author | paulson |
Thu, 15 Jul 1999 10:27:54 +0200 | |
changeset 7007 | b46ccfee8e59 |
parent 6865 | 5577ffe4c2f1 |
child 7029 | 08d4eb8500dd |
permissions | -rw-r--r-- |
3366 | 1 |
(* Title: HOL/Divides.ML |
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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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||
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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"; |
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||
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Goal "m<n ==> m mod n = (m::nat)"; |
3366 | 23 |
by (rtac (mod_eq RS wf_less_trans) 1); |
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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"; |
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||
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); |
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qed "mod_if"; |
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||
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"; |
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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); |
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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); |
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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"; |
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by (asm_simp_tac (simpset() addsimps [mult_commute, mod_mult_self1]) 1); |
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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); |
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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"; |
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||
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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); |
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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"; |
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||
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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) \ |
7007 | 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); |
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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"; |
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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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Goal "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))"; |
4774 | 120 |
by (asm_simp_tac (simpset() addsimps [div_less, div_geq]) 1); |
121 |
qed "div_if"; |
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||
3366 | 123 |
(*Main Result about quotient and remainder.*) |
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Goal "0<n ==> (m div n)*n + m mod n = m"; |
3366 | 125 |
by (res_inst_tac [("n","m")] less_induct 1); |
4774 | 126 |
by (stac mod_if 1); |
127 |
by (ALLGOALS (asm_simp_tac |
|
5537 | 128 |
(simpset() addsimps [add_assoc, div_less, div_geq, |
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add_diff_inverse, diff_less]))); |
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3366 | 130 |
qed "mod_div_equality"; |
131 |
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4358 | 132 |
(* a simple rearrangement of mod_div_equality: *) |
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Goal "0<k ==> k*(m div k) = m - (m mod k)"; |
4423 | 134 |
by (dres_inst_tac [("m","m")] mod_div_equality 1); |
4358 | 135 |
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 | 139 |
Goal "m div 1 = m"; |
3366 | 140 |
by (induct_tac "m" 1); |
4089 | 141 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [div_less, div_geq]))); |
3366 | 142 |
qed "div_1"; |
143 |
Addsimps [div_1]; |
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||
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Goal "0<n ==> n div n = 1"; |
4089 | 146 |
by (asm_simp_tac (simpset() addsimps [div_less, div_geq]) 1); |
3366 | 147 |
qed "div_self"; |
148 |
||
4811 | 149 |
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Goal "0<n ==> (m+n) div n = Suc (m div n)"; |
4811 | 151 |
by (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n)" 1); |
152 |
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 | 157 |
by (asm_simp_tac (simpset() addsimps [add_commute, div_add_self2]) 1); |
158 |
qed "div_add_self1"; |
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||
5069 | 160 |
Goal "!!n. 0<n ==> (m + k*n) div n = k + m div n"; |
4811 | 161 |
by (induct_tac "k" 1); |
5537 | 162 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps add_ac @ [div_add_self1]))); |
4811 | 163 |
qed "div_mult_self1"; |
164 |
||
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165 |
Goal "0<n ==> (m + n*k) div n = k + m div n"; |
4811 | 166 |
by (asm_simp_tac (simpset() addsimps [mult_commute, div_mult_self1]) 1); |
167 |
qed "div_mult_self2"; |
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||
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Addsimps [div_mult_self1, div_mult_self2]; |
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||
171 |
||
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|
172 |
(* Monotonicity of div in first argument *) |
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173 |
Goal "0<k ==> ALL m. m <= n --> (m div k) <= (n div k)"; |
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174 |
by (res_inst_tac [("n","n")] less_induct 1); |
3718 | 175 |
by (Clarify_tac 1); |
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by (case_tac "n<k" 1); |
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177 |
(* 1 case n<k *) |
4089 | 178 |
by (asm_simp_tac (simpset() addsimps [div_less]) 1); |
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|
179 |
(* 2 case n >= k *) |
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180 |
by (case_tac "m<k" 1); |
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181 |
(* 2.1 case m<k *) |
4089 | 182 |
by (asm_simp_tac (simpset() addsimps [div_less]) 1); |
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|
183 |
(* 2.2 case m>=k *) |
4089 | 184 |
by (asm_simp_tac (simpset() addsimps [div_geq, diff_less, diff_le_mono]) 1); |
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185 |
qed_spec_mp "div_le_mono"; |
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|
186 |
|
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|
187 |
|
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|
188 |
(* Antimonotonicity of div in second argument *) |
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189 |
Goal "[| 0<m; m<=n |] ==> (k div n) <= (k div m)"; |
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|
190 |
by (subgoal_tac "0<n" 1); |
6073 | 191 |
by (Asm_simp_tac 2); |
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|
192 |
by (res_inst_tac [("n","k")] less_induct 1); |
3496 | 193 |
by (rename_tac "k" 1); |
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|
194 |
by (case_tac "k<n" 1); |
4089 | 195 |
by (asm_simp_tac (simpset() addsimps [div_less]) 1); |
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|
196 |
by (subgoal_tac "~(k<m)" 1); |
6073 | 197 |
by (Asm_simp_tac 2); |
4089 | 198 |
by (asm_simp_tac (simpset() addsimps [div_geq]) 1); |
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|
199 |
by (subgoal_tac "(k-n) div n <= (k-m) div n" 1); |
5316 | 200 |
by (REPEAT (eresolve_tac [div_le_mono,diff_le_mono2] 2)); |
5318 | 201 |
by (rtac le_trans 1); |
5316 | 202 |
by (Asm_simp_tac 1); |
203 |
by (asm_simp_tac (simpset() addsimps [diff_less]) 1); |
|
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|
204 |
qed "div_le_mono2"; |
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|
205 |
|
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|
206 |
Goal "0<n ==> m div n <= m"; |
3484
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nipkow
parents:
3457
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|
207 |
by (subgoal_tac "m div n <= m div 1" 1); |
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nipkow
parents:
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|
208 |
by (Asm_full_simp_tac 1); |
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nipkow
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|
209 |
by (rtac div_le_mono2 1); |
6073 | 210 |
by (ALLGOALS Asm_simp_tac); |
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|
211 |
qed "div_le_dividend"; |
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|
212 |
Addsimps [div_le_dividend]; |
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nipkow
parents:
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|
213 |
|
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|
214 |
(* Similar for "less than" *) |
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|
215 |
Goal "1<n ==> (0 < m) --> (m div n < m)"; |
3484
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nipkow
parents:
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|
216 |
by (res_inst_tac [("n","m")] less_induct 1); |
3496 | 217 |
by (rename_tac "m" 1); |
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nipkow
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|
218 |
by (case_tac "m<n" 1); |
4089 | 219 |
by (asm_full_simp_tac (simpset() addsimps [div_less]) 1); |
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nipkow
parents:
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|
220 |
by (subgoal_tac "0<n" 1); |
6073 | 221 |
by (Asm_simp_tac 2); |
4089 | 222 |
by (asm_full_simp_tac (simpset() addsimps [div_geq]) 1); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
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|
223 |
by (case_tac "n<m" 1); |
1e93eb09ebb9
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nipkow
parents:
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changeset
|
224 |
by (subgoal_tac "(m-n) div n < (m-n)" 1); |
1e93eb09ebb9
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nipkow
parents:
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changeset
|
225 |
by (REPEAT (ares_tac [impI,less_trans_Suc] 1)); |
4089 | 226 |
by (asm_full_simp_tac (simpset() addsimps [diff_less]) 1); |
227 |
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
|
228 |
(* case n=m *) |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
229 |
by (subgoal_tac "m=n" 1); |
6073 | 230 |
by (Asm_simp_tac 2); |
4089 | 231 |
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
|
232 |
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
|
233 |
Addsimps [div_less_dividend]; |
3366 | 234 |
|
235 |
(*** Further facts about mod (mainly for the mutilated chess board ***) |
|
236 |
||
5278 | 237 |
Goal "0<n ==> Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"; |
3366 | 238 |
by (res_inst_tac [("n","m")] less_induct 1); |
239 |
by (excluded_middle_tac "Suc(na)<n" 1); |
|
240 |
(* case Suc(na) < n *) |
|
241 |
by (forward_tac [lessI RS less_trans] 2); |
|
5355 | 242 |
by (asm_simp_tac (simpset() addsimps [mod_less, less_not_refl3]) 2); |
3366 | 243 |
(* case n <= Suc(na) *) |
5415 | 244 |
by (asm_full_simp_tac (simpset() addsimps [not_less_iff_le, le_Suc_eq, |
245 |
mod_geq]) 1); |
|
246 |
by (etac disjE 1); |
|
247 |
by (asm_simp_tac (simpset() addsimps [mod_less]) 2); |
|
248 |
by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_diff_less, |
|
249 |
le_mod_geq]) 1); |
|
3366 | 250 |
qed "mod_Suc"; |
251 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
252 |
Goal "0<n ==> m mod n < n"; |
3366 | 253 |
by (res_inst_tac [("n","m")] less_induct 1); |
5498 | 254 |
by (case_tac "na<n" 1); |
255 |
(*case n le na*) |
|
256 |
by (asm_full_simp_tac (simpset() addsimps [mod_geq, diff_less]) 2); |
|
3366 | 257 |
(*case na<n*) |
5498 | 258 |
by (asm_simp_tac (simpset() addsimps [mod_less]) 1); |
3366 | 259 |
qed "mod_less_divisor"; |
260 |
||
261 |
||
262 |
(** Evens and Odds **) |
|
263 |
||
264 |
(*With less_zeroE, causes case analysis on b<2*) |
|
265 |
AddSEs [less_SucE]; |
|
266 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
267 |
Goal "b<2 ==> k mod 2 = b | k mod 2 = (if b=1 then 0 else 1)"; |
3366 | 268 |
by (subgoal_tac "k mod 2 < 2" 1); |
4089 | 269 |
by (asm_simp_tac (simpset() addsimps [mod_less_divisor]) 2); |
4686 | 270 |
by (Asm_simp_tac 1); |
4356 | 271 |
by Safe_tac; |
3366 | 272 |
qed "mod2_cases"; |
273 |
||
5069 | 274 |
Goal "Suc(Suc(m)) mod 2 = m mod 2"; |
3366 | 275 |
by (subgoal_tac "m mod 2 < 2" 1); |
4089 | 276 |
by (asm_simp_tac (simpset() addsimps [mod_less_divisor]) 2); |
3724 | 277 |
by Safe_tac; |
4089 | 278 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [mod_Suc]))); |
3366 | 279 |
qed "mod2_Suc_Suc"; |
280 |
Addsimps [mod2_Suc_Suc]; |
|
281 |
||
5069 | 282 |
Goal "(0 < m mod 2) = (m mod 2 = 1)"; |
3366 | 283 |
by (subgoal_tac "m mod 2 < 2" 1); |
4089 | 284 |
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
|
285 |
by Auto_tac; |
4356 | 286 |
qed "mod2_gr_0"; |
287 |
Addsimps [mod2_gr_0]; |
|
288 |
||
5069 | 289 |
Goal "(m+m) mod 2 = 0"; |
3366 | 290 |
by (induct_tac "m" 1); |
4089 | 291 |
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
|
292 |
by (Asm_simp_tac 1); |
4385 | 293 |
qed "mod2_add_self_eq_0"; |
294 |
Addsimps [mod2_add_self_eq_0]; |
|
295 |
||
5069 | 296 |
Goal "((m+m)+n) mod 2 = n mod 2"; |
4385 | 297 |
by (induct_tac "m" 1); |
298 |
by (simp_tac (simpset() addsimps [mod_less]) 1); |
|
299 |
by (Asm_simp_tac 1); |
|
3366 | 300 |
qed "mod2_add_self"; |
301 |
Addsimps [mod2_add_self]; |
|
302 |
||
5498 | 303 |
(*Restore the default*) |
3366 | 304 |
Delrules [less_SucE]; |
305 |
||
306 |
(*** More division laws ***) |
|
307 |
||
7007 | 308 |
Goal "0<n ==> (m*n) div n = m"; |
3366 | 309 |
by (cut_inst_tac [("m", "m*n")] mod_div_equality 1); |
3457 | 310 |
by (assume_tac 1); |
4089 | 311 |
by (asm_full_simp_tac (simpset() addsimps [mod_mult_self_is_0]) 1); |
3366 | 312 |
qed "div_mult_self_is_m"; |
313 |
Addsimps [div_mult_self_is_m]; |
|
314 |
||
315 |
(*Cancellation law for division*) |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
316 |
Goal "[| 0<n; 0<k |] ==> (k*m) div (k*n) = m div n"; |
3366 | 317 |
by (res_inst_tac [("n","m")] less_induct 1); |
318 |
by (case_tac "na<n" 1); |
|
4089 | 319 |
by (asm_simp_tac (simpset() addsimps [div_less, zero_less_mult_iff, |
3366 | 320 |
mult_less_mono2]) 1); |
321 |
by (subgoal_tac "~ k*na < k*n" 1); |
|
322 |
by (asm_simp_tac |
|
4089 | 323 |
(simpset() addsimps [zero_less_mult_iff, div_geq, |
5415 | 324 |
diff_mult_distrib2 RS sym, diff_less]) 1); |
4089 | 325 |
by (asm_full_simp_tac (simpset() addsimps [not_less_iff_le, |
3366 | 326 |
le_refl RS mult_le_mono]) 1); |
327 |
qed "div_cancel"; |
|
328 |
Addsimps [div_cancel]; |
|
329 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
330 |
Goal "[| 0<n; 0<k |] ==> (k*m) mod (k*n) = k * (m mod n)"; |
3366 | 331 |
by (res_inst_tac [("n","m")] less_induct 1); |
332 |
by (case_tac "na<n" 1); |
|
4089 | 333 |
by (asm_simp_tac (simpset() addsimps [mod_less, zero_less_mult_iff, |
3366 | 334 |
mult_less_mono2]) 1); |
335 |
by (subgoal_tac "~ k*na < k*n" 1); |
|
336 |
by (asm_simp_tac |
|
4089 | 337 |
(simpset() addsimps [zero_less_mult_iff, mod_geq, |
3366 | 338 |
diff_mult_distrib2 RS sym, diff_less]) 1); |
4089 | 339 |
by (asm_full_simp_tac (simpset() addsimps [not_less_iff_le, |
3366 | 340 |
le_refl RS mult_le_mono]) 1); |
341 |
qed "mult_mod_distrib"; |
|
342 |
||
343 |
||
344 |
(************************************************) |
|
345 |
(** Divides Relation **) |
|
346 |
(************************************************) |
|
347 |
||
5069 | 348 |
Goalw [dvd_def] "m dvd 0"; |
4089 | 349 |
by (blast_tac (claset() addIs [mult_0_right RS sym]) 1); |
3366 | 350 |
qed "dvd_0_right"; |
351 |
Addsimps [dvd_0_right]; |
|
352 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
353 |
Goalw [dvd_def] "0 dvd m ==> m = 0"; |
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
354 |
by Auto_tac; |
3366 | 355 |
qed "dvd_0_left"; |
356 |
||
5069 | 357 |
Goalw [dvd_def] "1 dvd k"; |
3366 | 358 |
by (Simp_tac 1); |
359 |
qed "dvd_1_left"; |
|
360 |
AddIffs [dvd_1_left]; |
|
361 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
362 |
Goalw [dvd_def] "m dvd (m::nat)"; |
4089 | 363 |
by (blast_tac (claset() addIs [mult_1_right RS sym]) 1); |
3366 | 364 |
qed "dvd_refl"; |
365 |
Addsimps [dvd_refl]; |
|
366 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
367 |
Goalw [dvd_def] "[| m dvd n; n dvd p |] ==> m dvd (p::nat)"; |
4089 | 368 |
by (blast_tac (claset() addIs [mult_assoc] ) 1); |
3366 | 369 |
qed "dvd_trans"; |
370 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
371 |
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
|
372 |
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
|
373 |
simpset() addsimps [mult_assoc, mult_eq_1_iff]) 1); |
3366 | 374 |
qed "dvd_anti_sym"; |
375 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
376 |
Goalw [dvd_def] "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"; |
4089 | 377 |
by (blast_tac (claset() addIs [add_mult_distrib2 RS sym]) 1); |
3366 | 378 |
qed "dvd_add"; |
379 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
380 |
Goalw [dvd_def] "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"; |
4089 | 381 |
by (blast_tac (claset() addIs [diff_mult_distrib2 RS sym]) 1); |
3366 | 382 |
qed "dvd_diff"; |
383 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
384 |
Goal "[| k dvd (m-n); k dvd n; n<=m |] ==> k dvd (m::nat)"; |
3457 | 385 |
by (etac (not_less_iff_le RS iffD2 RS add_diff_inverse RS subst) 1); |
4089 | 386 |
by (blast_tac (claset() addIs [dvd_add]) 1); |
3366 | 387 |
qed "dvd_diffD"; |
388 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
389 |
Goalw [dvd_def] "k dvd n ==> k dvd (m*n :: nat)"; |
4089 | 390 |
by (blast_tac (claset() addIs [mult_left_commute]) 1); |
3366 | 391 |
qed "dvd_mult"; |
392 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
393 |
Goal "k dvd m ==> k dvd (m*n :: nat)"; |
3366 | 394 |
by (stac mult_commute 1); |
395 |
by (etac dvd_mult 1); |
|
396 |
qed "dvd_mult2"; |
|
397 |
||
398 |
(* k dvd (m*k) *) |
|
399 |
AddIffs [dvd_refl RS dvd_mult, dvd_refl RS dvd_mult2]; |
|
400 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
401 |
Goalw [dvd_def] "[| f dvd m; f dvd n; 0<n |] ==> f dvd (m mod n)"; |
3718 | 402 |
by (Clarify_tac 1); |
4089 | 403 |
by (full_simp_tac (simpset() addsimps [zero_less_mult_iff]) 1); |
3366 | 404 |
by (res_inst_tac |
405 |
[("x", "(((k div ka)*ka + k mod ka) - ((f*k) div (f*ka)) * ka)")] |
|
406 |
exI 1); |
|
4089 | 407 |
by (asm_simp_tac (simpset() addsimps [diff_mult_distrib2, |
3366 | 408 |
mult_mod_distrib, add_mult_distrib2]) 1); |
409 |
qed "dvd_mod"; |
|
410 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
411 |
Goal "[| k dvd (m mod n); k dvd n; 0<n |] ==> k dvd m"; |
3366 | 412 |
by (subgoal_tac "k dvd ((m div n)*n + m mod n)" 1); |
4089 | 413 |
by (asm_simp_tac (simpset() addsimps [dvd_add, dvd_mult]) 2); |
4356 | 414 |
by (asm_full_simp_tac (simpset() addsimps [mod_div_equality]) 1); |
3366 | 415 |
qed "dvd_mod_imp_dvd"; |
416 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
417 |
Goalw [dvd_def] "!!k::nat. [| (k*m) dvd (k*n); 0<k |] ==> m dvd n"; |
3366 | 418 |
by (etac exE 1); |
4089 | 419 |
by (asm_full_simp_tac (simpset() addsimps mult_ac) 1); |
3366 | 420 |
by (Blast_tac 1); |
421 |
qed "dvd_mult_cancel"; |
|
422 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
423 |
Goalw [dvd_def] "[| i dvd m; j dvd n|] ==> (i*j) dvd (m*n :: nat)"; |
3718 | 424 |
by (Clarify_tac 1); |
3366 | 425 |
by (res_inst_tac [("x","k*ka")] exI 1); |
4089 | 426 |
by (asm_simp_tac (simpset() addsimps mult_ac) 1); |
3366 | 427 |
qed "mult_dvd_mono"; |
428 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
429 |
Goalw [dvd_def] "(i*j :: nat) dvd k ==> i dvd k"; |
4089 | 430 |
by (full_simp_tac (simpset() addsimps [mult_assoc]) 1); |
3366 | 431 |
by (Blast_tac 1); |
432 |
qed "dvd_mult_left"; |
|
433 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
434 |
Goalw [dvd_def] "[| k dvd n; 0 < n |] ==> k <= n"; |
3718 | 435 |
by (Clarify_tac 1); |
4089 | 436 |
by (ALLGOALS (full_simp_tac (simpset() addsimps [zero_less_mult_iff]))); |
3457 | 437 |
by (etac conjE 1); |
438 |
by (rtac le_trans 1); |
|
439 |
by (rtac (le_refl RS mult_le_mono) 2); |
|
3366 | 440 |
by (etac Suc_leI 2); |
441 |
by (Simp_tac 1); |
|
442 |
qed "dvd_imp_le"; |
|
443 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
444 |
Goalw [dvd_def] "0<k ==> (k dvd n) = (n mod k = 0)"; |
3724 | 445 |
by Safe_tac; |
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
446 |
by (asm_simp_tac (simpset() addsimps [mult_commute]) 1); |
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
447 |
by (eres_inst_tac [("t","n")] (mod_div_equality RS subst) 1); |
3366 | 448 |
by (stac mult_commute 1); |
449 |
by (Asm_simp_tac 1); |
|
450 |
by (Blast_tac 1); |
|
451 |
qed "dvd_eq_mod_eq_0"; |