src/HOL/Divides.thy
author huffman
Mon, 23 Feb 2009 13:55:36 -0800
changeset 30078 beee83623cc9
parent 30052 410fefc247aa
child 30079 293b896b9c25
permissions -rw-r--r--
move lemma dvd_mod_imp_dvd into class semiring_div
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
3366
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
     1
(*  Title:      HOL/Divides.thy
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
     2
    ID:         $Id$
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
     3
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
6865
5577ffe4c2f1 now div and mod are overloaded; dvd is polymorphic
paulson
parents: 3366
diff changeset
     4
    Copyright   1999  University of Cambridge
18154
0c05abaf6244 add header
huffman
parents: 17609
diff changeset
     5
*)
3366
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
     6
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
     7
header {* The division operators div and mod *}
3366
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
     8
15131
c69542757a4d New theory header syntax.
nipkow
parents: 14640
diff changeset
     9
theory Divides
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26300
diff changeset
    10
imports Nat Power Product_Type
22993
haftmann
parents: 22916
diff changeset
    11
uses "~~/src/Provers/Arith/cancel_div_mod.ML"
15131
c69542757a4d New theory header syntax.
nipkow
parents: 14640
diff changeset
    12
begin
3366
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
    13
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
    14
subsection {* Syntactic division operations *}
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
    15
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    16
class div = dvd +
27540
dc38e79f5a1c separate class dvd for divisibility predicate
haftmann
parents: 26748
diff changeset
    17
  fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    18
    and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
27540
dc38e79f5a1c separate class dvd for divisibility predicate
haftmann
parents: 26748
diff changeset
    19
dc38e79f5a1c separate class dvd for divisibility predicate
haftmann
parents: 26748
diff changeset
    20
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    21
subsection {* Abstract division in commutative semirings. *}
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
    22
29509
1ff0f3f08a7b migrated class package to new locale implementation
haftmann
parents: 29405
diff changeset
    23
class semiring_div = comm_semiring_1_cancel + div +
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
    24
  assumes mod_div_equality: "a div b * b + a mod b = a"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    25
    and div_by_0 [simp]: "a div 0 = 0"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    26
    and div_0 [simp]: "0 div a = 0"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    27
    and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
    28
begin
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
    29
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
    30
text {* @{const div} and @{const mod} *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
    31
26062
16f334d7156a more abstract lemmas
haftmann
parents: 25947
diff changeset
    32
lemma mod_div_equality2: "b * (a div b) + a mod b = a"
16f334d7156a more abstract lemmas
haftmann
parents: 25947
diff changeset
    33
  unfolding mult_commute [of b]
16f334d7156a more abstract lemmas
haftmann
parents: 25947
diff changeset
    34
  by (rule mod_div_equality)
16f334d7156a more abstract lemmas
haftmann
parents: 25947
diff changeset
    35
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
    36
lemma mod_div_equality': "a mod b + a div b * b = a"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
    37
  using mod_div_equality [of a b]
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
    38
  by (simp only: add_ac)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
    39
26062
16f334d7156a more abstract lemmas
haftmann
parents: 25947
diff changeset
    40
lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
    41
by (simp add: mod_div_equality)
26062
16f334d7156a more abstract lemmas
haftmann
parents: 25947
diff changeset
    42
16f334d7156a more abstract lemmas
haftmann
parents: 25947
diff changeset
    43
lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
    44
by (simp add: mod_div_equality2)
26062
16f334d7156a more abstract lemmas
haftmann
parents: 25947
diff changeset
    45
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    46
lemma mod_by_0 [simp]: "a mod 0 = a"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    47
  using mod_div_equality [of a zero] by simp
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    48
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    49
lemma mod_0 [simp]: "0 mod a = 0"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    50
  using mod_div_equality [of zero a] div_0 by simp 
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    51
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    52
lemma div_mult_self2 [simp]:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    53
  assumes "b \<noteq> 0"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    54
  shows "(a + b * c) div b = c + a div b"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    55
  using assms div_mult_self1 [of b a c] by (simp add: mult_commute)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
    56
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    57
lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    58
proof (cases "b = 0")
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    59
  case True then show ?thesis by simp
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    60
next
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    61
  case False
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    62
  have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    63
    by (simp add: mod_div_equality)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    64
  also from False div_mult_self1 [of b a c] have
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    65
    "\<dots> = (c + a div b) * b + (a + c * b) mod b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
    66
      by (simp add: algebra_simps)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    67
  finally have "a = a div b * b + (a + c * b) mod b"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    68
    by (simp add: add_commute [of a] add_assoc left_distrib)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    69
  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    70
    by (simp add: mod_div_equality)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    71
  then show ?thesis by simp
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    72
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    73
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    74
lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
    75
by (simp add: mult_commute [of b])
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    76
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    77
lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    78
  using div_mult_self2 [of b 0 a] by simp
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    79
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    80
lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    81
  using div_mult_self1 [of b 0 a] by simp
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    82
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    83
lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    84
  using mod_mult_self2 [of 0 b a] by simp
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    85
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    86
lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    87
  using mod_mult_self1 [of 0 a b] by simp
26062
16f334d7156a more abstract lemmas
haftmann
parents: 25947
diff changeset
    88
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    89
lemma div_by_1 [simp]: "a div 1 = a"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    90
  using div_mult_self2_is_id [of 1 a] zero_neq_one by simp
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    91
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    92
lemma mod_by_1 [simp]: "a mod 1 = 0"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    93
proof -
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    94
  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    95
  then have "a + a mod 1 = a + 0" by simp
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    96
  then show ?thesis by (rule add_left_imp_eq)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    97
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    98
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
    99
lemma mod_self [simp]: "a mod a = 0"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   100
  using mod_mult_self2_is_0 [of 1] by simp
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   101
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   102
lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   103
  using div_mult_self2_is_id [of _ 1] by simp
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   104
27676
55676111ed69 (re-)added simp rules for (_ + _) div/mod _
haftmann
parents: 27651
diff changeset
   105
lemma div_add_self1 [simp]:
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   106
  assumes "b \<noteq> 0"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   107
  shows "(b + a) div b = a div b + 1"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   108
  using assms div_mult_self1 [of b a 1] by (simp add: add_commute)
26062
16f334d7156a more abstract lemmas
haftmann
parents: 25947
diff changeset
   109
27676
55676111ed69 (re-)added simp rules for (_ + _) div/mod _
haftmann
parents: 27651
diff changeset
   110
lemma div_add_self2 [simp]:
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   111
  assumes "b \<noteq> 0"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   112
  shows "(a + b) div b = a div b + 1"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   113
  using assms div_add_self1 [of b a] by (simp add: add_commute)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   114
27676
55676111ed69 (re-)added simp rules for (_ + _) div/mod _
haftmann
parents: 27651
diff changeset
   115
lemma mod_add_self1 [simp]:
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   116
  "(b + 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
   117
  using mod_mult_self1 [of a 1 b] by (simp add: add_commute)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   118
27676
55676111ed69 (re-)added simp rules for (_ + _) div/mod _
haftmann
parents: 27651
diff changeset
   119
lemma mod_add_self2 [simp]:
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   120
  "(a + b) mod b = a mod b"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   121
  using mod_mult_self1 [of a 1 b] by simp
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   122
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   123
lemma mod_div_decomp:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   124
  fixes a b
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   125
  obtains q r where "q = a div b" and "r = a mod b"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   126
    and "a = q * b + r"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   127
proof -
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   128
  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
   129
  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
   130
  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
   131
  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
   132
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   133
29108
12ca66b887a0 codegen
nipkow
parents: 28823
diff changeset
   134
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
   135
proof
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   136
  assume "b mod a = 0"
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   137
  with mod_div_equality [of b a] have "b div a * a = b" by simp
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   138
  then have "b = a * (b div a)" unfolding mult_commute ..
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   139
  then have "\<exists>c. b = a * c" ..
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   140
  then show "a dvd b" unfolding dvd_def .
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   141
next
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   142
  assume "a dvd b"
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   143
  then have "\<exists>c. b = a * c" unfolding dvd_def .
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   144
  then obtain c where "b = a * c" ..
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   145
  then have "b mod a = a * c mod a" by simp
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   146
  then have "b mod a = c * a mod a" by (simp add: mult_commute)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   147
  then show "b mod a = 0" by simp
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   148
qed
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   149
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   150
lemma mod_div_trivial [simp]: "a mod b div b = 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   151
proof (cases "b = 0")
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   152
  assume "b = 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   153
  thus ?thesis by simp
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   154
next
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   155
  assume "b \<noteq> 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   156
  hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   157
    by (rule div_mult_self1 [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   158
  also have "\<dots> = a div b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   159
    by (simp only: mod_div_equality')
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   160
  also have "\<dots> = a div b + 0"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   161
    by simp
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   162
  finally show ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   163
    by (rule add_left_imp_eq)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   164
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   165
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   166
lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   167
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   168
  have "a mod b mod b = (a mod b + a div b * b) mod b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   169
    by (simp only: mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   170
  also have "\<dots> = a mod b"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   171
    by (simp only: mod_div_equality')
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   172
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   173
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   174
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   175
lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0"
29948
cdf12a1cb963 Cleaned up IntDiv and removed subsumed lemmas.
nipkow
parents: 29925
diff changeset
   176
by (rule dvd_eq_mod_eq_0[THEN iffD1])
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   177
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   178
lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   179
by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   180
30052
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   181
lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a"
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   182
apply (cases "a = 0")
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   183
 apply simp
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   184
apply (auto simp: dvd_def mult_assoc)
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   185
done
410fefc247aa added dvd_div_mult
nipkow
parents: 30042
diff changeset
   186
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   187
lemma div_dvd_div[simp]:
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   188
  "a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)"
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   189
apply (cases "a = 0")
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   190
 apply simp
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   191
apply (unfold dvd_def)
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   192
apply auto
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   193
 apply(blast intro:mult_assoc[symmetric])
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   194
apply(fastsimp simp add: mult_assoc)
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   195
done
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29667
diff changeset
   196
30078
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   197
lemma dvd_mod_imp_dvd: "[| k dvd m mod n;  k dvd n |] ==> k dvd m"
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   198
  apply (subgoal_tac "k dvd (m div n) *n + m mod n")
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   199
   apply (simp add: mod_div_equality)
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   200
  apply (simp only: dvd_add dvd_mult)
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   201
  done
beee83623cc9 move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents: 30052
diff changeset
   202
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   203
text {* Addition respects modular equivalence. *}
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   204
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   205
lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   206
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   207
  have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   208
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   209
  also have "\<dots> = (a mod c + b + a div c * c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   210
    by (simp only: add_ac)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   211
  also have "\<dots> = (a mod c + b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   212
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   213
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   214
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   215
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   216
lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   217
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   218
  have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   219
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   220
  also have "\<dots> = (a + b mod c + b div c * c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   221
    by (simp only: add_ac)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   222
  also have "\<dots> = (a + b mod c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   223
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   224
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   225
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   226
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   227
lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   228
by (rule trans [OF mod_add_left_eq mod_add_right_eq])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   229
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   230
lemma mod_add_cong:
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   231
  assumes "a mod c = a' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   232
  assumes "b mod c = b' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   233
  shows "(a + b) mod c = (a' + b') mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   234
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   235
  have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   236
    unfolding assms ..
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   237
  thus ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   238
    by (simp only: mod_add_eq [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   239
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   240
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   241
text {* Multiplication respects modular equivalence. *}
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   242
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   243
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
   244
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   245
  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
   246
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   247
  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
   248
    by (simp only: algebra_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   249
  also have "\<dots> = (a mod c * b) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   250
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   251
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   252
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   253
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   254
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
   255
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   256
  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
   257
    by (simp only: mod_div_equality)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   258
  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
   259
    by (simp only: algebra_simps)
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   260
  also have "\<dots> = (a * (b mod c)) mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   261
    by (rule mod_mult_self1)
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   262
  finally show ?thesis .
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   263
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   264
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   265
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
   266
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
   267
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   268
lemma mod_mult_cong:
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   269
  assumes "a mod c = a' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   270
  assumes "b mod c = b' mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   271
  shows "(a * b) mod c = (a' * b') mod c"
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   272
proof -
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   273
  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
   274
    unfolding assms ..
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   275
  thus ?thesis
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   276
    by (simp only: mod_mult_eq [symmetric])
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   277
qed
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29252
diff changeset
   278
29404
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   279
lemma mod_mod_cancel:
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   280
  assumes "c dvd b"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   281
  shows "a mod b mod c = a mod c"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   282
proof -
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   283
  from `c dvd b` obtain k where "b = c * k"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   284
    by (rule dvdE)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   285
  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
   286
    by (simp only: `b = c * k`)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   287
  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
   288
    by (simp only: mod_mult_self1)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   289
  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
   290
    by (simp only: add_ac mult_ac)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   291
  also have "\<dots> = a mod c"
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   292
    by (simp only: mod_div_equality)
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   293
  finally show ?thesis .
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   294
qed
ee15ccdeaa72 generalize zmod_zmod_cancel -> mod_mod_cancel
huffman
parents: 29403
diff changeset
   295
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   296
end
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   297
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   298
class ring_div = semiring_div + comm_ring_1
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   299
begin
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   300
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   301
text {* Negation respects modular equivalence. *}
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   302
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   303
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
   304
proof -
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   305
  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
   306
    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
   307
  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
   308
    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
   309
  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
   310
    by (rule mod_mult_self1)
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   311
  finally show ?thesis .
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   312
qed
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   313
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   314
lemma mod_minus_cong:
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   315
  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
   316
  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
   317
proof -
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   318
  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
   319
    unfolding assms ..
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   320
  thus ?thesis
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   321
    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
   322
qed
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   323
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   324
text {* Subtraction respects modular equivalence. *}
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   325
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   326
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
   327
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   328
  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
   329
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   330
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
   331
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   332
  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
   333
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   334
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
   335
  unfolding diff_minus
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   336
  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
   337
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   338
lemma mod_diff_cong:
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   339
  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
   340
  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
   341
  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
   342
  unfolding diff_minus using assms
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   343
  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
   344
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   345
end
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   346
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   347
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   348
subsection {* Division on @{typ nat} *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   349
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   350
text {*
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   351
  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
   352
  of a characteristic relation with two input arguments
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   353
  @{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
   354
  @{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
   355
*}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   356
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   357
definition divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   358
  "divmod_rel m n q r \<longleftrightarrow> m = q * n + r \<and> (if n > 0 then 0 \<le> r \<and> r < n else q = 0)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   359
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   360
text {* @{const divmod_rel} is total: *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   361
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   362
lemma divmod_rel_ex:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   363
  obtains q r where "divmod_rel m n q r"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   364
proof (cases "n = 0")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   365
  case True with that show thesis
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   366
    by (auto simp add: divmod_rel_def)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   367
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   368
  case False
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   369
  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
   370
  proof (induct m)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   371
    case 0 with `n \<noteq> 0`
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   372
    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
   373
    then show ?case by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   374
  next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   375
    case (Suc m) then obtain q' r'
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   376
      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
   377
    then show ?case proof (cases "Suc r' < n")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   378
      case True
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   379
      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
   380
      with True show ?thesis by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   381
    next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   382
      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
   383
      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
   384
      ultimately have "n = Suc r'" by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   385
      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
   386
      with `n \<noteq> 0` show ?thesis by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   387
    qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   388
  qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   389
  with that show thesis
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   390
    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
   391
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   392
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   393
text {* @{const divmod_rel} is injective: *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   394
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   395
lemma divmod_rel_unique_div:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   396
  assumes "divmod_rel m n q r"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   397
    and "divmod_rel m n q' r'"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   398
  shows "q = q'"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   399
proof (cases "n = 0")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   400
  case True with assms show ?thesis
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   401
    by (simp add: divmod_rel_def)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   402
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   403
  case False
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   404
  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
   405
  apply (rule leI)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   406
  apply (subst less_iff_Suc_add)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   407
  apply (auto simp add: add_mult_distrib)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   408
  done
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   409
  from `n \<noteq> 0` assms show ?thesis
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   410
    by (auto simp add: divmod_rel_def
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   411
      intro: order_antisym dest: aux sym)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   412
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   413
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   414
lemma divmod_rel_unique_mod:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   415
  assumes "divmod_rel m n q r"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   416
    and "divmod_rel m n q' r'"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   417
  shows "r = r'"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   418
proof -
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   419
  from assms have "q = q'" by (rule divmod_rel_unique_div)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   420
  with assms show ?thesis by (simp add: divmod_rel_def)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   421
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   422
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   423
text {*
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   424
  We instantiate divisibility on the natural numbers by
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   425
  means of @{const divmod_rel}:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   426
*}
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   427
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   428
instantiation nat :: semiring_div
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   429
begin
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   430
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   431
definition divmod :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28262
diff changeset
   432
  [code del]: "divmod m n = (THE (q, r). divmod_rel m n q r)"
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   433
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   434
definition div_nat where
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   435
  "m div n = fst (divmod m n)"
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
definition mod_nat where
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   438
  "m mod n = snd (divmod m n)"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   439
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   440
lemma divmod_div_mod:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   441
  "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
   442
  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
   443
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   444
lemma divmod_eq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   445
  assumes "divmod_rel m n q r" 
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   446
  shows "divmod m n = (q, r)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   447
  using assms by (auto simp add: divmod_def
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   448
    dest: divmod_rel_unique_div divmod_rel_unique_mod)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   449
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   450
lemma div_eq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   451
  assumes "divmod_rel m n q r" 
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   452
  shows "m div n = q"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   453
  using assms by (auto dest: divmod_eq simp add: div_nat_def)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   454
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   455
lemma mod_eq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   456
  assumes "divmod_rel m n q r" 
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   457
  shows "m mod n = r"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   458
  using assms by (auto dest: divmod_eq simp add: mod_nat_def)
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25162
diff changeset
   459
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   460
lemma divmod_rel: "divmod_rel m n (m div n) (m mod n)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   461
proof -
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   462
  from divmod_rel_ex
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   463
    obtain q r where rel: "divmod_rel m n q r" .
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   464
  moreover with div_eq mod_eq have "m div n = q" and "m mod n = r"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   465
    by simp_all
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   466
  ultimately show ?thesis by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   467
qed
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   468
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   469
lemma divmod_zero:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   470
  "divmod m 0 = (0, m)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   471
proof -
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   472
  from divmod_rel [of m 0] show ?thesis
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   473
    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
   474
qed
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   475
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   476
lemma divmod_base:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   477
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   478
  shows "divmod m n = (0, m)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   479
proof -
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   480
  from divmod_rel [of m n] show ?thesis
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   481
    unfolding divmod_div_mod divmod_rel_def
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   482
    using assms by (cases "m div n = 0")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   483
      (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
   484
qed
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   485
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   486
lemma divmod_step:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   487
  assumes "0 < n" and "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   488
  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
   489
proof -
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   490
  from divmod_rel have divmod_m_n: "divmod_rel m n (m div n) (m mod n)" .
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   491
  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
   492
    by (cases "m div n") (auto simp add: divmod_rel_def)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   493
  from assms divmod_m_n have "divmod_rel (m - n) n (m div n - 1) (m mod n)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   494
    by (cases "m div n") (auto simp add: divmod_rel_def)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   495
  with divmod_eq have "divmod (m - n) n = (m div n - 1, m mod n)" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   496
  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
   497
  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
   498
    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
   499
  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
   500
qed
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   501
26300
03def556e26e removed duplicate lemmas;
wenzelm
parents: 26100
diff changeset
   502
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
   503
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   504
lemma div_less [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   505
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   506
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   507
  shows "m div n = 0"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   508
  using assms divmod_base divmod_div_mod by simp
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   509
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   510
lemma le_div_geq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   511
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   512
  assumes "0 < n" and "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   513
  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
   514
  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
   515
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   516
lemma mod_less [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   517
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   518
  assumes "m < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   519
  shows "m mod n = m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   520
  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
   521
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   522
lemma le_mod_geq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   523
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   524
  assumes "n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   525
  shows "m mod n = (m - n) mod n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   526
  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
   527
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   528
instance proof
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   529
  fix m n :: nat show "m div n * n + m mod n = m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   530
    using divmod_rel [of m n] by (simp add: divmod_rel_def)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   531
next
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   532
  fix n :: nat show "n div 0 = 0"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   533
    using divmod_zero divmod_div_mod [of n 0] by simp
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   534
next
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   535
  fix n :: nat show "0 div n = 0"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   536
    using divmod_rel [of 0 n] by (cases n) (simp_all add: divmod_rel_def)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   537
next
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   538
  fix m n q :: nat assume "n \<noteq> 0" then show "(q + m * n) div n = m + q div n"
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   539
    by (induct m) (simp_all add: le_div_geq)
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   540
qed
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   541
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   542
end
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   543
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   544
text {* Simproc for cancelling @{const div} and @{const mod} *}
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   545
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   546
(*lemmas mod_div_equality_nat = semiring_div_class.times_div_mod_plus_zero_one.mod_div_equality [of "m\<Colon>nat" n, standard]
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   547
lemmas mod_div_equality2_nat = mod_div_equality2 [of "n\<Colon>nat" m, standard*)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   548
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   549
ML {*
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   550
structure CancelDivModData =
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   551
struct
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   552
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   553
val div_name = @{const_name div};
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   554
val mod_name = @{const_name mod};
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   555
val mk_binop = HOLogic.mk_binop;
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   556
val mk_sum = ArithData.mk_sum;
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   557
val dest_sum = ArithData.dest_sum;
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   558
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   559
(*logic*)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   560
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   561
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   562
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   563
val trans = trans
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   564
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   565
val prove_eq_sums =
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   566
  let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac}
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   567
  in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end;
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   568
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   569
end;
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   570
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   571
structure CancelDivMod = CancelDivModFun(CancelDivModData);
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   572
28262
aa7ca36d67fd back to dynamic the_context(), because static @{theory} is invalidated if ML environment changes within the same code block;
wenzelm
parents: 27676
diff changeset
   573
val cancel_div_mod_proc = Simplifier.simproc (the_context ())
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   574
  "cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   575
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   576
Addsimprocs[cancel_div_mod_proc];
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   577
*}
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   578
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   579
text {* code generator setup *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   580
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   581
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
   582
  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
   583
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
   584
    (simp add: divmod_div_mod)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   585
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   586
code_modulename SML
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   587
  Divides Nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   588
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   589
code_modulename OCaml
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   590
  Divides Nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   591
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   592
code_modulename Haskell
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   593
  Divides Nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   594
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   595
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   596
subsubsection {* Quotient *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   597
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   598
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
   599
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
   600
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   601
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
   602
by (simp add: div_geq)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   603
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   604
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
   605
by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   606
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   607
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
   608
by simp
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   609
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   610
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   611
subsubsection {* Remainder *}
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   612
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   613
lemma mod_less_divisor [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   614
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   615
  assumes "n > 0"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   616
  shows "m mod n < (n::nat)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   617
  using assms divmod_rel 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
   618
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   619
lemma mod_less_eq_dividend [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   620
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   621
  shows "m mod n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   622
proof (rule add_leD2)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   623
  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
   624
  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
   625
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   626
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   627
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
   628
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
   629
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   630
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
   631
by (simp add: le_mod_geq)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   632
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   633
lemma mod_1 [simp]: "m mod Suc 0 = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   634
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
   635
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   636
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   637
  apply (cases "n = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   638
  apply (cases "k = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   639
  apply (induct m rule: nat_less_induct)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   640
  apply (subst mod_if, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   641
  apply (simp add: mod_geq diff_mult_distrib)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   642
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   643
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   644
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
   645
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
   646
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   647
(* 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
   648
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
   649
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
   650
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   651
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   652
  apply (drule mod_less_divisor [where m = m])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   653
  apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   654
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   655
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   656
subsubsection {* Quotient and Remainder *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   657
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   658
lemma divmod_rel_mult1_eq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   659
  "[| divmod_rel b c q r; c > 0 |]
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   660
   ==> 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
   661
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
   662
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   663
lemma div_mult1_eq: "(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
   664
apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   665
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
   666
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   667
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   668
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   669
by (rule mod_mult_right_eq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   670
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   671
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   672
by (rule mod_mult_left_eq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   673
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   674
lemma mod_mult_distrib_mod:
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   675
  "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   676
by (rule mod_mult_eq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   677
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   678
lemma divmod_rel_add1_eq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   679
  "[| divmod_rel a c aq ar; divmod_rel b c bq br;  c > 0 |]
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   680
   ==> 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
   681
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
   682
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   683
(*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
   684
lemma div_add1_eq:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   685
  "(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
   686
apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   687
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
   688
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   689
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   690
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   691
by (rule mod_add_eq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   692
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   693
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   694
  apply (cut_tac m = q and n = c in mod_less_divisor)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   695
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   696
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   697
  apply (simp add: add_mult_distrib2)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   698
  done
10559
d3fd54fc659b many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents: 10214
diff changeset
   699
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   700
lemma divmod_rel_mult2_eq: "[| divmod_rel a b q r;  0 < b;  0 < c |]
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   701
      ==> 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
   702
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
   703
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   704
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   705
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   706
  apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   707
  apply (force simp add: divmod_rel [THEN divmod_rel_mult2_eq, THEN div_eq])
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   708
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   709
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   710
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
   711
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   712
  apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   713
  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
   714
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   715
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   716
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   717
subsubsection{*Cancellation of Common Factors in Division*}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   718
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   719
lemma div_mult_mult_lemma:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   720
    "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   721
by (auto simp add: div_mult2_eq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   722
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   723
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   724
  apply (cases "b = 0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   725
  apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   726
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   727
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   728
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   729
  apply (drule div_mult_mult1)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   730
  apply (auto simp add: mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   731
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   732
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   733
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   734
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
   735
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   736
lemma div_1 [simp]: "m div Suc 0 = m"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   737
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
   738
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   739
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   740
(* Monotonicity of div in first argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   741
lemma div_le_mono [rule_format (no_asm)]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   742
    "\<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
   743
apply (case_tac "k=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   744
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
   745
apply (case_tac "n<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   746
(* 1  case n<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   747
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   748
(* 2  case n >= k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   749
apply (case_tac "m<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   750
(* 2.1  case m<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   751
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   752
(* 2.2  case m>=k *)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   753
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
   754
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   755
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   756
(* Antimonotonicity of div in second argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   757
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
   758
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   759
 prefer 2 apply simp
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   760
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
   761
apply (rename_tac "k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   762
apply (case_tac "k<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   763
apply (subgoal_tac "~ (k<m) ")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   764
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   765
apply (simp add: div_geq)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   766
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
   767
 prefer 2
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   768
 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
   769
apply (rule le_trans, simp)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   770
apply (simp)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   771
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   772
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   773
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
   774
apply (case_tac "n=0", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   775
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
   776
apply (rule div_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   777
apply (simp_all (no_asm_simp))
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   778
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   779
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   780
(* Similar for "less than" *)
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   781
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
   782
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   783
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
   784
apply (rename_tac "m")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   785
apply (case_tac "m<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   786
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   787
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   788
apply (simp add: div_geq)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   789
apply (case_tac "n<m")
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   790
 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
   791
  apply (rule impI less_trans_Suc)+
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   792
apply assumption
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   793
  apply (simp_all)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   794
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   795
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   796
declare div_less_dividend [simp]
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   797
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   798
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
   799
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
   800
apply (case_tac "n=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   801
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
   802
apply (case_tac "Suc (na) <n")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   803
(* case Suc(na) < n *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   804
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
   805
(* case n \<le> Suc(na) *)
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16733
diff changeset
   806
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   807
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
   808
done
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
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   811
subsubsection {* The Divides Relation *}
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24268
diff changeset
   812
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   813
lemma dvd_1_left [iff]: "Suc 0 dvd k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   814
  unfolding dvd_def by simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   815
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   816
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   817
by (simp add: dvd_def)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   818
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   819
lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   820
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   821
  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   822
23684
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   823
text {* @{term "op dvd"} is a partial order *}
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   824
29509
1ff0f3f08a7b migrated class package to new locale implementation
haftmann
parents: 29405
diff changeset
   825
interpretation dvd!: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28562
diff changeset
   826
  proof qed (auto intro: dvd_refl dvd_trans dvd_anti_sym)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   827
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30027
diff changeset
   828
lemma nat_dvd_diff[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30027
diff changeset
   829
unfolding dvd_def
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30027
diff changeset
   830
by (blast intro: diff_mult_distrib2 [symmetric])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   831
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   832
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   833
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   834
  apply (blast intro: dvd_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   835
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   836
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   837
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30027
diff changeset
   838
by (drule_tac m = m in nat_dvd_diff, auto)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   839
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   840
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   841
  apply (rule iffI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   842
   apply (erule_tac [2] dvd_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   843
   apply (rule_tac [2] dvd_refl)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   844
  apply (subgoal_tac "n = (n+k) -k")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   845
   prefer 2 apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   846
  apply (erule ssubst)
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30027
diff changeset
   847
  apply (erule nat_dvd_diff)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   848
  apply (rule dvd_refl)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   849
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   850
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   851
lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   852
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   853
  apply (case_tac "n = 0", auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   854
  apply (blast intro: mod_mult_distrib2 [symmetric])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   855
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   856
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   857
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   858
by (blast intro: dvd_mod_imp_dvd dvd_mod)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   859
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   860
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   861
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   862
  apply (erule exE)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   863
  apply (simp add: mult_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   864
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   865
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   866
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   867
  apply auto
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   868
   apply (subgoal_tac "m*n dvd m*1")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   869
   apply (drule dvd_mult_cancel, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   870
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   871
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   872
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   873
  apply (subst mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   874
  apply (erule dvd_mult_cancel1)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   875
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   876
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   877
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   878
  apply (unfold dvd_def, clarify)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   879
  apply (simp_all (no_asm_use) add: zero_less_mult_iff)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   880
  apply (erule conjE)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   881
  apply (rule le_trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   882
   apply (rule_tac [2] le_refl [THEN mult_le_mono])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   883
   apply (erule_tac [2] Suc_leI, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   884
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   885
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   886
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   887
  apply (subgoal_tac "m mod n = 0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   888
   apply (simp add: mult_div_cancel)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   889
  apply (simp only: dvd_eq_mod_eq_0)
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
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   892
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   893
  by (induct n) auto
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   894
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   895
lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   896
  apply (rule power_le_imp_le_exp, assumption)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   897
  apply (erule dvd_imp_le, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   898
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   899
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   900
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
   901
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
   902
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   903
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
   904
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   905
(*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
   906
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
   907
  apply (cut_tac a = m in mod_div_equality)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   908
  apply (simp only: add_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   909
  apply (blast intro: sym)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   910
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   911
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
   912
lemma split_div:
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   913
 "P(n div k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   914
 ((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
   915
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   916
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   917
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   918
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   919
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   920
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   921
    with P show ?Q by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   922
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   923
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   924
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   925
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   926
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   927
      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
   928
      show "P i"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   929
      proof (cases)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   930
        assume "i = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   931
        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
   932
      next
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   933
        assume "i \<noteq> 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   934
        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
   935
      qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   936
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   937
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   938
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   939
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   940
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   941
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   942
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   943
    with Q show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   944
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   945
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   946
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   947
    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
   948
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   949
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   950
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   951
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   952
lemma split_div_lemma:
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   953
  assumes "0 < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   954
  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
   955
proof
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   956
  assume ?rhs
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   957
  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
   958
  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
   959
  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
   960
  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
   961
  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
   962
  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
   963
  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
   964
  from A B show ?lhs ..
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   965
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   966
  assume P: ?lhs
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   967
  then have "divmod_rel m n q (m - n * q)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   968
    unfolding divmod_rel_def by (auto simp add: mult_ac)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   969
  then show ?rhs using divmod_rel by (rule divmod_rel_unique_div)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   970
qed
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   971
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   972
theorem split_div':
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   973
  "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
   974
   (\<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
   975
  apply (case_tac "0 < n")
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   976
  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
   977
  apply simp_all
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   978
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   979
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   980
lemma split_mod:
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   981
 "P(n mod k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   982
 ((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
   983
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   984
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   985
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   986
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   987
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   988
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   989
    with P show ?Q by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   990
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   991
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   992
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   993
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   994
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   995
      assume "n = k*i + j" "j < k"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   996
      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
   997
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   998
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   999
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1000
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1001
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1002
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1003
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
  1004
    with Q show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1005
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1006
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1007
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1008
    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
  1009
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1010
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1011
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1012
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1013
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
  1014
  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
  1015
    subst [OF mod_div_equality [of _ n]])
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1016
  apply arith
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1017
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1018
22800
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1019
lemma div_mod_equality':
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1020
  fixes m n :: nat
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1021
  shows "m div n * n = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1022
proof -
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1023
  have "m mod n \<le> m mod n" ..
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1024
  from div_mod_equality have 
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1025
    "m div n * n + m mod n - m mod n = m - m mod n" by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1026
  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
  1027
    "m div n * n + (m mod n - m mod n) = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1028
    by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1029
  then show ?thesis by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1030
qed
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1031
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1032
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
  1033
subsubsection {*An ``induction'' law for modulus arithmetic.*}
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1034
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1035
lemma mod_induct_0:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1036
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1037
  and base: "P i" and i: "i<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1038
  shows "P 0"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1039
proof (rule ccontr)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1040
  assume contra: "\<not>(P 0)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1041
  from i have p: "0<p" by simp
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1042
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1043
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1044
    fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1045
    show "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1046
    proof (induct k)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1047
      show "?A 0" by simp  -- "by contradiction"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1048
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1049
      fix n
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1050
      assume ih: "?A n"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1051
      show "?A (Suc n)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1052
      proof (clarsimp)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1053
        assume y: "P (p - Suc n)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1054
        have n: "Suc n < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1055
        proof (rule ccontr)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1056
          assume "\<not>(Suc n < p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1057
          hence "p - Suc n = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1058
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1059
          with y contra show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1060
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1061
        qed
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1062
        hence n2: "Suc (p - Suc n) = p-n" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1063
        from p have "p - Suc n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1064
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1065
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1066
        show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1067
        proof (cases "n=0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1068
          case True
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1069
          with z n2 contra show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1070
        next
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1071
          case False
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1072
          with p have "p-n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1073
          with z n2 False ih show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1074
        qed
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1075
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1076
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1077
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1078
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1079
  from i obtain k where "0<k \<and> i+k=p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1080
    by (blast dest: less_imp_add_positive)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1081
  hence "0<k \<and> i=p-k" by auto
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1082
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1083
  note base
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1084
  ultimately
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1085
  show "False" by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1086
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1087
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1088
lemma mod_induct:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1089
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1090
  and base: "P i" and i: "i<p" and j: "j<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1091
  shows "P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1092
proof -
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1093
  have "\<forall>j<p. P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1094
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1095
    fix j
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1096
    show "j<p \<longrightarrow> P j" (is "?A j")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1097
    proof (induct j)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1098
      from step base i show "?A 0"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1099
        by (auto elim: mod_induct_0)
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1100
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1101
      fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1102
      assume ih: "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1103
      show "?A (Suc k)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1104
      proof
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1105
        assume suc: "Suc k < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1106
        hence k: "k<p" by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1107
        with ih have "P k" ..
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1108
        with step k have "P (Suc k mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1109
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1110
        moreover
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1111
        from suc have "Suc k mod p = Suc k"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1112
          by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1113
        ultimately
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1114
        show "P (Suc k)" by simp
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1115
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1116
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1117
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1118
  with j show ?thesis by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1119
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1120
3366
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
  1121
end