src/HOL/Divides.thy
author haftmann
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type class for generic division algorithm on numerals
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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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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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  assume "b mod a = 0"
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   136
  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
   137
  then have "b = a * (b div a)" unfolding mult_commute ..
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   138
  then have "\<exists>c. b = a * c" ..
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   139
  then show "a dvd b" unfolding dvd_def .
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   140
next
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   141
  assume "a dvd b"
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   142
  then have "\<exists>c. b = a * c" unfolding dvd_def .
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   143
  then obtain c where "b = a * c" ..
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   144
  then have "b mod a = a * c mod a" by simp
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   145
  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
   146
  then show "b mod a = 0" by simp
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   147
qed
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   148
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   149
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
   150
proof (cases "b = 0")
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   151
  assume "b = 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   152
  thus ?thesis by simp
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   153
next
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   154
  assume "b \<noteq> 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   155
  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
   156
    by (rule div_mult_self1 [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   157
  also have "\<dots> = a div b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   158
    by (simp only: mod_div_equality')
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   159
  also have "\<dots> = a div b + 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   160
    by simp
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   161
  finally show ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   162
    by (rule add_left_imp_eq)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   163
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   164
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   165
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
   166
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   167
  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
   168
    by (simp only: mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   169
  also have "\<dots> = a mod b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   170
    by (simp only: mod_div_equality')
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   171
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   172
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   173
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   174
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
   175
by (rule dvd_eq_mod_eq_0[THEN iffD1])
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   176
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   177
lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   178
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
   179
33274
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 32010
diff changeset
   180
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
   181
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
   182
30052
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   183
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
   184
apply (cases "a = 0")
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   185
 apply simp
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   186
apply (auto simp: dvd_def mult_assoc)
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   187
done
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   188
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   189
lemma div_dvd_div[simp]:
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   190
  "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
   191
apply (cases "a = 0")
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   192
 apply simp
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   193
apply (unfold dvd_def)
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   194
apply auto
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   195
 apply(blast intro:mult_assoc[symmetric])
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44766
diff changeset
   196
apply(fastforce simp add: mult_assoc)
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   197
done
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   198
30078
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   199
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
   200
  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
   201
   apply (simp add: mod_div_equality)
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   202
  apply (simp only: dvd_add dvd_mult)
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   203
  done
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   204
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   205
text {* Addition respects modular equivalence. *}
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   206
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   207
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
   208
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   209
  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
   210
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   211
  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
   212
    by (simp only: add_ac)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   213
  also have "\<dots> = (a mod c + b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   214
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   215
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   216
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   217
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   218
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
   219
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   220
  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
   221
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   222
  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
   223
    by (simp only: add_ac)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   224
  also have "\<dots> = (a + b mod c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   225
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   226
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   227
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   228
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   229
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
   230
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
   231
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   232
lemma mod_add_cong:
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   233
  assumes "a mod c = a' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   234
  assumes "b mod c = b' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   235
  shows "(a + b) mod c = (a' + b') mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   236
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   237
  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
   238
    unfolding assms ..
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   239
  thus ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   240
    by (simp only: mod_add_eq [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   241
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   242
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   243
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
   244
  \<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
   245
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
   246
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   247
text {* Multiplication respects modular equivalence. *}
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   248
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   249
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
   250
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   251
  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
   252
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   253
  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
   254
    by (simp only: algebra_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   255
  also have "\<dots> = (a mod c * b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   256
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   257
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   258
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   259
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   260
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
   261
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   262
  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
   263
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   264
  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
   265
    by (simp only: algebra_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   266
  also have "\<dots> = (a * (b mod c)) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   267
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   268
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   269
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   270
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   271
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
   272
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
   273
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   274
lemma mod_mult_cong:
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   275
  assumes "a mod c = a' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   276
  assumes "b mod c = b' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   277
  shows "(a * b) mod c = (a' * b') mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   278
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   279
  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
   280
    unfolding assms ..
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   281
  thus ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   282
    by (simp only: mod_mult_eq [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   283
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   284
47164
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   285
text {* Exponentiation respects modular equivalence. *}
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   286
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   287
lemma power_mod: "(a mod b)^n mod b = a^n mod b"
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   288
apply (induct n, simp_all)
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   289
apply (rule mod_mult_right_eq [THEN trans])
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   290
apply (simp (no_asm_simp))
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   291
apply (rule mod_mult_eq [symmetric])
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   292
done
6a4c479ba94f generalized lemma zpower_zmod
huffman
parents: 47163
diff changeset
   293
29404
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   294
lemma mod_mod_cancel:
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   295
  assumes "c dvd b"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   296
  shows "a mod b mod c = a mod c"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   297
proof -
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   298
  from `c dvd b` obtain k where "b = c * k"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   299
    by (rule dvdE)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   300
  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
   301
    by (simp only: `b = c * k`)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   302
  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
   303
    by (simp only: mod_mult_self1)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   304
  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
   305
    by (simp only: add_ac mult_ac)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   306
  also have "\<dots> = a mod c"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   307
    by (simp only: mod_div_equality)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   308
  finally show ?thesis .
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   309
qed
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   310
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   311
lemma div_mult_div_if_dvd:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   312
  "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
   313
  apply (cases "y = 0", simp)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   314
  apply (cases "z = 0", simp)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   315
  apply (auto elim!: dvdE simp add: algebra_simps)
30476
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   316
  apply (subst mult_assoc [symmetric])
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   317
  apply (simp add: no_zero_divisors)
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   318
  done
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   319
35367
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   320
lemma div_mult_swap:
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   321
  assumes "c dvd b"
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   322
  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
   323
proof -
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   324
  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
   325
    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
   326
  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
   327
qed
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   328
   
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   329
lemma div_mult_mult2 [simp]:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   330
  "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   331
  by (drule div_mult_mult1) (simp add: mult_commute)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   332
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   333
lemma div_mult_mult1_if [simp]:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   334
  "(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
   335
  by simp_all
30476
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   336
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   337
lemma mod_mult_mult1:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   338
  "(c * a) mod (c * b) = c * (a mod b)"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   339
proof (cases "c = 0")
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   340
  case True then show ?thesis by simp
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   341
next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   342
  case False
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   343
  from mod_div_equality
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   344
  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
   345
  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
   346
    = c * a + c * (a mod b)" by (simp add: algebra_simps)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   347
  with mod_div_equality show ?thesis by simp 
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   348
qed
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   349
  
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   350
lemma mod_mult_mult2:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   351
  "(a * c) mod (b * c) = (a mod b) * c"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   352
  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
   353
47159
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   354
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
   355
  by (fact mod_mult_mult2 [symmetric])
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   356
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   357
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
   358
  by (fact mod_mult_mult1 [symmetric])
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   359
31662
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   360
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
   361
  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
   362
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   363
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
   364
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
   365
31009
41fd307cab30 dropped reference to class recpower and lemma duplicate
haftmann
parents: 30934
diff changeset
   366
lemma div_power:
31661
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   367
  "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
30476
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   368
apply (induct n)
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   369
 apply simp
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   370
apply(simp add: div_mult_div_if_dvd dvd_power_same)
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   371
done
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   372
35367
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   373
lemma dvd_div_eq_mult:
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   374
  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
   375
  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
   376
proof
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   377
  assume "b = c * a"
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   378
  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
   379
next
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   380
  assume "b div a = c"
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   381
  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
   382
  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
   383
  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
   384
qed
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   385
   
45a193f0ed0c lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents: 35216
diff changeset
   386
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
   387
  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
   388
  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
   389
  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
   390
31661
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   391
end
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   392
35673
178caf872f95 weakend class ring_div; tuned
haftmann
parents: 35644
diff changeset
   393
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
   394
begin
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   395
36634
f9b43d197d16 a ring_div is a ring_1_no_zero_divisors
haftmann
parents: 35815
diff changeset
   396
subclass ring_1_no_zero_divisors ..
f9b43d197d16 a ring_div is a ring_1_no_zero_divisors
haftmann
parents: 35815
diff changeset
   397
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   398
text {* Negation respects modular equivalence. *}
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   399
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   400
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
   401
proof -
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   402
  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
   403
    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
   404
  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
   405
    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
   406
  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
   407
    by (rule mod_mult_self1)
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   408
  finally show ?thesis .
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   409
qed
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   410
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   411
lemma mod_minus_cong:
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   412
  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
   413
  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
   414
proof -
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   415
  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
   416
    unfolding assms ..
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   417
  thus ?thesis
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   418
    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
   419
qed
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   420
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   421
text {* Subtraction respects modular equivalence. *}
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
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
   424
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   425
  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
   426
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   427
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
   428
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   429
  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
   430
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   431
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
   432
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   433
  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
   434
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   435
lemma mod_diff_cong:
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   436
  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
   437
  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
   438
  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
   439
  unfolding diff_minus using assms
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   440
  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
   441
30180
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   442
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
   443
apply (case_tac "y = 0") apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   444
apply (auto simp add: dvd_def)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   445
apply (subgoal_tac "-(y * k) = y * - k")
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   446
 apply (erule ssubst)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   447
 apply (erule div_mult_self1_is_id)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   448
apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   449
done
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   450
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   451
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
   452
apply (case_tac "y = 0") apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   453
apply (auto simp add: dvd_def)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   454
apply (subgoal_tac "y * k = -y * -k")
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   455
 apply (erule ssubst)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   456
 apply (rule div_mult_self1_is_id)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   457
 apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   458
apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   459
done
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   460
47159
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   461
lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   462
  using div_mult_mult1 [of "- 1" a b]
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   463
  unfolding neg_equal_0_iff_equal by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   464
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   465
lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   466
  using mod_mult_mult1 [of "- 1" a b] by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   467
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   468
lemma div_minus_right: "a div (-b) = (-a) div b"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   469
  using div_minus_minus [of "-a" b] by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   470
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   471
lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   472
  using mod_minus_minus [of "-a" b] by simp
978c00c20a59 generalize some theorems about div/mod
huffman
parents: 47142
diff changeset
   473
47160
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   474
lemma div_minus1_right [simp]: "a div (-1) = -a"
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   475
  using div_minus_right [of a 1] by simp
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   476
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   477
lemma mod_minus1_right [simp]: "a mod (-1) = 0"
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   478
  using mod_minus_right [of a 1] by simp
8ada79014cb2 generalize more div/mod lemmas
huffman
parents: 47159
diff changeset
   479
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   480
end
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   481
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   482
53067
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   483
subsection {* Generic numeral division with a pragmatic type class *}
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   484
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   485
text {*
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   486
  The following type class contains everything necessary to formulate
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   487
  a division algorithm in ring structures with numerals, restricted
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   488
  to its positive segments.  This is its primary motiviation, and it
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   489
  could surely be formulated using a more fine-grained, more algebraic
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   490
  and less technical class hierarchy.
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   491
*}
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   492
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   493
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   494
class semiring_numeral_div = linordered_semidom + minus + semiring_div +
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   495
  assumes diff_invert_add1: "a + b = c \<Longrightarrow> a = c - b"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   496
    and le_add_diff_inverse2: "b \<le> a \<Longrightarrow> a - b + b = a"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   497
  assumes mult_div_cancel: "b * (a div b) = a - a mod b"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   498
    and div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   499
    and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   500
    and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   501
    and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   502
    and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   503
    and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   504
    and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   505
    and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   506
  assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   507
begin
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   508
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   509
lemma diff_zero [simp]:
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   510
  "a - 0 = a"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   511
  by (rule diff_invert_add1 [symmetric]) simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   512
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   513
lemma parity:
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   514
  "a mod 2 = 0 \<or> a mod 2 = 1"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   515
proof (rule ccontr)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   516
  assume "\<not> (a mod 2 = 0 \<or> a mod 2 = 1)"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   517
  then have "a mod 2 \<noteq> 0" and "a mod 2 \<noteq> 1" by simp_all
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   518
  have "0 < 2" by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   519
  with pos_mod_bound pos_mod_sign have "0 \<le> a mod 2" "a mod 2 < 2" by simp_all
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   520
  with `a mod 2 \<noteq> 0` have "0 < a mod 2" by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   521
  with discrete have "1 \<le> a mod 2" by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   522
  with `a mod 2 \<noteq> 1` have "1 < a mod 2" by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   523
  with discrete have "2 \<le> a mod 2" by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   524
  with `a mod 2 < 2` show False by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   525
qed
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   526
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   527
lemma divmod_digit_1:
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   528
  assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   529
  shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   530
    and "a mod (2 * b) - b = a mod b" (is "?Q")
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   531
proof -
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   532
  from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   533
    by (auto intro: trans)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   534
  with `0 < b` have "0 < a div b" by (auto intro: div_positive)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   535
  then have [simp]: "1 \<le> a div b" by (simp add: discrete)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   536
  with `0 < b` have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   537
  def w \<equiv> "a div b mod 2" with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   538
  have mod_w: "a mod (2 * b) = a mod b + b * w"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   539
    by (simp add: w_def mod_mult2_eq ac_simps)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   540
  from assms w_exhaust have "w = 1"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   541
    by (auto simp add: mod_w) (insert mod_less, auto)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   542
  with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   543
  have "2 * (a div (2 * b)) = a div b - w"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   544
    by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   545
  with `w = 1` have div: "2 * (a div (2 * b)) = a div b - 1" by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   546
  then show ?P and ?Q
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   547
    by (simp_all add: div mod diff_invert_add1 [symmetric] le_add_diff_inverse2)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   548
qed
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   549
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   550
lemma divmod_digit_0:
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   551
  assumes "0 < b" and "a mod (2 * b) < b"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   552
  shows "2 * (a div (2 * b)) = a div b" (is "?P")
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   553
    and "a mod (2 * b) = a mod b" (is "?Q")
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   554
proof -
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   555
  def w \<equiv> "a div b mod 2" with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   556
  have mod_w: "a mod (2 * b) = a mod b + b * w"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   557
    by (simp add: w_def mod_mult2_eq ac_simps)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   558
  moreover have "b \<le> a mod b + b"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   559
  proof -
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   560
    from `0 < b` pos_mod_sign have "0 \<le> a mod b" by blast
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   561
    then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   562
    then show ?thesis by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   563
  qed
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   564
  moreover note assms w_exhaust
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   565
  ultimately have "w = 0" by auto
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   566
  find_theorems "_ + ?b < _ + ?b"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   567
  with mod_w have mod: "a mod (2 * b) = a mod b" by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   568
  have "2 * (a div (2 * b)) = a div b - w"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   569
    by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   570
  with `w = 0` have div: "2 * (a div (2 * b)) = a div b" by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   571
  then show ?P and ?Q
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   572
    by (simp_all add: div mod)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   573
qed
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   574
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   575
definition divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   576
where
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   577
  "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   578
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   579
lemma fst_divmod [simp]:
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   580
  "fst (divmod m n) = numeral m div numeral n"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   581
  by (simp add: divmod_def)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   582
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   583
lemma snd_divmod [simp]:
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   584
  "snd (divmod m n) = numeral m mod numeral n"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   585
  by (simp add: divmod_def)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   586
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   587
definition divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   588
where
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   589
  "divmod_step l qr = (let (q, r) = qr
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   590
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   591
    else (2 * q, r))"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   592
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   593
text {*
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   594
  This is a formulation of one step (referring to one digit position)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   595
  in school-method division: compare the dividend at the current
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   596
  digit position with the remained from previous division steps
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   597
  and evaluate accordingly.
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   598
*}
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   599
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   600
lemma divmod_step_eq [code]:
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   601
  "divmod_step l (q, r) = (if numeral l \<le> r
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   602
    then (2 * q + 1, r - numeral l) else (2 * q, r))"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   603
  by (simp add: divmod_step_def)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   604
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   605
lemma divmod_step_simps [simp]:
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   606
  "r < numeral l \<Longrightarrow> divmod_step l (q, r) = (2 * q, r)"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   607
  "numeral l \<le> r \<Longrightarrow> divmod_step l (q, r) = (2 * q + 1, r - numeral l)"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   608
  by (auto simp add: divmod_step_eq not_le)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   609
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   610
text {*
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   611
  This is a formulation of school-method division.
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   612
  If the divisor is smaller than the dividend, terminate.
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   613
  If not, shift the dividend to the right until termination
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   614
  occurs and then reiterate single division steps in the
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   615
  opposite direction.
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   616
*}
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   617
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   618
lemma divmod_divmod_step [code]:
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   619
  "divmod m n = (if m < n then (0, numeral m)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   620
    else divmod_step n (divmod m (Num.Bit0 n)))"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   621
proof (cases "m < n")
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   622
  case True then have "numeral m < numeral n" by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   623
  then show ?thesis
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   624
    by (simp add: prod_eq_iff div_less mod_less)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   625
next
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   626
  case False
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   627
  have "divmod m n =
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   628
    divmod_step n (numeral m div (2 * numeral n),
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   629
      numeral m mod (2 * numeral n))"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   630
  proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   631
    case True
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   632
    with divmod_step_simps
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   633
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   634
        (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   635
        by blast
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   636
    moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   637
      have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   638
      and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   639
      by simp_all
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   640
    ultimately show ?thesis by (simp only: divmod_def)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   641
  next
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   642
    case False then have *: "numeral m mod (2 * numeral n) < numeral n"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   643
      by (simp add: not_le)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   644
    with divmod_step_simps
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   645
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   646
        (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   647
        by blast
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   648
    moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   649
      have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   650
      and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   651
      by (simp_all only: zero_less_numeral)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   652
    ultimately show ?thesis by (simp only: divmod_def)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   653
  qed
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   654
  then have "divmod m n =
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   655
    divmod_step n (numeral m div numeral (Num.Bit0 n),
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   656
      numeral m mod numeral (Num.Bit0 n))"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   657
    by (simp only: numeral.simps distrib mult_1) 
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   658
  then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   659
    by (simp add: divmod_def)
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   660
  with False show ?thesis by simp
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   661
qed
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   662
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   663
end
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   664
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   665
hide_fact (open) diff_invert_add1 le_add_diff_inverse2 diff_zero
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   666
  -- {* restore simple accesses for more general variants of theorems *}
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   667
ee0b7c2315d2 type class for generic division algorithm on numerals
haftmann
parents: 53066
diff changeset
   668
  
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   669
subsection {* Division on @{typ nat} *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   670
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   671
text {*
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   672
  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
   673
  of a characteristic relation with two input arguments
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   674
  @{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
   675
  @{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
   676
*}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   677
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   678
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
   679
  "divmod_nat_rel m n qr \<longleftrightarrow>
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   680
    m = fst qr * n + snd qr \<and>
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   681
      (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
   682
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   683
text {* @{const divmod_nat_rel} is total: *}
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   684
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   685
lemma divmod_nat_rel_ex:
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   686
  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
   687
proof (cases "n = 0")
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   688
  case True  with that show thesis
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   689
    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
   690
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   691
  case False
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   692
  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
   693
  proof (induct m)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   694
    case 0 with `n \<noteq> 0`
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   695
    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
   696
    then show ?case by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   697
  next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   698
    case (Suc m) then obtain q' r'
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   699
      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
   700
    then show ?case proof (cases "Suc r' < n")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   701
      case True
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   702
      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
   703
      with True show ?thesis by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   704
    next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   705
      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
   706
      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
   707
      ultimately have "n = Suc r'" by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   708
      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
   709
      with `n \<noteq> 0` show ?thesis by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   710
    qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   711
  qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   712
  with that show thesis
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   713
    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
   714
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   715
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   716
text {* @{const divmod_nat_rel} is injective: *}
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   717
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   718
lemma divmod_nat_rel_unique:
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   719
  assumes "divmod_nat_rel m n qr"
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   720
    and "divmod_nat_rel m n qr'"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   721
  shows "qr = qr'"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   722
proof (cases "n = 0")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   723
  case True with assms show ?thesis
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   724
    by (cases qr, cases qr')
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   725
      (simp add: divmod_nat_rel_def)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   726
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   727
  case False
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   728
  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
   729
  apply (rule leI)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   730
  apply (subst less_iff_Suc_add)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   731
  apply (auto simp add: add_mult_distrib)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   732
  done
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   733
  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
   734
    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
   735
  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
   736
    by (simp add: divmod_nat_rel_def)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   737
  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
   738
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   739
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   740
text {*
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   741
  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
   742
  means of @{const divmod_nat_rel}:
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   743
*}
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   744
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   745
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 36634
diff changeset
   746
  "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
   747
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   748
lemma divmod_nat_rel_divmod_nat:
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   749
  "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
   750
proof -
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   751
  from divmod_nat_rel_ex
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   752
    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
   753
  then show ?thesis
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   754
  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
   755
qed
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   756
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   757
lemma divmod_nat_unique:
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   758
  assumes "divmod_nat_rel m n qr" 
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   759
  shows "divmod_nat m n = qr"
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   760
  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
   761
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   762
instantiation nat :: semiring_div
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   763
begin
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   764
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   765
definition div_nat where
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   766
  "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
   767
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   768
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
   769
  "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
   770
  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
   771
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   772
definition mod_nat where
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   773
  "m mod n = snd (divmod_nat m n)"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   774
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
   775
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
   776
  "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
   777
  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
   778
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   779
lemma divmod_nat_div_mod:
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   780
  "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
   781
  by (simp add: prod_eq_iff)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   782
47135
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   783
lemma div_nat_unique:
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   784
  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
   785
  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
   786
  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
   787
fb67b596067f rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents: 47134
diff changeset
   788
lemma mod_nat_unique:
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   789
  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
   790
  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
   791
  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
   792
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   793
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
   794
  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
   795
47136
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   796
lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   797
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   798
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   799
lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   800
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   801
47137
7f5f0531cae6 shorten a proof
huffman
parents: 47136
diff changeset
   802
lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
7f5f0531cae6 shorten a proof
huffman
parents: 47136
diff changeset
   803
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   804
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   805
lemma divmod_nat_step:
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   806
  assumes "0 < n" and "n \<le> m"
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   807
  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
   808
proof (rule divmod_nat_unique)
47134
28c1db43d4d0 simplify some proofs
huffman
parents: 47108
diff changeset
   809
  have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"
28c1db43d4d0 simplify some proofs
huffman
parents: 47108
diff changeset
   810
    by (rule divmod_nat_rel)
28c1db43d4d0 simplify some proofs
huffman
parents: 47108
diff changeset
   811
  thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"
28c1db43d4d0 simplify some proofs
huffman
parents: 47108
diff changeset
   812
    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
   813
qed
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   814
26300
03def556e26e removed duplicate lemmas;
wenzelm
parents: 26100
diff changeset
   815
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
   816
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   817
lemma div_less [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   818
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   819
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   820
  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
   821
  using assms divmod_nat_base by (simp add: prod_eq_iff)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   822
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   823
lemma le_div_geq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   824
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   825
  assumes "0 < n" and "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   826
  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
   827
  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
   828
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   829
lemma mod_less [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   830
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   831
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   832
  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
   833
  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
   834
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   835
lemma le_mod_geq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   836
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   837
  assumes "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   838
  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
   839
  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
   840
47136
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   841
instance proof
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   842
  fix m n :: nat
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   843
  show "m div n * n + m mod n = m"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   844
    using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   845
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   846
  fix m n q :: nat
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   847
  assume "n \<noteq> 0"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   848
  then show "(q + m * n) div n = m + q div n"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   849
    by (induct m) (simp_all add: le_div_geq)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   850
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   851
  fix m n q :: nat
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   852
  assume "m \<noteq> 0"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   853
  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
   854
    unfolding divmod_nat_rel_def
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   855
    by (auto split: split_if_asm, simp_all add: algebra_simps)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   856
  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
   857
  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
   858
  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   859
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   860
  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
   861
    by (simp add: div_nat_def divmod_nat_zero)
47136
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   862
next
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   863
  fix n :: nat show "0 div n = 0"
5b6c5641498a simplify some proofs
huffman
parents: 47135
diff changeset
   864
    by (simp add: div_nat_def divmod_nat_zero_left)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   865
qed
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   866
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   867
end
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   868
33361
1f18de40b43f combined former theories Divides and IntDiv to one theory Divides
haftmann
parents: 33340
diff changeset
   869
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
   870
  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
   871
  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
   872
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   873
text {* Simproc for cancelling @{const div} and @{const mod} *}
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   874
51299
30b014246e21 proper place for cancel_div_mod.ML (see also ee729dbd1b7f and ec7f10155389);
wenzelm
parents: 51173
diff changeset
   875
ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"
30b014246e21 proper place for cancel_div_mod.ML (see also ee729dbd1b7f and ec7f10155389);
wenzelm
parents: 51173
diff changeset
   876
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   877
ML {*
43594
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 41792
diff changeset
   878
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
   879
(
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   880
  val div_name = @{const_name div};
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   881
  val mod_name = @{const_name mod};
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   882
  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
   883
  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
   884
  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
   885
  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
   886
    | 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
   887
    | 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
   888
  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
   889
    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
   890
    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
   891
      (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
   892
        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
   893
      | 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
   894
          (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
   895
            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
   896
          | NONE => [tm]));
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   897
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   898
  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
   899
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   900
  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
   901
    (@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39489
diff changeset
   902
)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   903
*}
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   904
43594
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 41792
diff changeset
   905
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
   906
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   907
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   908
subsubsection {* Quotient *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   909
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   910
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
   911
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
   912
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   913
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
   914
by (simp add: div_geq)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   915
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   916
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
   917
by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   918
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   919
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
   920
by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   921
53066
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
   922
lemma div_positive:
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
   923
  fixes m n :: nat
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
   924
  assumes "n > 0"
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
   925
  assumes "m \<ge> n"
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
   926
  shows "m div n > 0"
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
   927
proof -
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
   928
  from `m \<ge> n` obtain q where "m = n + q"
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
   929
    by (auto simp add: le_iff_add)
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
   930
  with `n > 0` show ?thesis by simp
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
   931
qed
1f61a923c2d6 added lemma
haftmann
parents: 52435
diff changeset
   932
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   933
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   934
subsubsection {* Remainder *}
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   935
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   936
lemma mod_less_divisor [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   937
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   938
  assumes "n > 0"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   939
  shows "m mod n < (n::nat)"
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   940
  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
   941
51173
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50422
diff changeset
   942
lemma mod_Suc_le_divisor [simp]:
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50422
diff changeset
   943
  "m mod Suc n \<le> n"
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50422
diff changeset
   944
  using mod_less_divisor [of "Suc n" m] by arith
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50422
diff changeset
   945
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   946
lemma mod_less_eq_dividend [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   947
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   948
  shows "m mod n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   949
proof (rule add_leD2)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   950
  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
   951
  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
   952
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   953
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   954
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
   955
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
   956
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   957
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
   958
by (simp add: le_mod_geq)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   959
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   960
lemma mod_1 [simp]: "m mod Suc 0 = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   961
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
   962
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   963
(* 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
   964
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
   965
  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
   966
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   967
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   968
  apply (drule mod_less_divisor [where m = m])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   969
  apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   970
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   971
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   972
subsubsection {* Quotient and Remainder *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   973
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   974
lemma divmod_nat_rel_mult1_eq:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
   975
  "divmod_nat_rel b c (q, r)
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   976
   \<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
   977
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
   978
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   979
lemma div_mult1_eq:
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   980
  "(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
   981
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
   982
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
   983
lemma divmod_nat_rel_add1_eq:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
   984
  "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
   985
   \<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
   986
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
   987
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   988
(*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
   989
lemma div_add1_eq:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   990
  "(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
   991
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
   992
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   993
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   994
  apply (cut_tac m = q and n = c in mod_less_divisor)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   995
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   996
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   997
  apply (simp add: add_mult_distrib2)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   998
  done
10559
d3fd54fc659b many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents: 10214
diff changeset
   999
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
  1000
lemma divmod_nat_rel_mult2_eq:
46552
5d33a3269029 removing some unnecessary premises from Divides
bulwahn
parents: 46551
diff changeset
  1001
  "divmod_nat_rel a b (q, r)
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
  1002
   \<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
  1003
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
  1004
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1005
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
  1006
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
  1007
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1008
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
  1009
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
  1010
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1011
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1012
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
  1013
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1014
lemma div_1 [simp]: "m div Suc 0 = m"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
  1015
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
  1016
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1017
(* Monotonicity of div in first argument *)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
  1018
lemma div_le_mono [rule_format (no_asm)]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1019
    "\<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
  1020
apply (case_tac "k=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
  1021
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
  1022
apply (case_tac "n<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1023
(* 1  case n<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1024
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1025
(* 2  case n >= k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1026
apply (case_tac "m<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1027
(* 2.1  case m<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1028
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1029
(* 2.2  case m>=k *)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
  1030
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
  1031
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1032
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1033
(* Antimonotonicity of div in second argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1034
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
  1035
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1036
 prefer 2 apply simp
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
  1037
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
  1038
apply (rename_tac "k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1039
apply (case_tac "k<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1040
apply (subgoal_tac "~ (k<m) ")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1041
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1042
apply (simp add: div_geq)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
  1043
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
  1044
 prefer 2
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1045
 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
  1046
apply (rule le_trans, simp)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
  1047
apply (simp)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1048
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1049
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1050
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
  1051
apply (case_tac "n=0", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1052
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
  1053
apply (rule div_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1054
apply (simp_all (no_asm_simp))
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1055
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1056
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1057
(* Similar for "less than" *)
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1058
lemma div_less_dividend [simp]:
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1059
  "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1060
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
  1061
apply (rename_tac "m")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1062
apply (case_tac "m<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1063
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1064
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1065
apply (simp add: div_geq)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1066
apply (case_tac "n<m")
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
  1067
 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
  1068
  apply (rule impI less_trans_Suc)+
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1069
apply assumption
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
  1070
  apply (simp_all)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1071
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1072
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1073
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
  1074
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
  1075
apply (case_tac "n=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
  1076
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
  1077
apply (case_tac "Suc (na) <n")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1078
(* case Suc(na) < n *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1079
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
  1080
(* case n \<le> Suc(na) *)
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16733
diff changeset
  1081
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
  1082
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
  1083
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1084
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1085
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
  1086
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
  1087
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1088
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
  1089
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1090
(*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
  1091
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
  1092
  apply (cut_tac a = m in mod_div_equality)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1093
  apply (simp only: add_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1094
  apply (blast intro: sym)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1095
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1096
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1097
lemma split_div:
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1098
 "P(n div k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1099
 ((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
  1100
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1101
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1102
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1103
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1104
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1105
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
  1106
    with P show ?Q by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1107
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1108
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1109
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1110
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1111
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1112
      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
  1113
      show "P i"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1114
      proof (cases)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1115
        assume "i = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1116
        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
  1117
      next
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1118
        assume "i \<noteq> 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1119
        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
  1120
      qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1121
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1122
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1123
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1124
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1125
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1126
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1127
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
  1128
    with Q show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1129
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1130
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1131
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1132
    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
  1133
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1134
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1135
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1136
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1137
lemma split_div_lemma:
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1138
  assumes "0 < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1139
  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
  1140
proof
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1141
  assume ?rhs
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1142
  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
  1143
  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
  1144
  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
  1145
  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
  1146
  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
  1147
  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
  1148
  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
  1149
  from A B show ?lhs ..
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1150
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1151
  assume P: ?lhs
33340
a165b97f3658 moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents: 33318
diff changeset
  1152
  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
  1153
    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
  1154
  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
  1155
  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
  1156
  then show ?rhs by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1157
qed
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1158
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1159
theorem split_div':
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1160
  "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
  1161
   (\<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
  1162
  apply (case_tac "0 < n")
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1163
  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
  1164
  apply simp_all
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1165
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1166
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1167
lemma split_mod:
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1168
 "P(n mod k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1169
 ((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
  1170
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1171
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1172
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1173
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1174
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1175
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
  1176
    with P show ?Q by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1177
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1178
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1179
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1180
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1181
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1182
      assume "n = k*i + j" "j < k"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1183
      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
  1184
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1185
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1186
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1187
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1188
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1189
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1190
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
  1191
    with Q show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1192
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1193
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1194
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1195
    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
  1196
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1197
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1198
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1199
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1200
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1201
  using mod_div_equality [of m n] by arith
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1202
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1203
lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1204
  using mod_div_equality [of m n] by arith
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1205
(* FIXME: very similar to mult_div_cancel *)
22800
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1206
52398
656e5e171f19 added lemma
noschinl
parents: 51717
diff changeset
  1207
lemma div_eq_dividend_iff: "a \<noteq> 0 \<Longrightarrow> (a :: nat) div b = a \<longleftrightarrow> b = 1"
656e5e171f19 added lemma
noschinl
parents: 51717
diff changeset
  1208
  apply rule
656e5e171f19 added lemma
noschinl
parents: 51717
diff changeset
  1209
  apply (cases "b = 0")
656e5e171f19 added lemma
noschinl
parents: 51717
diff changeset
  1210
  apply simp_all
656e5e171f19 added lemma
noschinl
parents: 51717
diff changeset
  1211
  apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)
656e5e171f19 added lemma
noschinl
parents: 51717
diff changeset
  1212
  done
656e5e171f19 added lemma
noschinl
parents: 51717
diff changeset
  1213
22800
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1214
46551
866bce5442a3 simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents: 46026
diff changeset
  1215
subsubsection {* An ``induction'' law for modulus arithmetic. *}
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1216
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1217
lemma mod_induct_0:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1218
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1219
  and base: "P i" and i: "i<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1220
  shows "P 0"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1221
proof (rule ccontr)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1222
  assume contra: "\<not>(P 0)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1223
  from i have p: "0<p" by simp
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1224
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1225
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1226
    fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1227
    show "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1228
    proof (induct k)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1229
      show "?A 0" by simp  -- "by contradiction"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1230
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1231
      fix n
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1232
      assume ih: "?A n"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1233
      show "?A (Suc n)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1234
      proof (clarsimp)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1235
        assume y: "P (p - Suc n)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1236
        have n: "Suc n < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1237
        proof (rule ccontr)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1238
          assume "\<not>(Suc n < p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1239
          hence "p - Suc n = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1240
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1241
          with y contra show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1242
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1243
        qed
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1244
        hence n2: "Suc (p - Suc n) = p-n" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1245
        from p have "p - Suc n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1246
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1247
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1248
        show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1249
        proof (cases "n=0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1250
          case True
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1251
          with z n2 contra show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1252
        next
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1253
          case False
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1254
          with p have "p-n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1255
          with z n2 False ih show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1256
        qed
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1257
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1258
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1259
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1260
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1261
  from i obtain k where "0<k \<and> i+k=p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1262
    by (blast dest: less_imp_add_positive)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1263
  hence "0<k \<and> i=p-k" by auto
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1264
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1265
  note base
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1266
  ultimately
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1267
  show "False" by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1268
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1269
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1270
lemma mod_induct:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1271
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1272
  and base: "P i" and i: "i<p" and j: "j<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1273
  shows "P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1274
proof -
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1275
  have "\<forall>j<p. P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1276
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1277
    fix j
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1278
    show "j<p \<longrightarrow> P j" (is "?A j")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1279
    proof (induct j)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1280
      from step base i show "?A 0"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1281
        by (auto elim: mod_induct_0)
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1282
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1283
      fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1284
      assume ih: "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1285
      show "?A (Suc k)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1286
      proof
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1287
        assume suc: "Suc k < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1288
        hence k: "k<p" by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1289
        with ih have "P k" ..
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1290
        with step k have "P (Suc k mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1291
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1292
        moreover
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1293
        from suc have "Suc k mod p = Suc k"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1294
          by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1295
        ultimately
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1296
        show "P (Suc k)" by simp
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1297
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1298
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1299
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1300
  with j show ?thesis by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1301
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1302
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1303
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
47138
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1304
  by (simp add: numeral_2_eq_2 le_div_geq)
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1305
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1306
lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"
f8cf96545eed tuned proofs
huffman
parents: 47137
diff changeset
  1307
  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
  1308
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1309
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