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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header {* The division operators div and mod *}
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theory Divides
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imports Nat Power Product_Type
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uses "~~/src/Provers/Arith/cancel_div_mod.ML"
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begin
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subsection {* Syntactic division operations *}
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class div = dvd +
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  fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
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    and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
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subsection {* Abstract division in commutative semirings. *}
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class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +
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  assumes mod_div_equality: "a div b * b + a mod b = a"
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    and div_by_0 [simp]: "a div 0 = 0"
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    and div_0 [simp]: "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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text {* @{const div} and @{const mod} *}
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lemma mod_div_equality2: "b * (a div b) + a mod b = a"
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  unfolding mult_commute [of b]
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  by (rule mod_div_equality)
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lemma mod_div_equality': "a mod b + a div b * b = a"
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  using mod_div_equality [of a b]
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  by (simp only: add_ac)
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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 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 left_distrib)
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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]: "(a + b * c) mod b = a mod b"
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  by (simp add: mult_commute [of b])
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lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
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  using div_mult_self2 [of b 0 a] by simp
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lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
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  using div_mult_self1 [of b 0 a] by simp
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lemma mod_mult_self1_is_0 [simp]: "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]: "a * b mod b = 0"
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  using mod_mult_self1 [of 0 a b] by simp
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lemma div_by_1 [simp]: "a div 1 = a"
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  using div_mult_self2_is_id [of 1 a] zero_neq_one by simp
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lemma mod_by_1 [simp]: "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]: "a mod a = 0"
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  using mod_mult_self2_is_0 [of 1] by simp
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lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
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  using div_mult_self2_is_id [of _ 1] by simp
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lemma div_add_self1 [simp]:
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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 [simp]:
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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 mod_div_decomp:
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  fixes a b
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  obtains q r where "q = a div b" and "r = a mod b"
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    and "a = q * b + r"
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proof -
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  from mod_div_equality have "a = a div b * b + a mod b" by simp
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  moreover have "a div b = a div b" ..
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  moreover have "a mod b = a mod b" ..
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  note that ultimately show thesis by blast
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qed
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lemma dvd_eq_mod_eq_0 [code unfold]: "a dvd b \<longleftrightarrow> b mod a = 0"
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proof
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  assume "b mod a = 0"
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   137
  with mod_div_equality [of b a] have "b div a * a = b" by simp
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   138
  then have "b = a * (b div a)" unfolding mult_commute ..
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   139
  then have "\<exists>c. b = a * c" ..
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   140
  then show "a dvd b" unfolding dvd_def .
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   141
next
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   142
  assume "a dvd b"
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   143
  then have "\<exists>c. b = a * c" unfolding dvd_def .
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   144
  then obtain c where "b = a * c" ..
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   145
  then have "b mod a = a * c mod a" by simp
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   146
  then have "b mod a = c * a mod a" by (simp add: mult_commute)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   147
  then show "b mod a = 0" by simp
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   148
qed
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   149
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   150
lemma mod_div_trivial [simp]: "a mod b div b = 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   151
proof (cases "b = 0")
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   152
  assume "b = 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   153
  thus ?thesis by simp
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   154
next
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   155
  assume "b \<noteq> 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   156
  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
   157
    by (rule div_mult_self1 [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   158
  also have "\<dots> = a div b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   159
    by (simp only: mod_div_equality')
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   160
  also have "\<dots> = a div b + 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   161
    by simp
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   162
  finally show ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   163
    by (rule add_left_imp_eq)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   164
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   165
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   166
lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   167
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   168
  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
   169
    by (simp only: mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   170
  also have "\<dots> = a mod b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   171
    by (simp only: mod_div_equality')
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   172
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   173
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   174
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   175
lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0"
29948
cdf12a1cb963 Cleaned up IntDiv and removed subsumed lemmas.
nipkow
parents: 29925
diff changeset
   176
by (rule dvd_eq_mod_eq_0[THEN iffD1])
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   177
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   178
lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   179
by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   180
30052
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   181
lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a"
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   182
apply (cases "a = 0")
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   183
 apply simp
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   184
apply (auto simp: dvd_def mult_assoc)
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   185
done
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   186
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   187
lemma div_dvd_div[simp]:
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   188
  "a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)"
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   189
apply (cases "a = 0")
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   190
 apply simp
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   191
apply (unfold dvd_def)
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   192
apply auto
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   193
 apply(blast intro:mult_assoc[symmetric])
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   194
apply(fastsimp simp add: mult_assoc)
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   195
done
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   196
30078
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   197
lemma dvd_mod_imp_dvd: "[| k dvd m mod n;  k dvd n |] ==> k dvd m"
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   198
  apply (subgoal_tac "k dvd (m div n) *n + m mod n")
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   199
   apply (simp add: mod_div_equality)
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   200
  apply (simp only: dvd_add dvd_mult)
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   201
  done
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   202
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   203
text {* Addition respects modular equivalence. *}
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   204
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   205
lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   206
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   207
  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
   208
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   209
  also have "\<dots> = (a mod c + b + a div c * c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   210
    by (simp only: add_ac)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   211
  also have "\<dots> = (a mod c + b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   212
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   213
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   214
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   215
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   216
lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   217
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   218
  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
   219
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   220
  also have "\<dots> = (a + b mod c + b div c * c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   221
    by (simp only: add_ac)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   222
  also have "\<dots> = (a + b mod c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   223
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   224
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   225
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   226
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   227
lemma mod_add_eq: "(a + b) 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
   228
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
   229
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   230
lemma mod_add_cong:
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   231
  assumes "a mod c = a' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   232
  assumes "b mod c = b' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   233
  shows "(a + b) mod c = (a' + b') mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   234
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   235
  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
   236
    unfolding assms ..
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   237
  thus ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   238
    by (simp only: mod_add_eq [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   239
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   240
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   241
lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y
30837
3d4832d9f7e4 added strong_setprod_cong[cong] (in analogy with setsum)
nipkow
parents: 30729
diff changeset
   242
  \<Longrightarrow> (x + y) div z = x div z + y div z"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   243
by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)
30837
3d4832d9f7e4 added strong_setprod_cong[cong] (in analogy with setsum)
nipkow
parents: 30729
diff changeset
   244
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   245
text {* Multiplication respects modular equivalence. *}
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   246
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   247
lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   248
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   249
  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
   250
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   251
  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
   252
    by (simp only: algebra_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   253
  also have "\<dots> = (a mod c * b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   254
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   255
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   256
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   257
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   258
lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   259
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   260
  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
   261
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   262
  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
   263
    by (simp only: algebra_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   264
  also have "\<dots> = (a * (b mod c)) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   265
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   266
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   267
qed
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_eq: "(a * b) 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
   270
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
   271
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   272
lemma mod_mult_cong:
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   273
  assumes "a mod c = a' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   274
  assumes "b mod c = b' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   275
  shows "(a * b) mod c = (a' * b') mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   276
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   277
  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
   278
    unfolding assms ..
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   279
  thus ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   280
    by (simp only: mod_mult_eq [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   281
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   282
29404
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   283
lemma mod_mod_cancel:
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   284
  assumes "c dvd b"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   285
  shows "a mod b mod c = a mod c"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   286
proof -
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   287
  from `c dvd b` obtain k where "b = c * k"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   288
    by (rule dvdE)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   289
  have "a mod b mod c = a mod (c * k) mod c"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   290
    by (simp only: `b = c * k`)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   291
  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
   292
    by (simp only: mod_mult_self1)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   293
  also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   294
    by (simp only: add_ac mult_ac)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   295
  also have "\<dots> = a mod c"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   296
    by (simp only: mod_div_equality)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   297
  finally show ?thesis .
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   298
qed
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   299
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   300
lemma div_mult_div_if_dvd:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   301
  "y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   302
  apply (cases "y = 0", simp)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   303
  apply (cases "z = 0", simp)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   304
  apply (auto elim!: dvdE simp add: algebra_simps)
30476
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   305
  apply (subst mult_assoc [symmetric])
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   306
  apply (simp add: no_zero_divisors)
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   307
  done
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   308
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   309
lemma div_mult_mult2 [simp]:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   310
  "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   311
  by (drule div_mult_mult1) (simp add: mult_commute)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   312
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   313
lemma div_mult_mult1_if [simp]:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   314
  "(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
   315
  by simp_all
30476
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   316
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   317
lemma mod_mult_mult1:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   318
  "(c * a) mod (c * b) = c * (a mod b)"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   319
proof (cases "c = 0")
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   320
  case True then show ?thesis by simp
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   321
next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   322
  case False
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   323
  from mod_div_equality
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   324
  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
   325
  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
   326
    = c * a + c * (a mod b)" by (simp add: algebra_simps)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   327
  with mod_div_equality show ?thesis by simp 
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   328
qed
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   329
  
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   330
lemma mod_mult_mult2:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   331
  "(a * c) mod (b * c) = (a mod b) * c"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   332
  using mod_mult_mult1 [of c a b] by (simp add: mult_commute)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   333
31009
41fd307cab30 dropped reference to class recpower and lemma duplicate
haftmann
parents: 30934
diff changeset
   334
lemma div_power:
31661
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   335
  "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
30476
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   336
apply (induct n)
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   337
 apply simp
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   338
apply(simp add: div_mult_div_if_dvd dvd_power_same)
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   339
done
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   340
31661
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   341
end
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   342
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   343
class ring_div = semiring_div + idom
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   344
begin
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   345
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   346
text {* Negation respects modular equivalence. *}
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   347
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   348
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
   349
proof -
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   350
  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
   351
    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
   352
  also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   353
    by (simp only: minus_add_distrib minus_mult_left add_ac)
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   354
  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
   355
    by (rule mod_mult_self1)
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   356
  finally show ?thesis .
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   357
qed
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   358
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   359
lemma mod_minus_cong:
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   360
  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
   361
  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
   362
proof -
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   363
  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
   364
    unfolding assms ..
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   365
  thus ?thesis
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   366
    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
   367
qed
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
text {* Subtraction respects modular equivalence. *}
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   370
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   371
lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   372
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   373
  by (intro mod_add_cong mod_minus_cong) simp_all
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   374
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   375
lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   376
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   377
  by (intro mod_add_cong mod_minus_cong) simp_all
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   378
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   379
lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   380
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   381
  by (intro mod_add_cong mod_minus_cong) simp_all
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   382
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   383
lemma mod_diff_cong:
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   384
  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
   385
  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
   386
  shows "(a - b) mod c = (a' - b') mod c"
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   387
  unfolding diff_minus using assms
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   388
  by (intro mod_add_cong mod_minus_cong)
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   389
30180
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   390
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
   391
apply (case_tac "y = 0") apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   392
apply (auto simp add: dvd_def)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   393
apply (subgoal_tac "-(y * k) = y * - k")
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   394
 apply (erule ssubst)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   395
 apply (erule div_mult_self1_is_id)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   396
apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   397
done
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   398
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   399
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
   400
apply (case_tac "y = 0") apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   401
apply (auto simp add: dvd_def)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   402
apply (subgoal_tac "y * k = -y * -k")
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   403
 apply (erule ssubst)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   404
 apply (rule div_mult_self1_is_id)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   405
 apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   406
apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   407
done
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   408
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   409
end
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   410
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   411
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   412
subsection {* Division on @{typ nat} *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   413
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   414
text {*
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   415
  We define @{const div} and @{const mod} on @{typ nat} by means
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   416
  of a characteristic relation with two input arguments
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   417
  @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   418
  @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   419
*}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   420
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   421
definition divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   422
  "divmod_rel m n qr \<longleftrightarrow>
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   423
    m = fst qr * n + snd qr \<and>
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   424
      (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
   425
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   426
text {* @{const divmod_rel} is total: *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   427
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   428
lemma divmod_rel_ex:
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   429
  obtains q r where "divmod_rel m n (q, r)"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   430
proof (cases "n = 0")
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   431
  case True  with that show thesis
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   432
    by (auto simp add: divmod_rel_def)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   433
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   434
  case False
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   435
  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
   436
  proof (induct m)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   437
    case 0 with `n \<noteq> 0`
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   438
    have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   439
    then show ?case by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   440
  next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   441
    case (Suc m) then obtain q' r'
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   442
      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
   443
    then show ?case proof (cases "Suc r' < n")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   444
      case True
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   445
      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
   446
      with True show ?thesis by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   447
    next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   448
      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
   449
      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
   450
      ultimately have "n = Suc r'" by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   451
      with m have "Suc m = Suc q' * n + 0" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   452
      with `n \<noteq> 0` show ?thesis by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   453
    qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   454
  qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   455
  with that show thesis
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   456
    using `n \<noteq> 0` by (auto simp add: divmod_rel_def)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   457
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   458
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   459
text {* @{const divmod_rel} is injective: *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   460
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   461
lemma divmod_rel_unique:
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   462
  assumes "divmod_rel m n qr"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   463
    and "divmod_rel m n qr'"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   464
  shows "qr = qr'"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   465
proof (cases "n = 0")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   466
  case True with assms show ?thesis
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   467
    by (cases qr, cases qr')
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   468
      (simp add: divmod_rel_def)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   469
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   470
  case False
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   471
  have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   472
  apply (rule leI)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   473
  apply (subst less_iff_Suc_add)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   474
  apply (auto simp add: add_mult_distrib)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   475
  done
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   476
  from `n \<noteq> 0` assms have "fst qr = fst qr'"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   477
    by (auto simp add: divmod_rel_def intro: order_antisym dest: aux sym)
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   478
  moreover from this assms have "snd qr = snd qr'"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   479
    by (simp add: divmod_rel_def)
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   480
  ultimately 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
   481
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   482
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   483
text {*
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   484
  We instantiate divisibility on the natural numbers by
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   485
  means of @{const divmod_rel}:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   486
*}
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   487
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   488
instantiation nat :: semiring_div
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   489
begin
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   490
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   491
definition divmod :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   492
  [code del]: "divmod m n = (THE qr. divmod_rel m n qr)"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   493
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   494
lemma divmod_rel_divmod:
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   495
  "divmod_rel m n (divmod m n)"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   496
proof -
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   497
  from divmod_rel_ex
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   498
    obtain qr where rel: "divmod_rel m n qr" .
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   499
  then show ?thesis
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   500
  by (auto simp add: divmod_def intro: theI elim: divmod_rel_unique)
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   501
qed
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   502
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   503
lemma divmod_eq:
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   504
  assumes "divmod_rel m n qr" 
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   505
  shows "divmod m n = qr"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   506
  using assms by (auto intro: divmod_rel_unique divmod_rel_divmod)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   507
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   508
definition div_nat where
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   509
  "m div n = fst (divmod m n)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   510
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   511
definition mod_nat where
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   512
  "m mod n = snd (divmod m n)"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   513
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   514
lemma divmod_div_mod:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   515
  "divmod m n = (m div n, m mod n)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   516
  unfolding div_nat_def mod_nat_def by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   517
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   518
lemma div_eq:
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   519
  assumes "divmod_rel m n (q, r)" 
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   520
  shows "m div n = q"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   521
  using assms by (auto dest: divmod_eq simp add: divmod_div_mod)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   522
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   523
lemma mod_eq:
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   524
  assumes "divmod_rel m n (q, r)" 
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   525
  shows "m mod n = r"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   526
  using assms by (auto dest: divmod_eq simp add: divmod_div_mod)
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   527
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   528
lemma divmod_rel: "divmod_rel m n (m div n, m mod n)"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   529
  by (simp add: div_nat_def mod_nat_def divmod_rel_divmod)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   530
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   531
lemma divmod_zero:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   532
  "divmod m 0 = (0, m)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   533
proof -
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   534
  from divmod_rel [of m 0] show ?thesis
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   535
    unfolding divmod_div_mod divmod_rel_def by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   536
qed
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   537
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   538
lemma divmod_base:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   539
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   540
  shows "divmod m n = (0, m)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   541
proof -
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   542
  from divmod_rel [of m n] show ?thesis
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   543
    unfolding divmod_div_mod divmod_rel_def
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   544
    using assms by (cases "m div n = 0")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   545
      (auto simp add: gr0_conv_Suc [of "m div n"])
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   546
qed
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   547
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   548
lemma divmod_step:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   549
  assumes "0 < n" and "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   550
  shows "divmod m n = (Suc ((m - n) div n), (m - n) mod n)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   551
proof -
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   552
  from divmod_rel have divmod_m_n: "divmod_rel m n (m div n, m mod n)" .
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   553
  with assms have m_div_n: "m div n \<ge> 1"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   554
    by (cases "m div n") (auto simp add: divmod_rel_def)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   555
  from assms divmod_m_n have "divmod_rel (m - n) n (m div n - Suc 0, m mod n)"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   556
    by (cases "m div n") (auto simp add: divmod_rel_def)
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30078
diff changeset
   557
  with divmod_eq have "divmod (m - n) n = (m div n - Suc 0, m mod n)" by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   558
  moreover from divmod_div_mod have "divmod (m - n) n = ((m - n) div n, (m - n) mod n)" .
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   559
  ultimately have "m div n = Suc ((m - n) div n)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   560
    and "m mod n = (m - n) mod n" using m_div_n by simp_all
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   561
  then show ?thesis using divmod_div_mod by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   562
qed
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   563
26300
03def556e26e removed duplicate lemmas;
wenzelm
parents: 26100
diff changeset
   564
text {* The ''recursion'' equations for @{const div} and @{const mod} *}
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   565
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   566
lemma div_less [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   567
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   568
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   569
  shows "m div n = 0"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   570
  using assms divmod_base divmod_div_mod by simp
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   571
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   572
lemma le_div_geq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   573
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   574
  assumes "0 < n" and "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   575
  shows "m div n = Suc ((m - n) div n)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   576
  using assms divmod_step divmod_div_mod by simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   577
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   578
lemma mod_less [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   579
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   580
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   581
  shows "m mod n = m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   582
  using assms divmod_base divmod_div_mod by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   583
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   584
lemma le_mod_geq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   585
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   586
  assumes "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   587
  shows "m mod n = (m - n) mod n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   588
  using assms divmod_step divmod_div_mod by (cases "n = 0") simp_all
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   589
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   590
instance proof -
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   591
  have [simp]: "\<And>n::nat. n div 0 = 0"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   592
    by (simp add: div_nat_def divmod_zero)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   593
  have [simp]: "\<And>n::nat. 0 div n = 0"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   594
  proof -
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   595
    fix n :: nat
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   596
    show "0 div n = 0"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   597
      by (cases "n = 0") simp_all
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   598
  qed
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   599
  show "OFCLASS(nat, semiring_div_class)" proof
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   600
    fix m n :: nat
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   601
    show "m div n * n + m mod n = m"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   602
      using divmod_rel [of m n] by (simp add: divmod_rel_def)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   603
  next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   604
    fix m n q :: nat
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   605
    assume "n \<noteq> 0"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   606
    then show "(q + m * n) div n = m + q div n"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   607
      by (induct m) (simp_all add: le_div_geq)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   608
  next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   609
    fix m n q :: nat
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   610
    assume "m \<noteq> 0"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   611
    then show "(m * n) div (m * q) = n div q"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   612
    proof (cases "n \<noteq> 0 \<and> q \<noteq> 0")
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   613
      case False then show ?thesis by auto
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   614
    next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   615
      case True with `m \<noteq> 0`
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   616
        have "m > 0" and "n > 0" and "q > 0" by auto
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   617
      then have "\<And>a b. divmod_rel n q (a, b) \<Longrightarrow> divmod_rel (m * n) (m * q) (a, m * b)"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   618
        by (auto simp add: divmod_rel_def) (simp_all add: algebra_simps)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   619
      moreover from divmod_rel have "divmod_rel n q (n div q, n mod q)" .
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   620
      ultimately have "divmod_rel (m * n) (m * q) (n div q, m * (n mod q))" .
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   621
      then show ?thesis by (simp add: div_eq)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   622
    qed
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   623
  qed simp_all
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   624
qed
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   625
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   626
end
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   627
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   628
text {* Simproc for cancelling @{const div} and @{const mod} *}
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   629
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   630
ML {*
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   631
local
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   632
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   633
structure CancelDivMod = CancelDivModFun(struct
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   634
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   635
  val div_name = @{const_name div};
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   636
  val mod_name = @{const_name mod};
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   637
  val mk_binop = HOLogic.mk_binop;
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   638
  val mk_sum = Nat_Arith.mk_sum;
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   639
  val dest_sum = Nat_Arith.dest_sum;
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   640
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   641
  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
   642
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   643
  val trans = trans;
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   644
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   645
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   646
    (@{thm monoid_add_class.add_0_left} :: @{thm monoid_add_class.add_0_right} :: @{thms add_ac}))
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   647
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   648
end)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   649
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   650
in
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   651
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   652
val cancel_div_mod_nat_proc = Simplifier.simproc (the_context ())
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   653
  "cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   654
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   655
val _ = Addsimprocs [cancel_div_mod_nat_proc];
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   656
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   657
end
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   658
*}
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   659
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   660
text {* code generator setup *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   661
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   662
lemma divmod_if [code]: "divmod m n = (if n = 0 \<or> m < n then (0, m) else
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   663
  let (q, r) = divmod (m - n) n in (Suc q, r))"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   664
by (simp add: divmod_zero divmod_base divmod_step)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   665
    (simp add: divmod_div_mod)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   666
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   667
code_modulename SML
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   668
  Divides Nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   669
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   670
code_modulename OCaml
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   671
  Divides Nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   672
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   673
code_modulename Haskell
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   674
  Divides Nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   675
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   676
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   677
subsubsection {* Quotient *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   678
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   679
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
   680
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
   681
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   682
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
   683
by (simp add: div_geq)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   684
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   685
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
   686
by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   687
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   688
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
   689
by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   690
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   691
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   692
subsubsection {* Remainder *}
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   693
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   694
lemma mod_less_divisor [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   695
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   696
  assumes "n > 0"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   697
  shows "m mod n < (n::nat)"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   698
  using assms divmod_rel [of m n] unfolding divmod_rel_def by auto
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   699
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   700
lemma mod_less_eq_dividend [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   701
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   702
  shows "m mod n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   703
proof (rule add_leD2)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   704
  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
   705
  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
   706
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   707
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   708
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   709
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
   710
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   711
lemma mod_if: "m mod (n\<Colon>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
   712
by (simp add: le_mod_geq)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   713
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   714
lemma mod_1 [simp]: "m mod Suc 0 = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   715
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
   716
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   717
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   718
  apply (cases "n = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   719
  apply (cases "k = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   720
  apply (induct m rule: nat_less_induct)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   721
  apply (subst mod_if, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   722
  apply (simp add: mod_geq diff_mult_distrib)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   723
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   724
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   725
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   726
by (simp add: mult_commute [of k] mod_mult_distrib)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   727
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   728
(* 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
   729
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   730
by (cut_tac a = m and b = n in mod_div_equality2, arith)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   731
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   732
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   733
  apply (drule mod_less_divisor [where m = m])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   734
  apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   735
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   736
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   737
subsubsection {* Quotient and Remainder *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   738
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   739
lemma divmod_rel_mult1_eq:
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   740
  "divmod_rel b c (q, r) \<Longrightarrow> c > 0
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   741
   \<Longrightarrow> divmod_rel (a * b) c (a * q + a * r div c, a * r mod c)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   742
by (auto simp add: split_ifs divmod_rel_def algebra_simps)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   743
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   744
lemma div_mult1_eq:
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   745
  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   746
apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   747
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN div_eq])
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   748
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   749
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   750
lemma divmod_rel_add1_eq:
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   751
  "divmod_rel a c (aq, ar) \<Longrightarrow> divmod_rel b c (bq, br) \<Longrightarrow>  c > 0
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   752
   \<Longrightarrow> divmod_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   753
by (auto simp add: split_ifs divmod_rel_def algebra_simps)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   754
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   755
(*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
   756
lemma div_add1_eq:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   757
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   758
apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   759
apply (blast intro: divmod_rel_add1_eq [THEN div_eq] divmod_rel)
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   760
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   761
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   762
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   763
  apply (cut_tac m = q and n = c in mod_less_divisor)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   764
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   765
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   766
  apply (simp add: add_mult_distrib2)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   767
  done
10559
d3fd54fc659b many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents: 10214
diff changeset
   768
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   769
lemma divmod_rel_mult2_eq:
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   770
  "divmod_rel a b (q, r) \<Longrightarrow> 0 < b \<Longrightarrow> 0 < c
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   771
   \<Longrightarrow> divmod_rel a (b * c) (q div c, b *(q mod c) + r)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   772
by (auto simp add: mult_ac divmod_rel_def add_mult_distrib2 [symmetric] mod_lemma)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   773
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   774
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   775
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   776
  apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   777
  apply (force simp add: divmod_rel [THEN divmod_rel_mult2_eq, THEN div_eq])
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   778
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   779
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   780
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   781
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   782
  apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   783
  apply (auto simp add: mult_commute divmod_rel [THEN divmod_rel_mult2_eq, THEN mod_eq])
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   784
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   785
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   786
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   787
subsubsection{*Further Facts about Quotient and Remainder*}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   788
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   789
lemma div_1 [simp]: "m div Suc 0 = m"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   790
by (induct m) (simp_all add: div_geq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   791
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   792
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   793
(* Monotonicity of div in first argument *)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   794
lemma div_le_mono [rule_format (no_asm)]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   795
    "\<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
   796
apply (case_tac "k=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   797
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
   798
apply (case_tac "n<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   799
(* 1  case n<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   800
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   801
(* 2  case n >= k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   802
apply (case_tac "m<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   803
(* 2.1  case m<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   804
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   805
(* 2.2  case m>=k *)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   806
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
   807
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   808
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   809
(* Antimonotonicity of div in second argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   810
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
   811
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   812
 prefer 2 apply simp
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   813
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
   814
apply (rename_tac "k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   815
apply (case_tac "k<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   816
apply (subgoal_tac "~ (k<m) ")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   817
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   818
apply (simp add: div_geq)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   819
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
   820
 prefer 2
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   821
 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
   822
apply (rule le_trans, simp)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   823
apply (simp)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   824
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   825
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   826
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
   827
apply (case_tac "n=0", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   828
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
   829
apply (rule div_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   830
apply (simp_all (no_asm_simp))
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   831
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   832
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   833
(* Similar for "less than" *)
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   834
lemma div_less_dividend [rule_format]:
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   835
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   836
apply (induct_tac m rule: nat_less_induct)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   837
apply (rename_tac "m")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   838
apply (case_tac "m<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   839
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   840
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   841
apply (simp add: div_geq)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   842
apply (case_tac "n<m")
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   843
 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
   844
  apply (rule impI less_trans_Suc)+
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   845
apply assumption
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   846
  apply (simp_all)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   847
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   848
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   849
declare div_less_dividend [simp]
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   850
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   851
text{*A fact for the mutilated chess board*}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   852
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
   853
apply (case_tac "n=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   854
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
   855
apply (case_tac "Suc (na) <n")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   856
(* case Suc(na) < n *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   857
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
   858
(* case n \<le> Suc(na) *)
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16733
diff changeset
   859
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   860
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
   861
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   862
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   863
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   864
subsubsection {* The Divides Relation *}
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24268
diff changeset
   865
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   866
lemma dvd_1_left [iff]: "Suc 0 dvd k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   867
  unfolding dvd_def by simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   868
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   869
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   870
by (simp add: dvd_def)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   871
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30078
diff changeset
   872
lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1"
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30078
diff changeset
   873
by (simp add: dvd_def)
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30078
diff changeset
   874
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   875
lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   876
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   877
  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   878
23684
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   879
text {* @{term "op dvd"} is a partial order *}
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   880
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 30653
diff changeset
   881
interpretation dvd: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28562
diff changeset
   882
  proof qed (auto intro: dvd_refl dvd_trans dvd_anti_sym)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   883
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30027
diff changeset
   884
lemma nat_dvd_diff[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30027
diff changeset
   885
unfolding dvd_def
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30027
diff changeset
   886
by (blast intro: diff_mult_distrib2 [symmetric])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   887
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   888
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   889
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   890
  apply (blast intro: dvd_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   891
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   892
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   893
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30027
diff changeset
   894
by (drule_tac m = m in nat_dvd_diff, auto)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   895
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   896
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   897
  apply (rule iffI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   898
   apply (erule_tac [2] dvd_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   899
   apply (rule_tac [2] dvd_refl)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   900
  apply (subgoal_tac "n = (n+k) -k")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   901
   prefer 2 apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   902
  apply (erule ssubst)
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30027
diff changeset
   903
  apply (erule nat_dvd_diff)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   904
  apply (rule dvd_refl)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   905
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   906
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   907
lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   908
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   909
  apply (case_tac "n = 0", auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   910
  apply (blast intro: mod_mult_distrib2 [symmetric])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   911
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   912
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   913
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   914
by (blast intro: dvd_mod_imp_dvd dvd_mod)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   915
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   916
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   917
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   918
  apply (erule exE)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   919
  apply (simp add: mult_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   920
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   921
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   922
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   923
  apply auto
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   924
   apply (subgoal_tac "m*n dvd m*1")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   925
   apply (drule dvd_mult_cancel, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   926
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   927
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   928
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   929
  apply (subst mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   930
  apply (erule dvd_mult_cancel1)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   931
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   932
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   933
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   934
  by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   935
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   936
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   937
  by (simp add: dvd_eq_mod_eq_0 mult_div_cancel)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   938
31009
41fd307cab30 dropped reference to class recpower and lemma duplicate
haftmann
parents: 30934
diff changeset
   939
lemma power_dvd_imp_le:
41fd307cab30 dropped reference to class recpower and lemma duplicate
haftmann
parents: 30934
diff changeset
   940
  "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   941
  apply (rule power_le_imp_le_exp, assumption)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   942
  apply (erule dvd_imp_le, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   943
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   944
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   945
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
   946
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
   947
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   948
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   949
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   950
(*Loses information, namely we also have r<d provided d is nonzero*)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   951
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   952
  apply (cut_tac a = m in mod_div_equality)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   953
  apply (simp only: add_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   954
  apply (blast intro: sym)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   955
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   956
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
   957
lemma split_div:
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   958
 "P(n div k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   959
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   960
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   961
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   962
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   963
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   964
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   965
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   966
    with P show ?Q by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   967
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   968
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   969
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   970
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   971
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   972
      assume n: "n = k*i + j" and j: "j < k"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   973
      show "P i"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   974
      proof (cases)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   975
        assume "i = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   976
        with n j P show "P i" by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   977
      next
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   978
        assume "i \<noteq> 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   979
        with not0 n j P show "P i" by(simp add:add_ac)
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   980
      qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   981
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   982
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   983
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   984
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   985
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   986
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   987
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   988
    with Q show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   989
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   990
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   991
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   992
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
13517
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13189
diff changeset
   993
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   994
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   995
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   996
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   997
lemma split_div_lemma:
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   998
  assumes "0 < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   999
  shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1000
proof
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1001
  assume ?rhs
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1002
  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1003
  then have A: "n * q \<le> m" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1004
  have "n - (m mod n) > 0" using mod_less_divisor assms by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1005
  then have "m < m + (n - (m mod n))" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1006
  then have "m < n + (m - (m mod n))" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1007
  with nq have "m < n + n * q" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1008
  then have B: "m < n * Suc q" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1009
  from A B show ?lhs ..
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1010
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1011
  assume P: ?lhs
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
  1012
  then have "divmod_rel m n (q, m - n * q)"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1013
    unfolding divmod_rel_def by (auto simp add: mult_ac)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
  1014
  with divmod_rel_unique divmod_rel [of m n]
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
  1015
  have "(q, m - n * q) = (m div n, m mod n)" by auto
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
  1016
  then show ?rhs by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1017
qed
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1018
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1019
theorem split_div':
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1020
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1021
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1022
  apply (case_tac "0 < n")
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1023
  apply (simp only: add: split_div_lemma)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
  1024
  apply simp_all
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1025
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1026
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1027
lemma split_mod:
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1028
 "P(n mod k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1029
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1030
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1031
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1032
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1033
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1034
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1035
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
  1036
    with P show ?Q by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1037
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1038
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1039
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1040
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1041
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1042
      assume "n = k*i + j" "j < k"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1043
      thus "P j" using not0 P by(simp add:add_ac mult_ac)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1044
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1045
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1046
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1047
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1048
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1049
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1050
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
  1051
    with Q show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1052
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1053
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1054
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1055
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
13517
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13189
diff changeset
  1056
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1057
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1058
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1059
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1060
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1061
  apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1062
    subst [OF mod_div_equality [of _ n]])
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1063
  apply arith
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1064
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1065
22800
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1066
lemma div_mod_equality':
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1067
  fixes m n :: nat
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1068
  shows "m div n * n = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1069
proof -
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1070
  have "m mod n \<le> m mod n" ..
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1071
  from div_mod_equality have 
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1072
    "m div n * n + m mod n - m mod n = m - m mod n" by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1073
  with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1074
    "m div n * n + (m mod n - m mod n) = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1075
    by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1076
  then show ?thesis by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1077
qed
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1078
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1079
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
  1080
subsubsection {*An ``induction'' law for modulus arithmetic.*}
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1081
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1082
lemma mod_induct_0:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1083
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1084
  and base: "P i" and i: "i<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1085
  shows "P 0"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1086
proof (rule ccontr)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1087
  assume contra: "\<not>(P 0)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1088
  from i have p: "0<p" by simp
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1089
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1090
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1091
    fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1092
    show "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1093
    proof (induct k)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1094
      show "?A 0" by simp  -- "by contradiction"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1095
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1096
      fix n
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1097
      assume ih: "?A n"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1098
      show "?A (Suc n)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1099
      proof (clarsimp)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1100
        assume y: "P (p - Suc n)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1101
        have n: "Suc n < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1102
        proof (rule ccontr)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1103
          assume "\<not>(Suc n < p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1104
          hence "p - Suc n = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1105
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1106
          with y contra show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1107
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1108
        qed
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1109
        hence n2: "Suc (p - Suc n) = p-n" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1110
        from p have "p - Suc n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1111
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1112
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1113
        show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1114
        proof (cases "n=0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1115
          case True
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1116
          with z n2 contra show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1117
        next
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1118
          case False
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1119
          with p have "p-n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1120
          with z n2 False ih show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1121
        qed
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1122
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1123
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1124
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1125
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1126
  from i obtain k where "0<k \<and> i+k=p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1127
    by (blast dest: less_imp_add_positive)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1128
  hence "0<k \<and> i=p-k" by auto
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1129
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1130
  note base
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1131
  ultimately
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1132
  show "False" by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1133
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1134
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1135
lemma mod_induct:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1136
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1137
  and base: "P i" and i: "i<p" and j: "j<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1138
  shows "P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1139
proof -
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1140
  have "\<forall>j<p. P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1141
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1142
    fix j
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1143
    show "j<p \<longrightarrow> P j" (is "?A j")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1144
    proof (induct j)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1145
      from step base i show "?A 0"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1146
        by (auto elim: mod_induct_0)
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1147
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1148
      fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1149
      assume ih: "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1150
      show "?A (Suc k)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1151
      proof
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1152
        assume suc: "Suc k < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1153
        hence k: "k<p" by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1154
        with ih have "P k" ..
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1155
        with step k have "P (Suc k mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1156
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1157
        moreover
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1158
        from suc have "Suc k mod p = Suc k"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1159
          by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1160
        ultimately
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1161
        show "P (Suc k)" by simp
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1162
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1163
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1164
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1165
  with j show ?thesis by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1166
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1167
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
  1168
lemma nat_dvd_not_less:
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
  1169
  fixes m n :: nat
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
  1170
  shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
  1171
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
  1172
3366
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
  1173
end