src/HOL/Divides.thy
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(*  Title:      HOL/Divides.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1999  University of Cambridge
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*)
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header {* The division operators div and mod *}
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theory Divides
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imports Nat_Numeral Nat_Transfer
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uses
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  "~~/src/Provers/Arith/assoc_fold.ML"
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  "~~/src/Provers/Arith/cancel_numerals.ML"
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  "~~/src/Provers/Arith/combine_numerals.ML"
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  "~~/src/Provers/Arith/cancel_numeral_factor.ML"
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  "~~/src/Provers/Arith/extract_common_term.ML"
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  ("Tools/numeral_simprocs.ML")
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  ("Tools/nat_numeral_simprocs.ML")
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  "~~/src/Provers/Arith/cancel_div_mod.ML"
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begin
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subsection {* Syntactic division operations *}
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class div = dvd +
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  fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
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    and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
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subsection {* Abstract division in commutative semirings. *}
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class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +
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  assumes mod_div_equality: "a div b * b + a mod b = a"
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    and div_by_0 [simp]: "a div 0 = 0"
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    and div_0 [simp]: "0 div a = 0"
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    and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
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    and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
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begin
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text {* @{const div} and @{const mod} *}
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lemma mod_div_equality2: "b * (a div b) + a mod b = a"
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  unfolding mult_commute [of b]
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  by (rule mod_div_equality)
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lemma mod_div_equality': "a mod b + a div b * b = a"
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  using mod_div_equality [of a b]
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  by (simp only: add_ac)
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lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
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  by (simp add: mod_div_equality)
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lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
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  by (simp add: mod_div_equality2)
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lemma mod_by_0 [simp]: "a mod 0 = a"
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  using mod_div_equality [of a zero] by simp
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lemma mod_0 [simp]: "0 mod a = 0"
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  using mod_div_equality [of zero a] div_0 by simp
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lemma div_mult_self2 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(a + b * c) div b = c + a div b"
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  using assms div_mult_self1 [of b a c] by (simp add: mult_commute)
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lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
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proof (cases "b = 0")
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  case True then show ?thesis by simp
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next
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  case False
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  have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
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    by (simp add: mod_div_equality)
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  also from False div_mult_self1 [of b a c] have
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    "\<dots> = (c + a div b) * b + (a + c * b) mod b"
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      by (simp add: algebra_simps)
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  finally have "a = a div b * b + (a + c * b) mod b"
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    by (simp add: add_commute [of a] add_assoc left_distrib)
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  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
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    by (simp add: mod_div_equality)
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  then show ?thesis by simp
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qed
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lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"
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  by (simp add: mult_commute [of b])
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lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
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  using div_mult_self2 [of b 0 a] by simp
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lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
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  using div_mult_self1 [of b 0 a] by simp
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lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"
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  using mod_mult_self2 [of 0 b a] by simp
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lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"
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  using mod_mult_self1 [of 0 a b] by simp
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lemma div_by_1 [simp]: "a div 1 = a"
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  using div_mult_self2_is_id [of 1 a] zero_neq_one by simp
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lemma mod_by_1 [simp]: "a mod 1 = 0"
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proof -
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  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
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  then have "a + a mod 1 = a + 0" by simp
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  then show ?thesis by (rule add_left_imp_eq)
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qed
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lemma mod_self [simp]: "a mod a = 0"
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  using mod_mult_self2_is_0 [of 1] by simp
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lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
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  using div_mult_self2_is_id [of _ 1] by simp
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lemma div_add_self1 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(b + a) div b = a div b + 1"
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  using assms div_mult_self1 [of b a 1] by (simp add: add_commute)
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lemma div_add_self2 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(a + b) div b = a div b + 1"
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  using assms div_add_self1 [of b a] by (simp add: add_commute)
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lemma mod_add_self1 [simp]:
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  "(b + a) mod b = a mod b"
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  using mod_mult_self1 [of a 1 b] by (simp add: add_commute)
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lemma mod_add_self2 [simp]:
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  "(a + b) mod b = a mod b"
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  using mod_mult_self1 [of a 1 b] by simp
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lemma mod_div_decomp:
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  fixes a b
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  obtains q r where "q = a div b" and "r = a mod b"
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    and "a = q * b + r"
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proof -
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diff changeset
   136
  from mod_div_equality have "a = a div b * b + a mod b" by simp
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   137
  moreover have "a div b = a div b" ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   138
  moreover have "a mod b = a mod b" ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   139
  note that ultimately show thesis by blast
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   140
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   141
31998
2c7a24f74db9 code attributes use common underscore convention
haftmann
parents: 31952
diff changeset
   142
lemma dvd_eq_mod_eq_0 [code_unfold]: "a dvd b \<longleftrightarrow> b mod a = 0"
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   143
proof
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   144
  assume "b mod a = 0"
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   145
  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
   146
  then have "b = a * (b div a)" unfolding mult_commute ..
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   147
  then have "\<exists>c. b = a * c" ..
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   148
  then show "a dvd b" unfolding dvd_def .
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   149
next
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   150
  assume "a dvd b"
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   151
  then have "\<exists>c. b = a * c" unfolding dvd_def .
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   152
  then obtain c where "b = a * c" ..
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   153
  then have "b mod a = a * c mod a" by simp
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   154
  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
   155
  then show "b mod a = 0" by simp
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   156
qed
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   157
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   158
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
   159
proof (cases "b = 0")
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   160
  assume "b = 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   161
  thus ?thesis by simp
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   162
next
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   163
  assume "b \<noteq> 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   164
  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
   165
    by (rule div_mult_self1 [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   166
  also have "\<dots> = a div b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   167
    by (simp only: mod_div_equality')
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   168
  also have "\<dots> = a div b + 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   169
    by simp
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   170
  finally show ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   171
    by (rule add_left_imp_eq)
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
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   174
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
   175
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   176
  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
   177
    by (simp only: mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   178
  also have "\<dots> = a mod b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   179
    by (simp only: mod_div_equality')
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   180
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   181
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   182
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   183
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
   184
by (rule dvd_eq_mod_eq_0[THEN iffD1])
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   185
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   186
lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   187
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
   188
33274
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 32010
diff changeset
   189
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
   190
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
   191
30052
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   192
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
   193
apply (cases "a = 0")
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   194
 apply simp
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   195
apply (auto simp: dvd_def mult_assoc)
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   196
done
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   197
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   198
lemma div_dvd_div[simp]:
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   199
  "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
   200
apply (cases "a = 0")
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   201
 apply simp
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   202
apply (unfold dvd_def)
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   203
apply auto
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   204
 apply(blast intro:mult_assoc[symmetric])
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   205
apply(fastsimp simp add: mult_assoc)
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   206
done
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   207
30078
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   208
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
   209
  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
   210
   apply (simp add: mod_div_equality)
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   211
  apply (simp only: dvd_add dvd_mult)
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   212
  done
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   213
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   214
text {* Addition respects modular equivalence. *}
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   215
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   216
lemma mod_add_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
   217
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   218
  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
   219
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   220
  also have "\<dots> = (a mod c + b + a div c * c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   221
    by (simp only: add_ac)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   222
  also have "\<dots> = (a mod c + b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   223
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   224
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   225
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   226
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   227
lemma mod_add_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
   228
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   229
  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
   230
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   231
  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
   232
    by (simp only: add_ac)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   233
  also have "\<dots> = (a + b mod c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   234
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   235
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   236
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   237
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   238
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
   239
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
   240
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   241
lemma mod_add_cong:
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   242
  assumes "a mod c = a' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   243
  assumes "b mod c = b' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   244
  shows "(a + b) mod c = (a' + b') mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   245
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   246
  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
   247
    unfolding assms ..
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   248
  thus ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   249
    by (simp only: mod_add_eq [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   250
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   251
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   252
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
   253
  \<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
   254
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
   255
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   256
text {* Multiplication respects modular equivalence. *}
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   257
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   258
lemma mod_mult_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
   259
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   260
  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
   261
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   262
  also have "\<dots> = (a 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
   263
    by (simp only: algebra_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   264
  also have "\<dots> = (a mod c * b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   265
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   266
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   267
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   268
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   269
lemma mod_mult_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
   270
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   271
  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
   272
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   273
  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
   274
    by (simp only: algebra_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   275
  also have "\<dots> = (a * (b mod c)) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   276
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   277
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   278
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   279
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   280
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
   281
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
   282
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   283
lemma mod_mult_cong:
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   284
  assumes "a mod c = a' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   285
  assumes "b mod c = b' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   286
  shows "(a * b) mod c = (a' * b') mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   287
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   288
  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
   289
    unfolding assms ..
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   290
  thus ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   291
    by (simp only: mod_mult_eq [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   292
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
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
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   320
lemma div_mult_mult2 [simp]:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   321
  "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   322
  by (drule div_mult_mult1) (simp add: mult_commute)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   323
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   324
lemma div_mult_mult1_if [simp]:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   325
  "(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
   326
  by simp_all
30476
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   327
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   328
lemma mod_mult_mult1:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   329
  "(c * a) mod (c * b) = c * (a mod b)"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   330
proof (cases "c = 0")
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   331
  case True then show ?thesis by simp
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   332
next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   333
  case False
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   334
  from mod_div_equality
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   335
  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
   336
  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
   337
    = c * a + c * (a mod b)" by (simp add: algebra_simps)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   338
  with mod_div_equality show ?thesis by simp 
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   339
qed
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   340
  
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   341
lemma mod_mult_mult2:
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   342
  "(a * c) mod (b * c) = (a mod b) * c"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   343
  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
   344
31662
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   345
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
   346
  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
   347
57f7ef0dba8e generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents: 31661
diff changeset
   348
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
   349
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
   350
31009
41fd307cab30 dropped reference to class recpower and lemma duplicate
haftmann
parents: 30934
diff changeset
   351
lemma div_power:
31661
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   352
  "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
30476
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   353
apply (induct n)
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   354
 apply simp
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   355
apply(simp add: div_mult_div_if_dvd dvd_power_same)
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   356
done
0a41b0662264 added div lemmas
nipkow
parents: 30242
diff changeset
   357
31661
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   358
end
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   359
1e252b8b2334 move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents: 31009
diff changeset
   360
class ring_div = semiring_div + idom
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   361
begin
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   362
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   363
text {* Negation respects modular equivalence. *}
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   364
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   365
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
   366
proof -
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   367
  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
   368
    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
   369
  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
   370
    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
   371
  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
   372
    by (rule mod_mult_self1)
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   373
  finally show ?thesis .
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   374
qed
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   375
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   376
lemma mod_minus_cong:
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   377
  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
   378
  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
   379
proof -
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   380
  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
   381
    unfolding assms ..
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   382
  thus ?thesis
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   383
    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
   384
qed
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   385
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   386
text {* Subtraction respects modular equivalence. *}
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   387
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   388
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
   389
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   390
  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
   391
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   392
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
   393
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   394
  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
   395
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   396
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
   397
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   398
  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
   399
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   400
lemma mod_diff_cong:
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   401
  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
   402
  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
   403
  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
   404
  unfolding diff_minus using assms
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   405
  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
   406
30180
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   407
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
   408
apply (case_tac "y = 0") apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   409
apply (auto simp add: dvd_def)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   410
apply (subgoal_tac "-(y * k) = y * - k")
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   411
 apply (erule ssubst)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   412
 apply (erule div_mult_self1_is_id)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   413
apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   414
done
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   415
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   416
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
   417
apply (case_tac "y = 0") apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   418
apply (auto simp add: dvd_def)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   419
apply (subgoal_tac "y * k = -y * -k")
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   420
 apply (erule ssubst)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   421
 apply (rule div_mult_self1_is_id)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   422
 apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   423
apply simp
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   424
done
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
   425
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   426
end
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   427
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   428
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   429
subsection {* Division on @{typ nat} *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   430
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   431
text {*
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   432
  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
   433
  of a characteristic relation with two input arguments
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   434
  @{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
   435
  @{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
   436
*}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   437
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   438
definition divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   439
  "divmod_rel m n qr \<longleftrightarrow>
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   440
    m = fst qr * n + snd qr \<and>
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   441
      (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
   442
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   443
text {* @{const divmod_rel} is total: *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   444
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   445
lemma divmod_rel_ex:
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   446
  obtains q r where "divmod_rel m n (q, r)"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   447
proof (cases "n = 0")
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   448
  case True  with that show thesis
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   449
    by (auto simp add: divmod_rel_def)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   450
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   451
  case False
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   452
  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
   453
  proof (induct m)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   454
    case 0 with `n \<noteq> 0`
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   455
    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
   456
    then show ?case by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   457
  next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   458
    case (Suc m) then obtain q' r'
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   459
      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
   460
    then show ?case proof (cases "Suc r' < n")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   461
      case True
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   462
      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
   463
      with True show ?thesis by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   464
    next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   465
      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
   466
      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
   467
      ultimately have "n = Suc r'" by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   468
      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
   469
      with `n \<noteq> 0` show ?thesis by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   470
    qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   471
  qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   472
  with that show thesis
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   473
    using `n \<noteq> 0` by (auto simp add: divmod_rel_def)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   474
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   475
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   476
text {* @{const divmod_rel} is injective: *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   477
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   478
lemma divmod_rel_unique:
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   479
  assumes "divmod_rel m n qr"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   480
    and "divmod_rel m n qr'"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   481
  shows "qr = qr'"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   482
proof (cases "n = 0")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   483
  case True with assms show ?thesis
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   484
    by (cases qr, cases qr')
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   485
      (simp add: divmod_rel_def)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   486
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   487
  case False
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   488
  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
   489
  apply (rule leI)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   490
  apply (subst less_iff_Suc_add)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   491
  apply (auto simp add: add_mult_distrib)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   492
  done
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   493
  from `n \<noteq> 0` assms have "fst qr = fst qr'"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   494
    by (auto simp add: divmod_rel_def intro: order_antisym dest: aux sym)
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   495
  moreover from this assms have "snd qr = snd qr'"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   496
    by (simp add: divmod_rel_def)
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   497
  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
   498
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   499
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   500
text {*
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   501
  We instantiate divisibility on the natural numbers by
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   502
  means of @{const divmod_rel}:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   503
*}
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   504
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   505
instantiation nat :: semiring_div
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   506
begin
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   507
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   508
definition divmod :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   509
  [code del]: "divmod m n = (THE qr. divmod_rel m n qr)"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   510
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   511
lemma divmod_rel_divmod:
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   512
  "divmod_rel m n (divmod m n)"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   513
proof -
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   514
  from divmod_rel_ex
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   515
    obtain qr where rel: "divmod_rel m n qr" .
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   516
  then show ?thesis
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   517
  by (auto simp add: divmod_def intro: theI elim: divmod_rel_unique)
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   518
qed
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   519
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   520
lemma divmod_eq:
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   521
  assumes "divmod_rel m n qr" 
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   522
  shows "divmod m n = qr"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   523
  using assms by (auto intro: divmod_rel_unique divmod_rel_divmod)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   524
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   525
definition div_nat where
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   526
  "m div n = fst (divmod m n)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   527
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   528
definition mod_nat where
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   529
  "m mod n = snd (divmod m n)"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   530
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   531
lemma divmod_div_mod:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   532
  "divmod m n = (m div n, m mod n)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   533
  unfolding div_nat_def mod_nat_def by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   534
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   535
lemma div_eq:
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   536
  assumes "divmod_rel m n (q, r)" 
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   537
  shows "m div n = q"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   538
  using assms by (auto dest: divmod_eq simp add: divmod_div_mod)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   539
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   540
lemma mod_eq:
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   541
  assumes "divmod_rel m n (q, r)" 
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   542
  shows "m mod n = r"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   543
  using assms by (auto dest: divmod_eq simp add: divmod_div_mod)
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   544
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   545
lemma divmod_rel: "divmod_rel m n (m div n, m mod n)"
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   546
  by (simp add: div_nat_def mod_nat_def divmod_rel_divmod)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   547
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   548
lemma divmod_zero:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   549
  "divmod m 0 = (0, m)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   550
proof -
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   551
  from divmod_rel [of m 0] show ?thesis
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   552
    unfolding divmod_div_mod divmod_rel_def by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   553
qed
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   554
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   555
lemma divmod_base:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   556
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   557
  shows "divmod m n = (0, m)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   558
proof -
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   559
  from divmod_rel [of m n] show ?thesis
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   560
    unfolding divmod_div_mod divmod_rel_def
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   561
    using assms by (cases "m div n = 0")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   562
      (auto simp add: gr0_conv_Suc [of "m div n"])
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   563
qed
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   564
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   565
lemma divmod_step:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   566
  assumes "0 < n" and "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   567
  shows "divmod m n = (Suc ((m - n) div n), (m - n) mod n)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   568
proof -
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   569
  from divmod_rel have divmod_m_n: "divmod_rel m n (m div n, m mod n)" .
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   570
  with assms have m_div_n: "m div n \<ge> 1"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   571
    by (cases "m div n") (auto simp add: divmod_rel_def)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   572
  from assms divmod_m_n have "divmod_rel (m - n) n (m div n - Suc 0, m mod n)"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   573
    by (cases "m div n") (auto simp add: divmod_rel_def)
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30078
diff changeset
   574
  with divmod_eq have "divmod (m - n) n = (m div n - Suc 0, m mod n)" by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   575
  moreover from divmod_div_mod have "divmod (m - n) n = ((m - n) div n, (m - n) mod n)" .
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   576
  ultimately have "m div n = Suc ((m - n) div n)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   577
    and "m mod n = (m - n) mod n" using m_div_n by simp_all
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   578
  then show ?thesis using divmod_div_mod by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   579
qed
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   580
26300
03def556e26e removed duplicate lemmas;
wenzelm
parents: 26100
diff changeset
   581
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
   582
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   583
lemma div_less [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   584
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   585
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   586
  shows "m div n = 0"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   587
  using assms divmod_base divmod_div_mod by simp
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   588
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   589
lemma le_div_geq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   590
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   591
  assumes "0 < n" and "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   592
  shows "m div n = Suc ((m - n) div n)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   593
  using assms divmod_step divmod_div_mod by simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   594
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   595
lemma mod_less [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   596
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   597
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   598
  shows "m mod n = m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   599
  using assms divmod_base divmod_div_mod by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   600
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   601
lemma le_mod_geq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   602
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   603
  assumes "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   604
  shows "m mod n = (m - n) mod n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   605
  using assms divmod_step divmod_div_mod by (cases "n = 0") simp_all
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   606
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   607
instance proof -
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   608
  have [simp]: "\<And>n::nat. n div 0 = 0"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   609
    by (simp add: div_nat_def divmod_zero)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   610
  have [simp]: "\<And>n::nat. 0 div n = 0"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   611
  proof -
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   612
    fix n :: nat
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   613
    show "0 div n = 0"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   614
      by (cases "n = 0") simp_all
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   615
  qed
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   616
  show "OFCLASS(nat, semiring_div_class)" proof
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   617
    fix m n :: nat
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   618
    show "m div n * n + m mod n = m"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   619
      using divmod_rel [of m n] by (simp add: divmod_rel_def)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   620
  next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   621
    fix m n q :: nat
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   622
    assume "n \<noteq> 0"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   623
    then show "(q + m * n) div n = m + q div n"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   624
      by (induct m) (simp_all add: le_div_geq)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   625
  next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   626
    fix m n q :: nat
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   627
    assume "m \<noteq> 0"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   628
    then show "(m * n) div (m * q) = n div q"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   629
    proof (cases "n \<noteq> 0 \<and> q \<noteq> 0")
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   630
      case False then show ?thesis by auto
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   631
    next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   632
      case True with `m \<noteq> 0`
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   633
        have "m > 0" and "n > 0" and "q > 0" by auto
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   634
      then have "\<And>a b. divmod_rel n q (a, b) \<Longrightarrow> divmod_rel (m * n) (m * q) (a, m * b)"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   635
        by (auto simp add: divmod_rel_def) (simp_all add: algebra_simps)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   636
      moreover from divmod_rel have "divmod_rel n q (n div q, n mod q)" .
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   637
      ultimately have "divmod_rel (m * n) (m * q) (n div q, m * (n mod q))" .
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   638
      then show ?thesis by (simp add: div_eq)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   639
    qed
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30923
diff changeset
   640
  qed simp_all
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   641
qed
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   642
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   643
end
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   644
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   645
text {* Simproc for cancelling @{const div} and @{const mod} *}
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   646
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   647
ML {*
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   648
local
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   649
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   650
structure CancelDivMod = CancelDivModFun(struct
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   651
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   652
  val div_name = @{const_name div};
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   653
  val mod_name = @{const_name mod};
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   654
  val mk_binop = HOLogic.mk_binop;
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   655
  val mk_sum = Nat_Arith.mk_sum;
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   656
  val dest_sum = Nat_Arith.dest_sum;
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   657
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   658
  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
   659
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   660
  val trans = trans;
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   661
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   662
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   663
    (@{thm monoid_add_class.add_0_left} :: @{thm monoid_add_class.add_0_right} :: @{thms add_ac}))
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   664
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   665
end)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   666
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   667
in
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   668
32010
cb1a1c94b4cd more antiquotations;
wenzelm
parents: 31998
diff changeset
   669
val cancel_div_mod_nat_proc = Simplifier.simproc @{theory}
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   670
  "cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   671
30934
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   672
val _ = Addsimprocs [cancel_div_mod_nat_proc];
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   673
ed5377c2b0a3 tuned setups of CancelDivMod
haftmann
parents: 30930
diff changeset
   674
end
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   675
*}
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   676
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   677
text {* code generator setup *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   678
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   679
lemma divmod_if [code]: "divmod m n = (if n = 0 \<or> m < n then (0, m) else
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   680
  let (q, r) = divmod (m - n) n in (Suc q, r))"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   681
by (simp add: divmod_zero divmod_base divmod_step)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   682
    (simp add: divmod_div_mod)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   683
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   684
code_modulename SML
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   685
  Divides Nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   686
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   687
code_modulename OCaml
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   688
  Divides Nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   689
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   690
code_modulename Haskell
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   691
  Divides Nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   692
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   693
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   694
subsubsection {* Quotient *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   695
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   696
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
   697
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
   698
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   699
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
   700
by (simp add: div_geq)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   701
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   702
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
   703
by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   704
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   705
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
   706
by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   707
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   708
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   709
subsubsection {* Remainder *}
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   710
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   711
lemma mod_less_divisor [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   712
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   713
  assumes "n > 0"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   714
  shows "m mod n < (n::nat)"
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   715
  using assms divmod_rel [of m n] unfolding divmod_rel_def by auto
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   716
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   717
lemma mod_less_eq_dividend [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   718
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   719
  shows "m mod n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   720
proof (rule add_leD2)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   721
  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
   722
  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
   723
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   724
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   725
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
   726
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
   727
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   728
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
   729
by (simp add: le_mod_geq)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   730
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   731
lemma mod_1 [simp]: "m mod Suc 0 = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   732
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
   733
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   734
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   735
  apply (cases "n = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   736
  apply (cases "k = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   737
  apply (induct m rule: nat_less_induct)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   738
  apply (subst mod_if, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   739
  apply (simp add: mod_geq diff_mult_distrib)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   740
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   741
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   742
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   743
by (simp add: mult_commute [of k] mod_mult_distrib)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   744
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   745
(* 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
   746
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   747
by (cut_tac a = m and b = n in mod_div_equality2, arith)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   748
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   749
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   750
  apply (drule mod_less_divisor [where m = m])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   751
  apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   752
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   753
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   754
subsubsection {* Quotient and Remainder *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   755
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   756
lemma divmod_rel_mult1_eq:
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   757
  "divmod_rel b c (q, r) \<Longrightarrow> c > 0
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   758
   \<Longrightarrow> divmod_rel (a * b) c (a * q + a * r div c, a * r mod c)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   759
by (auto simp add: split_ifs divmod_rel_def algebra_simps)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   760
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   761
lemma div_mult1_eq:
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   762
  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   763
apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   764
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN div_eq])
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   765
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   766
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   767
lemma divmod_rel_add1_eq:
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   768
  "divmod_rel a c (aq, ar) \<Longrightarrow> divmod_rel b c (bq, br) \<Longrightarrow>  c > 0
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   769
   \<Longrightarrow> divmod_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   770
by (auto simp add: split_ifs divmod_rel_def algebra_simps)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   771
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   772
(*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
   773
lemma div_add1_eq:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   774
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   775
apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   776
apply (blast intro: divmod_rel_add1_eq [THEN div_eq] divmod_rel)
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   777
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   778
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   779
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   780
  apply (cut_tac m = q and n = c in mod_less_divisor)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   781
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   782
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   783
  apply (simp add: add_mult_distrib2)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   784
  done
10559
d3fd54fc659b many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents: 10214
diff changeset
   785
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   786
lemma divmod_rel_mult2_eq:
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   787
  "divmod_rel a b (q, r) \<Longrightarrow> 0 < b \<Longrightarrow> 0 < c
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   788
   \<Longrightarrow> divmod_rel a (b * c) (q div c, b *(q mod c) + r)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   789
by (auto simp add: mult_ac divmod_rel_def add_mult_distrib2 [symmetric] mod_lemma)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   790
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   791
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   792
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   793
  apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   794
  apply (force simp add: divmod_rel [THEN divmod_rel_mult2_eq, THEN div_eq])
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   795
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   796
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   797
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   798
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   799
  apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   800
  apply (auto simp add: mult_commute divmod_rel [THEN divmod_rel_mult2_eq, THEN mod_eq])
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   801
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   802
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   803
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   804
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
   805
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   806
lemma div_1 [simp]: "m div Suc 0 = m"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   807
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
   808
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   809
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   810
(* Monotonicity of div in first argument *)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   811
lemma div_le_mono [rule_format (no_asm)]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   812
    "\<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
   813
apply (case_tac "k=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   814
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
   815
apply (case_tac "n<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   816
(* 1  case n<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   817
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   818
(* 2  case n >= k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   819
apply (case_tac "m<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   820
(* 2.1  case m<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   821
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   822
(* 2.2  case m>=k *)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   823
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
   824
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   825
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   826
(* Antimonotonicity of div in second argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   827
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
   828
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   829
 prefer 2 apply simp
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   830
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
   831
apply (rename_tac "k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   832
apply (case_tac "k<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   833
apply (subgoal_tac "~ (k<m) ")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   834
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   835
apply (simp add: div_geq)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   836
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
   837
 prefer 2
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   838
 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
   839
apply (rule le_trans, simp)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   840
apply (simp)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   841
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   842
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   843
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
   844
apply (case_tac "n=0", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   845
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
   846
apply (rule div_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   847
apply (simp_all (no_asm_simp))
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   848
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   849
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   850
(* Similar for "less than" *)
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   851
lemma div_less_dividend [rule_format]:
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   852
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   853
apply (induct_tac m rule: nat_less_induct)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   854
apply (rename_tac "m")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   855
apply (case_tac "m<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   856
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   857
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   858
apply (simp add: div_geq)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   859
apply (case_tac "n<m")
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   860
 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
   861
  apply (rule impI less_trans_Suc)+
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   862
apply assumption
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   863
  apply (simp_all)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   864
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   865
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   866
declare div_less_dividend [simp]
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   867
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   868
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
   869
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
   870
apply (case_tac "n=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   871
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
   872
apply (case_tac "Suc (na) <n")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   873
(* case Suc(na) < n *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   874
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
   875
(* case n \<le> Suc(na) *)
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16733
diff changeset
   876
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   877
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
   878
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   879
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   880
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
   881
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
   882
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   883
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
   884
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   885
(*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
   886
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
   887
  apply (cut_tac a = m in mod_div_equality)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   888
  apply (simp only: add_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   889
  apply (blast intro: sym)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   890
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   891
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
   892
lemma split_div:
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   893
 "P(n div k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   894
 ((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
   895
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   896
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   897
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   898
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   899
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   900
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   901
    with P show ?Q by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   902
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   903
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   904
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   905
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   906
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   907
      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
   908
      show "P i"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   909
      proof (cases)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   910
        assume "i = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   911
        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
   912
      next
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   913
        assume "i \<noteq> 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   914
        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
   915
      qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   916
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   917
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   918
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   919
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   920
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   921
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   922
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   923
    with Q show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   924
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   925
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   926
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   927
    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
   928
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   929
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   930
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   931
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   932
lemma split_div_lemma:
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   933
  assumes "0 < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   934
  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
   935
proof
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   936
  assume ?rhs
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   937
  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
   938
  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
   939
  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
   940
  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
   941
  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
   942
  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
   943
  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
   944
  from A B show ?lhs ..
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   945
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   946
  assume P: ?lhs
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   947
  then have "divmod_rel m n (q, m - n * q)"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   948
    unfolding divmod_rel_def by (auto simp add: mult_ac)
30923
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   949
  with divmod_rel_unique divmod_rel [of m n]
2697a1d1d34a more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents: 30840
diff changeset
   950
  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
   951
  then show ?rhs by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   952
qed
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   953
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   954
theorem split_div':
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   955
  "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
   956
   (\<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
   957
  apply (case_tac "0 < n")
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   958
  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
   959
  apply simp_all
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   960
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   961
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   962
lemma split_mod:
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   963
 "P(n mod k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   964
 ((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
   965
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   966
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   967
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   968
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   969
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   970
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   971
    with P show ?Q by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   972
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   973
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   974
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   975
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   976
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   977
      assume "n = k*i + j" "j < k"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   978
      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
   979
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   980
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   981
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   982
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   983
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   984
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   985
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   986
    with Q show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   987
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   988
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   989
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   990
    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
   991
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   992
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   993
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   994
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   995
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   996
  apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   997
    subst [OF mod_div_equality [of _ n]])
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   998
  apply arith
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   999
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1000
22800
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1001
lemma div_mod_equality':
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1002
  fixes m n :: nat
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1003
  shows "m div n * n = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1004
proof -
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1005
  have "m mod n \<le> m mod n" ..
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1006
  from div_mod_equality have 
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1007
    "m div n * n + m mod n - m mod n = m - m mod n" by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1008
  with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1009
    "m div n * n + (m mod n - m mod n) = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1010
    by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1011
  then show ?thesis by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1012
qed
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1013
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1014
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
  1015
subsubsection {*An ``induction'' law for modulus arithmetic.*}
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1016
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1017
lemma mod_induct_0:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1018
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1019
  and base: "P i" and i: "i<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1020
  shows "P 0"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1021
proof (rule ccontr)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1022
  assume contra: "\<not>(P 0)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1023
  from i have p: "0<p" by simp
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1024
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1025
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1026
    fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1027
    show "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1028
    proof (induct k)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1029
      show "?A 0" by simp  -- "by contradiction"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1030
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1031
      fix n
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1032
      assume ih: "?A n"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1033
      show "?A (Suc n)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1034
      proof (clarsimp)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1035
        assume y: "P (p - Suc n)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1036
        have n: "Suc n < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1037
        proof (rule ccontr)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1038
          assume "\<not>(Suc n < p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1039
          hence "p - Suc n = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1040
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1041
          with y contra show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1042
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1043
        qed
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1044
        hence n2: "Suc (p - Suc n) = p-n" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1045
        from p have "p - Suc n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1046
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1047
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1048
        show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1049
        proof (cases "n=0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1050
          case True
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1051
          with z n2 contra show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1052
        next
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1053
          case False
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1054
          with p have "p-n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1055
          with z n2 False ih show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1056
        qed
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1057
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1058
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1059
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1060
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1061
  from i obtain k where "0<k \<and> i+k=p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1062
    by (blast dest: less_imp_add_positive)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1063
  hence "0<k \<and> i=p-k" by auto
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1064
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1065
  note base
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1066
  ultimately
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1067
  show "False" by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1068
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1069
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1070
lemma mod_induct:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1071
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1072
  and base: "P i" and i: "i<p" and j: "j<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1073
  shows "P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1074
proof -
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1075
  have "\<forall>j<p. P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1076
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1077
    fix j
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1078
    show "j<p \<longrightarrow> P j" (is "?A j")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1079
    proof (induct j)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1080
      from step base i show "?A 0"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1081
        by (auto elim: mod_induct_0)
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1082
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1083
      fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1084
      assume ih: "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1085
      show "?A (Suc k)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1086
      proof
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1087
        assume suc: "Suc k < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1088
        hence k: "k<p" by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1089
        with ih have "P k" ..
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1090
        with step k have "P (Suc k mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1091
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1092
        moreover
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1093
        from suc have "Suc k mod p = Suc k"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1094
          by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1095
        ultimately
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1096
        show "P (Suc k)" by simp
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1097
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1098
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1099
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1100
  with j show ?thesis by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1101
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1102
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1103
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1104
by (auto simp add: numeral_2_eq_2 le_div_geq)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1105
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1106
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1107
by (simp add: nat_mult_2 [symmetric])
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1108
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1109
lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1110
apply (subgoal_tac "m mod 2 < 2")
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1111
apply (erule less_2_cases [THEN disjE])
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1112
apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1113
done
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1114
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1115
lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1116
proof -
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1117
  { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (induct n) simp_all }
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1118
  moreover have "m mod 2 < 2" by simp
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1119
  ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1120
  then show ?thesis by auto
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1121
qed
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1122
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1123
text{*These lemmas collapse some needless occurrences of Suc:
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1124
    at least three Sucs, since two and fewer are rewritten back to Suc again!
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1125
    We already have some rules to simplify operands smaller than 3.*}
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1126
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1127
lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1128
by (simp add: Suc3_eq_add_3)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1129
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1130
lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1131
by (simp add: Suc3_eq_add_3)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1132
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1133
lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1134
by (simp add: Suc3_eq_add_3)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1135
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1136
lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1137
by (simp add: Suc3_eq_add_3)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1138
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1139
lemmas Suc_div_eq_add3_div_number_of =
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1140
    Suc_div_eq_add3_div [of _ "number_of v", standard]
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1141
declare Suc_div_eq_add3_div_number_of [simp]
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1142
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1143
lemmas Suc_mod_eq_add3_mod_number_of =
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1144
    Suc_mod_eq_add3_mod [of _ "number_of v", standard]
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1145
declare Suc_mod_eq_add3_mod_number_of [simp]
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1146
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1147
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1148
subsection {* Proof Tools setup; Combination and Cancellation Simprocs *}
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1149
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1150
declare split_div[of _ _ "number_of k", standard, arith_split]
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1151
declare split_mod[of _ _ "number_of k", standard, arith_split]
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1152
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1153
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1154
subsubsection{*For @{text combine_numerals}*}
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1155
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1156
lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1157
by (simp add: add_mult_distrib)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1158
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1159
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1160
subsubsection{*For @{text cancel_numerals}*}
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1161
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1162
lemma nat_diff_add_eq1:
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1163
     "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1164
by (simp split add: nat_diff_split add: add_mult_distrib)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1165
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1166
lemma nat_diff_add_eq2:
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1167
     "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1168
by (simp split add: nat_diff_split add: add_mult_distrib)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1169
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1170
lemma nat_eq_add_iff1:
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1171
     "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1172
by (auto split add: nat_diff_split simp add: add_mult_distrib)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1173
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1174
lemma nat_eq_add_iff2:
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1175
     "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1176
by (auto split add: nat_diff_split simp add: add_mult_distrib)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1177
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1178
lemma nat_less_add_iff1:
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1179
     "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1180
by (auto split add: nat_diff_split simp add: add_mult_distrib)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1181
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1182
lemma nat_less_add_iff2:
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1183
     "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1184
by (auto split add: nat_diff_split simp add: add_mult_distrib)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1185
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1186
lemma nat_le_add_iff1:
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1187
     "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1188
by (auto split add: nat_diff_split simp add: add_mult_distrib)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1189
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1190
lemma nat_le_add_iff2:
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1191
     "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1192
by (auto split add: nat_diff_split simp add: add_mult_distrib)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1193
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1194
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1195
subsubsection{*For @{text cancel_numeral_factors} *}
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1196
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1197
lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1198
by auto
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1199
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1200
lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1201
by auto
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1202
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1203
lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1204
by auto
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1205
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1206
lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1207
by auto
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1208
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1209
lemma nat_mult_dvd_cancel_disj[simp]:
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1210
  "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1211
by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1212
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1213
lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1214
by(auto)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1215
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1216
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1217
subsubsection{*For @{text cancel_factor} *}
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1218
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1219
lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1220
by auto
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1221
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1222
lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1223
by auto
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1224
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1225
lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1226
by auto
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1227
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1228
lemma nat_mult_div_cancel_disj[simp]:
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1229
     "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1230
by (simp add: nat_mult_div_cancel1)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1231
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1232
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1233
use "Tools/numeral_simprocs.ML"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1234
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1235
use "Tools/nat_numeral_simprocs.ML"
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1236
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1237
declaration {* 
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1238
  K (Lin_Arith.add_simps (@{thms neg_simps} @ [@{thm Suc_nat_number_of}, @{thm int_nat_number_of}])
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1239
  #> Lin_Arith.add_simps (@{thms ring_distribs} @ [@{thm Let_number_of}, @{thm Let_0}, @{thm Let_1},
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1240
     @{thm nat_0}, @{thm nat_1},
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1241
     @{thm add_nat_number_of}, @{thm diff_nat_number_of}, @{thm mult_nat_number_of},
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1242
     @{thm eq_nat_number_of}, @{thm less_nat_number_of}, @{thm le_number_of_eq_not_less},
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1243
     @{thm le_Suc_number_of}, @{thm le_number_of_Suc},
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1244
     @{thm less_Suc_number_of}, @{thm less_number_of_Suc},
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1245
     @{thm Suc_eq_number_of}, @{thm eq_number_of_Suc},
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1246
     @{thm mult_Suc}, @{thm mult_Suc_right},
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1247
     @{thm add_Suc}, @{thm add_Suc_right},
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1248
     @{thm eq_number_of_0}, @{thm eq_0_number_of}, @{thm less_0_number_of},
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1249
     @{thm of_int_number_of_eq}, @{thm of_nat_number_of_eq}, @{thm nat_number_of},
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1250
     @{thm if_True}, @{thm if_False}])
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1251
  #> Lin_Arith.add_simprocs (Numeral_Simprocs.assoc_fold_simproc
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1252
      :: Numeral_Simprocs.combine_numerals
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1253
      :: Numeral_Simprocs.cancel_numerals)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1254
  #> Lin_Arith.add_simprocs (Nat_Numeral_Simprocs.combine_numerals :: Nat_Numeral_Simprocs.cancel_numerals))
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1255
*}
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33274
diff changeset
  1256
3366
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
  1257
end