author | nipkow |
Tue, 05 Jan 1999 17:27:59 +0100 | |
changeset 6059 | aa00e235ea27 |
parent 5983 | 79e301a6a51b |
child 6073 | fba734ba6894 |
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 |
|
4 |
Copyright 1993 University of Cambridge |
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||
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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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||
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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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14 |
||
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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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|
22 |
Goal "m<n ==> m mod n = m"; |
3366 | 23 |
by (rtac (mod_eq RS wf_less_trans) 1); |
24 |
by (Asm_simp_tac 1); |
|
25 |
qed "mod_less"; |
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26 |
||
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parents:
5069
diff
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27 |
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*) |
33 |
Goal "[| 0<n; n<=m |] ==> m mod n = (m-n) mod n"; |
|
34 |
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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36 |
||
4774 | 37 |
(*NOT suitable for rewriting: loops*) |
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5069
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38 |
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"; |
|
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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47 |
||
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|
48 |
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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parents:
5069
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|
52 |
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"; |
4811 | 59 |
by (asm_simp_tac (simpset() addsimps [add_commute, mod_add_self2]) 1); |
60 |
qed "mod_add_self1"; |
|
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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67 |
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"; |
|
4810 | 70 |
|
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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parents:
5069
diff
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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"; |
95 |
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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parents:
5069
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|
104 |
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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5069
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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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117 |
qed "le_div_geq"; |
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118 |
||
4774 | 119 |
(*NOT suitable for rewriting: loops*) |
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parents:
5069
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|
120 |
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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125 |
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]))); |
|
3366 | 131 |
qed "mod_div_equality"; |
132 |
||
4358 | 133 |
(* a simple rearrangement of mod_div_equality: *) |
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|
134 |
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), |
137 |
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]; |
|
145 |
||
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parents:
5069
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|
146 |
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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parents:
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|
151 |
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]))); |
|
155 |
qed "div_add_self2"; |
|
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||
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parents:
5069
diff
changeset
|
157 |
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"; |
|
160 |
||
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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parents:
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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"; |
|
169 |
||
170 |
Addsimps [div_mult_self1, div_mult_self2]; |
|
171 |
||
172 |
||
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Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
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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)"; |
3484
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nipkow
parents:
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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); |
3484
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|
178 |
(* 1 case n<k *) |
4089 | 179 |
by (asm_simp_tac (simpset() addsimps [div_less]) 1); |
3484
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nipkow
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|
180 |
(* 2 case n >= k *) |
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|
181 |
by (case_tac "m<k" 1); |
1e93eb09ebb9
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|
182 |
(* 2.1 case m<k *) |
4089 | 183 |
by (asm_simp_tac (simpset() addsimps [div_less]) 1); |
3484
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nipkow
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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); |
3484
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nipkow
parents:
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|
186 |
qed_spec_mp "div_le_mono"; |
1e93eb09ebb9
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|
187 |
|
1e93eb09ebb9
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parents:
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|
188 |
|
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parents:
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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)"; |
3484
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nipkow
parents:
3457
diff
changeset
|
191 |
by (subgoal_tac "0<n" 1); |
5983 | 192 |
by (Simp_tac 2); |
3484
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nipkow
parents:
3457
diff
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|
193 |
by (res_inst_tac [("n","k")] less_induct 1); |
3496 | 194 |
by (Simp_tac 1); |
195 |
by (rename_tac "k" 1); |
|
3484
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|
196 |
by (case_tac "k<n" 1); |
4089 | 197 |
by (asm_simp_tac (simpset() addsimps [div_less]) 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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|
198 |
by (subgoal_tac "~(k<m)" 1); |
5983 | 199 |
by (Simp_tac 2); |
4089 | 200 |
by (asm_simp_tac (simpset() addsimps [div_geq]) 1); |
3484
1e93eb09ebb9
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nipkow
parents:
3457
diff
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|
201 |
by (subgoal_tac "(k-n) div n <= (k-m) div n" 1); |
5316 | 202 |
by (REPEAT (eresolve_tac [div_le_mono,diff_le_mono2] 2)); |
5318 | 203 |
by (rtac le_trans 1); |
5316 | 204 |
by (Asm_simp_tac 1); |
205 |
by (asm_simp_tac (simpset() addsimps [diff_less]) 1); |
|
3484
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Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
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|
206 |
qed "div_le_mono2"; |
1e93eb09ebb9
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nipkow
parents:
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diff
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|
207 |
|
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paulson
parents:
5069
diff
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|
208 |
Goal "0<n ==> m div n <= m"; |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
209 |
by (subgoal_tac "m div n <= m div 1" 1); |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
210 |
by (Asm_full_simp_tac 1); |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
211 |
by (rtac div_le_mono2 1); |
5983 | 212 |
by (ALLGOALS Simp_tac); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
213 |
qed "div_le_dividend"; |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
214 |
Addsimps [div_le_dividend]; |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
215 |
|
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
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diff
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|
216 |
(* Similar for "less than" *) |
5143
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|
217 |
Goal "1<n ==> (0 < m) --> (m div n < m)"; |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
218 |
by (res_inst_tac [("n","m")] less_induct 1); |
3496 | 219 |
by (Simp_tac 1); |
220 |
by (rename_tac "m" 1); |
|
3484
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Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
221 |
by (case_tac "m<n" 1); |
4089 | 222 |
by (asm_full_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
|
223 |
by (subgoal_tac "0<n" 1); |
5983 | 224 |
by (Simp_tac 2); |
4089 | 225 |
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
diff
changeset
|
226 |
by (case_tac "n<m" 1); |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
227 |
by (subgoal_tac "(m-n) div n < (m-n)" 1); |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
228 |
by (REPEAT (ares_tac [impI,less_trans_Suc] 1)); |
4089 | 229 |
by (asm_full_simp_tac (simpset() addsimps [diff_less]) 1); |
230 |
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
|
231 |
(* case n=m *) |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
232 |
by (subgoal_tac "m=n" 1); |
5983 | 233 |
by (Simp_tac 2); |
4089 | 234 |
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
|
235 |
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
|
236 |
Addsimps [div_less_dividend]; |
3366 | 237 |
|
238 |
(*** Further facts about mod (mainly for the mutilated chess board ***) |
|
239 |
||
5278 | 240 |
Goal "0<n ==> Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"; |
3366 | 241 |
by (res_inst_tac [("n","m")] less_induct 1); |
242 |
by (excluded_middle_tac "Suc(na)<n" 1); |
|
243 |
(* case Suc(na) < n *) |
|
244 |
by (forward_tac [lessI RS less_trans] 2); |
|
5355 | 245 |
by (asm_simp_tac (simpset() addsimps [mod_less, less_not_refl3]) 2); |
3366 | 246 |
(* case n <= Suc(na) *) |
5415 | 247 |
by (asm_full_simp_tac (simpset() addsimps [not_less_iff_le, le_Suc_eq, |
248 |
mod_geq]) 1); |
|
249 |
by (etac disjE 1); |
|
250 |
by (asm_simp_tac (simpset() addsimps [mod_less]) 2); |
|
251 |
by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_diff_less, |
|
252 |
le_mod_geq]) 1); |
|
3366 | 253 |
qed "mod_Suc"; |
254 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
255 |
Goal "0<n ==> m mod n < n"; |
3366 | 256 |
by (res_inst_tac [("n","m")] less_induct 1); |
5498 | 257 |
by (case_tac "na<n" 1); |
258 |
(*case n le na*) |
|
259 |
by (asm_full_simp_tac (simpset() addsimps [mod_geq, diff_less]) 2); |
|
3366 | 260 |
(*case na<n*) |
5498 | 261 |
by (asm_simp_tac (simpset() addsimps [mod_less]) 1); |
3366 | 262 |
qed "mod_less_divisor"; |
263 |
||
264 |
||
265 |
(** Evens and Odds **) |
|
266 |
||
267 |
(*With less_zeroE, causes case analysis on b<2*) |
|
268 |
AddSEs [less_SucE]; |
|
269 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
270 |
Goal "b<2 ==> k mod 2 = b | k mod 2 = (if b=1 then 0 else 1)"; |
3366 | 271 |
by (subgoal_tac "k mod 2 < 2" 1); |
4089 | 272 |
by (asm_simp_tac (simpset() addsimps [mod_less_divisor]) 2); |
4686 | 273 |
by (Asm_simp_tac 1); |
4356 | 274 |
by Safe_tac; |
3366 | 275 |
qed "mod2_cases"; |
276 |
||
5069 | 277 |
Goal "Suc(Suc(m)) mod 2 = m mod 2"; |
3366 | 278 |
by (subgoal_tac "m mod 2 < 2" 1); |
4089 | 279 |
by (asm_simp_tac (simpset() addsimps [mod_less_divisor]) 2); |
3724 | 280 |
by Safe_tac; |
4089 | 281 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [mod_Suc]))); |
3366 | 282 |
qed "mod2_Suc_Suc"; |
283 |
Addsimps [mod2_Suc_Suc]; |
|
284 |
||
5069 | 285 |
Goal "(0 < m mod 2) = (m mod 2 = 1)"; |
3366 | 286 |
by (subgoal_tac "m mod 2 < 2" 1); |
4089 | 287 |
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
|
288 |
by Auto_tac; |
4356 | 289 |
qed "mod2_gr_0"; |
290 |
Addsimps [mod2_gr_0]; |
|
291 |
||
5069 | 292 |
Goal "(m+m) mod 2 = 0"; |
3366 | 293 |
by (induct_tac "m" 1); |
4089 | 294 |
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
|
295 |
by (Asm_simp_tac 1); |
4385 | 296 |
qed "mod2_add_self_eq_0"; |
297 |
Addsimps [mod2_add_self_eq_0]; |
|
298 |
||
5069 | 299 |
Goal "((m+m)+n) mod 2 = n mod 2"; |
4385 | 300 |
by (induct_tac "m" 1); |
301 |
by (simp_tac (simpset() addsimps [mod_less]) 1); |
|
302 |
by (Asm_simp_tac 1); |
|
3366 | 303 |
qed "mod2_add_self"; |
304 |
Addsimps [mod2_add_self]; |
|
305 |
||
5498 | 306 |
(*Restore the default*) |
3366 | 307 |
Delrules [less_SucE]; |
308 |
||
309 |
(*** More division laws ***) |
|
310 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
311 |
Goal "0<n ==> m*n div n = m"; |
3366 | 312 |
by (cut_inst_tac [("m", "m*n")] mod_div_equality 1); |
3457 | 313 |
by (assume_tac 1); |
4089 | 314 |
by (asm_full_simp_tac (simpset() addsimps [mod_mult_self_is_0]) 1); |
3366 | 315 |
qed "div_mult_self_is_m"; |
316 |
Addsimps [div_mult_self_is_m]; |
|
317 |
||
318 |
(*Cancellation law for division*) |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
319 |
Goal "[| 0<n; 0<k |] ==> (k*m) div (k*n) = m div n"; |
3366 | 320 |
by (res_inst_tac [("n","m")] less_induct 1); |
321 |
by (case_tac "na<n" 1); |
|
4089 | 322 |
by (asm_simp_tac (simpset() addsimps [div_less, zero_less_mult_iff, |
3366 | 323 |
mult_less_mono2]) 1); |
324 |
by (subgoal_tac "~ k*na < k*n" 1); |
|
325 |
by (asm_simp_tac |
|
4089 | 326 |
(simpset() addsimps [zero_less_mult_iff, div_geq, |
5415 | 327 |
diff_mult_distrib2 RS sym, diff_less]) 1); |
4089 | 328 |
by (asm_full_simp_tac (simpset() addsimps [not_less_iff_le, |
3366 | 329 |
le_refl RS mult_le_mono]) 1); |
330 |
qed "div_cancel"; |
|
331 |
Addsimps [div_cancel]; |
|
332 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
333 |
Goal "[| 0<n; 0<k |] ==> (k*m) mod (k*n) = k * (m mod n)"; |
3366 | 334 |
by (res_inst_tac [("n","m")] less_induct 1); |
335 |
by (case_tac "na<n" 1); |
|
4089 | 336 |
by (asm_simp_tac (simpset() addsimps [mod_less, zero_less_mult_iff, |
3366 | 337 |
mult_less_mono2]) 1); |
338 |
by (subgoal_tac "~ k*na < k*n" 1); |
|
339 |
by (asm_simp_tac |
|
4089 | 340 |
(simpset() addsimps [zero_less_mult_iff, mod_geq, |
3366 | 341 |
diff_mult_distrib2 RS sym, diff_less]) 1); |
4089 | 342 |
by (asm_full_simp_tac (simpset() addsimps [not_less_iff_le, |
3366 | 343 |
le_refl RS mult_le_mono]) 1); |
344 |
qed "mult_mod_distrib"; |
|
345 |
||
346 |
||
347 |
(************************************************) |
|
348 |
(** Divides Relation **) |
|
349 |
(************************************************) |
|
350 |
||
5069 | 351 |
Goalw [dvd_def] "m dvd 0"; |
4089 | 352 |
by (blast_tac (claset() addIs [mult_0_right RS sym]) 1); |
3366 | 353 |
qed "dvd_0_right"; |
354 |
Addsimps [dvd_0_right]; |
|
355 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
356 |
Goalw [dvd_def] "0 dvd m ==> m = 0"; |
4089 | 357 |
by (fast_tac (claset() addss simpset()) 1); |
3366 | 358 |
qed "dvd_0_left"; |
359 |
||
5069 | 360 |
Goalw [dvd_def] "1 dvd k"; |
3366 | 361 |
by (Simp_tac 1); |
362 |
qed "dvd_1_left"; |
|
363 |
AddIffs [dvd_1_left]; |
|
364 |
||
5069 | 365 |
Goalw [dvd_def] "m dvd m"; |
4089 | 366 |
by (blast_tac (claset() addIs [mult_1_right RS sym]) 1); |
3366 | 367 |
qed "dvd_refl"; |
368 |
Addsimps [dvd_refl]; |
|
369 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
370 |
Goalw [dvd_def] "[| m dvd n; n dvd p |] ==> m dvd p"; |
4089 | 371 |
by (blast_tac (claset() addIs [mult_assoc] ) 1); |
3366 | 372 |
qed "dvd_trans"; |
373 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
374 |
Goalw [dvd_def] "[| m dvd n; n dvd m |] ==> m=n"; |
4089 | 375 |
by (fast_tac (claset() addDs [mult_eq_self_implies_10] |
376 |
addss (simpset() addsimps [mult_assoc, mult_eq_1_iff])) 1); |
|
3366 | 377 |
qed "dvd_anti_sym"; |
378 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
379 |
Goalw [dvd_def] "[| k dvd m; k dvd n |] ==> k dvd (m + n)"; |
4089 | 380 |
by (blast_tac (claset() addIs [add_mult_distrib2 RS sym]) 1); |
3366 | 381 |
qed "dvd_add"; |
382 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
383 |
Goalw [dvd_def] "[| k dvd m; k dvd n |] ==> k dvd (m-n)"; |
4089 | 384 |
by (blast_tac (claset() addIs [diff_mult_distrib2 RS sym]) 1); |
3366 | 385 |
qed "dvd_diff"; |
386 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
387 |
Goal "[| k dvd (m-n); k dvd n; n<=m |] ==> k dvd m"; |
3457 | 388 |
by (etac (not_less_iff_le RS iffD2 RS add_diff_inverse RS subst) 1); |
4089 | 389 |
by (blast_tac (claset() addIs [dvd_add]) 1); |
3366 | 390 |
qed "dvd_diffD"; |
391 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
392 |
Goalw [dvd_def] "k dvd n ==> k dvd (m*n)"; |
4089 | 393 |
by (blast_tac (claset() addIs [mult_left_commute]) 1); |
3366 | 394 |
qed "dvd_mult"; |
395 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
396 |
Goal "k dvd m ==> k dvd (m*n)"; |
3366 | 397 |
by (stac mult_commute 1); |
398 |
by (etac dvd_mult 1); |
|
399 |
qed "dvd_mult2"; |
|
400 |
||
401 |
(* k dvd (m*k) *) |
|
402 |
AddIffs [dvd_refl RS dvd_mult, dvd_refl RS dvd_mult2]; |
|
403 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
404 |
Goalw [dvd_def] "[| f dvd m; f dvd n; 0<n |] ==> f dvd (m mod n)"; |
3718 | 405 |
by (Clarify_tac 1); |
4089 | 406 |
by (full_simp_tac (simpset() addsimps [zero_less_mult_iff]) 1); |
3366 | 407 |
by (res_inst_tac |
408 |
[("x", "(((k div ka)*ka + k mod ka) - ((f*k) div (f*ka)) * ka)")] |
|
409 |
exI 1); |
|
4089 | 410 |
by (asm_simp_tac (simpset() addsimps [diff_mult_distrib2, |
3366 | 411 |
mult_mod_distrib, add_mult_distrib2]) 1); |
412 |
qed "dvd_mod"; |
|
413 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
414 |
Goal "[| k dvd (m mod n); k dvd n; 0<n |] ==> k dvd m"; |
3366 | 415 |
by (subgoal_tac "k dvd ((m div n)*n + m mod n)" 1); |
4089 | 416 |
by (asm_simp_tac (simpset() addsimps [dvd_add, dvd_mult]) 2); |
4356 | 417 |
by (asm_full_simp_tac (simpset() addsimps [mod_div_equality]) 1); |
3366 | 418 |
qed "dvd_mod_imp_dvd"; |
419 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
420 |
Goalw [dvd_def] "!!k. [| (k*m) dvd (k*n); 0<k |] ==> m dvd n"; |
3366 | 421 |
by (etac exE 1); |
4089 | 422 |
by (asm_full_simp_tac (simpset() addsimps mult_ac) 1); |
3366 | 423 |
by (Blast_tac 1); |
424 |
qed "dvd_mult_cancel"; |
|
425 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
426 |
Goalw [dvd_def] "[| i dvd m; j dvd n|] ==> (i*j) dvd (m*n)"; |
3718 | 427 |
by (Clarify_tac 1); |
3366 | 428 |
by (res_inst_tac [("x","k*ka")] exI 1); |
4089 | 429 |
by (asm_simp_tac (simpset() addsimps mult_ac) 1); |
3366 | 430 |
qed "mult_dvd_mono"; |
431 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
432 |
Goalw [dvd_def] "(i*j) dvd k ==> i dvd k"; |
4089 | 433 |
by (full_simp_tac (simpset() addsimps [mult_assoc]) 1); |
3366 | 434 |
by (Blast_tac 1); |
435 |
qed "dvd_mult_left"; |
|
436 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
437 |
Goalw [dvd_def] "[| k dvd n; 0 < n |] ==> k <= n"; |
3718 | 438 |
by (Clarify_tac 1); |
4089 | 439 |
by (ALLGOALS (full_simp_tac (simpset() addsimps [zero_less_mult_iff]))); |
3457 | 440 |
by (etac conjE 1); |
441 |
by (rtac le_trans 1); |
|
442 |
by (rtac (le_refl RS mult_le_mono) 2); |
|
3366 | 443 |
by (etac Suc_leI 2); |
444 |
by (Simp_tac 1); |
|
445 |
qed "dvd_imp_le"; |
|
446 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
447 |
Goalw [dvd_def] "0<k ==> (k dvd n) = (n mod k = 0)"; |
3724 | 448 |
by Safe_tac; |
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
449 |
by (asm_simp_tac (simpset() addsimps [mult_commute]) 1); |
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
450 |
by (eres_inst_tac [("t","n")] (mod_div_equality RS subst) 1); |
3366 | 451 |
by (stac mult_commute 1); |
452 |
by (Asm_simp_tac 1); |
|
453 |
by (Blast_tac 1); |
|
454 |
qed "dvd_eq_mod_eq_0"; |