author | paulson |
Fri, 21 Apr 2000 11:28:18 +0200 | |
changeset 8756 | b03a0b219139 |
parent 8698 | 8812dad6ef12 |
child 8783 | 9edcc005ebd9 |
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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Goal "(%m. m mod n) = wfrec (trancl pred_nat) \ |
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\ (%f j. if j<n | n=0 then j else f (j-n))"; |
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by (simp_tac (simpset() addsimps [mod_def]) 1); |
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qed "mod_eq"; |
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Goal "(%m. m div n) = wfrec (trancl pred_nat) \ |
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\ (%f j. if j<n | n=0 then 0 else Suc (f (j-n)))"; |
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by (simp_tac (simpset() addsimps [div_def]) 1); |
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qed "div_eq"; |
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|
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(** Aribtrary definitions for division by zero. Useful to simplify |
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certain equations **) |
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|
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Goal "a div 0 = 0"; |
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by (rtac (div_eq RS wf_less_trans) 1); |
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by (Asm_simp_tac 1); |
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qed "DIVISION_BY_ZERO_DIV"; (*NOT for adding to default simpset*) |
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Goal "a mod 0 = a"; |
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by (rtac (mod_eq RS wf_less_trans) 1); |
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by (Asm_simp_tac 1); |
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qed "DIVISION_BY_ZERO_MOD"; (*NOT for adding to default simpset*) |
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fun div_undefined_case_tac s i = |
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case_tac s i THEN |
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Full_simp_tac (i+1) THEN |
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asm_simp_tac (simpset() addsimps [DIVISION_BY_ZERO_DIV, |
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DIVISION_BY_ZERO_MOD]) i; |
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|
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(*** Remainder ***) |
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Goal "m<n ==> m mod n = (m::nat)"; |
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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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Addsimps [mod_less]; |
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|
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Goal "~ m < (n::nat) ==> m mod n = (m-n) mod n"; |
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by (div_undefined_case_tac "n=0" 1); |
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by (rtac (mod_eq RS wf_less_trans) 1); |
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by (asm_simp_tac (simpset() addsimps [diff_less, cut_apply, less_eq]) 1); |
3366 | 57 |
qed "mod_geq"; |
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(*Avoids the ugly ~m<n above*) |
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Goal "(n::nat) <= 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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||
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Goal "m mod (n::nat) = (if m<n then m else (m-n) mod n)"; |
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by (asm_simp_tac (simpset() addsimps [mod_geq]) 1); |
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qed "mod_if"; |
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Goal "m mod 1 = 0"; |
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by (induct_tac "m" 1); |
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by (ALLGOALS (asm_simp_tac (simpset() addsimps [mod_geq]))); |
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qed "mod_1"; |
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Addsimps [mod_1]; |
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Goal "n mod n = 0"; |
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by (div_undefined_case_tac "n=0" 1); |
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by (asm_simp_tac (simpset() addsimps [mod_geq]) 1); |
3366 | 77 |
qed "mod_self"; |
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Goal "(m+n) mod n = m mod (n::nat)"; |
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by (subgoal_tac "(n + m) mod n = (n+m-n) mod n" 1); |
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by (stac (mod_geq RS sym) 2); |
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by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [add_commute]))); |
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qed "mod_add_self2"; |
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Goal "(n+m) mod n = m mod (n::nat)"; |
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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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Goal "(m + k*n) mod n = m mod (n::nat)"; |
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by (induct_tac "k" 1); |
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by (ALLGOALS |
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(asm_simp_tac |
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(simpset() addsimps [read_instantiate [("y","n")] add_left_commute, |
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mod_add_self1]))); |
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qed "mod_mult_self1"; |
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Goal "(m + n*k) mod n = m mod (n::nat)"; |
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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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Addsimps [mod_mult_self1, mod_mult_self2]; |
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Goal "(m mod n) * (k::nat) = (m*k) mod (n*k)"; |
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by (div_undefined_case_tac "n=0" 1); |
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by (div_undefined_case_tac "k=0" 1); |
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by (res_inst_tac [("n","m")] less_induct 1); |
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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_geq, |
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diff_less, diff_mult_distrib]) 1); |
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qed "mod_mult_distrib"; |
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Goal "(k::nat) * (m mod n) = (k*m) mod (k*n)"; |
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by (asm_simp_tac |
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(simpset() addsimps [read_instantiate [("m","k")] mult_commute, |
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mod_mult_distrib]) 1); |
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qed "mod_mult_distrib2"; |
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Goal "(m*n) mod n = 0"; |
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by (div_undefined_case_tac "n=0" 1); |
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by (induct_tac "m" 1); |
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by (Asm_simp_tac 1); |
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by (rename_tac "k" 1); |
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by (cut_inst_tac [("m","k*n"),("n","n")] mod_add_self2 1); |
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by (asm_full_simp_tac (simpset() addsimps [add_commute]) 1); |
3366 | 126 |
qed "mod_mult_self_is_0"; |
7082 | 127 |
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Goal "(n*m) mod n = 0"; |
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by (simp_tac (simpset() addsimps [mult_commute, mod_mult_self_is_0]) 1); |
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qed "mod_mult_self1_is_0"; |
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Addsimps [mod_mult_self_is_0, mod_mult_self1_is_0]; |
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(*** Quotient ***) |
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Goal "m<n ==> m div n = 0"; |
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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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Addsimps [div_less]; |
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Goal "[| 0<n; ~m<n |] ==> m div n = Suc((m-n) div n)"; |
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by (rtac (div_eq RS wf_less_trans) 1); |
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by (asm_simp_tac (simpset() addsimps [diff_less, cut_apply, less_eq]) 1); |
3366 | 145 |
qed "div_geq"; |
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(*Avoids the ugly ~m<n above*) |
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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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||
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Goal "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))"; |
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by (asm_simp_tac (simpset() addsimps [div_geq]) 1); |
4774 | 154 |
qed "div_if"; |
155 |
||
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(*Main Result about quotient and remainder.*) |
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Goal "(m div n)*n + m mod n = (m::nat)"; |
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by (div_undefined_case_tac "n=0" 1); |
3366 | 160 |
by (res_inst_tac [("n","m")] less_induct 1); |
4774 | 161 |
by (stac mod_if 1); |
162 |
by (ALLGOALS (asm_simp_tac |
|
8393 | 163 |
(simpset() addsimps [add_assoc, div_geq, |
5537 | 164 |
add_diff_inverse, diff_less]))); |
3366 | 165 |
qed "mod_div_equality"; |
166 |
||
4358 | 167 |
(* a simple rearrangement of mod_div_equality: *) |
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Goal "(n::nat) * (m div n) = m - (m mod n)"; |
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by (cut_inst_tac [("m","m"),("n","n")] mod_div_equality 1); |
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by (EVERY1[etac subst, simp_tac (simpset() addsimps mult_ac), |
171 |
K(IF_UNSOLVED no_tac)]); |
|
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qed "mult_div_cancel"; |
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173 |
||
5069 | 174 |
Goal "m div 1 = m"; |
3366 | 175 |
by (induct_tac "m" 1); |
8393 | 176 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [div_geq]))); |
3366 | 177 |
qed "div_1"; |
178 |
Addsimps [div_1]; |
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||
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Goal "0<n ==> n div n = 1"; |
8393 | 181 |
by (asm_simp_tac (simpset() addsimps [div_geq]) 1); |
3366 | 182 |
qed "div_self"; |
183 |
||
4811 | 184 |
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185 |
Goal "0<n ==> (m+n) div n = Suc (m div n)"; |
4811 | 186 |
by (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n)" 1); |
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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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191 |
Goal "0<n ==> (n+m) div n = Suc (m div n)"; |
4811 | 192 |
by (asm_simp_tac (simpset() addsimps [add_commute, div_add_self2]) 1); |
193 |
qed "div_add_self1"; |
|
194 |
||
5069 | 195 |
Goal "!!n. 0<n ==> (m + k*n) div n = k + m div n"; |
4811 | 196 |
by (induct_tac "k" 1); |
5537 | 197 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps add_ac @ [div_add_self1]))); |
4811 | 198 |
qed "div_mult_self1"; |
199 |
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200 |
Goal "0<n ==> (m + n*k) div n = k + m div n"; |
4811 | 201 |
by (asm_simp_tac (simpset() addsimps [mult_commute, div_mult_self1]) 1); |
202 |
qed "div_mult_self2"; |
|
203 |
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204 |
Addsimps [div_mult_self1, div_mult_self2]; |
|
205 |
||
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
206 |
(** A dividend of zero **) |
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
207 |
|
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
208 |
Goal "0 div m = 0"; |
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
209 |
by (div_undefined_case_tac "m=0" 1); |
8393 | 210 |
by (Asm_simp_tac 1); |
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
211 |
qed "div_0"; |
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
212 |
|
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
213 |
Goal "0 mod m = 0"; |
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
214 |
by (div_undefined_case_tac "m=0" 1); |
8393 | 215 |
by (Asm_simp_tac 1); |
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
216 |
qed "mod_0"; |
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
217 |
Addsimps [div_0, mod_0]; |
4811 | 218 |
|
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
219 |
(* Monotonicity of div in first argument *) |
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
220 |
Goal "ALL m::nat. m <= n --> (m div k) <= (n div k)"; |
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
221 |
by (div_undefined_case_tac "k=0" 1); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
222 |
by (res_inst_tac [("n","n")] less_induct 1); |
3718 | 223 |
by (Clarify_tac 1); |
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
224 |
by (case_tac "n<k" 1); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
225 |
(* 1 case n<k *) |
8393 | 226 |
by (Asm_simp_tac 1); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
227 |
(* 2 case n >= k *) |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
228 |
by (case_tac "m<k" 1); |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
229 |
(* 2.1 case m<k *) |
8393 | 230 |
by (Asm_simp_tac 1); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
231 |
(* 2.2 case m>=k *) |
4089 | 232 |
by (asm_simp_tac (simpset() addsimps [div_geq, diff_less, diff_le_mono]) 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_le_mono"; |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
234 |
|
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
235 |
(* Antimonotonicity of div in second argument *) |
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
236 |
Goal "[| 0<m; m<=n |] ==> (k div n) <= (k div m)"; |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
237 |
by (subgoal_tac "0<n" 1); |
6073 | 238 |
by (Asm_simp_tac 2); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
239 |
by (res_inst_tac [("n","k")] less_induct 1); |
3496 | 240 |
by (rename_tac "k" 1); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
241 |
by (case_tac "k<n" 1); |
8393 | 242 |
by (Asm_simp_tac 1); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
243 |
by (subgoal_tac "~(k<m)" 1); |
6073 | 244 |
by (Asm_simp_tac 2); |
4089 | 245 |
by (asm_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
|
246 |
by (subgoal_tac "(k-n) div n <= (k-m) div n" 1); |
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
247 |
by (REPEAT (ares_tac [div_le_mono,diff_le_mono2] 2)); |
5318 | 248 |
by (rtac le_trans 1); |
5316 | 249 |
by (Asm_simp_tac 1); |
250 |
by (asm_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
|
251 |
qed "div_le_mono2"; |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
252 |
|
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
253 |
Goal "m div n <= (m::nat)"; |
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
254 |
by (div_undefined_case_tac "n=0" 1); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
255 |
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
|
256 |
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
|
257 |
by (rtac div_le_mono2 1); |
6073 | 258 |
by (ALLGOALS Asm_simp_tac); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
259 |
qed "div_le_dividend"; |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
260 |
Addsimps [div_le_dividend]; |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
261 |
|
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
262 |
(* Similar for "less than" *) |
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
263 |
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
|
264 |
by (res_inst_tac [("n","m")] less_induct 1); |
3496 | 265 |
by (rename_tac "m" 1); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
266 |
by (case_tac "m<n" 1); |
8393 | 267 |
by (Asm_full_simp_tac 1); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
268 |
by (subgoal_tac "0<n" 1); |
6073 | 269 |
by (Asm_simp_tac 2); |
4089 | 270 |
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
|
271 |
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
|
272 |
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
|
273 |
by (REPEAT (ares_tac [impI,less_trans_Suc] 1)); |
4089 | 274 |
by (asm_full_simp_tac (simpset() addsimps [diff_less]) 1); |
275 |
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
|
276 |
(* case n=m *) |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
277 |
by (subgoal_tac "m=n" 1); |
6073 | 278 |
by (Asm_simp_tac 2); |
8393 | 279 |
by (Asm_simp_tac 1); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
280 |
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
|
281 |
Addsimps [div_less_dividend]; |
3366 | 282 |
|
283 |
(*** Further facts about mod (mainly for the mutilated chess board ***) |
|
284 |
||
5278 | 285 |
Goal "0<n ==> Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"; |
3366 | 286 |
by (res_inst_tac [("n","m")] less_induct 1); |
287 |
by (excluded_middle_tac "Suc(na)<n" 1); |
|
288 |
(* case Suc(na) < n *) |
|
289 |
by (forward_tac [lessI RS less_trans] 2); |
|
8393 | 290 |
by (asm_simp_tac (simpset() addsimps [less_not_refl3]) 2); |
3366 | 291 |
(* case n <= Suc(na) *) |
5415 | 292 |
by (asm_full_simp_tac (simpset() addsimps [not_less_iff_le, le_Suc_eq, |
293 |
mod_geq]) 1); |
|
294 |
by (etac disjE 1); |
|
8393 | 295 |
by (Asm_simp_tac 2); |
7059 | 296 |
by (asm_simp_tac (simpset() addsimps [Suc_diff_le, diff_less, |
5415 | 297 |
le_mod_geq]) 1); |
3366 | 298 |
qed "mod_Suc"; |
299 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
300 |
Goal "0<n ==> m mod n < n"; |
3366 | 301 |
by (res_inst_tac [("n","m")] less_induct 1); |
5498 | 302 |
by (case_tac "na<n" 1); |
303 |
(*case n le na*) |
|
304 |
by (asm_full_simp_tac (simpset() addsimps [mod_geq, diff_less]) 2); |
|
3366 | 305 |
(*case na<n*) |
8393 | 306 |
by (Asm_simp_tac 1); |
3366 | 307 |
qed "mod_less_divisor"; |
8698 | 308 |
Addsimps [mod_less_divisor]; |
3366 | 309 |
|
310 |
(** Evens and Odds **) |
|
311 |
||
312 |
(*With less_zeroE, causes case analysis on b<2*) |
|
313 |
AddSEs [less_SucE]; |
|
314 |
||
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
315 |
Goal "b<2 ==> (k mod 2 = b) | (k mod 2 = (if b=1 then 0 else 1))"; |
3366 | 316 |
by (subgoal_tac "k mod 2 < 2" 1); |
8698 | 317 |
by (Asm_simp_tac 2); |
4686 | 318 |
by (Asm_simp_tac 1); |
4356 | 319 |
by Safe_tac; |
3366 | 320 |
qed "mod2_cases"; |
321 |
||
5069 | 322 |
Goal "Suc(Suc(m)) mod 2 = m mod 2"; |
3366 | 323 |
by (subgoal_tac "m mod 2 < 2" 1); |
8698 | 324 |
by (Asm_simp_tac 2); |
3724 | 325 |
by Safe_tac; |
4089 | 326 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [mod_Suc]))); |
3366 | 327 |
qed "mod2_Suc_Suc"; |
328 |
Addsimps [mod2_Suc_Suc]; |
|
329 |
||
5069 | 330 |
Goal "(0 < m mod 2) = (m mod 2 = 1)"; |
3366 | 331 |
by (subgoal_tac "m mod 2 < 2" 1); |
8698 | 332 |
by (Asm_simp_tac 2); |
333 |
by (auto_tac (claset(), simpset() delsimps [mod_less_divisor])); |
|
4356 | 334 |
qed "mod2_gr_0"; |
335 |
Addsimps [mod2_gr_0]; |
|
336 |
||
5069 | 337 |
Goal "(m+m) mod 2 = 0"; |
3366 | 338 |
by (induct_tac "m" 1); |
8393 | 339 |
by Auto_tac; |
4385 | 340 |
qed "mod2_add_self_eq_0"; |
341 |
Addsimps [mod2_add_self_eq_0]; |
|
342 |
||
5069 | 343 |
Goal "((m+m)+n) mod 2 = n mod 2"; |
4385 | 344 |
by (induct_tac "m" 1); |
8393 | 345 |
by Auto_tac; |
3366 | 346 |
qed "mod2_add_self"; |
347 |
Addsimps [mod2_add_self]; |
|
348 |
||
5498 | 349 |
(*Restore the default*) |
3366 | 350 |
Delrules [less_SucE]; |
351 |
||
352 |
(*** More division laws ***) |
|
353 |
||
7007 | 354 |
Goal "0<n ==> (m*n) div n = m"; |
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
355 |
by (cut_inst_tac [("m", "m*n"),("n","n")] mod_div_equality 1); |
4089 | 356 |
by (asm_full_simp_tac (simpset() addsimps [mod_mult_self_is_0]) 1); |
3366 | 357 |
qed "div_mult_self_is_m"; |
7082 | 358 |
|
359 |
Goal "0<n ==> (n*m) div n = m"; |
|
360 |
by (asm_simp_tac (simpset() addsimps [mult_commute, div_mult_self_is_m]) 1); |
|
361 |
qed "div_mult_self1_is_m"; |
|
362 |
Addsimps [div_mult_self_is_m, div_mult_self1_is_m]; |
|
3366 | 363 |
|
364 |
(*Cancellation law for division*) |
|
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
365 |
Goal "0<k ==> (k*m) div (k*n) = m div n"; |
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
366 |
by (div_undefined_case_tac "n=0" 1); |
3366 | 367 |
by (res_inst_tac [("n","m")] less_induct 1); |
368 |
by (case_tac "na<n" 1); |
|
8393 | 369 |
by (asm_simp_tac (simpset() addsimps [zero_less_mult_iff, mult_less_mono2]) 1); |
3366 | 370 |
by (subgoal_tac "~ k*na < k*n" 1); |
371 |
by (asm_simp_tac |
|
4089 | 372 |
(simpset() addsimps [zero_less_mult_iff, div_geq, |
5415 | 373 |
diff_mult_distrib2 RS sym, diff_less]) 1); |
4089 | 374 |
by (asm_full_simp_tac (simpset() addsimps [not_less_iff_le, |
3366 | 375 |
le_refl RS mult_le_mono]) 1); |
376 |
qed "div_cancel"; |
|
377 |
Addsimps [div_cancel]; |
|
378 |
||
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
379 |
(*mod_mult_distrib2 above is the counterpart for remainder*) |
3366 | 380 |
|
381 |
||
382 |
(************************************************) |
|
383 |
(** Divides Relation **) |
|
384 |
(************************************************) |
|
385 |
||
5069 | 386 |
Goalw [dvd_def] "m dvd 0"; |
4089 | 387 |
by (blast_tac (claset() addIs [mult_0_right RS sym]) 1); |
3366 | 388 |
qed "dvd_0_right"; |
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
389 |
AddIffs [dvd_0_right]; |
3366 | 390 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
391 |
Goalw [dvd_def] "0 dvd m ==> m = 0"; |
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
392 |
by Auto_tac; |
3366 | 393 |
qed "dvd_0_left"; |
394 |
||
5069 | 395 |
Goalw [dvd_def] "1 dvd k"; |
3366 | 396 |
by (Simp_tac 1); |
397 |
qed "dvd_1_left"; |
|
398 |
AddIffs [dvd_1_left]; |
|
399 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
400 |
Goalw [dvd_def] "m dvd (m::nat)"; |
4089 | 401 |
by (blast_tac (claset() addIs [mult_1_right RS sym]) 1); |
3366 | 402 |
qed "dvd_refl"; |
403 |
Addsimps [dvd_refl]; |
|
404 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
405 |
Goalw [dvd_def] "[| m dvd n; n dvd p |] ==> m dvd (p::nat)"; |
4089 | 406 |
by (blast_tac (claset() addIs [mult_assoc] ) 1); |
3366 | 407 |
qed "dvd_trans"; |
408 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
409 |
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
|
410 |
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
|
411 |
simpset() addsimps [mult_assoc, mult_eq_1_iff]) 1); |
3366 | 412 |
qed "dvd_anti_sym"; |
413 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
414 |
Goalw [dvd_def] "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"; |
4089 | 415 |
by (blast_tac (claset() addIs [add_mult_distrib2 RS sym]) 1); |
3366 | 416 |
qed "dvd_add"; |
417 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
418 |
Goalw [dvd_def] "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"; |
4089 | 419 |
by (blast_tac (claset() addIs [diff_mult_distrib2 RS sym]) 1); |
3366 | 420 |
qed "dvd_diff"; |
421 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
422 |
Goal "[| k dvd (m-n); k dvd n; n<=m |] ==> k dvd (m::nat)"; |
3457 | 423 |
by (etac (not_less_iff_le RS iffD2 RS add_diff_inverse RS subst) 1); |
4089 | 424 |
by (blast_tac (claset() addIs [dvd_add]) 1); |
3366 | 425 |
qed "dvd_diffD"; |
426 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
427 |
Goalw [dvd_def] "k dvd n ==> k dvd (m*n :: nat)"; |
4089 | 428 |
by (blast_tac (claset() addIs [mult_left_commute]) 1); |
3366 | 429 |
qed "dvd_mult"; |
430 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
431 |
Goal "k dvd m ==> k dvd (m*n :: nat)"; |
3366 | 432 |
by (stac mult_commute 1); |
433 |
by (etac dvd_mult 1); |
|
434 |
qed "dvd_mult2"; |
|
435 |
||
436 |
(* k dvd (m*k) *) |
|
437 |
AddIffs [dvd_refl RS dvd_mult, dvd_refl RS dvd_mult2]; |
|
438 |
||
7493 | 439 |
Goal "k dvd (n + k) = k dvd (n::nat)"; |
7499 | 440 |
by (rtac iffI 1); |
441 |
by (etac dvd_add 2); |
|
442 |
by (rtac dvd_refl 2); |
|
7493 | 443 |
by (subgoal_tac "n = (n+k)-k" 1); |
444 |
by (Simp_tac 2); |
|
7499 | 445 |
by (etac ssubst 1); |
446 |
by (etac dvd_diff 1); |
|
447 |
by (rtac dvd_refl 1); |
|
7493 | 448 |
qed "dvd_reduce"; |
449 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
450 |
Goalw [dvd_def] "[| f dvd m; f dvd n; 0<n |] ==> f dvd (m mod n)"; |
3718 | 451 |
by (Clarify_tac 1); |
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
452 |
by (Full_simp_tac 1); |
3366 | 453 |
by (res_inst_tac |
454 |
[("x", "(((k div ka)*ka + k mod ka) - ((f*k) div (f*ka)) * ka)")] |
|
455 |
exI 1); |
|
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
456 |
by (asm_simp_tac |
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
457 |
(simpset() addsimps [diff_mult_distrib2, mod_mult_distrib2 RS sym, |
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
458 |
add_mult_distrib2]) 1); |
3366 | 459 |
qed "dvd_mod"; |
460 |
||
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
461 |
Goal "[| (k::nat) dvd (m mod n); k dvd n |] ==> k dvd m"; |
3366 | 462 |
by (subgoal_tac "k dvd ((m div n)*n + m mod n)" 1); |
4089 | 463 |
by (asm_simp_tac (simpset() addsimps [dvd_add, dvd_mult]) 2); |
4356 | 464 |
by (asm_full_simp_tac (simpset() addsimps [mod_div_equality]) 1); |
3366 | 465 |
qed "dvd_mod_imp_dvd"; |
466 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
467 |
Goalw [dvd_def] "!!k::nat. [| (k*m) dvd (k*n); 0<k |] ==> m dvd n"; |
3366 | 468 |
by (etac exE 1); |
4089 | 469 |
by (asm_full_simp_tac (simpset() addsimps mult_ac) 1); |
3366 | 470 |
by (Blast_tac 1); |
471 |
qed "dvd_mult_cancel"; |
|
472 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
473 |
Goalw [dvd_def] "[| i dvd m; j dvd n|] ==> (i*j) dvd (m*n :: nat)"; |
3718 | 474 |
by (Clarify_tac 1); |
3366 | 475 |
by (res_inst_tac [("x","k*ka")] exI 1); |
4089 | 476 |
by (asm_simp_tac (simpset() addsimps mult_ac) 1); |
3366 | 477 |
qed "mult_dvd_mono"; |
478 |
||
6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
6073
diff
changeset
|
479 |
Goalw [dvd_def] "(i*j :: nat) dvd k ==> i dvd k"; |
4089 | 480 |
by (full_simp_tac (simpset() addsimps [mult_assoc]) 1); |
3366 | 481 |
by (Blast_tac 1); |
482 |
qed "dvd_mult_left"; |
|
483 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
484 |
Goalw [dvd_def] "[| k dvd n; 0 < n |] ==> k <= n"; |
3718 | 485 |
by (Clarify_tac 1); |
4089 | 486 |
by (ALLGOALS (full_simp_tac (simpset() addsimps [zero_less_mult_iff]))); |
3457 | 487 |
by (etac conjE 1); |
488 |
by (rtac le_trans 1); |
|
489 |
by (rtac (le_refl RS mult_le_mono) 2); |
|
3366 | 490 |
by (etac Suc_leI 2); |
491 |
by (Simp_tac 1); |
|
492 |
qed "dvd_imp_le"; |
|
493 |
||
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
494 |
Goalw [dvd_def] "(k dvd n) = (n mod k = 0)"; |
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
495 |
by (div_undefined_case_tac "k=0" 1); |
3724 | 496 |
by Safe_tac; |
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
497 |
by (asm_simp_tac (simpset() addsimps [mult_commute]) 1); |
7029
08d4eb8500dd
new division laws taking advantage of (m div 0) = 0 and (m mod 0) = m
paulson
parents:
7007
diff
changeset
|
498 |
by (res_inst_tac [("t","n"),("n1","k")] (mod_div_equality RS subst) 1); |
3366 | 499 |
by (stac mult_commute 1); |
500 |
by (Asm_simp_tac 1); |
|
501 |
qed "dvd_eq_mod_eq_0"; |