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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begin
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ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"
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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 distrib_right)
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  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
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    by (simp add: mod_div_equality)
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  then show ?thesis by simp
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qed
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lemma mod_mult_self2 [simp]: "(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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diff changeset
   137
  assume "b mod a = 0"
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   138
  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
   139
  then have "b = a * (b div a)" unfolding mult_commute ..
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   140
  then have "\<exists>c. b = a * c" ..
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   141
  then show "a dvd b" unfolding dvd_def .
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   142
next
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   143
  assume "a dvd b"
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   144
  then have "\<exists>c. b = a * c" unfolding dvd_def .
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   145
  then obtain c where "b = a * c" ..
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   146
  then have "b mod a = a * c mod a" by simp
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   147
  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
   148
  then show "b mod a = 0" by simp
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   149
qed
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   150
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   151
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
   152
proof (cases "b = 0")
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   153
  assume "b = 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   154
  thus ?thesis by simp
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   155
next
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   156
  assume "b \<noteq> 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   157
  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
   158
    by (rule div_mult_self1 [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   159
  also have "\<dots> = a div b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   160
    by (simp only: mod_div_equality')
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   161
  also have "\<dots> = a div b + 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   162
    by simp
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   163
  finally show ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   164
    by (rule add_left_imp_eq)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   165
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   166
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   167
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
   168
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   169
  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
   170
    by (simp only: mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   171
  also have "\<dots> = a mod b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   172
    by (simp only: mod_div_equality')
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   173
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   174
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   175
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   176
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
   177
by (rule dvd_eq_mod_eq_0[THEN iffD1])
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   178
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   179
lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   180
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
   181
33274
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 32010
diff changeset
   182
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
   183
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
   184
30052
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   185
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
   186
apply (cases "a = 0")
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   187
 apply simp
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   188
apply (auto simp: dvd_def mult_assoc)
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   189
done
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   190
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   191
lemma div_dvd_div[simp]:
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   192
  "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
   193
apply (cases "a = 0")
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   194
 apply simp
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   195
apply (unfold dvd_def)
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   196
apply auto
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   197
 apply(blast intro:mult_assoc[symmetric])
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44766
diff changeset
   198
apply(fastforce simp add: mult_assoc)
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   199
done
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   200
30078
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   201
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
   202
  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
   203
   apply (simp add: mod_div_equality)
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   204
  apply (simp only: dvd_add dvd_mult)
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   205
  done
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   206
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   207
text {* Addition respects modular equivalence. *}
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   208
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   209
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
   210
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   211
  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
   212
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   213
  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
   214
    by (simp only: add_ac)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   215
  also have "\<dots> = (a mod c + b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   216
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   217
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   218
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   219
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   220
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
   221
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   222
  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
   223
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   224
  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
   225
    by (simp only: add_ac)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   226
  also have "\<dots> = (a + b mod c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   227
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   228
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   229
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   230
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   231
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
   232
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
   233
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   234
lemma mod_add_cong:
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   235
  assumes "a mod c = a' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   236
  assumes "b mod c = b' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   237
  shows "(a + b) mod c = (a' + b') mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   238
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   239
  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
   240
    unfolding assms ..
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   241
  thus ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   242
    by (simp only: mod_add_eq [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   243
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   244
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   245
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
   246
  \<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
   247
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
   248
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   249
text {* Multiplication respects modular equivalence. *}
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   250
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   251
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
   252
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   253
  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
   254
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   255
  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
   256
    by (simp only: algebra_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   257
  also have "\<dots> = (a mod c * b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   258
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   259
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   260
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   261
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   262
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
   263
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   264
  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
   265
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   266
  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
   267
    by (simp only: algebra_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   268
  also have "\<dots> = (a * (b mod c)) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   269
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   270
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   271
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   272
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   273
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
   274
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
   275
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   276
lemma mod_mult_cong:
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   277
  assumes "a mod c = a' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   278
  assumes "b mod c = b' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   279
  shows "(a * b) mod c = (a' * b') mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   280
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   281
  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
   282
    unfolding assms ..
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   283
  thus ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   284
    by (simp only: mod_mult_eq [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   285
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   286
47164
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   287
text {* Exponentiation respects modular equivalence. *}
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   288
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   289
lemma power_mod: "(a mod b)^n mod b = a^n mod b"
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   290
apply (induct n, simp_all)
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   291
apply (rule mod_mult_right_eq [THEN trans])
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   292
apply (simp (no_asm_simp))
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   293
apply (rule mod_mult_eq [symmetric])
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   294
done
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   295
29404
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   296
lemma mod_mod_cancel:
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   297
  assumes "c dvd b"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   298
  shows "a mod b mod c = a mod c"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   299
proof -
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   300
  from `c dvd b` obtain k where "b = c * k"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   301
    by (rule dvdE)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   302
  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
   303
    by (simp only: `b = c * k`)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   304
  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
   305
    by (simp only: mod_mult_self1)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   306
  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
   307
    by (simp only: add_ac mult_ac)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   308
  also have "\<dots> = a mod c"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   309
    by (simp only: mod_div_equality)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   310
  finally show ?thesis .
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   311
qed
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   312
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   313
lemma div_mult_div_if_dvd:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   314
  "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
   315
  apply (cases "y = 0", simp)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   316
  apply (cases "z = 0", simp)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   317
  apply (auto elim!: dvdE simp add: algebra_simps)
30476
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   318
  apply (subst mult_assoc [symmetric])
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   319
  apply (simp add: no_zero_divisors)
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   320
  done
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   321
35367
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   322
lemma div_mult_swap:
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   323
  assumes "c dvd b"
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   324
  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
   325
proof -
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   326
  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
   327
    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
   328
  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
   329
qed
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   330
   
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   331
lemma div_mult_mult2 [simp]:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   332
  "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   333
  by (drule div_mult_mult1) (simp add: mult_commute)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   334
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   335
lemma div_mult_mult1_if [simp]:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   336
  "(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
   337
  by simp_all
30476
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   338
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   339
lemma mod_mult_mult1:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   340
  "(c * a) mod (c * b) = c * (a mod b)"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   341
proof (cases "c = 0")
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   342
  case True then show ?thesis by simp
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   343
next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   344
  case False
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   345
  from mod_div_equality
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   346
  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
   347
  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
   348
    = c * a + c * (a mod b)" by (simp add: algebra_simps)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   349
  with mod_div_equality show ?thesis by simp 
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   350
qed
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   351
  
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   352
lemma mod_mult_mult2:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   353
  "(a * c) mod (b * c) = (a mod b) * c"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   354
  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
   355
47159
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   356
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
   357
  by (fact mod_mult_mult2 [symmetric])
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   358
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   359
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
   360
  by (fact mod_mult_mult1 [symmetric])
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   361
31662
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   362
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
   363
  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
   364
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   365
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
   366
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
   367
31009
41fd307cab30 dropped reference to class recpower and lemma duplicate
haftmann
parents: 30934
diff changeset
   368
lemma div_power:
31661
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   369
  "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
30476
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   370
apply (induct n)
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   371
 apply simp
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   372
apply(simp add: div_mult_div_if_dvd dvd_power_same)
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   373
done
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   374
35367
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   375
lemma dvd_div_eq_mult:
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   376
  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
   377
  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
   378
proof
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   379
  assume "b = c * a"
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   380
  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
   381
next
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   382
  assume "b div a = c"
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   383
  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
   384
  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
   385
  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
   386
qed
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   387
   
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   388
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
   389
  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
   390
  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
   391
  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
   392
31661
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   393
end
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   394
35673
178caf872f95 weakend class ring_div; tuned
haftmann
parents: 35644
diff changeset
   395
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
   396
begin
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   397
36634
f9b43d197d16 a ring_div is a ring_1_no_zero_divisors
haftmann
parents: 35815
diff changeset
   398
subclass ring_1_no_zero_divisors ..
f9b43d197d16 a ring_div is a ring_1_no_zero_divisors
haftmann
parents: 35815
diff changeset
   399
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   400
text {* Negation respects modular equivalence. *}
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   401
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   402
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
   403
proof -
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   404
  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
   405
    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
   406
  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
   407
    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
   408
  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
   409
    by (rule mod_mult_self1)
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   410
  finally show ?thesis .
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   411
qed
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   412
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   413
lemma mod_minus_cong:
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   414
  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
   415
  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
   416
proof -
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   417
  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
   418
    unfolding assms ..
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   419
  thus ?thesis
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   420
    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
   421
qed
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   422
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   423
text {* Subtraction respects modular equivalence. *}
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   424
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   425
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
   426
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   427
  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
   428
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   429
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
   430
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   431
  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
   432
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   433
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
   434
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   435
  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
   436
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   437
lemma mod_diff_cong:
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   438
  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
   439
  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
   440
  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
   441
  unfolding diff_minus using assms
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   442
  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
   443
30180
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   444
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
   445
apply (case_tac "y = 0") apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   446
apply (auto simp add: dvd_def)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   447
apply (subgoal_tac "-(y * k) = y * - k")
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   448
 apply (erule ssubst)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   449
 apply (erule div_mult_self1_is_id)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   450
apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   451
done
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   452
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   453
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
   454
apply (case_tac "y = 0") apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   455
apply (auto simp add: dvd_def)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   456
apply (subgoal_tac "y * k = -y * -k")
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   457
 apply (erule ssubst)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   458
 apply (rule div_mult_self1_is_id)
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
apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   461
done
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   462
47159
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   463
lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   464
  using div_mult_mult1 [of "- 1" a b]
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   465
  unfolding neg_equal_0_iff_equal by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   466
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   467
lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   468
  using mod_mult_mult1 [of "- 1" a b] by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   469
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   470
lemma div_minus_right: "a div (-b) = (-a) div b"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   471
  using div_minus_minus [of "-a" b] by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   472
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   473
lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   474
  using mod_minus_minus [of "-a" b] by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   475
47160
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   476
lemma div_minus1_right [simp]: "a div (-1) = -a"
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   477
  using div_minus_right [of a 1] by simp
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   478
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   479
lemma mod_minus1_right [simp]: "a mod (-1) = 0"
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   480
  using mod_minus_right [of a 1] by simp
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   481
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   482
end
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   483
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   484
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   485
subsection {* Division on @{typ nat} *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   486
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   487
text {*
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   488
  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
   489
  of a characteristic relation with two input arguments
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   490
  @{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
   491
  @{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
   492
*}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   493
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   494
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
   495
  "divmod_nat_rel m n qr \<longleftrightarrow>
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   496
    m = fst qr * n + snd qr \<and>
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   497
      (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
   498
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   499
text {* @{const divmod_nat_rel} is total: *}
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   500
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   501
lemma divmod_nat_rel_ex:
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   502
  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
   503
proof (cases "n = 0")
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   504
  case True  with that show thesis
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   505
    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
   506
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   507
  case False
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   508
  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
   509
  proof (induct m)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   510
    case 0 with `n \<noteq> 0`
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   511
    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
   512
    then show ?case by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   513
  next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   514
    case (Suc m) then obtain q' r'
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   515
      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
   516
    then show ?case proof (cases "Suc r' < n")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   517
      case True
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   518
      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
   519
      with True show ?thesis by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   520
    next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   521
      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
   522
      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
   523
      ultimately have "n = Suc r'" by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   524
      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
   525
      with `n \<noteq> 0` show ?thesis by blast
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
  qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   528
  with that show thesis
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   529
    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
   530
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   531
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   532
text {* @{const divmod_nat_rel} is injective: *}
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   533
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   534
lemma divmod_nat_rel_unique:
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   535
  assumes "divmod_nat_rel m n qr"
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   536
    and "divmod_nat_rel m n qr'"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   537
  shows "qr = qr'"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   538
proof (cases "n = 0")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   539
  case True with assms show ?thesis
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   540
    by (cases qr, cases qr')
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   541
      (simp add: divmod_nat_rel_def)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   542
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   543
  case False
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   544
  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
   545
  apply (rule leI)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   546
  apply (subst less_iff_Suc_add)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   547
  apply (auto simp add: add_mult_distrib)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   548
  done
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   549
  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
   550
    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
   551
  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
   552
    by (simp add: divmod_nat_rel_def)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   553
  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
   554
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   555
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   556
text {*
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   557
  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
   558
  means of @{const divmod_nat_rel}:
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   559
*}
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   560
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   561
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 36634
diff changeset
   562
  "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
   563
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   564
lemma divmod_nat_rel_divmod_nat:
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   565
  "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
   566
proof -
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   567
  from divmod_nat_rel_ex
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   568
    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
   569
  then show ?thesis
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   570
  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
   571
qed
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   572
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   573
lemma divmod_nat_unique:
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   574
  assumes "divmod_nat_rel m n qr" 
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   575
  shows "divmod_nat m n = qr"
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   576
  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
   577
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   578
instantiation nat :: semiring_div
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   579
begin
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   580
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   581
definition div_nat where
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   582
  "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
   583
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   584
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
   585
  "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
   586
  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
   587
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   588
definition mod_nat where
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   589
  "m mod n = snd (divmod_nat m n)"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   590
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   591
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
   592
  "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
   593
  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
   594
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   595
lemma divmod_nat_div_mod:
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   596
  "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
   597
  by (simp add: prod_eq_iff)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   598
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   599
lemma div_nat_unique:
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   600
  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
   601
  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
   602
  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
   603
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   604
lemma mod_nat_unique:
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   605
  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
   606
  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
   607
  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
   608
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   609
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
   610
  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
   611
47136
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   612
lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   613
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   614
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   615
lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   616
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   617
47137
7f5f0531cae6 shorten a proof
huffman
parents: 47136
diff changeset
   618
lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
7f5f0531cae6 shorten a proof
huffman
parents: 47136
diff changeset
   619
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   620
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   621
lemma divmod_nat_step:
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   622
  assumes "0 < n" and "n \<le> m"
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   623
  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
   624
proof (rule divmod_nat_unique)
47134
28c1db43d4d0 simplify some proofs
huffman
parents: 47108
diff changeset
   625
  have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"
28c1db43d4d0 simplify some proofs
huffman
parents: 47108
diff changeset
   626
    by (rule divmod_nat_rel)
28c1db43d4d0 simplify some proofs
huffman
parents: 47108
diff changeset
   627
  thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"
28c1db43d4d0 simplify some proofs
huffman
parents: 47108
diff changeset
   628
    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
   629
qed
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   630
26300
03def556e26e removed duplicate lemmas;
wenzelm
parents: 26100
diff changeset
   631
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
   632
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   633
lemma div_less [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   634
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   635
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   636
  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
   637
  using assms divmod_nat_base by (simp add: prod_eq_iff)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   638
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   639
lemma le_div_geq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   640
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   641
  assumes "0 < n" and "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   642
  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
   643
  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
   644
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   645
lemma mod_less [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   646
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   647
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   648
  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
   649
  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
   650
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   651
lemma le_mod_geq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   652
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   653
  assumes "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   654
  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
   655
  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
   656
47136
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   657
instance proof
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   658
  fix m n :: nat
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   659
  show "m div n * n + m mod n = m"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   660
    using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   661
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   662
  fix m n q :: nat
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   663
  assume "n \<noteq> 0"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   664
  then show "(q + m * n) div n = m + q div n"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   665
    by (induct m) (simp_all add: le_div_geq)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   666
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   667
  fix m n q :: nat
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   668
  assume "m \<noteq> 0"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   669
  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
   670
    unfolding divmod_nat_rel_def
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   671
    by (auto split: split_if_asm, simp_all add: algebra_simps)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   672
  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
   673
  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
   674
  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   675
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   676
  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
   677
    by (simp add: div_nat_def divmod_nat_zero)
47136
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   678
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   679
  fix n :: nat show "0 div n = 0"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   680
    by (simp add: div_nat_def divmod_nat_zero_left)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   681
qed
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   682
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   683
end
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   684
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
   685
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
   686
  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
   687
  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
   688
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   689
text {* Simproc for cancelling @{const div} and @{const mod} *}
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   690
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   691
ML {*
43594
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 41792
diff changeset
   692
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
   693
(
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   694
  val div_name = @{const_name div};
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   695
  val mod_name = @{const_name mod};
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   696
  val mk_binop = HOLogic.mk_binop;
48561
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
   697
  val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
   698
  val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
   699
  fun mk_sum [] = HOLogic.zero
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
   700
    | mk_sum [t] = t
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
   701
    | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
   702
  fun dest_sum tm =
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
   703
    if HOLogic.is_zero tm then []
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
   704
    else
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
   705
      (case try HOLogic.dest_Suc tm of
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
   706
        SOME t => HOLogic.Suc_zero :: dest_sum t
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
   707
      | NONE =>
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
   708
          (case try dest_plus tm of
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
   709
            SOME (t, u) => dest_sum t @ dest_sum u
12aa0cb2b447 move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents: 47268
diff changeset
   710
          | NONE => [tm]));
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   711
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   712
  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
   713
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   714
  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
   715
    (@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
   716
)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   717
*}
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   718
43594
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 41792
diff changeset
   719
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
   720
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   721
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   722
subsubsection {* Quotient *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   723
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   724
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
   725
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
   726
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   727
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
   728
by (simp add: div_geq)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   729
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   730
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
   731
by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   732
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   733
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
   734
by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   735
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   736
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   737
subsubsection {* Remainder *}
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   738
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   739
lemma mod_less_divisor [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   740
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   741
  assumes "n > 0"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   742
  shows "m mod n < (n::nat)"
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   743
  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
   744
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   745
lemma mod_less_eq_dividend [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   746
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   747
  shows "m mod n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   748
proof (rule add_leD2)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   749
  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
   750
  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
   751
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   752
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   753
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
   754
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
   755
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   756
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
   757
by (simp add: le_mod_geq)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   758
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   759
lemma mod_1 [simp]: "m mod Suc 0 = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   760
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
   761
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   762
(* 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
   763
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
   764
  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
   765
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   766
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   767
  apply (drule mod_less_divisor [where m = m])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   768
  apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   769
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   770
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   771
subsubsection {* Quotient and Remainder *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   772
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   773
lemma divmod_nat_rel_mult1_eq:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
   774
  "divmod_nat_rel b c (q, r)
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   775
   \<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
   776
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
   777
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   778
lemma div_mult1_eq:
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   779
  "(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
   780
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
   781
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   782
lemma divmod_nat_rel_add1_eq:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
   783
  "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
   784
   \<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
   785
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
   786
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   787
(*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
   788
lemma div_add1_eq:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   789
  "(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
   790
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
   791
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   792
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   793
  apply (cut_tac m = q and n = c in mod_less_divisor)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   794
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   795
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   796
  apply (simp add: add_mult_distrib2)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   797
  done
10559
d3fd54fc659b many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents: 10214
diff changeset
   798
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   799
lemma divmod_nat_rel_mult2_eq:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
   800
  "divmod_nat_rel a b (q, r)
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   801
   \<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
   802
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
   803
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   804
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
   805
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
   806
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   807
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
   808
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
   809
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   810
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   811
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
   812
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   813
lemma div_1 [simp]: "m div Suc 0 = m"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   814
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
   815
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   816
(* Monotonicity of div in first argument *)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   817
lemma div_le_mono [rule_format (no_asm)]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   818
    "\<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
   819
apply (case_tac "k=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   820
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
   821
apply (case_tac "n<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   822
(* 1  case n<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   823
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   824
(* 2  case n >= k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   825
apply (case_tac "m<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   826
(* 2.1  case m<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   827
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   828
(* 2.2  case m>=k *)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   829
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
   830
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   831
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   832
(* Antimonotonicity of div in second argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   833
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
   834
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   835
 prefer 2 apply simp
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   836
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
   837
apply (rename_tac "k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   838
apply (case_tac "k<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   839
apply (subgoal_tac "~ (k<m) ")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   840
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   841
apply (simp add: div_geq)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   842
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
   843
 prefer 2
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   844
 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
   845
apply (rule le_trans, simp)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   846
apply (simp)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   847
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   848
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   849
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
   850
apply (case_tac "n=0", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   851
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
   852
apply (rule div_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   853
apply (simp_all (no_asm_simp))
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   854
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   855
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   856
(* Similar for "less than" *)
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
   857
lemma div_less_dividend [simp]:
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
   858
  "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
   859
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
   860
apply (rename_tac "m")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   861
apply (case_tac "m<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   862
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   863
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   864
apply (simp add: div_geq)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   865
apply (case_tac "n<m")
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   866
 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
   867
  apply (rule impI less_trans_Suc)+
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   868
apply assumption
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   869
  apply (simp_all)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   870
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   871
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   872
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
   873
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
   874
apply (case_tac "n=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   875
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
   876
apply (case_tac "Suc (na) <n")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   877
(* case Suc(na) < n *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   878
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
   879
(* case n \<le> Suc(na) *)
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16733
diff changeset
   880
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   881
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
   882
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   883
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   884
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
   885
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
   886
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   887
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
   888
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   889
(*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
   890
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
   891
  apply (cut_tac a = m in mod_div_equality)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   892
  apply (simp only: add_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   893
  apply (blast intro: sym)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   894
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   895
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
   896
lemma split_div:
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   897
 "P(n div k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   898
 ((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
   899
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   900
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   901
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   902
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   903
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   904
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   905
    with P show ?Q by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   906
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   907
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   908
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   909
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   910
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   911
      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
   912
      show "P i"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   913
      proof (cases)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   914
        assume "i = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   915
        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
   916
      next
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   917
        assume "i \<noteq> 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   918
        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
   919
      qed
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
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   923
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   924
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   925
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   926
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   927
    with Q show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   928
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   929
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   930
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   931
    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
   932
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   933
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   934
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   935
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   936
lemma split_div_lemma:
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   937
  assumes "0 < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   938
  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
   939
proof
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   940
  assume ?rhs
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   941
  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
   942
  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
   943
  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
   944
  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
   945
  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
   946
  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
   947
  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
   948
  from A B show ?lhs ..
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   949
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   950
  assume P: ?lhs
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   951
  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
   952
    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
   953
  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
   954
  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
   955
  then show ?rhs by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   956
qed
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   957
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   958
theorem split_div':
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   959
  "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
   960
   (\<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
   961
  apply (case_tac "0 < n")
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   962
  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
   963
  apply simp_all
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   964
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   965
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   966
lemma split_mod:
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   967
 "P(n mod k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   968
 ((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
   969
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   970
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   971
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   972
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   973
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   974
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   975
    with P show ?Q by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   976
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   977
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   978
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   979
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   980
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   981
      assume "n = k*i + j" "j < k"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   982
      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
   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
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   986
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   987
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   988
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   989
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   990
    with Q show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   991
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   992
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   993
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   994
    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
   995
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   996
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   997
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   998
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   999
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1000
  using mod_div_equality [of m n] by arith
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1001
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1002
lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1003
  using mod_div_equality [of m n] by arith
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1004
(* FIXME: very similar to mult_div_cancel *)
22800
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1005
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1006
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1007
subsubsection {* An ``induction'' law for modulus arithmetic. *}
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1008
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1009
lemma mod_induct_0:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1010
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1011
  and base: "P i" and i: "i<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1012
  shows "P 0"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1013
proof (rule ccontr)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1014
  assume contra: "\<not>(P 0)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1015
  from i have p: "0<p" by simp
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1016
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1017
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1018
    fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1019
    show "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1020
    proof (induct k)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1021
      show "?A 0" by simp  -- "by contradiction"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1022
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1023
      fix n
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1024
      assume ih: "?A n"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1025
      show "?A (Suc n)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1026
      proof (clarsimp)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1027
        assume y: "P (p - Suc n)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1028
        have n: "Suc n < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1029
        proof (rule ccontr)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1030
          assume "\<not>(Suc n < p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1031
          hence "p - Suc n = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1032
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1033
          with y contra show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1034
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1035
        qed
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1036
        hence n2: "Suc (p - Suc n) = p-n" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1037
        from p have "p - Suc n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1038
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1039
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1040
        show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1041
        proof (cases "n=0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1042
          case True
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1043
          with z n2 contra show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1044
        next
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1045
          case False
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1046
          with p have "p-n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1047
          with z n2 False ih show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1048
        qed
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1049
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1050
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1051
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1052
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1053
  from i obtain k where "0<k \<and> i+k=p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1054
    by (blast dest: less_imp_add_positive)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1055
  hence "0<k \<and> i=p-k" by auto
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1056
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1057
  note base
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1058
  ultimately
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1059
  show "False" by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1060
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1061
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1062
lemma mod_induct:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1063
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1064
  and base: "P i" and i: "i<p" and j: "j<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1065
  shows "P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1066
proof -
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1067
  have "\<forall>j<p. P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1068
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1069
    fix j
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1070
    show "j<p \<longrightarrow> P j" (is "?A j")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1071
    proof (induct j)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1072
      from step base i show "?A 0"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1073
        by (auto elim: mod_induct_0)
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1074
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1075
      fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1076
      assume ih: "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1077
      show "?A (Suc k)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1078
      proof
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1079
        assume suc: "Suc k < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1080
        hence k: "k<p" by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1081
        with ih have "P k" ..
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1082
        with step k have "P (Suc k mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1083
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1084
        moreover
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1085
        from suc have "Suc k mod p = Suc k"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1086
          by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1087
        ultimately
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1088
        show "P (Suc k)" by simp
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1089
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1090
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1091
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1092
  with j show ?thesis by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1093
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1094
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1095
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1096
  by (simp add: numeral_2_eq_2 le_div_geq)
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1097
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1098
lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1099
  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
  1100
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1101
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
  1102
by (simp add: mult_2 [symmetric])
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1103
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1104
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
  1105
proof -
35815
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35673
diff changeset
  1106
  { 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
  1107
  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
  1108
  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
  1109
  then show ?thesis by auto
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1110
qed
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
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
  1113
    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
  1114
    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
  1115
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1116
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
  1117
by (simp add: Suc3_eq_add_3)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1118
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1119
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
  1120
by (simp add: Suc3_eq_add_3)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1121
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1122
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
  1123
by (simp add: Suc3_eq_add_3)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1124
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1125
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
  1126
by (simp add: Suc3_eq_add_3)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1127
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1128
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
  1129
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
  1130
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1131
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1132
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
  1133
apply (induct "m")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1134
apply (simp_all add: mod_Suc)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1135
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1136
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1137
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
  1138
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1139
lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1140
by (simp add: div_le_mono)
33361
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 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
  1143
by (cases n) simp_all
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1144
35815
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35673
diff changeset
  1145
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
  1146
proof -
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35673
diff changeset
  1147
  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
  1148
  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
  1149
qed
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1150
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1151
  (* Potential use of algebra : Equality modulo n*)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1152
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
  1153
by (simp add: mult_ac add_ac)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1154
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1155
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
  1156
proof -
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1157
  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
  1158
  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
  1159
  finally show ?thesis .
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1160
qed
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
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
  1163
apply (subst mod_Suc [of m]) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1164
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
  1165
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1166
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1167
lemma mod_2_not_eq_zero_eq_one_nat:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1168
  fixes n :: nat
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1169
  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
  1170
  by simp
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1171
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1172
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1173
subsection {* Division on @{typ int} *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1174
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1175
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
  1176
    --{*definition of quotient and remainder*}
47139
98bddfa0f717 extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents: 47138
diff changeset
  1177
  "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
  1178
    (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
  1179
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1180
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
  1181
    --{*for the division algorithm*}
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1182
    "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
  1183
                         else (2 * q, r))"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1184
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1185
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
  1186
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
  1187
  "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
  1188
     else adjust b (posDivAlg a (2 * b)))"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1189
by auto
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1190
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
  1191
  (auto simp add: mult_2)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1192
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1193
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
  1194
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
  1195
  "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
  1196
     else adjust b (negDivAlg a (2 * b)))"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1197
by auto
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1198
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
  1199
  (auto simp add: mult_2)
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
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
  1202
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1203
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
  1204
    --{*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
  1205
       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
  1206
       @{term negDivAlg} requires @{term "a<0"}.*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1207
  "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
  1208
                  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
  1209
                       else apsnd uminus (negDivAlg (-a) (-b))
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1210
               else 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1211
                  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
  1212
                  else apsnd uminus (posDivAlg (-a) (-b)))"
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1213
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1214
instantiation int :: Divides.div
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1215
begin
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1216
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1217
definition div_int where
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1218
  "a div b = fst (divmod_int a b)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1219
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1220
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
  1221
  "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
  1222
  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
  1223
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1224
definition mod_int where
46560
8e252a608765 remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents: 46552
diff changeset
  1225
  "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
  1226
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1227
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
  1228
  "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
  1229
  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
  1230
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1231
instance ..
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1232
3366
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
  1233
end
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1234
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1235
lemma divmod_int_mod_div:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1236
  "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
  1237
  by (simp add: prod_eq_iff)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1238
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1239
text{*
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1240
Here is the division algorithm in ML:
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
\begin{verbatim}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1243
    fun posDivAlg (a,b) =
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1244
      if a<b then (0,a)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1245
      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
  1246
               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
  1247
           end
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1248
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1249
    fun negDivAlg (a,b) =
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1250
      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
  1251
      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
  1252
               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
  1253
           end;
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1254
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1255
    fun negateSnd (q,r:int) = (q,~r);
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
    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
  1258
                          if b>0 then posDivAlg (a,b) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1259
                           else if a=0 then (0,0)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1260
                                else negateSnd (negDivAlg (~a,~b))
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1261
                       else 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1262
                          if 0<b then negDivAlg (a,b)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1263
                          else        negateSnd (posDivAlg (~a,~b));
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1264
\end{verbatim}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1265
*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1266
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1267
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1268
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
  1269
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1270
lemma unique_quotient_lemma:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1271
     "[| 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
  1272
      ==> q' \<le> (q::int)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1273
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
  1274
 prefer 2 apply (simp add: right_diff_distrib)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1275
apply (subgoal_tac "0 < b * (1 + q - q') ")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1276
apply (erule_tac [2] order_le_less_trans)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1277
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1278
apply (subgoal_tac "b * q' < b * (1 + q) ")
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1279
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1280
apply (simp add: mult_less_cancel_left)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1281
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1282
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1283
lemma unique_quotient_lemma_neg:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1284
     "[| 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
  1285
      ==> q \<le> (q'::int)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1286
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
  1287
    auto)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1288
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1289
lemma unique_quotient:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
  1290
     "[| 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
  1291
      ==> q = q'"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1292
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
  1293
apply (blast intro: order_antisym
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1294
             dest: order_eq_refl [THEN unique_quotient_lemma] 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1295
             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
  1296
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1297
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 unique_remainder:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
  1300
     "[| 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
  1301
      ==> r = r'"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1302
apply (subgoal_tac "q = q'")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1303
 apply (simp add: divmod_int_rel_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1304
apply (blast intro: unique_quotient)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1305
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1306
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1307
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1308
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
  1309
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1310
text{*And positive divisors*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1311
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1312
lemma adjust_eq [simp]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1313
     "adjust b (q, r) = 
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1314
      (let diff = r - b in  
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1315
        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
  1316
                     else (2*q, r))"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1317
  by (simp add: Let_def adjust_def)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1318
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1319
declare posDivAlg.simps [simp del]
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1320
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1321
text{*use with a simproc to avoid repeatedly proving the premise*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1322
lemma posDivAlg_eqn:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1323
     "0 < b ==>  
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1324
      posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1325
by (rule posDivAlg.simps [THEN trans], simp)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1326
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1327
text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1328
theorem posDivAlg_correct:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1329
  assumes "0 \<le> a" and "0 < b"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1330
  shows "divmod_int_rel a b (posDivAlg a b)"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1331
  using assms
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1332
  apply (induct a b rule: posDivAlg.induct)
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1333
  apply auto
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1334
  apply (simp add: divmod_int_rel_def)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1335
  apply (subst posDivAlg_eqn, simp add: distrib_left)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1336
  apply (case_tac "a < b")
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1337
  apply simp_all
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1338
  apply (erule splitE)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1339
  apply (auto simp add: distrib_left Let_def mult_ac mult_2_right)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1340
  done
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1341
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1342
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1343
subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1344
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1345
text{*And positive divisors*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1346
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1347
declare negDivAlg.simps [simp del]
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1348
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1349
text{*use with a simproc to avoid repeatedly proving the premise*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1350
lemma negDivAlg_eqn:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1351
     "0 < b ==>  
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1352
      negDivAlg a b =       
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1353
       (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1354
by (rule negDivAlg.simps [THEN trans], simp)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1355
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1356
(*Correctness of negDivAlg: it computes quotients correctly
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1357
  It doesn't work if a=0 because the 0/b equals 0, not -1*)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1358
lemma negDivAlg_correct:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1359
  assumes "a < 0" and "b > 0"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1360
  shows "divmod_int_rel a b (negDivAlg a b)"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1361
  using assms
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1362
  apply (induct a b rule: negDivAlg.induct)
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1363
  apply (auto simp add: linorder_not_le)
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1364
  apply (simp add: divmod_int_rel_def)
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1365
  apply (subst negDivAlg_eqn, assumption)
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1366
  apply (case_tac "a + b < (0\<Colon>int)")
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1367
  apply simp_all
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1368
  apply (erule splitE)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1369
  apply (auto simp add: distrib_left Let_def mult_ac mult_2_right)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1370
  done
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1371
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1372
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1373
subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1374
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1375
(*the case a=0*)
47139
98bddfa0f717 extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents: 47138
diff changeset
  1376
lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)"
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1377
by (auto simp add: divmod_int_rel_def linorder_neq_iff)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1378
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1379
lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1380
by (subst posDivAlg.simps, auto)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1381
47139
98bddfa0f717 extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents: 47138
diff changeset
  1382
lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"
98bddfa0f717 extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents: 47138
diff changeset
  1383
by (subst posDivAlg.simps, auto)
98bddfa0f717 extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents: 47138
diff changeset
  1384
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1385
lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1386
by (subst negDivAlg.simps, auto)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1387
46560
8e252a608765 remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents: 46552
diff changeset
  1388
lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"
47139
98bddfa0f717 extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents: 47138
diff changeset
  1389
by (auto simp add: divmod_int_rel_def)
98bddfa0f717 extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents: 47138
diff changeset
  1390
98bddfa0f717 extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents: 47138
diff changeset
  1391
lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"
98bddfa0f717 extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents: 47138
diff changeset
  1392
apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1393
by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1394
                    posDivAlg_correct negDivAlg_correct)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1395
47141
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1396
lemma divmod_int_unique:
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1397
  assumes "divmod_int_rel a b qr" 
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1398
  shows "divmod_int a b = qr"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1399
  using assms divmod_int_correct [of a b]
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1400
  using unique_quotient [of a b] unique_remainder [of a b]
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1401
  by (metis pair_collapse)
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1402
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1403
lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1404
  using divmod_int_correct by (simp add: divmod_int_mod_div)
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1405
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1406
lemma div_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a div b = q"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1407
  by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1408
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1409
lemma mod_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a mod b = r"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1410
  by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1411
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1412
instance int :: ring_div
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1413
proof
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1414
  fix a b :: int
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1415
  show "a div b * b + a mod b = a"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1416
    using divmod_int_rel_div_mod [of a b]
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1417
    unfolding divmod_int_rel_def by (simp add: mult_commute)
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1418
next
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1419
  fix a b c :: int
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1420
  assume "b \<noteq> 0"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1421
  hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1422
    using divmod_int_rel_div_mod [of a b]
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1423
    unfolding divmod_int_rel_def by (auto simp: algebra_simps)
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1424
  thus "(a + c * b) div b = c + a div b"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1425
    by (rule div_int_unique)
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1426
next
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1427
  fix a b c :: int
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1428
  assume "c \<noteq> 0"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1429
  hence "\<And>q r. divmod_int_rel a b (q, r)
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1430
    \<Longrightarrow> divmod_int_rel (c * a) (c * b) (q, c * r)"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1431
    unfolding divmod_int_rel_def
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1432
    by - (rule linorder_cases [of 0 b], auto simp: algebra_simps
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1433
      mult_less_0_iff zero_less_mult_iff mult_strict_right_mono
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1434
      mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1435
  hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1436
    using divmod_int_rel_div_mod [of a b] .
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1437
  thus "(c * a) div (c * b) = a div b"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1438
    by (rule div_int_unique)
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1439
next
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1440
  fix a :: int show "a div 0 = 0"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1441
    by (rule div_int_unique, simp add: divmod_int_rel_def)
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1442
next
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1443
  fix a :: int show "0 div a = 0"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1444
    by (rule div_int_unique, auto simp add: divmod_int_rel_def)
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1445
qed
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1446
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1447
text{*Basic laws about division and remainder*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1448
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1449
lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
47141
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1450
  by (fact mod_div_equality2 [symmetric])
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1451
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1452
text {* Tool setup *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1453
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1454
(* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1455
lemmas add_0s = add_0_left add_0_right
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1456
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1457
ML {*
43594
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 41792
diff changeset
  1458
structure Cancel_Div_Mod_Int = Cancel_Div_Mod
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1459
(
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1460
  val div_name = @{const_name div};
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1461
  val mod_name = @{const_name mod};
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1462
  val mk_binop = HOLogic.mk_binop;
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1463
  val mk_sum = Arith_Data.mk_sum HOLogic.intT;
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1464
  val dest_sum = Arith_Data.dest_sum;
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1465
47165
9344891b504b remove redundant lemmas
huffman
parents: 47164
diff changeset
  1466
  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1467
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1468
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1469
    (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
  1470
)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1471
*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1472
43594
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 41792
diff changeset
  1473
simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 41792
diff changeset
  1474
47141
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1475
lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1476
  using divmod_int_correct [of a b]
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1477
  by (auto simp add: divmod_int_rel_def prod_eq_iff)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1478
45607
16b4f5774621 eliminated obsolete "standard";
wenzelm
parents: 45530
diff changeset
  1479
lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
16b4f5774621 eliminated obsolete "standard";
wenzelm
parents: 45530
diff changeset
  1480
   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1481
47141
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1482
lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1483
  using divmod_int_correct [of a b]
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1484
  by (auto simp add: divmod_int_rel_def prod_eq_iff)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1485
45607
16b4f5774621 eliminated obsolete "standard";
wenzelm
parents: 45530
diff changeset
  1486
lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
16b4f5774621 eliminated obsolete "standard";
wenzelm
parents: 45530
diff changeset
  1487
   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1488
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1489
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1490
subsubsection {* General Properties of div and mod *}
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1491
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1492
lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1493
apply (rule div_int_unique)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1494
apply (auto simp add: divmod_int_rel_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1495
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1496
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1497
lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1498
apply (rule div_int_unique)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1499
apply (auto simp add: divmod_int_rel_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1500
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1501
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1502
lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1503
apply (rule div_int_unique)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1504
apply (auto simp add: divmod_int_rel_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1505
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1506
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1507
(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1508
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1509
lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1510
apply (rule_tac q = 0 in mod_int_unique)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1511
apply (auto simp add: divmod_int_rel_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1512
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1513
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1514
lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1515
apply (rule_tac q = 0 in mod_int_unique)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1516
apply (auto simp add: divmod_int_rel_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1517
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1518
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1519
lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1520
apply (rule_tac q = "-1" in mod_int_unique)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1521
apply (auto simp add: divmod_int_rel_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1522
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1523
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1524
text{*There is no @{text mod_neg_pos_trivial}.*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1525
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1526
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1527
subsubsection {* Laws for div and mod with Unary Minus *}
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1528
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1529
lemma zminus1_lemma:
47139
98bddfa0f717 extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents: 47138
diff changeset
  1530
     "divmod_int_rel a b (q, r) ==> b \<noteq> 0
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1531
      ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1532
                          if r=0 then 0 else b-r)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1533
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1534
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1535
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1536
lemma zdiv_zminus1_eq_if:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1537
     "b \<noteq> (0::int)  
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1538
      ==> (-a) div b =  
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1539
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1540
by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique])
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1541
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1542
lemma zmod_zminus1_eq_if:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1543
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1544
apply (case_tac "b = 0", simp)
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1545
apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique])
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1546
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1547
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1548
lemma zmod_zminus1_not_zero:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1549
  fixes k l :: int
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1550
  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1551
  unfolding zmod_zminus1_eq_if by auto
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1552
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1553
lemma zdiv_zminus2_eq_if:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1554
     "b \<noteq> (0::int)  
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1555
      ==> a div (-b) =  
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1556
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
47159
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
  1557
by (simp add: zdiv_zminus1_eq_if div_minus_right)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1558
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1559
lemma zmod_zminus2_eq_if:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1560
     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
47159
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
  1561
by (simp add: zmod_zminus1_eq_if mod_minus_right)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1562
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1563
lemma zmod_zminus2_not_zero:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1564
  fixes k l :: int
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1565
  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1566
  unfolding zmod_zminus2_eq_if by auto 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1567
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1568
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1569
subsubsection {* Computation of Division and Remainder *}
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1570
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1571
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1572
by (simp add: div_int_def divmod_int_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1573
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1574
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1575
by (simp add: mod_int_def divmod_int_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1576
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1577
text{*a positive, b positive *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1578
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1579
lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1580
by (simp add: div_int_def divmod_int_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1581
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1582
lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1583
by (simp add: mod_int_def divmod_int_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1584
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1585
text{*a negative, b positive *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1586
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1587
lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1588
by (simp add: div_int_def divmod_int_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1589
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1590
lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1591
by (simp add: mod_int_def divmod_int_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1592
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1593
text{*a positive, b negative *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1594
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1595
lemma div_pos_neg:
46560
8e252a608765 remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents: 46552
diff changeset
  1596
     "[| 0 < a;  b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1597
by (simp add: div_int_def divmod_int_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1598
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1599
lemma mod_pos_neg:
46560
8e252a608765 remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents: 46552
diff changeset
  1600
     "[| 0 < a;  b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1601
by (simp add: mod_int_def divmod_int_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1602
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1603
text{*a negative, b negative *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1604
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1605
lemma div_neg_neg:
46560
8e252a608765 remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents: 46552
diff changeset
  1606
     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1607
by (simp add: div_int_def divmod_int_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1608
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1609
lemma mod_neg_neg:
46560
8e252a608765 remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents: 46552
diff changeset
  1610
     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1611
by (simp add: mod_int_def divmod_int_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1612
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1613
text {*Simplify expresions in which div and mod combine numerical constants*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1614
45530
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1615
lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1616
  by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)
45530
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1617
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1618
lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1619
  by (rule div_int_unique [of a b q r],
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
  1620
    simp add: divmod_int_rel_def)
45530
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1621
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1622
lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1623
  by (rule mod_int_unique [of a b q r],
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
  1624
    simp add: divmod_int_rel_def)
45530
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1625
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1626
lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1627
  by (rule mod_int_unique [of a b q r],
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
  1628
    simp add: divmod_int_rel_def)
45530
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1629
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1630
(* simprocs adapted from HOL/ex/Binary.thy *)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1631
ML {*
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1632
local
45530
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1633
  val mk_number = HOLogic.mk_number HOLogic.intT
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1634
  val plus = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"}
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1635
  val times = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"}
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1636
  val zero = @{term "0 :: int"}
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1637
  val less = @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"}
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1638
  val le = @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"}
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1639
  val simps = @{thms arith_simps} @ @{thms rel_simps} @
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1640
    map (fn th => th RS sym) [@{thm numeral_1_eq_1}]
45530
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1641
  fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1642
    (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps simps))));
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1643
  fun binary_proc proc ss ct =
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1644
    (case Thm.term_of ct of
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1645
      _ $ t $ u =>
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1646
      (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1647
        SOME args => proc (Simplifier.the_context ss) args
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1648
      | NONE => NONE)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1649
    | _ => NONE);
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1650
in
45530
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1651
  fun divmod_proc posrule negrule =
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1652
    binary_proc (fn ctxt => fn ((a, t), (b, u)) =>
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1653
      if b = 0 then NONE else let
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1654
        val (q, r) = pairself mk_number (Integer.div_mod a b)
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1655
        val goal1 = HOLogic.mk_eq (t, plus $ (times $ u $ q) $ r)
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1656
        val (goal2, goal3, rule) = if b > 0
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1657
          then (le $ zero $ r, less $ r $ u, posrule RS eq_reflection)
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1658
          else (le $ r $ zero, less $ u $ r, negrule RS eq_reflection)
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1659
      in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1660
end
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1661
*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1662
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1663
simproc_setup binary_int_div
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1664
  ("numeral m div numeral n :: int" |
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1665
   "numeral m div neg_numeral n :: int" |
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1666
   "neg_numeral m div numeral n :: int" |
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1667
   "neg_numeral m div neg_numeral n :: int") =
45530
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1668
  {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1669
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1670
simproc_setup binary_int_mod
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1671
  ("numeral m mod numeral n :: int" |
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1672
   "numeral m mod neg_numeral n :: int" |
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1673
   "neg_numeral m mod numeral n :: int" |
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1674
   "neg_numeral m mod neg_numeral n :: int") =
45530
0c4853bb77bf rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents: 45231
diff changeset
  1675
  {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1676
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1677
lemmas posDivAlg_eqn_numeral [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1678
    posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1679
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1680
lemmas negDivAlg_eqn_numeral [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1681
    negDivAlg_eqn [of "numeral v" "neg_numeral w", OF zero_less_numeral] for v w
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1682
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1683
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1684
text{*Special-case simplification *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1685
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1686
(** The last remaining special cases for constant arithmetic:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1687
    1 div z and 1 mod z **)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1688
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1689
lemmas div_pos_pos_1_numeral [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1690
  div_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1691
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1692
lemmas div_pos_neg_1_numeral [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1693
  div_pos_neg [OF zero_less_one, of "neg_numeral w",
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1694
  OF neg_numeral_less_zero] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1695
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1696
lemmas mod_pos_pos_1_numeral [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1697
  mod_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1698
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1699
lemmas mod_pos_neg_1_numeral [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1700
  mod_pos_neg [OF zero_less_one, of "neg_numeral w",
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1701
  OF neg_numeral_less_zero] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1702
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1703
lemmas posDivAlg_eqn_1_numeral [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1704
    posDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1705
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1706
lemmas negDivAlg_eqn_1_numeral [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1707
    negDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1708
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1709
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1710
subsubsection {* Monotonicity in the First Argument (Dividend) *}
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1711
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1712
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1713
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1714
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1715
apply (rule unique_quotient_lemma)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1716
apply (erule subst)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1717
apply (erule subst, simp_all)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1718
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1719
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1720
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1721
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1722
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1723
apply (rule unique_quotient_lemma_neg)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1724
apply (erule subst)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1725
apply (erule subst, simp_all)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1726
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1727
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1728
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1729
subsubsection {* Monotonicity in the Second Argument (Divisor) *}
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1730
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1731
lemma q_pos_lemma:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1732
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1733
apply (subgoal_tac "0 < b'* (q' + 1) ")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1734
 apply (simp add: zero_less_mult_iff)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1735
apply (simp add: distrib_left)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1736
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1737
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1738
lemma zdiv_mono2_lemma:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1739
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1740
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1741
      ==> q \<le> (q'::int)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1742
apply (frule q_pos_lemma, assumption+) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1743
apply (subgoal_tac "b*q < b* (q' + 1) ")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1744
 apply (simp add: mult_less_cancel_left)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1745
apply (subgoal_tac "b*q = r' - r + b'*q'")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1746
 prefer 2 apply simp
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1747
apply (simp (no_asm_simp) add: distrib_left)
44766
d4d33a4d7548 avoid using legacy theorem names
huffman
parents: 43594
diff changeset
  1748
apply (subst add_commute, rule add_less_le_mono, arith)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1749
apply (rule mult_right_mono, auto)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1750
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1751
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1752
lemma zdiv_mono2:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1753
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1754
apply (subgoal_tac "b \<noteq> 0")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1755
 prefer 2 apply arith
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1756
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1757
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1758
apply (rule zdiv_mono2_lemma)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1759
apply (erule subst)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1760
apply (erule subst, simp_all)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1761
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1762
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1763
lemma q_neg_lemma:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1764
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1765
apply (subgoal_tac "b'*q' < 0")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1766
 apply (simp add: mult_less_0_iff, arith)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1767
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1768
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1769
lemma zdiv_mono2_neg_lemma:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1770
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1771
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1772
      ==> q' \<le> (q::int)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1773
apply (frule q_neg_lemma, assumption+) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1774
apply (subgoal_tac "b*q' < b* (q + 1) ")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1775
 apply (simp add: mult_less_cancel_left)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1776
apply (simp add: distrib_left)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1777
apply (subgoal_tac "b*q' \<le> b'*q'")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1778
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1779
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1780
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1781
lemma zdiv_mono2_neg:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1782
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1783
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1784
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1785
apply (rule zdiv_mono2_neg_lemma)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1786
apply (erule subst)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1787
apply (erule subst, simp_all)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1788
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1789
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1790
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1791
subsubsection {* More Algebraic Laws for div and mod *}
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1792
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1793
text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1794
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1795
lemma zmult1_lemma:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
  1796
     "[| divmod_int_rel b c (q, r) |]  
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1797
      ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1798
by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left mult_ac)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1799
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1800
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1801
apply (case_tac "c = 0", simp)
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1802
apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique])
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1803
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1804
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1805
text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1806
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1807
lemma zadd1_lemma:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
  1808
     "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br) |]  
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1809
      ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1810
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1811
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1812
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1813
lemma zdiv_zadd1_eq:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1814
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1815
apply (case_tac "c = 0", simp)
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1816
apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1817
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1818
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1819
lemma posDivAlg_div_mod:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1820
  assumes "k \<ge> 0"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1821
  and "l \<ge> 0"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1822
  shows "posDivAlg k l = (k div l, k mod l)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1823
proof (cases "l = 0")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1824
  case True then show ?thesis by (simp add: posDivAlg.simps)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1825
next
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1826
  case False with assms posDivAlg_correct
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1827
    have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1828
    by simp
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1829
  from div_int_unique [OF this] mod_int_unique [OF this]
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1830
  show ?thesis by simp
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1831
qed
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1832
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1833
lemma negDivAlg_div_mod:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1834
  assumes "k < 0"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1835
  and "l > 0"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1836
  shows "negDivAlg k l = (k div l, k mod l)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1837
proof -
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1838
  from assms have "l \<noteq> 0" by simp
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1839
  from assms negDivAlg_correct
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1840
    have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1841
    by simp
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1842
  from div_int_unique [OF this] mod_int_unique [OF this]
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1843
  show ?thesis by simp
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1844
qed
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1845
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1846
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1847
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1848
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1849
(* REVISIT: should this be generalized to all semiring_div types? *)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1850
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1851
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1852
lemma zmod_zdiv_equality':
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1853
  "(m\<Colon>int) mod n = m - (m div n) * n"
47141
02d6b816e4b3 move int::ring_div instance upward, simplify several proofs
huffman
parents: 47140
diff changeset
  1854
  using mod_div_equality [of m n] by arith
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1855
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1856
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1857
subsubsection {* Proving  @{term "a div (b*c) = (a div b) div c"} *}
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1858
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1859
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1860
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1861
  to cause particular problems.*)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1862
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1863
text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1864
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1865
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1866
apply (subgoal_tac "b * (c - q mod c) < r * 1")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1867
 apply (simp add: algebra_simps)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1868
apply (rule order_le_less_trans)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1869
 apply (erule_tac [2] mult_strict_right_mono)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1870
 apply (rule mult_left_mono_neg)
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35050
diff changeset
  1871
  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1872
 apply (simp)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1873
apply (simp)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1874
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1875
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1876
lemma zmult2_lemma_aux2:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1877
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1878
apply (subgoal_tac "b * (q mod c) \<le> 0")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1879
 apply arith
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1880
apply (simp add: mult_le_0_iff)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1881
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1882
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1883
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1884
apply (subgoal_tac "0 \<le> b * (q mod c) ")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1885
apply arith
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1886
apply (simp add: zero_le_mult_iff)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1887
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1888
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1889
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1890
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1891
 apply (simp add: right_diff_distrib)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1892
apply (rule order_less_le_trans)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1893
 apply (erule mult_strict_right_mono)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1894
 apply (rule_tac [2] mult_left_mono)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1895
  apply simp
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35050
diff changeset
  1896
 using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1897
apply simp
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1898
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1899
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
  1900
lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]  
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1901
      ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1902
by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1903
                   zero_less_mult_iff distrib_left [symmetric] 
47139
98bddfa0f717 extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents: 47138
diff changeset
  1904
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1905
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1906
lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1907
apply (case_tac "b = 0", simp)
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1908
apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique])
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1909
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1910
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1911
lemma zmod_zmult2_eq:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1912
     "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1913
apply (case_tac "b = 0", simp)
47140
97c3676c5c94 rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents: 47139
diff changeset
  1914
apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique])
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1915
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1916
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1917
lemma div_pos_geq:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1918
  fixes k l :: int
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1919
  assumes "0 < l" and "l \<le> k"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1920
  shows "k div l = (k - l) div l + 1"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1921
proof -
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1922
  have "k = (k - l) + l" by simp
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1923
  then obtain j where k: "k = j + l" ..
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1924
  with assms show ?thesis by simp
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1925
qed
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1926
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1927
lemma mod_pos_geq:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1928
  fixes k l :: int
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1929
  assumes "0 < l" and "l \<le> k"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1930
  shows "k mod l = (k - l) mod l"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1931
proof -
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1932
  have "k = (k - l) + l" by simp
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1933
  then obtain j where k: "k = j + l" ..
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1934
  with assms show ?thesis by simp
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1935
qed
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1936
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1937
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1938
subsubsection {* Splitting Rules for div and mod *}
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1939
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1940
text{*The proofs of the two lemmas below are essentially identical*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1941
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1942
lemma split_pos_lemma:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1943
 "0<k ==> 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1944
    P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1945
apply (rule iffI, clarify)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1946
 apply (erule_tac P="P ?x ?y" in rev_mp)  
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1947
 apply (subst mod_add_eq) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1948
 apply (subst zdiv_zadd1_eq) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1949
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1950
txt{*converse direction*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1951
apply (drule_tac x = "n div k" in spec) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1952
apply (drule_tac x = "n mod k" in spec, simp)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1953
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1954
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1955
lemma split_neg_lemma:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1956
 "k<0 ==>
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1957
    P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1958
apply (rule iffI, clarify)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1959
 apply (erule_tac P="P ?x ?y" in rev_mp)  
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1960
 apply (subst mod_add_eq) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1961
 apply (subst zdiv_zadd1_eq) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1962
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1963
txt{*converse direction*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1964
apply (drule_tac x = "n div k" in spec) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1965
apply (drule_tac x = "n mod k" in spec, simp)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1966
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1967
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1968
lemma split_zdiv:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1969
 "P(n div k :: int) =
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1970
  ((k = 0 --> P 0) & 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1971
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1972
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1973
apply (case_tac "k=0", simp)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1974
apply (simp only: linorder_neq_iff)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1975
apply (erule disjE) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1976
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1977
                      split_neg_lemma [of concl: "%x y. P x"])
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1978
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1979
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1980
lemma split_zmod:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1981
 "P(n mod k :: int) =
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1982
  ((k = 0 --> P n) & 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1983
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1984
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1985
apply (case_tac "k=0", simp)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1986
apply (simp only: linorder_neq_iff)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1987
apply (erule disjE) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1988
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1989
                      split_neg_lemma [of concl: "%x y. P y"])
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1990
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1991
33730
1755ca4ec022 Fixed splitting of div and mod on integers (split theorem differed from implementation).
webertj
parents: 33728
diff changeset
  1992
text {* Enable (lin)arith to deal with @{const div} and @{const mod}
1755ca4ec022 Fixed splitting of div and mod on integers (split theorem differed from implementation).
webertj
parents: 33728
diff changeset
  1993
  when these are applied to some constant that is of the form
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1994
  @{term "numeral k"}: *}
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1995
declare split_zdiv [of _ _ "numeral k", arith_split] for k
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  1996
declare split_zmod [of _ _ "numeral k", arith_split] for k
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1997
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  1998
47166
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  1999
subsubsection {* Computing @{text "div"} and @{text "mod"} with shifting *}
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2000
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2001
lemma pos_divmod_int_rel_mult_2:
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2002
  assumes "0 \<le> b"
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2003
  assumes "divmod_int_rel a b (q, r)"
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2004
  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2005
  using assms unfolding divmod_int_rel_def by auto
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2006
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2007
lemma neg_divmod_int_rel_mult_2:
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2008
  assumes "b \<le> 0"
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2009
  assumes "divmod_int_rel (a + 1) b (q, r)"
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2010
  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2011
  using assms unfolding divmod_int_rel_def by auto
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2012
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2013
text{*computing div by shifting *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2014
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2015
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
47166
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2016
  using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel_div_mod]
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2017
  by (rule div_int_unique)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2018
35815
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35673
diff changeset
  2019
lemma neg_zdiv_mult_2: 
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35673
diff changeset
  2020
  assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
47166
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2021
  using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel_div_mod]
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2022
  by (rule div_int_unique)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2023
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2024
(* FIXME: add rules for negative numerals *)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2025
lemma zdiv_numeral_Bit0 [simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2026
  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2027
    numeral v div (numeral w :: int)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2028
  unfolding numeral.simps unfolding mult_2 [symmetric]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2029
  by (rule div_mult_mult1, simp)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2030
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2031
lemma zdiv_numeral_Bit1 [simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2032
  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =  
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2033
    (numeral v div (numeral w :: int))"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2034
  unfolding numeral.simps
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2035
  unfolding mult_2 [symmetric] add_commute [of _ 1]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2036
  by (rule pos_zdiv_mult_2, simp)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2037
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2038
lemma pos_zmod_mult_2:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2039
  fixes a b :: int
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2040
  assumes "0 \<le> a"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2041
  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
47166
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2042
  using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2043
  by (rule mod_int_unique)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2044
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2045
lemma neg_zmod_mult_2:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2046
  fixes a b :: int
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2047
  assumes "a \<le> 0"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2048
  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
47166
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2049
  using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]
108bf76ca00c tuned proofs
huffman
parents: 47165
diff changeset
  2050
  by (rule mod_int_unique)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2051
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2052
(* FIXME: add rules for negative numerals *)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2053
lemma zmod_numeral_Bit0 [simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2054
  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =  
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2055
    (2::int) * (numeral v mod numeral w)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2056
  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2057
  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2058
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2059
lemma zmod_numeral_Bit1 [simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2060
  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2061
    2 * (numeral v mod numeral w) + (1::int)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2062
  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2063
  unfolding mult_2 [symmetric] add_commute [of _ 1]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2064
  by (rule pos_zmod_mult_2, simp)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2065
39489
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2066
lemma zdiv_eq_0_iff:
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2067
 "(i::int) div k = 0 \<longleftrightarrow> k=0 \<or> 0\<le>i \<and> i<k \<or> i\<le>0 \<and> k<i" (is "?L = ?R")
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2068
proof
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2069
  assume ?L
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2070
  have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2071
  with `?L` show ?R by blast
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2072
next
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2073
  assume ?R thus ?L
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2074
    by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2075
qed
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2076
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2077
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  2078
subsubsection {* Quotients of Signs *}
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2079
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2080
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2081
apply (subgoal_tac "a div b \<le> -1", force)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2082
apply (rule order_trans)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2083
apply (rule_tac a' = "-1" in zdiv_mono1)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2084
apply (auto simp add: div_eq_minus1)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2085
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2086
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2087
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2088
by (drule zdiv_mono1_neg, auto)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2089
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2090
lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2091
by (drule zdiv_mono1, auto)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2092
33804
39b494e8c055 added lemma
nipkow
parents: 33730
diff changeset
  2093
text{* Now for some equivalences of the form @{text"a div b >=< 0 \<longleftrightarrow> \<dots>"}
39b494e8c055 added lemma
nipkow
parents: 33730
diff changeset
  2094
conditional upon the sign of @{text a} or @{text b}. There are many more.
39b494e8c055 added lemma
nipkow
parents: 33730
diff changeset
  2095
They should all be simp rules unless that causes too much search. *}
39b494e8c055 added lemma
nipkow
parents: 33730
diff changeset
  2096
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2097
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2098
apply auto
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2099
apply (drule_tac [2] zdiv_mono1)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2100
apply (auto simp add: linorder_neq_iff)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2101
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2102
apply (blast intro: div_neg_pos_less0)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2103
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2104
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2105
lemma neg_imp_zdiv_nonneg_iff:
33804
39b494e8c055 added lemma
nipkow
parents: 33730
diff changeset
  2106
  "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
47159
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
  2107
apply (subst div_minus_minus [symmetric])
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2108
apply (subst pos_imp_zdiv_nonneg_iff, auto)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2109
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2110
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2111
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2112
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2113
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2114
39489
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2115
lemma pos_imp_zdiv_pos_iff:
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2116
  "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2117
using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2118
by arith
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2119
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2120
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2121
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2122
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2123
33804
39b494e8c055 added lemma
nipkow
parents: 33730
diff changeset
  2124
lemma nonneg1_imp_zdiv_pos_iff:
39b494e8c055 added lemma
nipkow
parents: 33730
diff changeset
  2125
  "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
39b494e8c055 added lemma
nipkow
parents: 33730
diff changeset
  2126
apply rule
39b494e8c055 added lemma
nipkow
parents: 33730
diff changeset
  2127
 apply rule
39b494e8c055 added lemma
nipkow
parents: 33730
diff changeset
  2128
  using div_pos_pos_trivial[of a b]apply arith
39b494e8c055 added lemma
nipkow
parents: 33730
diff changeset
  2129
 apply(cases "b=0")apply simp
39b494e8c055 added lemma
nipkow
parents: 33730
diff changeset
  2130
 using div_nonneg_neg_le0[of a b]apply arith
39b494e8c055 added lemma
nipkow
parents: 33730
diff changeset
  2131
using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
39b494e8c055 added lemma
nipkow
parents: 33730
diff changeset
  2132
done
39b494e8c055 added lemma
nipkow
parents: 33730
diff changeset
  2133
39489
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2134
lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2135
apply (rule split_zmod[THEN iffD2])
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44766
diff changeset
  2136
apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
39489
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2137
done
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2138
8bb7f32a3a08 added lemmas
nipkow
parents: 38715
diff changeset
  2139
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2140
subsubsection {* The Divides Relation *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2141
47268
262d96552e50 add simp rules for dvd on negative numerals
huffman
parents: 47255
diff changeset
  2142
lemma dvd_neg_numeral_left [simp]:
262d96552e50 add simp rules for dvd on negative numerals
huffman
parents: 47255
diff changeset
  2143
  fixes y :: "'a::comm_ring_1"
262d96552e50 add simp rules for dvd on negative numerals
huffman
parents: 47255
diff changeset
  2144
  shows "(neg_numeral k) dvd y \<longleftrightarrow> (numeral k) dvd y"
262d96552e50 add simp rules for dvd on negative numerals
huffman
parents: 47255
diff changeset
  2145
  unfolding neg_numeral_def minus_dvd_iff ..
262d96552e50 add simp rules for dvd on negative numerals
huffman
parents: 47255
diff changeset
  2146
262d96552e50 add simp rules for dvd on negative numerals
huffman
parents: 47255
diff changeset
  2147
lemma dvd_neg_numeral_right [simp]:
262d96552e50 add simp rules for dvd on negative numerals
huffman
parents: 47255
diff changeset
  2148
  fixes x :: "'a::comm_ring_1"
262d96552e50 add simp rules for dvd on negative numerals
huffman
parents: 47255
diff changeset
  2149
  shows "x dvd (neg_numeral k) \<longleftrightarrow> x dvd (numeral k)"
262d96552e50 add simp rules for dvd on negative numerals
huffman
parents: 47255
diff changeset
  2150
  unfolding neg_numeral_def dvd_minus_iff ..
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2151
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2152
lemmas dvd_eq_mod_eq_0_numeral [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2153
  dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2154
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2155
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2156
subsubsection {* Further properties *}
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2157
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2158
lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2159
  using zmod_zdiv_equality[where a="m" and b="n"]
47142
d64fa2ca54b8 remove redundant lemmas
huffman
parents: 47141
diff changeset
  2160
  by (simp add: algebra_simps) (* FIXME: generalize *)
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2161
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2162
lemma zdiv_int: "int (a div b) = (int a) div (int b)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2163
apply (subst split_div, auto)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2164
apply (subst split_zdiv, auto)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2165
apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2166
apply (auto simp add: divmod_int_rel_def of_nat_mult)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2167
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2168
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2169
lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2170
apply (subst split_mod, auto)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2171
apply (subst split_zmod, auto)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2172
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2173
       in unique_remainder)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2174
apply (auto simp add: divmod_int_rel_def of_nat_mult)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2175
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2176
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2177
lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2178
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2179
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2180
text{*Suggested by Matthias Daum*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2181
lemma int_power_div_base:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2182
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2183
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2184
 apply (erule ssubst)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2185
 apply (simp only: power_add)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2186
 apply simp_all
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2187
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2188
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2189
text {* by Brian Huffman *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2190
lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2191
by (rule mod_minus_eq [symmetric])
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2192
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2193
lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2194
by (rule mod_diff_left_eq [symmetric])
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2195
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2196
lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2197
by (rule mod_diff_right_eq [symmetric])
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2198
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2199
lemmas zmod_simps =
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2200
  mod_add_left_eq  [symmetric]
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2201
  mod_add_right_eq [symmetric]
47142
d64fa2ca54b8 remove redundant lemmas
huffman
parents: 47141
diff changeset
  2202
  mod_mult_right_eq[symmetric]
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2203
  mod_mult_left_eq [symmetric]
47164
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
  2204
  power_mod
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2205
  zminus_zmod zdiff_zmod_left zdiff_zmod_right
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2206
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2207
text {* Distributive laws for function @{text nat}. *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2208
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2209
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2210
apply (rule linorder_cases [of y 0])
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2211
apply (simp add: div_nonneg_neg_le0)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2212
apply simp
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2213
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2214
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2215
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2216
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2217
lemma nat_mod_distrib:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2218
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2219
apply (case_tac "y = 0", simp)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2220
apply (simp add: nat_eq_iff zmod_int)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2221
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2222
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2223
text  {* transfer setup *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2224
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2225
lemma transfer_nat_int_functions:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2226
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2227
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2228
  by (auto simp add: nat_div_distrib nat_mod_distrib)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2229
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2230
lemma transfer_nat_int_function_closures:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2231
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2232
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2233
  apply (cases "y = 0")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2234
  apply (auto simp add: pos_imp_zdiv_nonneg_iff)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2235
  apply (cases "y = 0")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2236
  apply auto
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2237
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2238
35644
d20cf282342e transfer: avoid camel case
haftmann
parents: 35367
diff changeset
  2239
declare transfer_morphism_nat_int [transfer add return:
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2240
  transfer_nat_int_functions
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2241
  transfer_nat_int_function_closures
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2242
]
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2243
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2244
lemma transfer_int_nat_functions:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2245
    "(int x) div (int y) = int (x div y)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2246
    "(int x) mod (int y) = int (x mod y)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2247
  by (auto simp add: zdiv_int zmod_int)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2248
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2249
lemma transfer_int_nat_function_closures:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2250
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2251
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2252
  by (simp_all only: is_nat_def transfer_nat_int_function_closures)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2253
35644
d20cf282342e transfer: avoid camel case
haftmann
parents: 35367
diff changeset
  2254
declare transfer_morphism_int_nat [transfer add return:
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2255
  transfer_int_nat_functions
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2256
  transfer_int_nat_function_closures
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2257
]
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2258
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2259
text{*Suggested by Matthias Daum*}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2260
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2261
apply (subgoal_tac "nat x div nat k < nat x")
34225
21c5405deb6b removed legacy asm_lr
nipkow
parents: 34126
diff changeset
  2262
 apply (simp add: nat_div_distrib [symmetric])
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2263
apply (rule Divides.div_less_dividend, simp_all)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2264
done
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2265
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2266
lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2267
proof
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2268
  assume H: "x mod n = y mod n"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2269
  hence "x mod n - y mod n = 0" by simp
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2270
  hence "(x mod n - y mod n) mod n = 0" by simp 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2271
  hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2272
  thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2273
next
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2274
  assume H: "n dvd x - y"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2275
  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2276
  hence "x = n*k + y" by simp
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2277
  hence "x mod n = (n*k + y) mod n" by simp
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2278
  thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2279
qed
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2280
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2281
lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2282
  shows "\<exists>q. x = y + n * q"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2283
proof-
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2284
  from xy have th: "int x - int y = int (x - y)" by simp 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2285
  from xyn have "int x mod int n = int y mod int n" 
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  2286
    by (simp add: zmod_int [symmetric])
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2287
  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2288
  hence "n dvd x - y" by (simp add: th zdvd_int)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2289
  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2290
qed
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2291
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2292
lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2293
  (is "?lhs = ?rhs")
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2294
proof
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2295
  assume H: "x mod n = y mod n"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2296
  {assume xy: "x \<le> y"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2297
    from H have th: "y mod n = x mod n" by simp
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2298
    from nat_mod_eq_lemma[OF th xy] have ?rhs 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2299
      apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2300
  moreover
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2301
  {assume xy: "y \<le> x"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2302
    from nat_mod_eq_lemma[OF H xy] have ?rhs 
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2303
      apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2304
  ultimately  show ?rhs using linear[of x y] by blast  
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2305
next
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2306
  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2307
  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2308
  thus  ?lhs by simp
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2309
qed
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2310
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2311
lemma div_nat_numeral [simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2312
  "(numeral v :: nat) div numeral v' = nat (numeral v div numeral v')"
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2313
  by (simp add: nat_div_distrib)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2314
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2315
lemma one_div_nat_numeral [simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2316
  "Suc 0 div numeral v' = nat (1 div numeral v')"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2317
  by (subst nat_div_distrib, simp_all)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2318
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2319
lemma mod_nat_numeral [simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2320
  "(numeral v :: nat) mod numeral v' = nat (numeral v mod numeral v')"
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2321
  by (simp add: nat_mod_distrib)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2322
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2323
lemma one_mod_nat_numeral [simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2324
  "Suc 0 mod numeral v' = nat (1 mod numeral v')"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2325
  by (subst nat_mod_distrib) simp_all
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2326
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2327
lemma mod_2_not_eq_zero_eq_one_int:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2328
  fixes k :: int
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2329
  shows "k mod 2 \<noteq> 0 \<longleftrightarrow> k mod 2 = 1"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2330
  by auto
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2331
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2332
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2333
subsubsection {* Tools setup *}
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2334
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2335
text {* Nitpick *}
34126
8a2c5d7aff51 polished Nitpick's binary integer support etc.;
blanchet
parents: 33804
diff changeset
  2336
41792
ff3cb0c418b7 renamed "nitpick\_def" to "nitpick_unfold" to reflect its new semantics
blanchet
parents: 41550
diff changeset
  2337
lemmas [nitpick_unfold] = dvd_eq_mod_eq_0 mod_div_equality' zmod_zdiv_equality'
34126
8a2c5d7aff51 polished Nitpick's binary integer support etc.;
blanchet
parents: 33804
diff changeset
  2338
35673
178caf872f95 weakend class ring_div; tuned
haftmann
parents: 35644
diff changeset
  2339
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2340
subsubsection {* Code generation *}
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2341
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2342
definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2343
  "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2344
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2345
lemma pdivmod_posDivAlg [code]:
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2346
  "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2347
by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2348
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2349
lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2350
  apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2351
    then pdivmod k l
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2352
    else (let (r, s) = pdivmod k l in
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46560
diff changeset
  2353
       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2354
proof -
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2355
  have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2356
  show ?thesis
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2357
    by (simp add: divmod_int_mod_div pdivmod_def)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2358
      (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2359
      zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2360
qed
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2362
lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2363
  apsnd ((op *) (sgn l)) (if sgn k = sgn l
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2364
    then pdivmod k l
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2365
    else (let (r, s) = pdivmod k l in
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2366
      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2367
proof -
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2368
  have "k \<noteq> 0 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 \<longleftrightarrow> sgn k = sgn l"
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2369
    by (auto simp add: not_less sgn_if)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2370
  then show ?thesis by (simp add: divmod_int_pdivmod)
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2371
qed
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2372
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33361
diff changeset
  2373
code_modulename SML
2bd12592c5e8 tuned code setup
haftmann
parents: 33361
diff changeset
  2374
  Divides Arith
2bd12592c5e8 tuned code setup
haftmann
parents: 33361
diff changeset
  2375
2bd12592c5e8 tuned code setup
haftmann
parents: 33361
diff changeset
  2376
code_modulename OCaml
2bd12592c5e8 tuned code setup
haftmann
parents: 33361
diff changeset
  2377
  Divides Arith
2bd12592c5e8 tuned code setup
haftmann
parents: 33361
diff changeset
  2378
2bd12592c5e8 tuned code setup
haftmann
parents: 33361
diff changeset
  2379
code_modulename Haskell
2bd12592c5e8 tuned code setup
haftmann
parents: 33361
diff changeset
  2380
  Divides Arith
2bd12592c5e8 tuned code setup
haftmann
parents: 33361
diff changeset
  2381
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
  2382
end