src/HOL/Divides.thy
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(*  Title:      HOL/Divides.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1999  University of Cambridge
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*)
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section \<open>The division operators div and mod\<close>
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theory Divides
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imports Parity
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begin
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subsection \<open>Abstract division in commutative semirings.\<close>
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class semiring_div = semidom + semiring_modulo +
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  assumes div_by_0: "a div 0 = 0"
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    and div_0: "0 div a = 0"
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    and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
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    and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
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begin
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subclass algebraic_semidom
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proof
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  fix b a
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  assume "b \<noteq> 0"
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  then show "a * b div b = a"
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    using div_mult_self1 [of b 0 a] by (simp add: ac_simps div_0)
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qed (simp add: div_by_0)
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lemma div_by_1:
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  "a div 1 = a"
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  by (fact divide_1)
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lemma div_mult_self1_is_id:
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  "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
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  by (fact nonzero_mult_divide_cancel_left)
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lemma div_mult_self2_is_id:
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  "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
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  by (fact nonzero_mult_divide_cancel_right)
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text \<open>@{const divide} and @{const modulo}\<close>
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lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
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  by (simp add: mod_div_equality)
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lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
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  by (simp add: mod_div_equality2)
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lemma mod_by_0 [simp]: "a mod 0 = a"
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  using mod_div_equality [of a zero] by simp
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lemma mod_0 [simp]: "0 mod a = 0"
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  using mod_div_equality [of zero a] div_0 by simp
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lemma div_mult_self2 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(a + b * c) div b = c + a div b"
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  using assms div_mult_self1 [of b a c] by (simp add: mult.commute)
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lemma div_mult_self3 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(c * b + a) div b = c + a div b"
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  using assms by (simp add: add.commute)
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lemma div_mult_self4 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(b * c + a) div b = c + a div b"
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  using assms by (simp add: add.commute)
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lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
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proof (cases "b = 0")
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  case True then show ?thesis by simp
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next
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  case False
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  have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
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    by (simp add: mod_div_equality)
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  also from False div_mult_self1 [of b a c] have
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    "\<dots> = (c + a div b) * b + (a + c * b) mod b"
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      by (simp add: algebra_simps)
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  finally have "a = a div b * b + (a + c * b) mod b"
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    by (simp add: add.commute [of a] add.assoc distrib_right)
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  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
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    by (simp add: mod_div_equality)
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  then show ?thesis by simp
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qed
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lemma mod_mult_self2 [simp]:
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  "(a + b * c) mod b = a mod b"
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  by (simp add: mult.commute [of b])
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lemma mod_mult_self3 [simp]:
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  "(c * b + a) mod b = a mod b"
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  by (simp add: add.commute)
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lemma mod_mult_self4 [simp]:
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  "(b * c + a) mod b = a mod b"
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  by (simp add: add.commute)
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lemma mod_mult_self1_is_0 [simp]:
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  "b * a mod b = 0"
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  using mod_mult_self2 [of 0 b a] by simp
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lemma mod_mult_self2_is_0 [simp]:
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  "a * b mod b = 0"
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  using mod_mult_self1 [of 0 a b] by simp
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lemma mod_by_1 [simp]:
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  "a mod 1 = 0"
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proof -
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  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
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  then have "a + a mod 1 = a + 0" by simp
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  then show ?thesis by (rule add_left_imp_eq)
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qed
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lemma mod_self [simp]:
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  "a mod a = 0"
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  using mod_mult_self2_is_0 [of 1] by simp
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lemma div_add_self1:
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  assumes "b \<noteq> 0"
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  shows "(b + a) div b = a div b + 1"
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  using assms div_mult_self1 [of b a 1] by (simp add: add.commute)
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lemma div_add_self2:
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  assumes "b \<noteq> 0"
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  shows "(a + b) div b = a div b + 1"
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  using assms div_add_self1 [of b a] by (simp add: add.commute)
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lemma mod_add_self1 [simp]:
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  "(b + a) mod b = a mod b"
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  using mod_mult_self1 [of a 1 b] by (simp add: add.commute)
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lemma mod_add_self2 [simp]:
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  "(a + b) mod b = a mod b"
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  using mod_mult_self1 [of a 1 b] by simp
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lemma dvd_imp_mod_0 [simp]:
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  assumes "a dvd b"
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  shows "b mod a = 0"
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proof -
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  from assms obtain c where "b = a * c" ..
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  then have "b mod a = a * c mod a" by simp
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  then show "b mod a = 0" by simp
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qed
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lemma mod_eq_0_iff_dvd:
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   147
  "a mod b = 0 \<longleftrightarrow> b dvd a"
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   148
proof
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   149
  assume "b dvd a"
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   150
  then show "a mod b = 0" by simp
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   151
next
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   152
  assume "a mod b = 0"
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   153
  with mod_div_equality [of a b] have "a div b * b = a" by simp
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   154
  then have "a = b * (a div b)" by (simp add: ac_simps)
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   155
  then show "b dvd a" ..
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   156
qed
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   157
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   158
lemma dvd_eq_mod_eq_0 [nitpick_unfold, code]:
58834
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58786
diff changeset
   159
  "a dvd b \<longleftrightarrow> b mod a = 0"
58911
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   160
  by (simp add: mod_eq_0_iff_dvd)
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   161
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   162
lemma mod_div_trivial [simp]:
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   163
  "a mod b div b = 0"
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   164
proof (cases "b = 0")
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   165
  assume "b = 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   166
  thus ?thesis by simp
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   167
next
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   168
  assume "b \<noteq> 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   169
  hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   170
    by (rule div_mult_self1 [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   171
  also have "\<dots> = a div b"
64164
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 64014
diff changeset
   172
    by (simp only: mod_div_equality3)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   173
  also have "\<dots> = a div b + 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   174
    by simp
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   175
  finally show ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   176
    by (rule add_left_imp_eq)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   177
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   178
58911
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   179
lemma mod_mod_trivial [simp]:
2cf595ee508b proper oriented equivalence of dvd predicate and mod
haftmann
parents: 58889
diff changeset
   180
  "a mod b mod b = a mod b"
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   181
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   182
  have "a mod b mod b = (a mod b + a div b * b) mod b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   183
    by (simp only: mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   184
  also have "\<dots> = a mod b"
64164
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 64014
diff changeset
   185
    by (simp only: mod_div_equality3)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   186
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   187
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   188
58834
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58786
diff changeset
   189
lemma dvd_mod_imp_dvd:
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58786
diff changeset
   190
  assumes "k dvd m mod n" and "k dvd n"
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58786
diff changeset
   191
  shows "k dvd m"
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58786
diff changeset
   192
proof -
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58786
diff changeset
   193
  from assms have "k dvd (m div n) * n + m mod n"
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58786
diff changeset
   194
    by (simp only: dvd_add dvd_mult)
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58786
diff changeset
   195
  then show ?thesis by (simp add: mod_div_equality)
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58786
diff changeset
   196
qed
30078
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   197
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   198
text \<open>Addition respects modular equivalence.\<close>
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   199
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   200
lemma mod_add_left_eq: \<comment> \<open>FIXME reorient\<close>
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   201
  "(a + b) mod c = (a mod c + b) mod c"
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   202
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   203
  have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   204
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   205
  also have "\<dots> = (a mod c + b + a div c * c) mod c"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   206
    by (simp only: ac_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   207
  also have "\<dots> = (a mod c + b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   208
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   209
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   210
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   211
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   212
lemma mod_add_right_eq: \<comment> \<open>FIXME reorient\<close>
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   213
  "(a + b) mod c = (a + b mod c) mod c"
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   214
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   215
  have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   216
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   217
  also have "\<dots> = (a + b mod c + b div c * c) mod c"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   218
    by (simp only: ac_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   219
  also have "\<dots> = (a + b mod c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   220
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   221
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   222
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   223
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   224
lemma mod_add_eq: \<comment> \<open>FIXME reorient\<close>
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   225
  "(a + b) mod c = (a mod c + b mod c) mod c"
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   226
by (rule trans [OF mod_add_left_eq mod_add_right_eq])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   227
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   228
lemma mod_add_cong:
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   229
  assumes "a mod c = a' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   230
  assumes "b mod c = b' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   231
  shows "(a + b) mod c = (a' + b') mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   232
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   233
  have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   234
    unfolding assms ..
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   235
  thus ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   236
    by (simp only: mod_add_eq [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   237
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   238
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   239
text \<open>Multiplication respects modular equivalence.\<close>
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   240
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   241
lemma mod_mult_left_eq: \<comment> \<open>FIXME reorient\<close>
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   242
  "(a * b) mod c = ((a mod c) * b) mod c"
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   243
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   244
  have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   245
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   246
  also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   247
    by (simp only: algebra_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   248
  also have "\<dots> = (a mod c * b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   249
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   250
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   251
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   252
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   253
lemma mod_mult_right_eq: \<comment> \<open>FIXME reorient\<close>
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   254
  "(a * b) mod c = (a * (b mod c)) mod c"
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   255
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   256
  have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   257
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   258
  also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   259
    by (simp only: algebra_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   260
  also have "\<dots> = (a * (b mod c)) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   261
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   262
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   263
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   264
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   265
lemma mod_mult_eq: \<comment> \<open>FIXME reorient\<close>
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   266
  "(a * b) mod c = ((a mod c) * (b mod c)) mod c"
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   267
by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   268
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   269
lemma mod_mult_cong:
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   270
  assumes "a mod c = a' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   271
  assumes "b mod c = b' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   272
  shows "(a * b) mod c = (a' * b') mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   273
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   274
  have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   275
    unfolding assms ..
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   276
  thus ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   277
    by (simp only: mod_mult_eq [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   278
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   279
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   280
text \<open>Exponentiation respects modular equivalence.\<close>
47164
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   281
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   282
lemma power_mod: "(a mod b) ^ n mod b = a ^ n mod b"
47164
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   283
apply (induct n, simp_all)
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   284
apply (rule mod_mult_right_eq [THEN trans])
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   285
apply (simp (no_asm_simp))
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   286
apply (rule mod_mult_eq [symmetric])
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   287
done
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   288
29404
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   289
lemma mod_mod_cancel:
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   290
  assumes "c dvd b"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   291
  shows "a mod b mod c = a mod c"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   292
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   293
  from \<open>c dvd b\<close> obtain k where "b = c * k"
29404
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   294
    by (rule dvdE)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   295
  have "a mod b mod c = a mod (c * k) mod c"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   296
    by (simp only: \<open>b = c * k\<close>)
29404
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   297
  also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   298
    by (simp only: mod_mult_self1)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   299
  also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
58786
fa5b67fb70ad more simp rules;
haftmann
parents: 58778
diff changeset
   300
    by (simp only: ac_simps)
29404
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   301
  also have "\<dots> = a mod c"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   302
    by (simp only: mod_div_equality)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   303
  finally show ?thesis .
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   304
qed
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   305
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   306
lemma div_mult_mult2 [simp]:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   307
  "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   308
  by (drule div_mult_mult1) (simp add: mult.commute)
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   309
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   310
lemma div_mult_mult1_if [simp]:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   311
  "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   312
  by simp_all
30476
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   313
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   314
lemma mod_mult_mult1:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   315
  "(c * a) mod (c * b) = c * (a mod b)"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   316
proof (cases "c = 0")
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   317
  case True then show ?thesis by simp
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   318
next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   319
  case False
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   320
  from mod_div_equality
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   321
  have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   322
  with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   323
    = c * a + c * (a mod b)" by (simp add: algebra_simps)
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60517
diff changeset
   324
  with mod_div_equality show ?thesis by simp
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   325
qed
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60517
diff changeset
   326
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   327
lemma mod_mult_mult2:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   328
  "(a * c) mod (b * c) = (a mod b) * c"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   329
  using mod_mult_mult1 [of c a b] by (simp add: mult.commute)
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   330
47159
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   331
lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   332
  by (fact mod_mult_mult2 [symmetric])
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   333
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   334
lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   335
  by (fact mod_mult_mult1 [symmetric])
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   336
31662
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   337
lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   338
  unfolding dvd_def by (auto simp add: mod_mult_mult1)
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   339
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   340
lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   341
by (blast intro: dvd_mod_imp_dvd dvd_mod)
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   342
63317
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   343
lemma div_div_eq_right:
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   344
  assumes "c dvd b" "b dvd a"
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   345
  shows   "a div (b div c) = a div b * c"
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   346
proof -
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   347
  from assms have "a div b * c = (a * c) div b"
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   348
    by (subst dvd_div_mult) simp_all
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   349
  also from assms have "\<dots> = (a * c) div ((b div c) * c)" by simp
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   350
  also have "a * c div (b div c * c) = a div (b div c)"
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   351
    by (cases "c = 0") simp_all
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   352
  finally show ?thesis ..
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   353
qed
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   354
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   355
lemma div_div_div_same:
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   356
  assumes "d dvd a" "d dvd b" "b dvd a"
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   357
  shows   "(a div d) div (b div d) = a div b"
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   358
  using assms by (subst dvd_div_mult2_eq [symmetric]) simp_all
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63145
diff changeset
   359
31661
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   360
end
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   361
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59816
diff changeset
   362
class ring_div = comm_ring_1 + semiring_div
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   363
begin
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   364
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   365
subclass idom_divide ..
36634
f9b43d197d16 a ring_div is a ring_1_no_zero_divisors
haftmann
parents: 35815
diff changeset
   366
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   367
text \<open>Negation respects modular equivalence.\<close>
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   368
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   369
lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   370
proof -
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   371
  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   372
    by (simp only: mod_div_equality)
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   373
  also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   374
    by (simp add: ac_simps)
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   375
  also have "\<dots> = (- (a mod b)) mod b"
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   376
    by (rule mod_mult_self1)
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   377
  finally show ?thesis .
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   378
qed
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   379
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   380
lemma mod_minus_cong:
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   381
  assumes "a mod b = a' mod b"
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   382
  shows "(- a) mod b = (- a') mod b"
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   383
proof -
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   384
  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   385
    unfolding assms ..
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   386
  thus ?thesis
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   387
    by (simp only: mod_minus_eq [symmetric])
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   388
qed
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   389
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   390
text \<open>Subtraction respects modular equivalence.\<close>
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   391
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54227
diff changeset
   392
lemma mod_diff_left_eq:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54227
diff changeset
   393
  "(a - b) mod c = (a mod c - b) mod c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54227
diff changeset
   394
  using mod_add_cong [of a c "a mod c" "- b" "- b"] by simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54227
diff changeset
   395
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54227
diff changeset
   396
lemma mod_diff_right_eq:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54227
diff changeset
   397
  "(a - b) mod c = (a - b mod c) mod c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54227
diff changeset
   398
  using mod_add_cong [of a c a "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b] by simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54227
diff changeset
   399
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54227
diff changeset
   400
lemma mod_diff_eq:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54227
diff changeset
   401
  "(a - b) mod c = (a mod c - b mod c) mod c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54227
diff changeset
   402
  using mod_add_cong [of a c "a mod c" "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b] by simp
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   403
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   404
lemma mod_diff_cong:
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   405
  assumes "a mod c = a' mod c"
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   406
  assumes "b mod c = b' mod c"
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   407
  shows "(a - b) mod c = (a' - b') mod c"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54227
diff changeset
   408
  using assms mod_add_cong [of a c a' "- b" "- b'"] mod_minus_cong [of b c "b'"] by simp
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   409
30180
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   410
lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   411
apply (case_tac "y = 0") apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   412
apply (auto simp add: dvd_def)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   413
apply (subgoal_tac "-(y * k) = y * - k")
57492
74bf65a1910a Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
parents: 55440
diff changeset
   414
 apply (simp only:)
30180
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   415
 apply (erule div_mult_self1_is_id)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   416
apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   417
done
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   418
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   419
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   420
apply (case_tac "y = 0") apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   421
apply (auto simp add: dvd_def)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   422
apply (subgoal_tac "y * k = -y * -k")
57492
74bf65a1910a Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
parents: 55440
diff changeset
   423
 apply (erule ssubst, rule div_mult_self1_is_id)
30180
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   424
 apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   425
apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   426
done
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   427
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   428
lemma div_diff [simp]:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   429
  "z dvd x \<Longrightarrow> z dvd y \<Longrightarrow> (x - y) div z = x div z - y div z"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   430
  using div_add [of _ _ "- y"] by (simp add: dvd_neg_div)
59380
e7d237c2ce93 added simp lemma
nipkow
parents: 59058
diff changeset
   431
47159
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   432
lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   433
  using div_mult_mult1 [of "- 1" a b]
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   434
  unfolding neg_equal_0_iff_equal by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   435
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   436
lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   437
  using mod_mult_mult1 [of "- 1" a b] by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   438
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   439
lemma div_minus_right: "a div (-b) = (-a) div b"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   440
  using div_minus_minus [of "-a" b] by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   441
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   442
lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   443
  using mod_minus_minus [of "-a" b] by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   444
47160
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   445
lemma div_minus1_right [simp]: "a div (-1) = -a"
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   446
  using div_minus_right [of a 1] by simp
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   447
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   448
lemma mod_minus1_right [simp]: "a mod (-1) = 0"
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   449
  using mod_minus_right [of a 1] by simp
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   450
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60517
diff changeset
   451
lemma minus_mod_self2 [simp]:
54221
56587960e444 more lemmas on division
haftmann
parents: 53374
diff changeset
   452
  "(a - b) mod b = a mod b"
56587960e444 more lemmas on division
haftmann
parents: 53374
diff changeset
   453
  by (simp add: mod_diff_right_eq)
56587960e444 more lemmas on division
haftmann
parents: 53374
diff changeset
   454
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60517
diff changeset
   455
lemma minus_mod_self1 [simp]:
54221
56587960e444 more lemmas on division
haftmann
parents: 53374
diff changeset
   456
  "(b - a) mod b = - a mod b"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54227
diff changeset
   457
  using mod_add_self2 [of "- a" b] by simp
54221
56587960e444 more lemmas on division
haftmann
parents: 53374
diff changeset
   458
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   459
end
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   460
58778
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   461
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   462
subsubsection \<open>Parity and division\<close>
58778
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   463
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60517
diff changeset
   464
class semiring_div_parity = semiring_div + comm_semiring_1_cancel + numeral +
54226
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   465
  assumes parity: "a mod 2 = 0 \<or> a mod 2 = 1"
58786
fa5b67fb70ad more simp rules;
haftmann
parents: 58778
diff changeset
   466
  assumes one_mod_two_eq_one [simp]: "1 mod 2 = 1"
58710
7216a10d69ba augmented and tuned facts on even/odd and division
haftmann
parents: 58646
diff changeset
   467
  assumes zero_not_eq_two: "0 \<noteq> 2"
54226
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   468
begin
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   469
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   470
lemma parity_cases [case_names even odd]:
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   471
  assumes "a mod 2 = 0 \<Longrightarrow> P"
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   472
  assumes "a mod 2 = 1 \<Longrightarrow> P"
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   473
  shows P
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   474
  using assms parity by blast
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   475
58786
fa5b67fb70ad more simp rules;
haftmann
parents: 58778
diff changeset
   476
lemma one_div_two_eq_zero [simp]:
58778
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   477
  "1 div 2 = 0"
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   478
proof (cases "2 = 0")
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   479
  case True then show ?thesis by simp
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   480
next
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   481
  case False
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   482
  from mod_div_equality have "1 div 2 * 2 + 1 mod 2 = 1" .
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   483
  with one_mod_two_eq_one have "1 div 2 * 2 + 1 = 1" by simp
58953
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   484
  then have "1 div 2 * 2 = 0" by (simp add: ac_simps add_left_imp_eq del: mult_eq_0_iff)
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   485
  then have "1 div 2 = 0 \<or> 2 = 0" by simp
58778
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   486
  with False show ?thesis by auto
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   487
qed
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   488
58786
fa5b67fb70ad more simp rules;
haftmann
parents: 58778
diff changeset
   489
lemma not_mod_2_eq_0_eq_1 [simp]:
fa5b67fb70ad more simp rules;
haftmann
parents: 58778
diff changeset
   490
  "a mod 2 \<noteq> 0 \<longleftrightarrow> a mod 2 = 1"
fa5b67fb70ad more simp rules;
haftmann
parents: 58778
diff changeset
   491
  by (cases a rule: parity_cases) simp_all
fa5b67fb70ad more simp rules;
haftmann
parents: 58778
diff changeset
   492
fa5b67fb70ad more simp rules;
haftmann
parents: 58778
diff changeset
   493
lemma not_mod_2_eq_1_eq_0 [simp]:
fa5b67fb70ad more simp rules;
haftmann
parents: 58778
diff changeset
   494
  "a mod 2 \<noteq> 1 \<longleftrightarrow> a mod 2 = 0"
fa5b67fb70ad more simp rules;
haftmann
parents: 58778
diff changeset
   495
  by (cases a rule: parity_cases) simp_all
fa5b67fb70ad more simp rules;
haftmann
parents: 58778
diff changeset
   496
58778
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   497
subclass semiring_parity
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   498
proof (unfold_locales, unfold dvd_eq_mod_eq_0 not_mod_2_eq_0_eq_1)
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   499
  show "1 mod 2 = 1"
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   500
    by (fact one_mod_two_eq_one)
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   501
next
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   502
  fix a b
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   503
  assume "a mod 2 = 1"
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   504
  moreover assume "b mod 2 = 1"
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   505
  ultimately show "(a + b) mod 2 = 0"
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   506
    using mod_add_eq [of a b 2] by simp
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   507
next
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   508
  fix a b
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   509
  assume "(a * b) mod 2 = 0"
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   510
  then have "(a mod 2) * (b mod 2) = 0"
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   511
    by (cases "a mod 2 = 0") (simp_all add: mod_mult_eq [of a b 2])
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   512
  then show "a mod 2 = 0 \<or> b mod 2 = 0"
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   513
    by (rule divisors_zero)
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   514
next
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   515
  fix a
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   516
  assume "a mod 2 = 1"
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   517
  then have "a = a div 2 * 2 + 1" using mod_div_equality [of a 2] by simp
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   518
  then show "\<exists>b. a = b + 1" ..
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   519
qed
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   520
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   521
lemma even_iff_mod_2_eq_zero:
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   522
  "even a \<longleftrightarrow> a mod 2 = 0"
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   523
  by (fact dvd_eq_mod_eq_0)
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   524
64014
ca1239a3277b more lemmas
haftmann
parents: 63950
diff changeset
   525
lemma odd_iff_mod_2_eq_one:
ca1239a3277b more lemmas
haftmann
parents: 63950
diff changeset
   526
  "odd a \<longleftrightarrow> a mod 2 = 1"
ca1239a3277b more lemmas
haftmann
parents: 63950
diff changeset
   527
  by (auto simp add: even_iff_mod_2_eq_zero)
ca1239a3277b more lemmas
haftmann
parents: 63950
diff changeset
   528
58778
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   529
lemma even_succ_div_two [simp]:
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   530
  "even a \<Longrightarrow> (a + 1) div 2 = a div 2"
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   531
  by (cases "a = 0") (auto elim!: evenE dest: mult_not_zero)
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   532
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   533
lemma odd_succ_div_two [simp]:
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   534
  "odd a \<Longrightarrow> (a + 1) div 2 = a div 2 + 1"
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   535
  by (auto elim!: oddE simp add: zero_not_eq_two [symmetric] add.assoc)
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   536
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   537
lemma even_two_times_div_two:
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   538
  "even a \<Longrightarrow> 2 * (a div 2) = a"
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   539
  by (fact dvd_mult_div_cancel)
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   540
58834
773b378d9313 more simp rules concerning dvd and even/odd
haftmann
parents: 58786
diff changeset
   541
lemma odd_two_times_div_two_succ [simp]:
58778
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   542
  "odd a \<Longrightarrow> 2 * (a div 2) + 1 = a"
e29cae8eab1f even further downshift of theory Parity in the hierarchy
haftmann
parents: 58710
diff changeset
   543
  using mod_div_equality2 [of 2 a] by (simp add: even_iff_mod_2_eq_zero)
60868
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60867
diff changeset
   544
 
54226
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   545
end
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   546
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   547
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   548
subsection \<open>Generic numeral division with a pragmatic type class\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   549
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   550
text \<open>
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   551
  The following type class contains everything necessary to formulate
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   552
  a division algorithm in ring structures with numerals, restricted
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   553
  to its positive segments.  This is its primary motiviation, and it
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   554
  could surely be formulated using a more fine-grained, more algebraic
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   555
  and less technical class hierarchy.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   556
\<close>
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   557
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60517
diff changeset
   558
class semiring_numeral_div = semiring_div + comm_semiring_1_cancel + linordered_semidom +
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59807
diff changeset
   559
  assumes div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   560
    and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   561
    and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   562
    and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   563
    and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   564
    and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   565
    and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   566
    and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   567
  assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
61275
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
   568
  fixes divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
   569
    and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
   570
  assumes divmod_def: "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
   571
    and divmod_step_def: "divmod_step l qr = (let (q, r) = qr
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
   572
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
   573
    else (2 * q, r))"
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   574
    \<comment> \<open>These are conceptually definitions but force generated code
61275
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
   575
    to be monomorphic wrt. particular instances of this class which
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
   576
    yields a significant speedup.\<close>
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
   577
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   578
begin
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   579
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59807
diff changeset
   580
lemma mult_div_cancel:
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59807
diff changeset
   581
  "b * (a div b) = a - a mod b"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59807
diff changeset
   582
proof -
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59807
diff changeset
   583
  have "b * (a div b) + a mod b = a"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59807
diff changeset
   584
    using mod_div_equality [of a b] by (simp add: ac_simps)
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59807
diff changeset
   585
  then have "b * (a div b) + a mod b - a mod b = a - a mod b"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59807
diff changeset
   586
    by simp
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59807
diff changeset
   587
  then show ?thesis
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59807
diff changeset
   588
    by simp
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59807
diff changeset
   589
qed
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   590
54226
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   591
subclass semiring_div_parity
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   592
proof
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   593
  fix a
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   594
  show "a mod 2 = 0 \<or> a mod 2 = 1"
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   595
  proof (rule ccontr)
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   596
    assume "\<not> (a mod 2 = 0 \<or> a mod 2 = 1)"
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   597
    then have "a mod 2 \<noteq> 0" and "a mod 2 \<noteq> 1" by simp_all
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   598
    have "0 < 2" by simp
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   599
    with pos_mod_bound pos_mod_sign have "0 \<le> a mod 2" "a mod 2 < 2" by simp_all
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   600
    with \<open>a mod 2 \<noteq> 0\<close> have "0 < a mod 2" by simp
54226
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   601
    with discrete have "1 \<le> a mod 2" by simp
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   602
    with \<open>a mod 2 \<noteq> 1\<close> have "1 < a mod 2" by simp
54226
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   603
    with discrete have "2 \<le> a mod 2" by simp
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   604
    with \<open>a mod 2 < 2\<close> show False by simp
54226
e3df2a4e02fc explicit type class for modelling even/odd parity
haftmann
parents: 54221
diff changeset
   605
  qed
58646
cd63a4b12a33 specialized specification: avoid trivial instances
haftmann
parents: 58511
diff changeset
   606
next
cd63a4b12a33 specialized specification: avoid trivial instances
haftmann
parents: 58511
diff changeset
   607
  show "1 mod 2 = 1"
cd63a4b12a33 specialized specification: avoid trivial instances
haftmann
parents: 58511
diff changeset
   608
    by (rule mod_less) simp_all
58710
7216a10d69ba augmented and tuned facts on even/odd and division
haftmann
parents: 58646
diff changeset
   609
next
7216a10d69ba augmented and tuned facts on even/odd and division
haftmann
parents: 58646
diff changeset
   610
  show "0 \<noteq> 2"
7216a10d69ba augmented and tuned facts on even/odd and division
haftmann
parents: 58646
diff changeset
   611
    by simp
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   612
qed
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   613
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   614
lemma divmod_digit_1:
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   615
  assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   616
  shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   617
    and "a mod (2 * b) - b = a mod b" (is "?Q")
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   618
proof -
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   619
  from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   620
    by (auto intro: trans)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   621
  with \<open>0 < b\<close> have "0 < a div b" by (auto intro: div_positive)
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   622
  then have [simp]: "1 \<le> a div b" by (simp add: discrete)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   623
  with \<open>0 < b\<close> have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62597
diff changeset
   624
  define w where "w = a div b mod 2"
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62597
diff changeset
   625
  with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   626
  have mod_w: "a mod (2 * b) = a mod b + b * w"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   627
    by (simp add: w_def mod_mult2_eq ac_simps)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   628
  from assms w_exhaust have "w = 1"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   629
    by (auto simp add: mod_w) (insert mod_less, auto)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   630
  with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   631
  have "2 * (a div (2 * b)) = a div b - w"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   632
    by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   633
  with \<open>w = 1\<close> have div: "2 * (a div (2 * b)) = a div b - 1" by simp
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   634
  then show ?P and ?Q
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   635
    by (simp_all add: div mod add_implies_diff [symmetric])
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   636
qed
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   637
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   638
lemma divmod_digit_0:
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   639
  assumes "0 < b" and "a mod (2 * b) < b"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   640
  shows "2 * (a div (2 * b)) = a div b" (is "?P")
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   641
    and "a mod (2 * b) = a mod b" (is "?Q")
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   642
proof -
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62597
diff changeset
   643
  define w where "w = a div b mod 2"
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62597
diff changeset
   644
  with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   645
  have mod_w: "a mod (2 * b) = a mod b + b * w"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   646
    by (simp add: w_def mod_mult2_eq ac_simps)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   647
  moreover have "b \<le> a mod b + b"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   648
  proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   649
    from \<open>0 < b\<close> pos_mod_sign have "0 \<le> a mod b" by blast
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   650
    then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   651
    then show ?thesis by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   652
  qed
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   653
  moreover note assms w_exhaust
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   654
  ultimately have "w = 0" by auto
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   655
  with mod_w have mod: "a mod (2 * b) = a mod b" by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   656
  have "2 * (a div (2 * b)) = a div b - w"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   657
    by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   658
  with \<open>w = 0\<close> have div: "2 * (a div (2 * b)) = a div b" by simp
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   659
  then show ?P and ?Q
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   660
    by (simp_all add: div mod)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   661
qed
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   662
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   663
lemma fst_divmod:
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   664
  "fst (divmod m n) = numeral m div numeral n"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   665
  by (simp add: divmod_def)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   666
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   667
lemma snd_divmod:
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   668
  "snd (divmod m n) = numeral m mod numeral n"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   669
  by (simp add: divmod_def)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   670
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   671
text \<open>
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   672
  This is a formulation of one step (referring to one digit position)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   673
  in school-method division: compare the dividend at the current
53070
6a3410845bb2 spelling and typos
haftmann
parents: 53069
diff changeset
   674
  digit position with the remainder from previous division steps
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   675
  and evaluate accordingly.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   676
\<close>
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   677
61275
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
   678
lemma divmod_step_eq [simp]:
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   679
  "divmod_step l (q, r) = (if numeral l \<le> r
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   680
    then (2 * q + 1, r - numeral l) else (2 * q, r))"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   681
  by (simp add: divmod_step_def)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   682
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   683
text \<open>
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   684
  This is a formulation of school-method division.
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   685
  If the divisor is smaller than the dividend, terminate.
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   686
  If not, shift the dividend to the right until termination
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   687
  occurs and then reiterate single division steps in the
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   688
  opposite direction.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   689
\<close>
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   690
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   691
lemma divmod_divmod_step:
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   692
  "divmod m n = (if m < n then (0, numeral m)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   693
    else divmod_step n (divmod m (Num.Bit0 n)))"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   694
proof (cases "m < n")
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   695
  case True then have "numeral m < numeral n" by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   696
  then show ?thesis
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   697
    by (simp add: prod_eq_iff div_less mod_less fst_divmod snd_divmod)
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   698
next
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   699
  case False
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   700
  have "divmod m n =
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   701
    divmod_step n (numeral m div (2 * numeral n),
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   702
      numeral m mod (2 * numeral n))"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   703
  proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   704
    case True
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   705
    with divmod_step_eq
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   706
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   707
        (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   708
        by simp
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   709
    moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   710
      have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   711
      and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   712
      by simp_all
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   713
    ultimately show ?thesis by (simp only: divmod_def)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   714
  next
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   715
    case False then have *: "numeral m mod (2 * numeral n) < numeral n"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   716
      by (simp add: not_le)
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   717
    with divmod_step_eq
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   718
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   719
        (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   720
        by auto
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   721
    moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   722
      have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   723
      and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   724
      by (simp_all only: zero_less_numeral)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   725
    ultimately show ?thesis by (simp only: divmod_def)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   726
  qed
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   727
  then have "divmod m n =
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   728
    divmod_step n (numeral m div numeral (Num.Bit0 n),
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   729
      numeral m mod numeral (Num.Bit0 n))"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60517
diff changeset
   730
    by (simp only: numeral.simps distrib mult_1)
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   731
  then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   732
    by (simp add: divmod_def)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   733
  with False show ?thesis by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   734
qed
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   735
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   736
text \<open>The division rewrite proper -- first, trivial results involving \<open>1\<close>\<close>
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   737
61275
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
   738
lemma divmod_trivial [simp]:
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   739
  "divmod Num.One Num.One = (numeral Num.One, 0)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   740
  "divmod (Num.Bit0 m) Num.One = (numeral (Num.Bit0 m), 0)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   741
  "divmod (Num.Bit1 m) Num.One = (numeral (Num.Bit1 m), 0)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   742
  "divmod num.One (num.Bit0 n) = (0, Numeral1)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   743
  "divmod num.One (num.Bit1 n) = (0, Numeral1)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   744
  using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   745
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   746
text \<open>Division by an even number is a right-shift\<close>
58953
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   747
61275
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
   748
lemma divmod_cancel [simp]:
53069
d165213e3924 execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents: 53068
diff changeset
   749
  "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))" (is ?P)
d165213e3924 execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents: 53068
diff changeset
   750
  "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))" (is ?Q)
d165213e3924 execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents: 53068
diff changeset
   751
proof -
d165213e3924 execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents: 53068
diff changeset
   752
  have *: "\<And>q. numeral (Num.Bit0 q) = 2 * numeral q"
d165213e3924 execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents: 53068
diff changeset
   753
    "\<And>q. numeral (Num.Bit1 q) = 2 * numeral q + 1"
d165213e3924 execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents: 53068
diff changeset
   754
    by (simp_all only: numeral_mult numeral.simps distrib) simp_all
d165213e3924 execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents: 53068
diff changeset
   755
  have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)
d165213e3924 execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents: 53068
diff changeset
   756
  then show ?P and ?Q
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   757
    by (simp_all add: fst_divmod snd_divmod prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   758
      div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2]
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   759
      add.commute del: numeral_times_numeral)
58953
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   760
qed
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   761
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   762
text \<open>The really hard work\<close>
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   763
61275
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
   764
lemma divmod_steps [simp]:
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   765
  "divmod (num.Bit0 m) (num.Bit1 n) =
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   766
      (if m \<le> n then (0, numeral (num.Bit0 m))
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   767
       else divmod_step (num.Bit1 n)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   768
             (divmod (num.Bit0 m)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   769
               (num.Bit0 (num.Bit1 n))))"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   770
  "divmod (num.Bit1 m) (num.Bit1 n) =
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   771
      (if m < n then (0, numeral (num.Bit1 m))
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   772
       else divmod_step (num.Bit1 n)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   773
             (divmod (num.Bit1 m)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   774
               (num.Bit0 (num.Bit1 n))))"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   775
  by (simp_all add: divmod_divmod_step)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   776
61275
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
   777
lemmas divmod_algorithm_code = divmod_step_eq divmod_trivial divmod_cancel divmod_steps  
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
   778
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   779
text \<open>Special case: divisibility\<close>
58953
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   780
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   781
definition divides_aux :: "'a \<times> 'a \<Rightarrow> bool"
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   782
where
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   783
  "divides_aux qr \<longleftrightarrow> snd qr = 0"
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   784
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   785
lemma divides_aux_eq [simp]:
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   786
  "divides_aux (q, r) \<longleftrightarrow> r = 0"
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   787
  by (simp add: divides_aux_def)
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   788
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   789
lemma dvd_numeral_simp [simp]:
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   790
  "numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
2e19b392d9e3 self-contained simp rules for dvd on numerals
haftmann
parents: 58911
diff changeset
   791
  by (simp add: divmod_def mod_eq_0_iff_dvd)
53069
d165213e3924 execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents: 53068
diff changeset
   792
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   793
text \<open>Generic computation of quotient and remainder\<close>  
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   794
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   795
lemma numeral_div_numeral [simp]: 
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   796
  "numeral k div numeral l = fst (divmod k l)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   797
  by (simp add: fst_divmod)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   798
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   799
lemma numeral_mod_numeral [simp]: 
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   800
  "numeral k mod numeral l = snd (divmod k l)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   801
  by (simp add: snd_divmod)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   802
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   803
lemma one_div_numeral [simp]:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   804
  "1 div numeral n = fst (divmod num.One n)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   805
  by (simp add: fst_divmod)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   806
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   807
lemma one_mod_numeral [simp]:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   808
  "1 mod numeral n = snd (divmod num.One n)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   809
  by (simp add: snd_divmod)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   810
  
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   811
end
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   812
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60517
diff changeset
   813
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   814
subsection \<open>Division on @{typ nat}\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   815
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   816
context
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   817
begin
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   818
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   819
text \<open>
63950
cdc1e59aa513 syntactic type class for operation mod named after mod;
haftmann
parents: 63947
diff changeset
   820
  We define @{const divide} and @{const modulo} on @{typ nat} by means
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   821
  of a characteristic relation with two input arguments
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60930
diff changeset
   822
  @{term "m::nat"}, @{term "n::nat"} and two output arguments
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60930
diff changeset
   823
  @{term "q::nat"}(uotient) and @{term "r::nat"}(emainder).
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   824
\<close>
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   825
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   826
definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   827
  "divmod_nat_rel m n qr \<longleftrightarrow>
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   828
    m = fst qr * n + snd qr \<and>
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   829
      (if n = 0 then fst qr = 0 else if n > 0 then 0 \<le> snd qr \<and> snd qr < n else n < snd qr \<and> snd qr \<le> 0)"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   830
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   831
text \<open>@{const divmod_nat_rel} is total:\<close>
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   832
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   833
qualified lemma divmod_nat_rel_ex:
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   834
  obtains q r where "divmod_nat_rel m n (q, r)"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   835
proof (cases "n = 0")
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   836
  case True  with that show thesis
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   837
    by (auto simp add: divmod_nat_rel_def)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   838
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   839
  case False
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   840
  have "\<exists>q r. m = q * n + r \<and> r < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   841
  proof (induct m)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   842
    case 0 with \<open>n \<noteq> 0\<close>
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60930
diff changeset
   843
    have "(0::nat) = 0 * n + 0 \<and> 0 < n" by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   844
    then show ?case by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   845
  next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   846
    case (Suc m) then obtain q' r'
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   847
      where m: "m = q' * n + r'" and n: "r' < n" by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   848
    then show ?case proof (cases "Suc r' < n")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   849
      case True
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   850
      from m n have "Suc m = q' * n + Suc r'" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   851
      with True show ?thesis by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   852
    next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   853
      case False then have "n \<le> Suc r'" by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   854
      moreover from n have "Suc r' \<le> n" by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   855
      ultimately have "n = Suc r'" by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   856
      with m have "Suc m = Suc q' * n + 0" by simp
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   857
      with \<open>n \<noteq> 0\<close> show ?thesis by blast
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   858
    qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   859
  qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   860
  with that show thesis
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   861
    using \<open>n \<noteq> 0\<close> by (auto simp add: divmod_nat_rel_def)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   862
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   863
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   864
text \<open>@{const divmod_nat_rel} is injective:\<close>
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   865
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   866
qualified lemma divmod_nat_rel_unique:
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   867
  assumes "divmod_nat_rel m n qr"
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   868
    and "divmod_nat_rel m n qr'"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   869
  shows "qr = qr'"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   870
proof (cases "n = 0")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   871
  case True with assms show ?thesis
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   872
    by (cases qr, cases qr')
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   873
      (simp add: divmod_nat_rel_def)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   874
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   875
  case False
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60930
diff changeset
   876
  have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q::nat)"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   877
  apply (rule leI)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   878
  apply (subst less_iff_Suc_add)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   879
  apply (auto simp add: add_mult_distrib)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   880
  done
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   881
  from \<open>n \<noteq> 0\<close> assms have *: "fst qr = fst qr'"
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   882
    by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53199
diff changeset
   883
  with assms have "snd qr = snd qr'"
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   884
    by (simp add: divmod_nat_rel_def)
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53199
diff changeset
   885
  with * show ?thesis by (cases qr, cases qr') simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   886
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   887
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   888
text \<open>
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   889
  We instantiate divisibility on the natural numbers by
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   890
  means of @{const divmod_nat_rel}:
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   891
\<close>
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   892
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   893
qualified definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 36634
diff changeset
   894
  "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   895
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   896
qualified lemma divmod_nat_rel_divmod_nat:
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   897
  "divmod_nat_rel m n (divmod_nat m n)"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   898
proof -
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   899
  from divmod_nat_rel_ex
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   900
    obtain qr where rel: "divmod_nat_rel m n qr" .
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   901
  then show ?thesis
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   902
  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   903
qed
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   904
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   905
qualified lemma divmod_nat_unique:
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60517
diff changeset
   906
  assumes "divmod_nat_rel m n qr"
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   907
  shows "divmod_nat m n = qr"
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   908
  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   909
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   910
qualified lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   911
  by (simp add: Divides.divmod_nat_unique divmod_nat_rel_def)
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   912
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   913
qualified lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   914
  by (simp add: Divides.divmod_nat_unique divmod_nat_rel_def)
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   915
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   916
qualified lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   917
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   918
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   919
qualified lemma divmod_nat_step:
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   920
  assumes "0 < n" and "n \<le> m"
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   921
  shows "divmod_nat m n = apfst Suc (divmod_nat (m - n) n)"
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   922
proof (rule divmod_nat_unique)
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   923
  have "divmod_nat_rel (m - n) n (divmod_nat (m - n) n)"
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   924
    by (fact divmod_nat_rel_divmod_nat)
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   925
  then show "divmod_nat_rel m n (apfst Suc (divmod_nat (m - n) n))"
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   926
    unfolding divmod_nat_rel_def using assms by auto
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   927
qed
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   928
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   929
end
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   930
  
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   931
instantiation nat :: semiring_div
60352
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
   932
begin
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
   933
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
   934
definition divide_nat where
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   935
  div_nat_def: "m div n = fst (Divides.divmod_nat m n)"
60352
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
   936
63950
cdc1e59aa513 syntactic type class for operation mod named after mod;
haftmann
parents: 63947
diff changeset
   937
definition modulo_nat where
cdc1e59aa513 syntactic type class for operation mod named after mod;
haftmann
parents: 63947
diff changeset
   938
  mod_nat_def: "m mod n = snd (Divides.divmod_nat m n)"
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   939
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   940
lemma fst_divmod_nat [simp]:
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   941
  "fst (Divides.divmod_nat m n) = m div n"
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   942
  by (simp add: div_nat_def)
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   943
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   944
lemma snd_divmod_nat [simp]:
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   945
  "snd (Divides.divmod_nat m n) = m mod n"
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   946
  by (simp add: mod_nat_def)
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   947
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   948
lemma divmod_nat_div_mod:
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   949
  "Divides.divmod_nat m n = (m div n, m mod n)"
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   950
  by (simp add: prod_eq_iff)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   951
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   952
lemma div_nat_unique:
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60517
diff changeset
   953
  assumes "divmod_nat_rel m n (q, r)"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   954
  shows "m div n = q"
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   955
  using assms by (auto dest!: Divides.divmod_nat_unique simp add: prod_eq_iff)
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   956
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   957
lemma mod_nat_unique:
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60517
diff changeset
   958
  assumes "divmod_nat_rel m n (q, r)"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   959
  shows "m mod n = r"
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   960
  using assms by (auto dest!: Divides.divmod_nat_unique simp add: prod_eq_iff)
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   961
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   962
lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   963
  using Divides.divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   964
63950
cdc1e59aa513 syntactic type class for operation mod named after mod;
haftmann
parents: 63947
diff changeset
   965
text \<open>The ''recursion'' equations for @{const divide} and @{const modulo}\<close>
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   966
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   967
lemma div_less [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   968
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   969
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   970
  shows "m div n = 0"
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   971
  using assms Divides.divmod_nat_base by (simp add: prod_eq_iff)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   972
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   973
lemma le_div_geq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   974
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   975
  assumes "0 < n" and "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   976
  shows "m div n = Suc ((m - n) div n)"
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   977
  using assms Divides.divmod_nat_step by (simp add: prod_eq_iff)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   978
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   979
lemma mod_less [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   980
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   981
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   982
  shows "m mod n = m"
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   983
  using assms Divides.divmod_nat_base by (simp add: prod_eq_iff)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   984
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   985
lemma le_mod_geq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   986
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   987
  assumes "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   988
  shows "m mod n = (m - n) mod n"
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
   989
  using assms Divides.divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   990
47136
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   991
instance proof
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   992
  fix m n :: nat
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   993
  show "m div n * n + m mod n = m"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   994
    using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   995
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   996
  fix m n q :: nat
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   997
  assume "n \<noteq> 0"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   998
  then show "(q + m * n) div n = m + q div n"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   999
    by (induct m) (simp_all add: le_div_geq)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
  1000
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
  1001
  fix m n q :: nat
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
  1002
  assume "m \<noteq> 0"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
  1003
  hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
  1004
    unfolding divmod_nat_rel_def
62390
842917225d56 more canonical names
nipkow
parents: 61944
diff changeset
  1005
    by (auto split: if_split_asm, simp_all add: algebra_simps)
47136
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
  1006
  moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
  1007
  ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
  1008
  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
  1009
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
  1010
  fix n :: nat show "n div 0 = 0"
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
  1011
    by (simp add: div_nat_def Divides.divmod_nat_zero)
47136
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
  1012
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
  1013
  fix n :: nat show "0 div n = 0"
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
  1014
    by (simp add: div_nat_def Divides.divmod_nat_zero_left)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
  1015
qed
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1016
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
  1017
end
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1018
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1019
instantiation nat :: normalization_semidom
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1020
begin
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1021
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1022
definition normalize_nat
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1023
  where [simp]: "normalize = (id :: nat \<Rightarrow> nat)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1024
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1025
definition unit_factor_nat
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1026
  where "unit_factor n = (if n = 0 then 0 else 1 :: nat)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1027
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1028
lemma unit_factor_simps [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1029
  "unit_factor 0 = (0::nat)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1030
  "unit_factor (Suc n) = 1"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1031
  by (simp_all add: unit_factor_nat_def)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1032
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1033
instance
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1034
  by standard (simp_all add: unit_factor_nat_def)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1035
  
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1036
end
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60562
diff changeset
  1037
61433
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
  1038
lemma divmod_nat_if [code]:
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
  1039
  "Divides.divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
a4c0de1df3d8 qualify some names stemming from internal bootstrap constructions
haftmann
parents: 61275
diff changeset
  1040
    let (q, r) = Divides.divmod_nat (m - n) n in (Suc q, r))"
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55172
diff changeset
  1041
  by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1042
63950
cdc1e59aa513 syntactic type class for operation mod named after mod;
haftmann
parents: 63947
diff changeset
  1043
text \<open>Simproc for cancelling @{const divide} and @{const modulo}\<close>
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
  1044
51299
30b014246e21 proper place for cancel_div_mod.ML (see also ee729dbd1b7f and ec7f10155389);
wenzelm
parents: 51173
diff changeset
  1045
ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"
30b014246e21 proper place for cancel_div_mod.ML (see also ee729dbd1b7f and ec7f10155389);
wenzelm
parents: 51173
diff changeset
  1046
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1047
ML \<open>
43594
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 41792
diff changeset
  1048
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1049
(
60352
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
  1050
  val div_name = @{const_name divide};
63950
cdc1e59aa513 syntactic type class for operation mod named after mod;
haftmann
parents: 63947
diff changeset
  1051
  val mod_name = @{const_name modulo};
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
  1052
  val mk_binop = HOLogic.mk_binop;
48561
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
  1053
  val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
  1054
  val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
  1055
  fun mk_sum [] = HOLogic.zero
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
  1056
    | mk_sum [t] = t
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
  1057
    | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
  1058
  fun dest_sum tm =
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
  1059
    if HOLogic.is_zero tm then []
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
  1060
    else
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
  1061
      (case try HOLogic.dest_Suc tm of
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
  1062
        SOME t => HOLogic.Suc_zero :: dest_sum t
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
  1063
      | NONE =>
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
  1064
          (case try dest_plus tm of
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
  1065
            SOME (t, u) => dest_sum t @ dest_sum u
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
  1066
          | NONE => [tm]));
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
  1067
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
  1068
  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1069
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
  1070
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
  1071
    (@{thm add_0_left} :: @{thm add_0_right} :: @{thms ac_simps}))
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1072
)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1073
\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1074
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1075
simproc_setup cancel_div_mod_nat ("(m::nat) + n") = \<open>K Cancel_Div_Mod_Nat.proc\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1076
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1077
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1078
subsubsection \<open>Quotient\<close>
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1079
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1080
lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
  1081
by (simp add: le_div_geq linorder_not_less)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1082
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1083
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
  1084
by (simp add: div_geq)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1085
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1086
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
  1087
by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1088
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1089
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
  1090
by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1091
53066
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
  1092
lemma div_positive:
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
  1093
  fixes m n :: nat
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
  1094
  assumes "n > 0"
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
  1095
  assumes "m \<ge> n"
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
  1096
  shows "m div n > 0"
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
  1097
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1098
  from \<open>m \<ge> n\<close> obtain q where "m = n + q"
53066
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
  1099
    by (auto simp add: le_iff_add)
63499
9c9a59949887 Tuned looping simp rules in semiring_div
eberlm <eberlm@in.tum.de>
parents: 63417
diff changeset
  1100
  with \<open>n > 0\<close> show ?thesis by (simp add: div_add_self1)
53066
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
  1101
qed
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
  1102
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58953
diff changeset
  1103
lemma div_eq_0_iff: "(a div b::nat) = 0 \<longleftrightarrow> a < b \<or> b = 0"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58953
diff changeset
  1104
  by (metis div_less div_positive div_by_0 gr0I less_numeral_extra(3) not_less)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
  1105
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1106
subsubsection \<open>Remainder\<close>
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
  1107
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1108
lemma mod_less_divisor [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1109
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1110
  assumes "n > 0"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1111
  shows "m mod n < (n::nat)"
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
  1112
  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1113
51173
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50422
diff changeset
  1114
lemma mod_Suc_le_divisor [simp]:
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50422
diff changeset
  1115
  "m mod Suc n \<le> n"
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50422
diff changeset
  1116
  using mod_less_divisor [of "Suc n" m] by arith
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50422
diff changeset
  1117
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1118
lemma mod_less_eq_dividend [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1119
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1120
  shows "m mod n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1121
proof (rule add_leD2)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1122
  from mod_div_equality have "m div n * n + m mod n = m" .
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1123
  then show "m div n * n + m mod n \<le> m" by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1124
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1125
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60930
diff changeset
  1126
lemma mod_geq: "\<not> m < (n::nat) \<Longrightarrow> m mod n = (m - n) mod n"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
  1127
by (simp add: le_mod_geq linorder_not_less)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1128
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60930
diff changeset
  1129
lemma mod_if: "m mod (n::nat) = (if m < n then m else (m - n) mod n)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
  1130
by (simp add: le_mod_geq)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1131
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1132
lemma mod_1 [simp]: "m mod Suc 0 = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
  1133
by (induct m) (simp_all add: mod_geq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1134
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1135
(* a simple rearrangement of mod_div_equality: *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1136
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1137
  using mod_div_equality2 [of n m] by arith
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1138
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
  1139
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1140
  apply (drule mod_less_divisor [where m = m])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1141
  apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1142
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1143
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1144
subsubsection \<open>Quotient and Remainder\<close>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1145
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
  1146
lemma divmod_nat_rel_mult1_eq:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
  1147
  "divmod_nat_rel b c (q, r)
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
  1148
   \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
  1149
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1150
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
  1151
lemma div_mult1_eq:
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
  1152
  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
  1153
by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1154
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
  1155
lemma divmod_nat_rel_add1_eq:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
  1156
  "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br)
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
  1157
   \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
  1158
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1159
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1160
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1161
lemma div_add1_eq:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
  1162
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
  1163
by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1164
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
  1165
lemma divmod_nat_rel_mult2_eq:
60352
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
  1166
  assumes "divmod_nat_rel a b (q, r)"
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
  1167
  shows "divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
  1168
proof -
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60517
diff changeset
  1169
  { assume "r < b" and "0 < c"
60352
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
  1170
    then have "b * (q mod c) + r < b * c"
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
  1171
      apply (cut_tac m = q and n = c in mod_less_divisor)
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
  1172
      apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
  1173
      apply (erule_tac P = "%x. lhs < rhs x" for lhs rhs in ssubst)
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
  1174
      apply (simp add: add_mult_distrib2)
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
  1175
      done
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
  1176
    then have "r + b * (q mod c) < b * c"
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
  1177
      by (simp add: ac_simps)
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
  1178
  } with assms show ?thesis
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
  1179
    by (auto simp add: divmod_nat_rel_def algebra_simps add_mult_distrib2 [symmetric])
d46de31a50c4 separate class for division operator, with particular syntax added in more specific classes
haftmann
parents: 59833
diff changeset
  1180
qed
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60517
diff changeset
  1181
55085
0e8e4dc55866 moved 'fundef_cong' attribute (and other basic 'fun' stuff) up the dependency chain
blanchet
parents: 54489
diff changeset
  1182
lemma div_mult2_eq: "a div (b * c) = (a div b) div (c::nat)"
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
  1183
by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1184
55085
0e8e4dc55866 moved 'fundef_cong' attribute (and other basic 'fun' stuff) up the dependency chain
blanchet
parents: 54489
diff changeset
  1185
lemma mod_mult2_eq: "a mod (b * c) = b * (a div b mod c) + a mod (b::nat)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
  1186
by (auto simp add: mult.commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1187
61275
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1188
instantiation nat :: semiring_numeral_div
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1189
begin
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1190
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1191
definition divmod_nat :: "num \<Rightarrow> num \<Rightarrow> nat \<times> nat"
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1192
where
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1193
  divmod'_nat_def: "divmod_nat m n = (numeral m div numeral n, numeral m mod numeral n)"
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1194
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1195
definition divmod_step_nat :: "num \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1196
where
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1197
  "divmod_step_nat l qr = (let (q, r) = qr
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1198
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1199
    else (2 * q, r))"
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1200
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1201
instance
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1202
  by standard (auto intro: div_positive simp add: divmod'_nat_def divmod_step_nat_def mod_mult2_eq div_mult2_eq)
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1203
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1204
end
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1205
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1206
declare divmod_algorithm_code [where ?'a = nat, code]
053ec04ea866 monomorphization of divmod wrt. code generation avoids costly dictionary unpacking at runtime
haftmann
parents: 61201
diff changeset
  1207
  
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1208
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1209
subsubsection \<open>Further Facts about Quotient and Remainder\<close>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1210
58786
fa5b67fb70ad more simp rules;
haftmann
parents: 58778
diff changeset
  1211
lemma div_1 [simp]:
fa5b67fb70ad more simp rules;
haftmann
parents: 58778
diff changeset
  1212
  "m div Suc 0 = m"
fa5b67fb70ad more simp rules;
haftmann
parents: 58778
diff changeset
  1213
  using div_by_1 [of m] by simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1214
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1215
(* Monotonicity of div in first argument *)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
  1216
lemma div_le_mono [rule_format (no_asm)]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1217
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1218
apply (case_tac "k=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
  1219
apply (induct "n" rule: nat_less_induct, clarify)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1220
apply (case_tac "n<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1221
(* 1  case n<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1222
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1223
(* 2  case n >= k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1224
apply (case_tac "m<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1225
(* 2.1  case m<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1226
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1227
(* 2.2  case m>=k *)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
  1228
apply (simp add: div_geq diff_le_mono)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1229
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1230
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1231
(* Antimonotonicity of div in second argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1232
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1233
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1234
 prefer 2 apply simp
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
  1235
apply (induct_tac k rule: nat_less_induct)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1236
apply (rename_tac "k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1237
apply (case_tac "k<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1238
apply (subgoal_tac "~ (k<m) ")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1239
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1240
apply (simp add: div_geq)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
  1241
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1242
 prefer 2
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1243
 apply (blast intro: div_le_mono diff_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1244
apply (rule le_trans, simp)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
  1245
apply (simp)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1246
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1247
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1248
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1249
apply (case_tac "n=0", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1250
apply (subgoal_tac "m div n \<le> m div 1", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1251
apply (rule div_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1252
apply (simp_all (no_asm_simp))
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1253
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1254
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1255
(* Similar for "less than" *)
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1256
lemma div_less_dividend [simp]:
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1257
  "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1258
apply (induct m rule: nat_less_induct)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1259
apply (rename_tac "m")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1260
apply (case_tac "m<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1261
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1262
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1263
apply (simp add: div_geq)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1264
apply (case_tac "n<m")
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
  1265
 apply (subgoal_tac "(m-n) div n < (m-n) ")
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1266
  apply (rule impI less_trans_Suc)+
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1267
apply assumption
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
  1268
  apply (simp_all)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1269
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1270
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1271
text\<open>A fact for the mutilated chess board\<close>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1272
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1273
apply (case_tac "n=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
  1274
apply (induct "m" rule: nat_less_induct)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1275
apply (case_tac "Suc (na) <n")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1276
(* case Suc(na) < n *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1277
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1278
(* case n \<le> Suc(na) *)
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16733
diff changeset
  1279
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
  1280
apply (auto simp add: Suc_diff_le le_mod_geq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1281
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1282
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1283
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
  1284
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
  1285