | 277 |
by (asm_simp_tac (simpset() addsimps [mod_less_divisor]) 2); |
3724 | 278 |
by Safe_tac; |
4089 | 279 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [mod_Suc]))); |
3366 | 280 |
qed "mod2_Suc_Suc"; |
281 |
Addsimps [mod2_Suc_Suc]; |
|
282 |
||
5069 | 283 |
Goal "(0 < m mod 2) = (m mod 2 = 1)"; |
3366 | 284 |
by (subgoal_tac "m mod 2 < 2" 1); |
4089 | 285 |
by (asm_simp_tac (simpset() addsimps [mod_less_divisor]) 2); |
4477
b3e5857d8d99
New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4423
diff
changeset
|
286 |
by Auto_tac; |
4356 | 287 |
qed "mod2_gr_0"; |
288 |
Addsimps [mod2_gr_0]; |
|
289 |
||
5069 | 290 |
Goal "(m+m) mod 2 = 0"; |
3366 | 291 |
by (induct_tac "m" 1); |
4089 | 292 |
by (simp_tac (simpset() addsimps [mod_less]) 1); |
3427
e7cef2081106
Removed a few redundant additions of simprules or classical rules
paulson
parents:
3366
diff
changeset
|
293 |
by (Asm_simp_tac 1); |
4385 | 294 |
qed "mod2_add_self_eq_0"; |
295 |
Addsimps [mod2_add_self_eq_0]; |
|
296 |
||
5069 | 297 |
Goal "((m+m)+n) mod 2 = n mod 2"; |
4385 | 298 |
by (induct_tac "m" 1); |
299 |
by (simp_tac (simpset() addsimps [mod_less]) 1); |
|
300 |
by (Asm_simp_tac 1); |
|
3366 | 301 |
qed "mod2_add_self"; |
302 |
Addsimps [mod2_add_self]; |
|
303 |
||
5498 | 304 |
(*Restore the default*) |
3366 | 305 |
Delrules [less_SucE]; |
306 |
||
307 |
(*** More division laws ***) |
|
308 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
309 |
Goal "0<n ==> m*n div n = m"; |
3366 | 310 |
by (cut_inst_tac [("m", "m*n")] mod_div_equality 1); |
3457 | 311 |
by (assume_tac 1); |
4089 | 312 |
by (asm_full_simp_tac (simpset() addsimps [mod_mult_self_is_0]) 1); |
3366 | 313 |
qed "div_mult_self_is_m"; |
314 |
Addsimps [div_mult_self_is_m]; |
|
315 |
||
316 |
(*Cancellation law for division*) |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
317 |
Goal "[| 0<n; 0<k |] ==> (k*m) div (k*n) = m div n"; |
3366 | 318 |
by (res_inst_tac [("n","m")] less_induct 1); |
319 |
by (case_tac "na<n" 1); |
|
4089 | 320 |
by (asm_simp_tac (simpset() addsimps [div_less, zero_less_mult_iff, |
3366 | 321 |
mult_less_mono2]) 1); |
322 |
by (subgoal_tac "~ k*na < k*n" 1); |
|
323 |
by (asm_simp_tac |
|
4089 | 324 |
(simpset() addsimps [zero_less_mult_iff, div_geq, |
5415 | 325 |
diff_mult_distrib2 RS sym, diff_less]) 1); |
4089 | 326 |
by (asm_full_simp_tac (simpset() addsimps [not_less_iff_le, |
3366 | 327 |
le_refl RS mult_le_mono]) 1); |
328 |
qed "div_cancel"; |
|
329 |
Addsimps [div_cancel]; |
|
330 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
331 |
Goal "[| 0<n; 0<k |] ==> (k*m) mod (k*n) = k * (m mod n)"; |
3366 | 332 |
by (res_inst_tac [("n","m")] less_induct 1); |
333 |
by (case_tac "na<n" 1); |
|
4089 | 334 |
by (asm_simp_tac (simpset() addsimps [mod_less, zero_less_mult_iff, |
3366 | 335 |
mult_less_mono2]) 1); |
336 |
by (subgoal_tac "~ k*na < k*n" 1); |
|
337 |
by (asm_simp_tac |
|
4089 | 338 |
(simpset() addsimps [zero_less_mult_iff, mod_geq, |
3366 | 339 |
diff_mult_distrib2 RS sym, diff_less]) 1); |
4089 | 340 |
by (asm_full_simp_tac (simpset() addsimps [not_less_iff_le, |
3366 | 341 |
le_refl RS mult_le_mono]) 1); |
342 |
qed "mult_mod_distrib"; |
|
343 |
||
344 |
||
345 |
(************************************************) |
|
346 |
(** Divides Relation **) |
|
347 |
(************************************************) |
|
348 |
||
5069 | 349 |
Goalw [dvd_def] "m dvd 0"; |
4089 | 350 |
by (blast_tac (claset() addIs [mult_0_right RS sym]) 1); |
3366 | 351 |
qed "dvd_0_right"; |
352 |
Addsimps [dvd_0_right]; |
|
353 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
354 |
Goalw [dvd_def] "0 dvd m ==> m = 0"; |
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
355 |
by Auto_tac; |
3366 | 356 |
qed "dvd_0_left"; |
357 |
||
5069 | 358 |
Goalw [dvd_def] "1 dvd k"; |
3366 | 359 |
by (Simp_tac 1); |
360 |
qed "dvd_1_left"; |
|
361 |
AddIffs [dvd_1_left]; |
|
362 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
363 |
Goalw [dvd_def] "m dvd (m::nat)"; |
4089 | 364 |
by (blast_tac (claset() addIs [mult_1_right RS sym]) 1); |
3366 | 365 |
qed "dvd_refl"; |
366 |
Addsimps [dvd_refl]; |
|
367 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
368 |
Goalw [dvd_def] "[| m dvd n; n dvd p |] ==> m dvd (p::nat)"; |
4089 | 369 |
by (blast_tac (claset() addIs [mult_assoc] ) 1); |
3366 | 370 |
qed "dvd_trans"; |
371 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
372 |
Goalw [dvd_def] "[| m dvd n; n dvd m |] ==> m = (n::nat)"; |
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
373 |
by (force_tac (claset() addDs [mult_eq_self_implies_10], |
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
374 |
simpset() addsimps [mult_assoc, mult_eq_1_iff]) 1); |
3366 | 375 |
qed "dvd_anti_sym"; |
376 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
377 |
Goalw [dvd_def] "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"; |
4089 | 378 |
by (blast_tac (claset() addIs [add_mult_distrib2 RS sym]) 1); |
3366 | 379 |
qed "dvd_add"; |
380 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
381 |
Goalw [dvd_def] "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"; |
4089 | 382 |
by (blast_tac (claset() addIs [diff_mult_distrib2 RS sym]) 1); |
3366 | 383 |
qed "dvd_diff"; |
384 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
385 |
Goal "[| k dvd (m-n); k dvd n; n<=m |] ==> k dvd (m::nat)"; |
3457 | 386 |
by (etac (not_less_iff_le RS iffD2 RS add_diff_inverse RS subst) 1); |
4089 | 387 |
by (blast_tac (claset() addIs [dvd_add]) 1); |
3366 | 388 |
qed "dvd_diffD"; |
389 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
390 |
Goalw [dvd_def] "k dvd n ==> k dvd (m*n :: nat)"; |
4089 | 391 |
by (blast_tac (claset() addIs [mult_left_commute]) 1); |
3366 | 392 |
qed "dvd_mult"; |
393 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
394 |
Goal "k dvd m ==> k dvd (m*n :: nat)"; |
3366 | 395 |
by (stac mult_commute 1); |
396 |
by (etac dvd_mult 1); |
|
397 |
qed "dvd_mult2"; |
|
398 |
||
399 |
(* k dvd (m*k) *) |
|
400 |
AddIffs [dvd_refl RS dvd_mult, dvd_refl RS dvd_mult2]; |
|
401 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
402 |
Goalw [dvd_def] "[| f dvd m; f dvd n; 0<n |] ==> f dvd (m mod n)"; |
3718 | 403 |
by (Clarify_tac 1); |
4089 | 404 |
by (full_simp_tac (simpset() addsimps [zero_less_mult_iff]) 1); |
3366 | 405 |
by (res_inst_tac |
406 |
[("x", "(((k div ka)*ka + k mod ka) - ((f*k) div (f*ka)) * ka)")] |
|
407 |
exI 1); |
|
4089 | 408 |
by (asm_simp_tac (simpset() addsimps [diff_mult_distrib2, |
3366 | 409 |
mult_mod_distrib, add_mult_distrib2]) 1); |
410 |
qed "dvd_mod"; |
|
411 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
412 |
Goal "[| k dvd (m mod n); k dvd n; 0<n |] ==> k dvd m"; |
3366 | 413 |
by (subgoal_tac "k dvd ((m div n)*n + m mod n)" 1); |
4089 | 414 |
by (asm_simp_tac (simpset() addsimps [dvd_add, dvd_mult]) 2); |
4356 | 415 |
by (asm_full_simp_tac (simpset() addsimps [mod_div_equality]) 1); |
3366 | 416 |
qed "dvd_mod_imp_dvd"; |
417 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
418 |
Goalw [dvd_def] "!!k::nat. [| (k*m) dvd (k*n); 0<k |] ==> m dvd n"; |
3366 | 419 |
by (etac exE 1); |
4089 | 420 |
by (asm_full_simp_tac (simpset() addsimps mult_ac) 1); |
3366 | 421 |
by (Blast_tac 1); |
422 |
qed "dvd_mult_cancel"; |
|
423 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
424 |
Goalw [dvd_def] "[| i dvd m; j dvd n|] ==> (i*j) dvd (m*n :: nat)"; |
3718 | 425 |
by (Clarify_tac 1); |
3366 | 426 |
by (res_inst_tac [("x","k*ka")] exI 1); |
4089 | 427 |
by (asm_simp_tac (simpset() addsimps mult_ac) 1); |
3366 | 428 |
qed "mult_dvd_mono"; |
429 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
430 |
Goalw [dvd_def] "(i*j :: nat) dvd k ==> i dvd k"; |
4089 | 431 |
by (full_simp_tac (simpset() addsimps [mult_assoc]) 1); |
3366 | 432 |
by (Blast_tac 1); |
433 |
qed "dvd_mult_left"; |
|
434 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
435 |
Goalw [dvd_def] "[| k dvd n; 0 < n |] ==> k <= n"; |
3718 | 436 |
by (Clarify_tac 1); |
4089 | 437 |
by (ALLGOALS (full_simp_tac (simpset() addsimps [zero_less_mult_iff]))); |
3457 | 438 |
by (etac conjE 1); |
439 |
by (rtac le_trans 1); |
|
440 |
by (rtac (le_refl RS mult_le_mono) 2); |
|
3366 | 441 |
by (etac Suc_leI 2); |
442 |
by (Simp_tac 1); |
|
443 |
qed "dvd_imp_le"; |
|
444 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
445 |
Goalw [dvd_def] "0<k ==> (k dvd n) = (n mod k = 0)"; |
3724 | 446 |
by Safe_tac; |
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
447 |
by (asm_simp_tac (simpset() addsimps [mult_commute]) 1); |
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
448 |
by (eres_inst_tac [("t","n")] (mod_div_equality RS subst) 1); |
3366 | 449 |
by (stac mult_commute 1); |
450 |
by (Asm_simp_tac 1); |
|
451 |
by (Blast_tac 1); |
|
452 |
qed "dvd_eq_mod_eq_0"; |