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_Transfer
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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]: "a dvd b \<longleftrightarrow> b mod a = 0"
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proof
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   136
  assume "b mod a = 0"
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   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
33274
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 32010
diff changeset
   181
lemma dvd_mult_div_cancel: "a dvd b \<Longrightarrow> a * (b div a) = b"
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 32010
diff changeset
   182
by (drule dvd_div_mult_self) (simp add: mult_commute)
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 32010
diff changeset
   183
30052
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   184
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
   185
apply (cases "a = 0")
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   186
 apply simp
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   187
apply (auto simp: dvd_def mult_assoc)
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   188
done
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   189
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   190
lemma div_dvd_div[simp]:
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   191
  "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
   192
apply (cases "a = 0")
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   193
 apply simp
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   194
apply (unfold dvd_def)
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   195
apply auto
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   196
 apply(blast intro:mult_assoc[symmetric])
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44766
diff changeset
   197
apply(fastforce simp add: mult_assoc)
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   198
done
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   199
30078
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   200
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
   201
  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
   202
   apply (simp add: mod_div_equality)
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   203
  apply (simp only: dvd_add dvd_mult)
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   204
  done
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   205
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   206
text {* Addition respects modular equivalence. *}
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   207
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   208
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
   209
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   210
  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
   211
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   212
  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
   213
    by (simp only: add_ac)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   214
  also have "\<dots> = (a mod c + b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   215
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   216
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   217
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   218
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   219
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
   220
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   221
  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
   222
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   223
  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
   224
    by (simp only: add_ac)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   225
  also have "\<dots> = (a + b mod c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   226
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   227
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   228
qed
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_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
   231
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
   232
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   233
lemma mod_add_cong:
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   234
  assumes "a mod c = a' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   235
  assumes "b mod c = b' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   236
  shows "(a + b) mod c = (a' + b') mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   237
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   238
  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
   239
    unfolding assms ..
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   240
  thus ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   241
    by (simp only: mod_add_eq [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   242
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   243
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   244
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
   245
  \<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
   246
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
   247
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   248
text {* Multiplication respects modular equivalence. *}
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   249
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   250
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
   251
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   252
  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
   253
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   254
  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
   255
    by (simp only: algebra_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   256
  also have "\<dots> = (a mod c * b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   257
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   258
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   259
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   260
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   261
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
   262
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   263
  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
   264
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   265
  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
   266
    by (simp only: algebra_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   267
  also have "\<dots> = (a * (b mod c)) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   268
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   269
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   270
qed
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_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
   273
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
   274
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   275
lemma mod_mult_cong:
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   276
  assumes "a mod c = a' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   277
  assumes "b mod c = b' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   278
  shows "(a * b) mod c = (a' * b') mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   279
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   280
  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
   281
    unfolding assms ..
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   282
  thus ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   283
    by (simp only: mod_mult_eq [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   284
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   285
47164
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   286
text {* Exponentiation respects modular equivalence. *}
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   287
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   288
lemma power_mod: "(a mod b)^n mod b = a^n mod b"
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   289
apply (induct n, simp_all)
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   290
apply (rule mod_mult_right_eq [THEN trans])
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   291
apply (simp (no_asm_simp))
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   292
apply (rule mod_mult_eq [symmetric])
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   293
done
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   294
29404
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   295
lemma mod_mod_cancel:
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   296
  assumes "c dvd b"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   297
  shows "a mod b mod c = a mod c"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   298
proof -
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   299
  from `c dvd b` obtain k where "b = c * k"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   300
    by (rule dvdE)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   301
  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
   302
    by (simp only: `b = c * k`)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   303
  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
   304
    by (simp only: mod_mult_self1)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   305
  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
   306
    by (simp only: add_ac mult_ac)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   307
  also have "\<dots> = a mod c"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   308
    by (simp only: mod_div_equality)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   309
  finally show ?thesis .
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   310
qed
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   311
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   312
lemma div_mult_div_if_dvd:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   313
  "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
   314
  apply (cases "y = 0", simp)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   315
  apply (cases "z = 0", simp)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   316
  apply (auto elim!: dvdE simp add: algebra_simps)
30476
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   317
  apply (subst mult_assoc [symmetric])
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   318
  apply (simp add: no_zero_divisors)
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   319
  done
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   320
35367
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   321
lemma div_mult_swap:
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   322
  assumes "c dvd b"
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   323
  shows "a * (b div c) = (a * b) div c"
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   324
proof -
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   325
  from assms have "b div c * (a div 1) = b * a div (c * 1)"
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   326
    by (simp only: div_mult_div_if_dvd one_dvd)
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   327
  then show ?thesis by (simp add: mult_commute)
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   328
qed
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   329
   
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   330
lemma div_mult_mult2 [simp]:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   331
  "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   332
  by (drule div_mult_mult1) (simp add: mult_commute)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   333
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   334
lemma div_mult_mult1_if [simp]:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   335
  "(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
   336
  by simp_all
30476
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   337
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   338
lemma mod_mult_mult1:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   339
  "(c * a) mod (c * b) = c * (a mod b)"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   340
proof (cases "c = 0")
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   341
  case True then show ?thesis by simp
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   342
next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   343
  case False
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   344
  from mod_div_equality
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   345
  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
   346
  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
   347
    = c * a + c * (a mod b)" by (simp add: algebra_simps)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   348
  with mod_div_equality show ?thesis by simp 
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   349
qed
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   350
  
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   351
lemma mod_mult_mult2:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   352
  "(a * c) mod (b * c) = (a mod b) * c"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   353
  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
   354
47159
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   355
lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   356
  by (fact mod_mult_mult2 [symmetric])
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   357
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   358
lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   359
  by (fact mod_mult_mult1 [symmetric])
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   360
31662
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   361
lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   362
  unfolding dvd_def by (auto simp add: mod_mult_mult1)
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   363
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   364
lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   365
by (blast intro: dvd_mod_imp_dvd dvd_mod)
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   366
31009
41fd307cab30 dropped reference to class recpower and lemma duplicate
haftmann
parents: 30934
diff changeset
   367
lemma div_power:
31661
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   368
  "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
30476
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   369
apply (induct n)
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   370
 apply simp
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   371
apply(simp add: div_mult_div_if_dvd dvd_power_same)
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   372
done
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   373
35367
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   374
lemma dvd_div_eq_mult:
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   375
  assumes "a \<noteq> 0" and "a dvd b"  
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   376
  shows "b div a = c \<longleftrightarrow> b = c * a"
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   377
proof
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   378
  assume "b = c * a"
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   379
  then show "b div a = c" by (simp add: assms)
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   380
next
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   381
  assume "b div a = c"
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   382
  then have "b div a * a = c * a" by simp
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   383
  moreover from `a dvd b` have "b div a * a = b" by (simp add: dvd_div_mult_self)
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   384
  ultimately show "b = c * a" by simp
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   385
qed
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   386
   
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   387
lemma dvd_div_div_eq_mult:
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   388
  assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   389
  shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   390
  using assms by (auto simp add: mult_commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym)
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   391
31661
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   392
end
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   393
35673
178caf872f95 weakend class ring_div; tuned
haftmann
parents: 35644
diff changeset
   394
class ring_div = semiring_div + comm_ring_1
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   395
begin
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   396
36634
f9b43d197d16 a ring_div is a ring_1_no_zero_divisors
haftmann
parents: 35815
diff changeset
   397
subclass ring_1_no_zero_divisors ..
f9b43d197d16 a ring_div is a ring_1_no_zero_divisors
haftmann
parents: 35815
diff changeset
   398
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   399
text {* Negation respects modular equivalence. *}
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   400
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   401
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
   402
proof -
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   403
  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
   404
    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
   405
  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
   406
    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
   407
  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
   408
    by (rule mod_mult_self1)
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   409
  finally show ?thesis .
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   410
qed
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   411
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   412
lemma mod_minus_cong:
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   413
  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
   414
  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
   415
proof -
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   416
  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
   417
    unfolding assms ..
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   418
  thus ?thesis
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   419
    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
   420
qed
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   421
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   422
text {* Subtraction respects modular equivalence. *}
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   423
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   424
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
   425
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   426
  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
   427
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   428
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
   429
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   430
  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
   431
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   432
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
   433
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   434
  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
   435
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   436
lemma mod_diff_cong:
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   437
  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
   438
  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
   439
  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
   440
  unfolding diff_minus using assms
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   441
  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
   442
30180
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   443
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
   444
apply (case_tac "y = 0") apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   445
apply (auto simp add: dvd_def)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   446
apply (subgoal_tac "-(y * k) = y * - k")
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   447
 apply (erule ssubst)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   448
 apply (erule div_mult_self1_is_id)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   449
apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   450
done
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   451
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   452
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
   453
apply (case_tac "y = 0") apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   454
apply (auto simp add: dvd_def)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   455
apply (subgoal_tac "y * k = -y * -k")
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   456
 apply (erule ssubst)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   457
 apply (rule div_mult_self1_is_id)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   458
 apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   459
apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   460
done
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   461
47159
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   462
lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   463
  using div_mult_mult1 [of "- 1" a b]
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   464
  unfolding neg_equal_0_iff_equal by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   465
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   466
lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   467
  using mod_mult_mult1 [of "- 1" a b] by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   468
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   469
lemma div_minus_right: "a div (-b) = (-a) div b"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   470
  using div_minus_minus [of "-a" b] by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   471
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   472
lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   473
  using mod_minus_minus [of "-a" b] by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   474
47160
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   475
lemma div_minus1_right [simp]: "a div (-1) = -a"
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   476
  using div_minus_right [of a 1] by simp
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   477
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   478
lemma mod_minus1_right [simp]: "a mod (-1) = 0"
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   479
  using mod_minus_right [of a 1] by simp
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   480
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   481
end
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   482
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   483
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   484
subsection {* Division on @{typ nat} *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   485
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   486
text {*
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   487
  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
   488
  of a characteristic relation with two input arguments
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   489
  @{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
   490
  @{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
   491
*}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   492
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   493
definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   494
  "divmod_nat_rel m n qr \<longleftrightarrow>
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   495
    m = fst qr * n + snd qr \<and>
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   496
      (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
   497
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   498
text {* @{const divmod_nat_rel} is total: *}
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   499
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   500
lemma divmod_nat_rel_ex:
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   501
  obtains q r where "divmod_nat_rel m n (q, r)"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   502
proof (cases "n = 0")
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   503
  case True  with that show thesis
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   504
    by (auto simp add: divmod_nat_rel_def)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   505
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   506
  case False
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   507
  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
   508
  proof (induct m)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   509
    case 0 with `n \<noteq> 0`
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   510
    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
   511
    then show ?case by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   512
  next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   513
    case (Suc m) then obtain q' r'
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   514
      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
   515
    then show ?case proof (cases "Suc r' < n")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   516
      case True
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   517
      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
   518
      with True show ?thesis by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   519
    next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   520
      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
   521
      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
   522
      ultimately have "n = Suc r'" by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   523
      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
   524
      with `n \<noteq> 0` show ?thesis by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   525
    qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   526
  qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   527
  with that show thesis
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   528
    using `n \<noteq> 0` by (auto simp add: divmod_nat_rel_def)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   529
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   530
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   531
text {* @{const divmod_nat_rel} is injective: *}
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   532
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   533
lemma divmod_nat_rel_unique:
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   534
  assumes "divmod_nat_rel m n qr"
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   535
    and "divmod_nat_rel m n qr'"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   536
  shows "qr = qr'"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   537
proof (cases "n = 0")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   538
  case True with assms show ?thesis
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   539
    by (cases qr, cases qr')
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   540
      (simp add: divmod_nat_rel_def)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   541
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   542
  case False
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   543
  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
   544
  apply (rule leI)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   545
  apply (subst less_iff_Suc_add)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   546
  apply (auto simp add: add_mult_distrib)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   547
  done
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   548
  from `n \<noteq> 0` assms have "fst qr = fst qr'"
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   549
    by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   550
  moreover from this assms have "snd qr = snd qr'"
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   551
    by (simp add: divmod_nat_rel_def)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   552
  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
   553
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   554
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   555
text {*
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   556
  We instantiate divisibility on the natural numbers by
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   557
  means of @{const divmod_nat_rel}:
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   558
*}
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   559
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   560
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 36634
diff changeset
   561
  "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   562
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   563
lemma divmod_nat_rel_divmod_nat:
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   564
  "divmod_nat_rel m n (divmod_nat m n)"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   565
proof -
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   566
  from divmod_nat_rel_ex
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   567
    obtain qr where rel: "divmod_nat_rel m n qr" .
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   568
  then show ?thesis
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   569
  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   570
qed
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   571
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   572
lemma divmod_nat_unique:
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   573
  assumes "divmod_nat_rel m n qr" 
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   574
  shows "divmod_nat m n = qr"
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   575
  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   576
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   577
instantiation nat :: semiring_div
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   578
begin
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   579
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   580
definition div_nat where
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   581
  "m div n = fst (divmod_nat m n)"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   582
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   583
lemma fst_divmod_nat [simp]:
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   584
  "fst (divmod_nat m n) = m div n"
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   585
  by (simp add: div_nat_def)
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   586
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   587
definition mod_nat where
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   588
  "m mod n = snd (divmod_nat m n)"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   589
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   590
lemma snd_divmod_nat [simp]:
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   591
  "snd (divmod_nat m n) = m mod n"
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   592
  by (simp add: mod_nat_def)
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   593
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   594
lemma divmod_nat_div_mod:
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   595
  "divmod_nat m n = (m div n, m mod n)"
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   596
  by (simp add: prod_eq_iff)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   597
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   598
lemma div_nat_unique:
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   599
  assumes "divmod_nat_rel m n (q, r)" 
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   600
  shows "m div n = q"
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   601
  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   602
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   603
lemma mod_nat_unique:
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   604
  assumes "divmod_nat_rel m n (q, r)" 
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   605
  shows "m mod n = r"
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   606
  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   607
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   608
lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   609
  using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   610
47136
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   611
lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   612
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   613
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   614
lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   615
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   616
47137
7f5f0531cae6 shorten a proof
huffman
parents: 47136
diff changeset
   617
lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
7f5f0531cae6 shorten a proof
huffman
parents: 47136
diff changeset
   618
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   619
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   620
lemma divmod_nat_step:
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   621
  assumes "0 < n" and "n \<le> m"
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   622
  shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   623
proof (rule divmod_nat_unique)
47134
28c1db43d4d0 simplify some proofs
huffman
parents: 47108
diff changeset
   624
  have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"
28c1db43d4d0 simplify some proofs
huffman
parents: 47108
diff changeset
   625
    by (rule divmod_nat_rel)
28c1db43d4d0 simplify some proofs
huffman
parents: 47108
diff changeset
   626
  thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"
28c1db43d4d0 simplify some proofs
huffman
parents: 47108
diff changeset
   627
    unfolding divmod_nat_rel_def using assms by auto
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   628
qed
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   629
26300
03def556e26e removed duplicate lemmas;
wenzelm
parents: 26100
diff changeset
   630
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
   631
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   632
lemma div_less [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   633
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   634
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   635
  shows "m div n = 0"
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   636
  using assms divmod_nat_base by (simp add: prod_eq_iff)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   637
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   638
lemma le_div_geq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   639
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   640
  assumes "0 < n" and "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   641
  shows "m div n = Suc ((m - n) div n)"
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   642
  using assms divmod_nat_step by (simp add: prod_eq_iff)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   643
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   644
lemma mod_less [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   645
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   646
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   647
  shows "m mod n = m"
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   648
  using assms divmod_nat_base by (simp add: prod_eq_iff)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   649
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   650
lemma le_mod_geq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   651
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   652
  assumes "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   653
  shows "m mod n = (m - n) mod n"
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   654
  using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   655
47136
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   656
instance proof
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   657
  fix m n :: nat
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   658
  show "m div n * n + m mod n = m"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   659
    using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   660
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   661
  fix m n q :: nat
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   662
  assume "n \<noteq> 0"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   663
  then show "(q + m * n) div n = m + q div n"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   664
    by (induct m) (simp_all add: le_div_geq)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   665
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   666
  fix m n q :: nat
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   667
  assume "m \<noteq> 0"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   668
  hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   669
    unfolding divmod_nat_rel_def
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   670
    by (auto split: split_if_asm, simp_all add: algebra_simps)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   671
  moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   672
  ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   673
  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   674
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   675
  fix n :: nat show "n div 0 = 0"
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   676
    by (simp add: div_nat_def divmod_nat_zero)
47136
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   677
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   678
  fix n :: nat show "0 div n = 0"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   679
    by (simp add: div_nat_def divmod_nat_zero_left)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   680
qed
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   681
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   682
end
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   683
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
   684
lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
   685
  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   686
  by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
   687
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   688
text {* Simproc for cancelling @{const div} and @{const mod} *}
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   689
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   690
ML {*
43594
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 41792
diff changeset
   691
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
   692
(
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   693
  val div_name = @{const_name div};
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   694
  val mod_name = @{const_name mod};
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   695
  val mk_binop = HOLogic.mk_binop;
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   696
  val mk_sum = Nat_Arith.mk_sum;
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   697
  val dest_sum = Nat_Arith.dest_sum;
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   698
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   699
  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
   700
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   701
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
35050
9f841f20dca6 renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents: 34982
diff changeset
   702
    (@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
   703
)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   704
*}
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   705
43594
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 41792
diff changeset
   706
simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 41792
diff changeset
   707
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   708
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   709
subsubsection {* Quotient *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   710
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   711
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
   712
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
   713
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   714
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
   715
by (simp add: div_geq)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   716
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   717
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
   718
by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   719
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   720
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
   721
by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   722
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   723
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   724
subsubsection {* Remainder *}
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   725
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   726
lemma mod_less_divisor [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   727
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   728
  assumes "n > 0"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   729
  shows "m mod n < (n::nat)"
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   730
  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   731
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   732
lemma mod_less_eq_dividend [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   733
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   734
  shows "m mod n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   735
proof (rule add_leD2)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   736
  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
   737
  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
   738
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   739
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   740
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
   741
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
   742
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   743
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
   744
by (simp add: le_mod_geq)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   745
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   746
lemma mod_1 [simp]: "m mod Suc 0 = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   747
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
   748
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   749
(* 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
   750
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
   751
  using mod_div_equality2 [of n m] by arith
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   752
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   753
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   754
  apply (drule mod_less_divisor [where m = m])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   755
  apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   756
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   757
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   758
subsubsection {* Quotient and Remainder *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   759
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   760
lemma divmod_nat_rel_mult1_eq:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
   761
  "divmod_nat_rel b c (q, r)
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   762
   \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   763
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   764
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   765
lemma div_mult1_eq:
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   766
  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   767
by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   768
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   769
lemma divmod_nat_rel_add1_eq:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
   770
  "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br)
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   771
   \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   772
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   773
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   774
(*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
   775
lemma div_add1_eq:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   776
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   777
by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   778
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   779
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   780
  apply (cut_tac m = q and n = c in mod_less_divisor)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   781
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   782
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   783
  apply (simp add: add_mult_distrib2)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   784
  done
10559
d3fd54fc659b many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents: 10214
diff changeset
   785
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   786
lemma divmod_nat_rel_mult2_eq:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
   787
  "divmod_nat_rel a b (q, r)
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   788
   \<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   789
by (auto simp add: mult_ac divmod_nat_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
   790
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   791
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   792
by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   793
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   794
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   795
by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   796
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   797
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   798
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
   799
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   800
lemma div_1 [simp]: "m div Suc 0 = m"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   801
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
   802
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   803
(* Monotonicity of div in first argument *)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   804
lemma div_le_mono [rule_format (no_asm)]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   805
    "\<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
   806
apply (case_tac "k=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   807
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
   808
apply (case_tac "n<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   809
(* 1  case n<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   810
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   811
(* 2  case n >= k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   812
apply (case_tac "m<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   813
(* 2.1  case m<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   814
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   815
(* 2.2  case m>=k *)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   816
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
   817
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   818
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   819
(* Antimonotonicity of div in second argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   820
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
   821
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   822
 prefer 2 apply simp
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   823
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
   824
apply (rename_tac "k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   825
apply (case_tac "k<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   826
apply (subgoal_tac "~ (k<m) ")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   827
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   828
apply (simp add: div_geq)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   829
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
   830
 prefer 2
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   831
 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
   832
apply (rule le_trans, simp)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   833
apply (simp)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   834
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   835
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   836
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
   837
apply (case_tac "n=0", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   838
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
   839
apply (rule div_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   840
apply (simp_all (no_asm_simp))
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   841
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   842
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   843
(* Similar for "less than" *)
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
   844
lemma div_less_dividend [simp]:
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
   845
  "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
   846
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
   847
apply (rename_tac "m")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   848
apply (case_tac "m<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   849
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   850
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   851
apply (simp add: div_geq)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   852
apply (case_tac "n<m")
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   853
 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
   854
  apply (rule impI less_trans_Suc)+
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   855
apply assumption
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   856
  apply (simp_all)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   857
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   858
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   859
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
   860
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
   861
apply (case_tac "n=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   862
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
   863
apply (case_tac "Suc (na) <n")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   864
(* case Suc(na) < n *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   865
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
   866
(* case n \<le> Suc(na) *)
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16733
diff changeset
   867
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   868
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
   869
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   870
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   871
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
   872
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
   873
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   874
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
   875
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   876
(*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
   877
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
   878
  apply (cut_tac a = m in mod_div_equality)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   879
  apply (simp only: add_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   880
  apply (blast intro: sym)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   881
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   882
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
   883
lemma split_div:
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   884
 "P(n div k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   885
 ((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
   886
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   887
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   888
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   889
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   890
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   891
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   892
    with P show ?Q by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   893
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   894
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   895
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   896
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   897
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   898
      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
   899
      show "P i"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   900
      proof (cases)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   901
        assume "i = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   902
        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
   903
      next
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   904
        assume "i \<noteq> 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   905
        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
   906
      qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   907
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   908
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   909
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   910
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   911
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   912
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   913
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   914
    with Q show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   915
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   916
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   917
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   918
    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
   919
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   920
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   921
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   922
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   923
lemma split_div_lemma:
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   924
  assumes "0 < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   925
  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
   926
proof
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   927
  assume ?rhs
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   928
  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
   929
  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
   930
  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
   931
  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
   932
  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
   933
  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
   934
  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
   935
  from A B show ?lhs ..
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   936
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   937
  assume P: ?lhs
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   938
  then have "divmod_nat_rel m n (q, m - n * q)"
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   939
    unfolding divmod_nat_rel_def by (auto simp add: mult_ac)
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   940
  with divmod_nat_rel_unique divmod_nat_rel [of m n]
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   941
  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
   942
  then show ?rhs by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   943
qed
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   944
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   945
theorem split_div':
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   946
  "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
   947
   (\<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
   948
  apply (case_tac "0 < n")
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   949
  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
   950
  apply simp_all
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   951
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   952
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   953
lemma split_mod:
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   954
 "P(n mod k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   955
 ((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
   956
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   957
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   958
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   959
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   960
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   961
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   962
    with P show ?Q by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   963
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   964
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   965
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   966
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   967
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   968
      assume "n = k*i + j" "j < k"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   969
      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
   970
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   971
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   972
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   973
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   974
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   975
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   976
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   977
    with Q show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   978
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   979
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   980
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   981
    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
   982
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   983
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   984
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   985
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   986
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
   987
  using mod_div_equality [of m n] by arith
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
   988
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
   989
lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
   990
  using mod_div_equality [of m n] by arith
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
   991
(* FIXME: very similar to mult_div_cancel *)
22800
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   992
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   993
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   994
subsubsection {* An ``induction'' law for modulus arithmetic. *}
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   995
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   996
lemma mod_induct_0:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   997
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   998
  and base: "P i" and i: "i<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   999
  shows "P 0"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1000
proof (rule ccontr)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1001
  assume contra: "\<not>(P 0)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1002
  from i have p: "0<p" by simp
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1003
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1004
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1005
    fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1006
    show "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1007
    proof (induct k)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1008
      show "?A 0" by simp  -- "by contradiction"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1009
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1010
      fix n
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1011
      assume ih: "?A n"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1012
      show "?A (Suc n)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1013
      proof (clarsimp)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1014
        assume y: "P (p - Suc n)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1015
        have n: "Suc n < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1016
        proof (rule ccontr)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1017
          assume "\<not>(Suc n < p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1018
          hence "p - Suc n = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1019
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1020
          with y contra show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1021
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1022
        qed
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1023
        hence n2: "Suc (p - Suc n) = p-n" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1024
        from p have "p - Suc n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1025
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1026
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1027
        show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1028
        proof (cases "n=0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1029
          case True
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1030
          with z n2 contra show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1031
        next
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1032
          case False
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1033
          with p have "p-n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1034
          with z n2 False ih show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1035
        qed
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1036
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1037
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1038
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1039
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1040
  from i obtain k where "0<k \<and> i+k=p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1041
    by (blast dest: less_imp_add_positive)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1042
  hence "0<k \<and> i=p-k" by auto
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1043
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1044
  note base
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1045
  ultimately
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1046
  show "False" by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1047
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1048
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1049
lemma mod_induct:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1050
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1051
  and base: "P i" and i: "i<p" and j: "j<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1052
  shows "P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1053
proof -
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1054
  have "\<forall>j<p. P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1055
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1056
    fix j
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1057
    show "j<p \<longrightarrow> P j" (is "?A j")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1058
    proof (induct j)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1059
      from step base i show "?A 0"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1060
        by (auto elim: mod_induct_0)
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1061
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1062
      fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1063
      assume ih: "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1064
      show "?A (Suc k)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1065
      proof
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1066
        assume suc: "Suc k < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1067
        hence k: "k<p" by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1068
        with ih have "P k" ..
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1069
        with step k have "P (Suc k mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1070
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1071
        moreover
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1072
        from suc have "Suc k mod p = Suc k"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1073
          by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1074
        ultimately
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1075
        show "P (Suc k)" by simp
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1076
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1077
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1078
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1079
  with j show ?thesis by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1080
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1081
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1082
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1083
  by (simp add: numeral_2_eq_2 le_div_geq)
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1084
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1085
lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1086
  by (simp add: numeral_2_eq_2 le_mod_geq)
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1087
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1088
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
47217
501b9bbd0d6e removed redundant nat-specific copies of theorems
huffman
parents: 47167
diff changeset
  1089
by (simp add: mult_2 [symmetric])
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1090
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1091
lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1092
proof -
35815
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35673
diff changeset
  1093
  { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1094
  moreover have "m mod 2 < 2" by simp
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1095
  ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1096
  then show ?thesis by auto
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1097
qed
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1098
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1099
text{*These lemmas collapse some needless occurrences of Suc:
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1100
    at least three Sucs, since two and fewer are rewritten back to Suc again!
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1101
    We already have some rules to simplify operands smaller than 3.*}
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1102
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1103
lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1104
by (simp add: Suc3_eq_add_3)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1105
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1106
lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1107
by (simp add: Suc3_eq_add_3)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1108
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1109
lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1110
by (simp add: Suc3_eq_add_3)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1111
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1112
lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1113
by (simp add: Suc3_eq_add_3)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1114
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1115
lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1116
lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1117
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1118
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1119
lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1120
apply (induct "m")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1121
apply (simp_all add: mod_Suc)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1122
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1123
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1124
declare Suc_times_mod_eq [of "numeral w", simp] for w
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1125
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1126
lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1127
by (simp add: div_le_mono)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1128
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1129
lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1130
by (cases n) simp_all
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1131
35815
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35673
diff changeset
  1132
lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35673
diff changeset
  1133
proof -
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35673
diff changeset
  1134
  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35673
diff changeset
  1135
  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp 
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35673
diff changeset
  1136
qed
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1137
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1138
  (* Potential use of algebra : Equality modulo n*)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1139
lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1140
by (simp add: mult_ac add_ac)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1141
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1142
lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1143
proof -
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1144
  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1145
  also have "... = Suc m mod n" by (rule mod_mult_self3) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1146
  finally show ?thesis .
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1147
qed
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1148
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1149
lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1150
apply (subst mod_Suc [of m]) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1151
apply (subst mod_Suc [of "m mod n"], simp) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1152
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1153
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1154
lemma mod_2_not_eq_zero_eq_one_nat:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1155
  fixes n :: nat
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1156
  shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1157
  by simp
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1158
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1159
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1160
subsection {* Division on @{typ int} *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1161
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1162
definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1163
    --{*definition of quotient and remainder*}
47139
98bddfa0f717 extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents: 47138
diff changeset
  1164
  "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
98bddfa0f717 extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents: 47138
diff changeset
  1165
    (if 0 < b then 0 \<le> r \<and> r < b else if b < 0 then b < r \<and> r \<le> 0 else q = 0))"
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1166
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1167
definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1168
    --{*for the division algorithm*}
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1169
    "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1170
                         else (2 * q, r))"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1171
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1172
text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1173
function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1174
  "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1175
     else adjust b (posDivAlg a (2 * b)))"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1176
by auto
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1177
termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1178
  (auto simp add: mult_2)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1179
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1180
text{*algorithm for the case @{text "a<0, b>0"}*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1181
function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1182
  "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1183
     else adjust b (negDivAlg a (2 * b)))"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1184
by auto
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1185
termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1186
  (auto simp add: mult_2)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1187
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1188
text{*algorithm for the general case @{term "b\<noteq>0"}*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1189
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1190
definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1191
    --{*The full division algorithm considers all possible signs for a, b
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1192
       including the special case @{text "a=0, b<0"} because 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1193
       @{term negDivAlg} requires @{term "a<0"}.*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1194
  "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1195
                  else if a = 0 then (0, 0)
46560
8e252a608765 remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents: 46552
diff changeset
  1196
                       else apsnd uminus (negDivAlg (-a) (-b))
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1197
               else 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1198
                  if 0 < b then negDivAlg a b
46560
8e252a608765 remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents: 46552
diff changeset
  1199
                  else apsnd uminus (posDivAlg (-a) (-b)))"
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1200
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1201
instantiation int :: Divides.div
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1202
begin
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1203
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1204
definition div_int where
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1205
  "a div b = fst (divmod_int a b)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1206
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1207
lemma fst_divmod_int [simp]:
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1208
  "fst (divmod_int a b) = a div b"
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1209
  by (simp add: div_int_def)
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1210
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1211
definition mod_int where
46560
8e252a608765 remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents: 46552
diff changeset
  1212
  "a mod b = snd (divmod_int a b)"
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1213
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1214
lemma snd_divmod_int [simp]:
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1215
  "snd (divmod_int a b) = a mod b"
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1216
  by (simp add: mod_int_def)
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1217
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1218
instance ..
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1219
3366
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
  1220
end
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1221
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1222
lemma divmod_int_mod_div:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1223
  "divmod_int p q = (p div q, p mod q)"
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1224
  by (simp add: prod_eq_iff)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1225
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1226
text{*
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1227
Here is the division algorithm in ML:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1228
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1229
\begin{verbatim}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1230
    fun posDivAlg (a,b) =
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1231
      if a<b then (0,a)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1232
      else let val (q,r) = posDivAlg(a, 2*b)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1233
               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1234
           end
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1235
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1236
    fun negDivAlg (a,b) =
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1237
      if 0\<le>a+b then (~1,a+b)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1238
      else let val (q,r) = negDivAlg(a, 2*b)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1239
               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1240
           end;
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1241
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1242
    fun negateSnd (q,r:int) = (q,~r);
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1243
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1244
    fun divmod (a,b) = if 0\<le>a then 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1245
                          if b>0 then posDivAlg (a,b) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1246
                           else if a=0 then (0,0)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1247
                                else negateSnd (negDivAlg (~a,~b))
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1248
                       else 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1249
                          if 0<b then negDivAlg (a,b)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1250
                          else        negateSnd (posDivAlg (~a,~b));
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1251
\end{verbatim}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1252
*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1253
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1254
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1255
subsubsection {* Uniqueness and Monotonicity of Quotients and Remainders *}
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1256
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1257
lemma unique_quotient_lemma:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1258
     "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1259
      ==> q' \<le> (q::int)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1260
apply (subgoal_tac "r' + b * (q'-q) \<le> r")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1261
 prefer 2 apply (simp add: right_diff_distrib)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1262
apply (subgoal_tac "0 < b * (1 + q - q') ")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1263
apply (erule_tac [2] order_le_less_trans)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1264
 prefer 2 apply (simp add: right_diff_distrib right_distrib)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1265
apply (subgoal_tac "b * q' < b * (1 + q) ")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1266
 prefer 2 apply (simp add: right_diff_distrib right_distrib)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1267
apply (simp add: mult_less_cancel_left)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1268
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1269
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1270
lemma unique_quotient_lemma_neg:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1271
     "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1272
      ==> q \<le> (q'::int)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1273
by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1274
    auto)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1275
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1276
lemma unique_quotient:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
  1277
     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1278
      ==> q = q'"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1279
apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1280
apply (blast intro: order_antisym
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1281
             dest: order_eq_refl [THEN unique_quotient_lemma] 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1282
             order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1283
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1284
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1285
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1286
lemma unique_remainder:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
  1287
     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1288
      ==> r = r'"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1289
apply (subgoal_tac "q = q'")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1290
 apply (simp add: divmod_int_rel_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1291
apply (blast intro: unique_quotient)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1292
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1293
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1294
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1295
subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1296
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1297
text{*And positive divisors*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1298
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1299
lemma adjust_eq [simp]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1300
     "adjust b (q, r) = 
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1301
      (let diff = r - b in  
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1302
        if 0 \<le> diff then (2 * q + 1, diff)   
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1303
                     else (2*q, r))"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1304
  by (simp add: Let_def adjust_def)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1305