src/HOL/Integ/IntDef.thy
author obua
Tue, 11 May 2004 20:11:08 +0200
changeset 14738 83f1a514dcb4
parent 14691 e1eedc8cad37
child 14740 c8e1937110c2
permissions -rw-r--r--
changes made due to new Ring_and_Field theory
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
5508
691c70898518 new files in Integ
paulson
parents:
diff changeset
     1
(*  Title:      IntDef.thy
691c70898518 new files in Integ
paulson
parents:
diff changeset
     2
    ID:         $Id$
691c70898518 new files in Integ
paulson
parents:
diff changeset
     3
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
691c70898518 new files in Integ
paulson
parents:
diff changeset
     4
    Copyright   1996  University of Cambridge
691c70898518 new files in Integ
paulson
parents:
diff changeset
     5
691c70898518 new files in Integ
paulson
parents:
diff changeset
     6
*)
691c70898518 new files in Integ
paulson
parents:
diff changeset
     7
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 14269
diff changeset
     8
header{*The Integers as Equivalence Classes over Pairs of Natural Numbers*}
8ed6989228bb Simplification of the development of Integers
paulson
parents: 14269
diff changeset
     9
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    10
theory IntDef = Equiv + NatArith:
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    11
5508
691c70898518 new files in Integ
paulson
parents:
diff changeset
    12
constdefs
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 14269
diff changeset
    13
  intrel :: "((nat * nat) * (nat * nat)) set"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    14
    --{*the equivalence relation underlying the integers*}
14496
paulson
parents: 14485
diff changeset
    15
    "intrel == {((x,y),(u,v)) | x y u v. x+v = u+y}"
5508
691c70898518 new files in Integ
paulson
parents:
diff changeset
    16
691c70898518 new files in Integ
paulson
parents:
diff changeset
    17
typedef (Integ)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    18
  int = "UNIV//intrel"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    19
    by (auto simp add: quotient_def)
5508
691c70898518 new files in Integ
paulson
parents:
diff changeset
    20
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14658
diff changeset
    21
instance int :: "{ord, zero, one, plus, times, minus}" ..
5508
691c70898518 new files in Integ
paulson
parents:
diff changeset
    22
691c70898518 new files in Integ
paulson
parents:
diff changeset
    23
constdefs
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    24
  int :: "nat => int"
10834
a7897aebbffc *** empty log message ***
nipkow
parents: 10797
diff changeset
    25
  "int m == Abs_Integ(intrel `` {(m,0)})"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    26
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    27
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14259
diff changeset
    28
defs (overloaded)
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 14269
diff changeset
    29
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    30
  Zero_int_def:  "0 == int 0"
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 14269
diff changeset
    31
  One_int_def:   "1 == int 1"
8937
7328d7c8d201 defining 0::int to be (int 0)
paulson
parents: 7375
diff changeset
    32
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    33
  minus_int_def:
14532
paulson
parents: 14496
diff changeset
    34
    "- z == Abs_Integ (\<Union>(x,y) \<in> Rep_Integ z. intrel``{(y,x)})"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    35
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    36
  add_int_def:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    37
   "z + w ==
14532
paulson
parents: 14496
diff changeset
    38
       Abs_Integ (\<Union>(x,y) \<in> Rep_Integ z. \<Union>(u,v) \<in> Rep_Integ w.
paulson
parents: 14496
diff changeset
    39
		 intrel``{(x+u, y+v)})"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    40
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    41
  diff_int_def:  "z - (w::int) == z + (-w)"
5508
691c70898518 new files in Integ
paulson
parents:
diff changeset
    42
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    43
  mult_int_def:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    44
   "z * w ==
14532
paulson
parents: 14496
diff changeset
    45
       Abs_Integ (\<Union>(x,y) \<in> Rep_Integ z. \<Union>(u,v) \<in> Rep_Integ w.
paulson
parents: 14496
diff changeset
    46
		  intrel``{(x*u + y*v, x*v + y*u)})"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    47
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    48
  le_int_def:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    49
   "z \<le> (w::int) == 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    50
    \<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Integ z & (u,v) \<in> Rep_Integ w"
5508
691c70898518 new files in Integ
paulson
parents:
diff changeset
    51
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    52
  less_int_def: "(z < (w::int)) == (z \<le> w & z \<noteq> w)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    53
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    54
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    55
subsection{*Construction of the Integers*}
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
    56
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    57
subsubsection{*Preliminary Lemmas about the Equivalence Relation*}
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
    58
14496
paulson
parents: 14485
diff changeset
    59
lemma intrel_iff [simp]: "(((x,y),(u,v)) \<in> intrel) = (x+v = u+y)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    60
by (simp add: intrel_def)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    61
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    62
lemma equiv_intrel: "equiv UNIV intrel"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    63
by (simp add: intrel_def equiv_def refl_def sym_def trans_def)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    64
14496
paulson
parents: 14485
diff changeset
    65
text{*Reduces equality of equivalence classes to the @{term intrel} relation:
paulson
parents: 14485
diff changeset
    66
  @{term "(intrel `` {x} = intrel `` {y}) = ((x,y) \<in> intrel)"} *}
paulson
parents: 14485
diff changeset
    67
lemmas equiv_intrel_iff = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I]
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    68
14496
paulson
parents: 14485
diff changeset
    69
declare equiv_intrel_iff [simp]
paulson
parents: 14485
diff changeset
    70
paulson
parents: 14485
diff changeset
    71
paulson
parents: 14485
diff changeset
    72
text{*All equivalence classes belong to set of representatives*}
14532
paulson
parents: 14496
diff changeset
    73
lemma [simp]: "intrel``{(x,y)} \<in> Integ"
14496
paulson
parents: 14485
diff changeset
    74
by (auto simp add: Integ_def intrel_def quotient_def)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    75
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    76
lemma inj_on_Abs_Integ: "inj_on Abs_Integ Integ"
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    77
apply (rule inj_on_inverseI)
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    78
apply (erule Abs_Integ_inverse)
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    79
done
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    80
14496
paulson
parents: 14485
diff changeset
    81
text{*This theorem reduces equality on abstractions to equality on 
paulson
parents: 14485
diff changeset
    82
      representatives:
paulson
parents: 14485
diff changeset
    83
  @{term "\<lbrakk>x \<in> Integ; y \<in> Integ\<rbrakk> \<Longrightarrow> (Abs_Integ x = Abs_Integ y) = (x=y)"} *}
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    84
declare inj_on_Abs_Integ [THEN inj_on_iff, simp]
14496
paulson
parents: 14485
diff changeset
    85
paulson
parents: 14485
diff changeset
    86
declare Abs_Integ_inverse [simp]
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    87
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    88
text{*Case analysis on the representation of an integer as an equivalence
14485
ea2707645af8 new material from Avigad
paulson
parents: 14479
diff changeset
    89
      class of pairs of naturals.*}
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    90
lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    91
     "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    92
apply (rule Rep_Integ [of z, unfolded Integ_def, THEN quotientE])
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    93
apply (drule arg_cong [where f=Abs_Integ])
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    94
apply (auto simp add: Rep_Integ_inverse)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    95
done
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    96
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    97
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
    98
subsubsection{*@{term int}: Embedding the Naturals into the Integers*}
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
    99
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   100
lemma inj_int: "inj int"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   101
by (simp add: inj_on_def int_def)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   102
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   103
lemma int_int_eq [iff]: "(int m = int n) = (m = n)"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   104
by (fast elim!: inj_int [THEN injD])
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   105
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   106
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   107
subsubsection{*Integer Unary Negation*}
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   108
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   109
lemma minus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   110
proof -
14532
paulson
parents: 14496
diff changeset
   111
  have "congruent intrel (\<lambda>(x,y). intrel``{(y,x)})"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   112
    by (simp add: congruent_def) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   113
  thus ?thesis
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   114
    by (simp add: minus_int_def UN_equiv_class [OF equiv_intrel])
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   115
qed
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   116
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   117
lemma zminus_zminus: "- (- z) = (z::int)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   118
by (cases z, simp add: minus)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   119
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   120
lemma zminus_0: "- 0 = (0::int)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   121
by (simp add: int_def Zero_int_def minus)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   122
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   123
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   124
subsection{*Integer Addition*}
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   125
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   126
lemma add:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   127
     "Abs_Integ (intrel``{(x,y)}) + Abs_Integ (intrel``{(u,v)}) =
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   128
      Abs_Integ (intrel``{(x+u, y+v)})"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   129
proof -
14658
b1293d0f8d5f congruent2 now allows different equiv relations
paulson
parents: 14532
diff changeset
   130
  have "congruent2 intrel intrel
14532
paulson
parents: 14496
diff changeset
   131
        (\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). intrel `` {(x+u, y+v)}) w) z)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   132
    by (simp add: congruent2_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   133
  thus ?thesis
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   134
    by (simp add: add_int_def UN_UN_split_split_eq
14658
b1293d0f8d5f congruent2 now allows different equiv relations
paulson
parents: 14532
diff changeset
   135
                  UN_equiv_class2 [OF equiv_intrel equiv_intrel])
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   136
qed
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   137
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   138
lemma zminus_zadd_distrib: "- (z + w) = (- z) + (- w::int)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   139
by (cases z, cases w, simp add: minus add)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   140
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   141
lemma zadd_commute: "(z::int) + w = w + z"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   142
by (cases z, cases w, simp add: add_ac add)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   143
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   144
lemma zadd_assoc: "((z1::int) + z2) + z3 = z1 + (z2 + z3)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   145
by (cases z1, cases z2, cases z3, simp add: add add_assoc)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   146
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   147
(*For AC rewriting*)
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 14269
diff changeset
   148
lemma zadd_left_commute: "x + (y + z) = y + ((x + z)  ::int)"
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   149
  apply (rule mk_left_commute [of "op +"])
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   150
  apply (rule zadd_assoc)
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   151
  apply (rule zadd_commute)
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   152
  done
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   153
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   154
lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   155
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   156
lemmas zmult_ac = OrderedGroup.mult_ac
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 14269
diff changeset
   157
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   158
lemma zadd_int: "(int m) + (int n) = int (m + n)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   159
by (simp add: int_def add)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   160
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   161
lemma zadd_int_left: "(int m) + (int n + z) = int (m + n) + z"
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   162
by (simp add: zadd_int zadd_assoc [symmetric])
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   163
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   164
lemma int_Suc: "int (Suc m) = 1 + (int m)"
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   165
by (simp add: One_int_def zadd_int)
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   166
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   167
(*also for the instance declaration int :: comm_monoid_add*)
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   168
lemma zadd_0: "(0::int) + z = z"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   169
apply (simp add: Zero_int_def int_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   170
apply (cases z, simp add: add)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   171
done
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   172
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   173
lemma zadd_0_right: "z + (0::int) = z"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   174
by (rule trans [OF zadd_commute zadd_0])
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   175
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   176
lemma zadd_zminus_inverse2: "(- z) + z = (0::int)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   177
by (cases z, simp add: int_def Zero_int_def minus add)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   178
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   179
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   180
subsection{*Integer Multiplication*}
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   181
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   182
text{*Congruence property for multiplication*}
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   183
lemma mult_congruent2:
14658
b1293d0f8d5f congruent2 now allows different equiv relations
paulson
parents: 14532
diff changeset
   184
     "congruent2 intrel intrel
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   185
        (%p1 p2. (%(x,y). (%(u,v).
14532
paulson
parents: 14496
diff changeset
   186
                    intrel``{(x*u + y*v, x*v + y*u)}) p2) p1)"
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   187
apply (rule equiv_intrel [THEN congruent2_commuteI])
14532
paulson
parents: 14496
diff changeset
   188
 apply (force simp add: mult_ac, clarify) 
paulson
parents: 14496
diff changeset
   189
apply (simp add: congruent_def mult_ac)  
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   190
apply (rename_tac u v w x y z)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   191
apply (subgoal_tac "u*y + x*y = w*y + v*y  &  u*z + x*z = w*z + v*z")
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   192
apply (simp add: mult_ac, arith)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   193
apply (simp add: add_mult_distrib [symmetric])
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   194
done
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   195
14532
paulson
parents: 14496
diff changeset
   196
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   197
lemma mult:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   198
     "Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) =
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   199
      Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   200
by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2
14658
b1293d0f8d5f congruent2 now allows different equiv relations
paulson
parents: 14532
diff changeset
   201
              UN_equiv_class2 [OF equiv_intrel equiv_intrel])
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   202
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   203
lemma zmult_zminus: "(- z) * w = - (z * (w::int))"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   204
by (cases z, cases w, simp add: minus mult add_ac)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   205
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   206
lemma zmult_commute: "(z::int) * w = w * z"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   207
by (cases z, cases w, simp add: mult add_ac mult_ac)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   208
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   209
lemma zmult_assoc: "((z1::int) * z2) * z3 = z1 * (z2 * z3)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   210
by (cases z1, cases z2, cases z3, simp add: mult add_mult_distrib2 mult_ac)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   211
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   212
lemma zadd_zmult_distrib: "((z1::int) + z2) * w = (z1 * w) + (z2 * w)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   213
by (cases z1, cases z2, cases w, simp add: add mult add_mult_distrib2 mult_ac)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   214
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   215
lemma zadd_zmult_distrib2: "(w::int) * (z1 + z2) = (w * z1) + (w * z2)"
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   216
by (simp add: zmult_commute [of w] zadd_zmult_distrib)
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   217
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   218
lemma zdiff_zmult_distrib: "((z1::int) - z2) * w = (z1 * w) - (z2 * w)"
14496
paulson
parents: 14485
diff changeset
   219
by (simp add: diff_int_def zadd_zmult_distrib zmult_zminus)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   220
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   221
lemma zdiff_zmult_distrib2: "(w::int) * (z1 - z2) = (w * z1) - (w * z2)"
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   222
by (simp add: zmult_commute [of w] zdiff_zmult_distrib)
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   223
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   224
lemmas int_distrib =
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   225
  zadd_zmult_distrib zadd_zmult_distrib2
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   226
  zdiff_zmult_distrib zdiff_zmult_distrib2
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   227
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   228
lemma zmult_int: "(int m) * (int n) = int (m * n)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   229
by (simp add: int_def mult)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   230
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   231
lemma zmult_1: "(1::int) * z = z"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   232
by (cases z, simp add: One_int_def int_def mult)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   233
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   234
lemma zmult_1_right: "z * (1::int) = z"
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   235
by (rule trans [OF zmult_commute zmult_1])
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   236
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   237
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   238
text{*The Integers Form A comm_ring_1*}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   239
instance int :: comm_ring_1
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   240
proof
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   241
  fix i j k :: int
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   242
  show "(i + j) + k = i + (j + k)" by (simp add: zadd_assoc)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   243
  show "i + j = j + i" by (simp add: zadd_commute)
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   244
  show "0 + i = i" by (rule zadd_0)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   245
  show "- i + i = 0" by (rule zadd_zminus_inverse2)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   246
  show "i - j = i + (-j)" by (simp add: diff_int_def)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   247
  show "(i * j) * k = i * (j * k)" by (rule zmult_assoc)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   248
  show "i * j = j * i" by (rule zmult_commute)
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   249
  show "1 * i = i" by (rule zmult_1) 
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   250
  show "(i + j) * k = i * k + j * k" by (simp add: int_distrib)
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   251
  show "0 \<noteq> (1::int)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   252
    by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   253
qed
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   254
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   255
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   256
subsection{*The @{text "\<le>"} Ordering*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   257
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   258
lemma le:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   259
  "(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   260
by (force simp add: le_int_def)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   261
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   262
lemma zle_refl: "w \<le> (w::int)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   263
by (cases w, simp add: le)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   264
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   265
lemma zle_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::int)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   266
by (cases i, cases j, cases k, simp add: le)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   267
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   268
lemma zle_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::int)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   269
by (cases w, cases z, simp add: le)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   270
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   271
(* Axiom 'order_less_le' of class 'order': *)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   272
lemma zless_le: "((w::int) < z) = (w \<le> z & w \<noteq> z)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   273
by (simp add: less_int_def)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   274
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   275
instance int :: order
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14658
diff changeset
   276
  by intro_classes
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14658
diff changeset
   277
    (assumption |
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14658
diff changeset
   278
      rule zle_refl zle_trans zle_anti_sym zless_le)+
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   279
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   280
(* Axiom 'linorder_linear' of class 'linorder': *)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   281
lemma zle_linear: "(z::int) \<le> w | w \<le> z"
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14658
diff changeset
   282
by (cases z, cases w) (simp add: le linorder_linear)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   283
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   284
instance int :: linorder
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14658
diff changeset
   285
  by intro_classes (rule zle_linear)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   286
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   287
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   288
lemmas zless_linear = linorder_less_linear [where 'a = int]
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   289
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   290
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   291
lemma int_eq_0_conv [simp]: "(int n = 0) = (n = 0)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   292
by (simp add: Zero_int_def)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   293
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   294
lemma zless_int [simp]: "(int m < int n) = (m<n)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   295
by (simp add: le add int_def linorder_not_le [symmetric]) 
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   296
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   297
lemma int_less_0_conv [simp]: "~ (int k < 0)"
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   298
by (simp add: Zero_int_def)
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   299
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   300
lemma zero_less_int_conv [simp]: "(0 < int n) = (0 < n)"
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   301
by (simp add: Zero_int_def)
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   302
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   303
lemma int_0_less_1: "0 < (1::int)"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   304
by (simp only: Zero_int_def One_int_def One_nat_def zless_int)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   305
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   306
lemma int_0_neq_1 [simp]: "0 \<noteq> (1::int)"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   307
by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   308
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   309
lemma zle_int [simp]: "(int m \<le> int n) = (m\<le>n)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   310
by (simp add: linorder_not_less [symmetric])
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   311
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   312
lemma zero_zle_int [simp]: "(0 \<le> int n)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   313
by (simp add: Zero_int_def)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   314
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   315
lemma int_le_0_conv [simp]: "(int n \<le> 0) = (n = 0)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   316
by (simp add: Zero_int_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   317
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   318
lemma int_0 [simp]: "int 0 = (0::int)"
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   319
by (simp add: Zero_int_def)
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   320
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   321
lemma int_1 [simp]: "int 1 = 1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   322
by (simp add: One_int_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   323
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   324
lemma int_Suc0_eq_1: "int (Suc 0) = 1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   325
by (simp add: One_int_def One_nat_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   326
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   327
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   328
subsection{*Monotonicity results*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   329
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   330
lemma zadd_left_mono: "i \<le> j ==> k + i \<le> k + (j::int)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   331
by (cases i, cases j, cases k, simp add: le add)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   332
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   333
lemma zadd_strict_right_mono: "i < j ==> i + k < j + (k::int)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   334
apply (cases i, cases j, cases k)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   335
apply (simp add: linorder_not_le [where 'a = int, symmetric]
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   336
                 linorder_not_le [where 'a = nat]  le add)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   337
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   338
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   339
lemma zadd_zless_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::int)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   340
by (rule order_less_le_trans [OF zadd_strict_right_mono zadd_left_mono])
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   341
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   342
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   343
subsection{*Strict Monotonicity of Multiplication*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   344
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   345
text{*strict, in 1st argument; proof is by induction on k>0*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   346
lemma zmult_zless_mono2_lemma [rule_format]:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   347
     "i<j ==> 0<k --> int k * i < int k * j"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   348
apply (induct_tac "k", simp)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   349
apply (simp add: int_Suc)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   350
apply (case_tac "n=0")
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   351
apply (simp_all add: zadd_zmult_distrib int_Suc0_eq_1 order_le_less)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   352
apply (simp add: zadd_zless_mono int_Suc0_eq_1 order_le_less)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   353
done
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   354
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   355
lemma zero_le_imp_eq_int: "0 \<le> k ==> \<exists>n. k = int n"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   356
apply (cases k)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   357
apply (auto simp add: le add int_def Zero_int_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   358
apply (rule_tac x="x-y" in exI, simp)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   359
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   360
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   361
lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   362
apply (frule order_less_imp_le [THEN zero_le_imp_eq_int])
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   363
apply (auto simp add: zmult_zless_mono2_lemma)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   364
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   365
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   366
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   367
defs (overloaded)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   368
    zabs_def:  "abs(i::int) == if i < 0 then -i else i"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   369
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   370
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   371
text{*The Integers Form an Ordered comm_ring_1*}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   372
instance int :: ordered_idom
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   373
proof
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   374
  fix i j k :: int
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   375
  show "i \<le> j ==> k + i \<le> k + j" by (rule zadd_left_mono)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   376
  show "i < j ==> 0 < k ==> k * i < k * j" by (rule zmult_zless_mono2)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   377
  show "\<bar>i\<bar> = (if i < 0 then -i else i)" by (simp only: zabs_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   378
qed
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   379
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   380
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   381
lemma zless_imp_add1_zle: "w<z ==> w + (1::int) \<le> z"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   382
apply (cases w, cases z) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   383
apply (simp add: linorder_not_le [symmetric] le int_def add One_int_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   384
done
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   385
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   386
subsection{*Magnitide of an Integer, as a Natural Number: @{term nat}*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   387
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   388
constdefs
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   389
   nat  :: "int => nat"
14532
paulson
parents: 14496
diff changeset
   390
    "nat z == contents (\<Union>(x,y) \<in> Rep_Integ z. { x-y })"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   391
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   392
lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   393
proof -
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   394
  have "congruent intrel (\<lambda>(x,y). {x-y})"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   395
    by (simp add: congruent_def, arith) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   396
  thus ?thesis
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   397
    by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   398
qed
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   399
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   400
lemma nat_int [simp]: "nat(int n) = n"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   401
by (simp add: nat int_def) 
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   402
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   403
lemma nat_zero [simp]: "nat 0 = 0"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   404
by (simp only: Zero_int_def nat_int)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   405
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   406
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   407
by (cases z, simp add: nat le int_def Zero_int_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   408
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   409
corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   410
apply simp 
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   411
done
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   412
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   413
lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   414
by (cases z, simp add: nat le int_def Zero_int_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   415
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   416
lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   417
apply (cases w, cases z) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   418
apply (simp add: nat le linorder_not_le [symmetric] int_def Zero_int_def, arith)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   419
done
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   420
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   421
text{*An alternative condition is @{term "0 \<le> w"} *}
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   422
corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   423
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   424
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   425
corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   426
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   427
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   428
lemma zless_nat_conj: "(nat w < nat z) = (0 < z & w < z)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   429
apply (cases w, cases z) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   430
apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   431
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   432
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   433
lemma nonneg_eq_int: "[| 0 \<le> z;  !!m. z = int m ==> P |] ==> P"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   434
by (blast dest: nat_0_le sym)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   435
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   436
lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   437
by (cases w, simp add: nat le int_def Zero_int_def, arith)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   438
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   439
corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int m else m=0)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   440
by (simp only: eq_commute [of m] nat_eq_iff) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   441
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   442
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < int m)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   443
apply (cases w)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   444
apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   445
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   446
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   447
lemma int_eq_iff: "(int m = z) = (m = nat z & 0 \<le> z)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   448
by (auto simp add: nat_eq_iff2)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   449
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   450
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   451
by (insert zless_nat_conj [of 0], auto)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   452
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   453
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   454
lemma nat_add_distrib:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   455
     "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   456
by (cases z, cases z', simp add: nat add le int_def Zero_int_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   457
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   458
lemma nat_diff_distrib:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   459
     "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   460
by (cases z, cases z', 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   461
    simp add: nat add minus diff_minus le int_def Zero_int_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   462
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   463
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   464
lemma nat_zminus_int [simp]: "nat (- (int n)) = 0"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   465
by (simp add: int_def minus nat Zero_int_def) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   466
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   467
lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   468
by (cases z, simp add: nat le int_def  linorder_not_le [symmetric], arith)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   469
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   470
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   471
subsection{*Lemmas about the Function @{term int} and Orderings*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   472
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   473
lemma negative_zless_0: "- (int (Suc n)) < 0"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   474
by (simp add: order_less_le)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   475
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   476
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   477
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   478
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   479
lemma negative_zle_0: "- int n \<le> 0"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   480
by (simp add: minus_le_iff)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   481
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   482
lemma negative_zle [iff]: "- int n \<le> int m"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   483
by (rule order_trans [OF negative_zle_0 zero_zle_int])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   484
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   485
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   486
by (subst le_minus_iff, simp)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   487
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   488
lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   489
by (simp add: int_def le minus Zero_int_def) 
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   490
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   491
lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   492
by (simp add: linorder_not_less)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   493
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   494
lemma negative_eq_positive [simp]: "(- int n = int m) = (n = 0 & m = 0)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   495
by (force simp add: order_eq_iff [of "- int n"] int_zle_neg)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   496
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   497
lemma zle_iff_zadd: "(w \<le> z) = (\<exists>n. z = w + int n)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   498
apply (cases w, cases z)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   499
apply (auto simp add: le add int_def) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   500
apply (rename_tac a b c d) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   501
apply (rule_tac x="c+b - (a+d)" in exI) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   502
apply arith
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   503
done
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   504
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   505
lemma abs_int_eq [simp]: "abs (int m) = int m"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   506
by (simp add: zabs_def)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   507
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   508
text{*This version is proved for all ordered rings, not just integers!
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   509
      It is proved here because attribute @{text arith_split} is not available
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   510
      in theory @{text Ring_and_Field}.
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   511
      But is it really better than just rewriting with @{text abs_if}?*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   512
lemma abs_split [arith_split]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   513
     "P(abs(a::'a::ordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   514
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   515
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   516
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   517
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   518
subsection{*The Constants @{term neg} and @{term iszero}*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   519
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   520
constdefs
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   521
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   522
  neg   :: "'a::ordered_idom => bool"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   523
  "neg(Z) == Z < 0"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   524
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   525
  (*For simplifying equalities*)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   526
  iszero :: "'a::comm_semiring_1_cancel => bool"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   527
  "iszero z == z = (0)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   528
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   529
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   530
lemma not_neg_int [simp]: "~ neg(int n)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   531
by (simp add: neg_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   532
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   533
lemma neg_zminus_int [simp]: "neg(- (int (Suc n)))"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   534
by (simp add: neg_def neg_less_0_iff_less)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   535
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   536
lemmas neg_eq_less_0 = neg_def
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   537
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   538
lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   539
by (simp add: neg_def linorder_not_less)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   540
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   541
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   542
subsection{*To simplify inequalities when Numeral1 can get simplified to 1*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   543
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   544
lemma not_neg_0: "~ neg 0"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   545
by (simp add: One_int_def neg_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   546
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   547
lemma not_neg_1: "~ neg 1"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   548
by (simp add: neg_def linorder_not_less zero_le_one)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   549
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   550
lemma iszero_0: "iszero 0"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   551
by (simp add: iszero_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   552
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   553
lemma not_iszero_1: "~ iszero 1"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   554
by (simp add: iszero_def eq_commute)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   555
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   556
lemma neg_nat: "neg z ==> nat z = 0"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   557
by (simp add: neg_def order_less_imp_le) 
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   558
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   559
lemma not_neg_nat: "~ neg z ==> int (nat z) = z"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   560
by (simp add: linorder_not_less neg_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   561
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   562
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   563
subsection{*Embedding of the Naturals into any comm_semiring_1_cancel: @{term of_nat}*}
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   564
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   565
consts of_nat :: "nat => 'a::comm_semiring_1_cancel"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   566
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   567
primrec
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   568
  of_nat_0:   "of_nat 0 = 0"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   569
  of_nat_Suc: "of_nat (Suc m) = of_nat m + 1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   570
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   571
lemma of_nat_1 [simp]: "of_nat 1 = 1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   572
by simp
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   573
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   574
lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   575
apply (induct m)
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   576
apply (simp_all add: add_ac)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   577
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   578
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   579
lemma of_nat_mult [simp]: "of_nat (m*n) = of_nat m * of_nat n"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   580
apply (induct m)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   581
apply (simp_all add: mult_ac add_ac right_distrib)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   582
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   583
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   584
lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semidom)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   585
apply (induct m, simp_all)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   586
apply (erule order_trans)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   587
apply (rule less_add_one [THEN order_less_imp_le])
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   588
done
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   589
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   590
lemma less_imp_of_nat_less:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   591
     "m < n ==> of_nat m < (of_nat n::'a::ordered_semidom)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   592
apply (induct m n rule: diff_induct, simp_all)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   593
apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   594
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   595
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   596
lemma of_nat_less_imp_less:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   597
     "of_nat m < (of_nat n::'a::ordered_semidom) ==> m < n"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   598
apply (induct m n rule: diff_induct, simp_all)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   599
apply (insert zero_le_imp_of_nat)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   600
apply (force simp add: linorder_not_less [symmetric])
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   601
done
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   602
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   603
lemma of_nat_less_iff [simp]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   604
     "(of_nat m < (of_nat n::'a::ordered_semidom)) = (m<n)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   605
by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   606
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   607
text{*Special cases where either operand is zero*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   608
declare of_nat_less_iff [of 0, simplified, simp]
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   609
declare of_nat_less_iff [of _ 0, simplified, simp]
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   610
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   611
lemma of_nat_le_iff [simp]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   612
     "(of_nat m \<le> (of_nat n::'a::ordered_semidom)) = (m \<le> n)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   613
by (simp add: linorder_not_less [symmetric])
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   614
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   615
text{*Special cases where either operand is zero*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   616
declare of_nat_le_iff [of 0, simplified, simp]
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   617
declare of_nat_le_iff [of _ 0, simplified, simp]
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   618
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   619
text{*The ordering on the comm_semiring_1_cancel is necessary to exclude the possibility of
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   620
a finite field, which indeed wraps back to zero.*}
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   621
lemma of_nat_eq_iff [simp]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   622
     "(of_nat m = (of_nat n::'a::ordered_semidom)) = (m = n)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   623
by (simp add: order_eq_iff)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   624
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   625
text{*Special cases where either operand is zero*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   626
declare of_nat_eq_iff [of 0, simplified, simp]
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   627
declare of_nat_eq_iff [of _ 0, simplified, simp]
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   628
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   629
lemma of_nat_diff [simp]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   630
     "n \<le> m ==> of_nat (m - n) = of_nat m - (of_nat n :: 'a::comm_ring_1)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   631
by (simp del: of_nat_add
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   632
	 add: compare_rls of_nat_add [symmetric] split add: nat_diff_split)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   633
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   634
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   635
subsection{*The Set of Natural Numbers*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   636
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   637
constdefs
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   638
   Nats  :: "'a::comm_semiring_1_cancel set"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   639
    "Nats == range of_nat"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   640
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   641
syntax (xsymbols)    Nats :: "'a set"   ("\<nat>")
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   642
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   643
lemma of_nat_in_Nats [simp]: "of_nat n \<in> Nats"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   644
by (simp add: Nats_def)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   645
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   646
lemma Nats_0 [simp]: "0 \<in> Nats"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   647
apply (simp add: Nats_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   648
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   649
apply (rule of_nat_0 [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   650
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   651
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   652
lemma Nats_1 [simp]: "1 \<in> Nats"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   653
apply (simp add: Nats_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   654
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   655
apply (rule of_nat_1 [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   656
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   657
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   658
lemma Nats_add [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a+b \<in> Nats"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   659
apply (auto simp add: Nats_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   660
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   661
apply (rule of_nat_add [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   662
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   663
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   664
lemma Nats_mult [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a*b \<in> Nats"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   665
apply (auto simp add: Nats_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   666
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   667
apply (rule of_nat_mult [symmetric])
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   668
done
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   669
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   670
text{*Agreement with the specific embedding for the integers*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   671
lemma int_eq_of_nat: "int = (of_nat :: nat => int)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   672
proof
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   673
  fix n
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   674
  show "int n = of_nat n"  by (induct n, simp_all add: int_Suc add_ac)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   675
qed
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   676
14496
paulson
parents: 14485
diff changeset
   677
lemma of_nat_eq_id [simp]: "of_nat = (id :: nat => nat)"
paulson
parents: 14485
diff changeset
   678
proof
paulson
parents: 14485
diff changeset
   679
  fix n
paulson
parents: 14485
diff changeset
   680
  show "of_nat n = id n"  by (induct n, simp_all)
paulson
parents: 14485
diff changeset
   681
qed
paulson
parents: 14485
diff changeset
   682
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   683
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   684
subsection{*Embedding of the Integers into any comm_ring_1: @{term of_int}*}
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   685
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   686
constdefs
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   687
   of_int :: "int => 'a::comm_ring_1"
14532
paulson
parents: 14496
diff changeset
   688
   "of_int z == contents (\<Union>(i,j) \<in> Rep_Integ z. { of_nat i - of_nat j })"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   689
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   690
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   691
lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
14496
paulson
parents: 14485
diff changeset
   692
proof -
paulson
parents: 14485
diff changeset
   693
  have "congruent intrel (\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) })"
paulson
parents: 14485
diff changeset
   694
    by (simp add: congruent_def compare_rls of_nat_add [symmetric]
paulson
parents: 14485
diff changeset
   695
            del: of_nat_add) 
paulson
parents: 14485
diff changeset
   696
  thus ?thesis
paulson
parents: 14485
diff changeset
   697
    by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])
paulson
parents: 14485
diff changeset
   698
qed
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   699
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   700
lemma of_int_0 [simp]: "of_int 0 = 0"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   701
by (simp add: of_int Zero_int_def int_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   702
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   703
lemma of_int_1 [simp]: "of_int 1 = 1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   704
by (simp add: of_int One_int_def int_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   705
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   706
lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   707
by (cases w, cases z, simp add: compare_rls of_int add)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   708
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   709
lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   710
by (cases z, simp add: compare_rls of_int minus)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   711
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   712
lemma of_int_diff [simp]: "of_int (w-z) = of_int w - of_int z"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   713
by (simp add: diff_minus)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   714
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   715
lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   716
apply (cases w, cases z)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   717
apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   718
                 mult add_ac)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   719
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   720
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   721
lemma of_int_le_iff [simp]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   722
     "(of_int w \<le> (of_int z::'a::ordered_idom)) = (w \<le> z)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   723
apply (cases w)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   724
apply (cases z)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   725
apply (simp add: compare_rls of_int le diff_int_def add minus
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   726
                 of_nat_add [symmetric]   del: of_nat_add)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   727
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   728
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   729
text{*Special cases where either operand is zero*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   730
declare of_int_le_iff [of 0, simplified, simp]
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   731
declare of_int_le_iff [of _ 0, simplified, simp]
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   732
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   733
lemma of_int_less_iff [simp]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   734
     "(of_int w < (of_int z::'a::ordered_idom)) = (w < z)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   735
by (simp add: linorder_not_le [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   736
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   737
text{*Special cases where either operand is zero*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   738
declare of_int_less_iff [of 0, simplified, simp]
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   739
declare of_int_less_iff [of _ 0, simplified, simp]
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   740
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   741
text{*The ordering on the comm_ring_1 is necessary. See @{text of_nat_eq_iff} above.*}
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   742
lemma of_int_eq_iff [simp]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   743
     "(of_int w = (of_int z::'a::ordered_idom)) = (w = z)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   744
by (simp add: order_eq_iff)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   745
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   746
text{*Special cases where either operand is zero*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   747
declare of_int_eq_iff [of 0, simplified, simp]
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   748
declare of_int_eq_iff [of _ 0, simplified, simp]
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   749
14496
paulson
parents: 14485
diff changeset
   750
lemma of_int_eq_id [simp]: "of_int = (id :: int => int)"
paulson
parents: 14485
diff changeset
   751
proof
paulson
parents: 14485
diff changeset
   752
 fix z
paulson
parents: 14485
diff changeset
   753
 show "of_int z = id z"  
paulson
parents: 14485
diff changeset
   754
  by (cases z,
paulson
parents: 14485
diff changeset
   755
      simp add: of_int add minus int_eq_of_nat [symmetric] int_def diff_minus)
paulson
parents: 14485
diff changeset
   756
qed
paulson
parents: 14485
diff changeset
   757
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   758
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   759
subsection{*The Set of Integers*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   760
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   761
constdefs
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   762
   Ints  :: "'a::comm_ring_1 set"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   763
    "Ints == range of_int"
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 14269
diff changeset
   764
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   765
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   766
syntax (xsymbols)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   767
  Ints      :: "'a set"                   ("\<int>")
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   768
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   769
lemma Ints_0 [simp]: "0 \<in> Ints"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   770
apply (simp add: Ints_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   771
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   772
apply (rule of_int_0 [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   773
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   774
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   775
lemma Ints_1 [simp]: "1 \<in> Ints"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   776
apply (simp add: Ints_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   777
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   778
apply (rule of_int_1 [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   779
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   780
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   781
lemma Ints_add [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a+b \<in> Ints"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   782
apply (auto simp add: Ints_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   783
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   784
apply (rule of_int_add [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   785
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   786
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   787
lemma Ints_minus [simp]: "a \<in> Ints ==> -a \<in> Ints"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   788
apply (auto simp add: Ints_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   789
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   790
apply (rule of_int_minus [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   791
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   792
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   793
lemma Ints_diff [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a-b \<in> Ints"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   794
apply (auto simp add: Ints_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   795
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   796
apply (rule of_int_diff [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   797
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   798
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   799
lemma Ints_mult [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a*b \<in> Ints"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   800
apply (auto simp add: Ints_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   801
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   802
apply (rule of_int_mult [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   803
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   804
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   805
text{*Collapse nested embeddings*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   806
lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   807
by (induct n, auto)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   808
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   809
lemma of_int_int_eq [simp]: "of_int (int n) = int n"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   810
by (simp add: int_eq_of_nat)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   811
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   812
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   813
lemma Ints_cases [case_names of_int, cases set: Ints]:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   814
  "q \<in> \<int> ==> (!!z. q = of_int z ==> C) ==> C"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   815
proof (simp add: Ints_def)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   816
  assume "!!z. q = of_int z ==> C"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   817
  assume "q \<in> range of_int" thus C ..
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   818
qed
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   819
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   820
lemma Ints_induct [case_names of_int, induct set: Ints]:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   821
  "q \<in> \<int> ==> (!!z. P (of_int z)) ==> P q"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   822
  by (rule Ints_cases) auto
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   823
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   824
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   825
(* int (Suc n) = 1 + int n *)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   826
declare int_Suc [simp]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   827
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   828
text{*Simplification of @{term "x-y < 0"}, etc.*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   829
declare less_iff_diff_less_0 [symmetric, simp]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   830
declare eq_iff_diff_eq_0 [symmetric, simp]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   831
declare le_iff_diff_le_0 [symmetric, simp]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   832
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   833
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   834
subsection{*More Properties of @{term setsum} and  @{term setprod}*}
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   835
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   836
text{*By Jeremy Avigad*}
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   837
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   838
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   839
lemma setsum_of_nat: "of_nat (setsum f A) = setsum (of_nat \<circ> f) A"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   840
  apply (case_tac "finite A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   841
  apply (erule finite_induct, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   842
  apply (simp add: setsum_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   843
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   844
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   845
lemma setsum_of_int: "of_int (setsum f A) = setsum (of_int \<circ> f) A"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   846
  apply (case_tac "finite A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   847
  apply (erule finite_induct, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   848
  apply (simp add: setsum_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   849
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   850
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   851
lemma int_setsum: "int (setsum f A) = setsum (int \<circ> f) A"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   852
  by (subst int_eq_of_nat, rule setsum_of_nat)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   853
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   854
lemma setprod_of_nat: "of_nat (setprod f A) = setprod (of_nat \<circ> f) A"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   855
  apply (case_tac "finite A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   856
  apply (erule finite_induct, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   857
  apply (simp add: setprod_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   858
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   859
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   860
lemma setprod_of_int: "of_int (setprod f A) = setprod (of_int \<circ> f) A"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   861
  apply (case_tac "finite A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   862
  apply (erule finite_induct, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   863
  apply (simp add: setprod_def)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   864
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   865
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   866
lemma int_setprod: "int (setprod f A) = setprod (int \<circ> f) A"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   867
  by (subst int_eq_of_nat, rule setprod_of_nat)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   868
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   869
lemma setsum_constant: "finite A ==> (\<Sum>x \<in> A. y) = of_nat(card A) * y"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   870
  apply (erule finite_induct)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   871
  apply (auto simp add: ring_distrib add_ac)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   872
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   873
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   874
lemma setprod_nonzero_nat:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   875
    "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::nat)) ==> setprod f A \<noteq> 0"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   876
  by (rule setprod_nonzero, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   877
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   878
lemma setprod_zero_eq_nat:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   879
    "finite A ==> (setprod f A = (0::nat)) = (\<exists>x \<in> A. f x = 0)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   880
  by (rule setprod_zero_eq, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   881
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   882
lemma setprod_nonzero_int:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   883
    "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::int)) ==> setprod f A \<noteq> 0"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   884
  by (rule setprod_nonzero, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   885
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   886
lemma setprod_zero_eq_int:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   887
    "finite A ==> (setprod f A = (0::int)) = (\<exists>x \<in> A. f x = 0)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   888
  by (rule setprod_zero_eq, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   889
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   890
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   891
text{*Now we replace the case analysis rule by a more conventional one:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   892
whether an integer is negative or not.*}
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   893
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   894
lemma zless_iff_Suc_zadd:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   895
    "(w < z) = (\<exists>n. z = w + int(Suc n))"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   896
apply (cases z, cases w)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   897
apply (auto simp add: le add int_def linorder_not_le [symmetric]) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   898
apply (rename_tac a b c d) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   899
apply (rule_tac x="a+d - Suc(c+b)" in exI) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   900
apply arith
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   901
done
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   902
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   903
lemma negD: "x<0 ==> \<exists>n. x = - (int (Suc n))"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   904
apply (cases x)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   905
apply (auto simp add: le minus Zero_int_def int_def order_less_le) 
14496
paulson
parents: 14485
diff changeset
   906
apply (rule_tac x="y - Suc x" in exI, arith)
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   907
done
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   908
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   909
theorem int_cases [cases type: int, case_names nonneg neg]:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   910
     "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   911
apply (case_tac "z < 0", blast dest!: negD)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   912
apply (simp add: linorder_not_less)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   913
apply (blast dest: nat_0_le [THEN sym])
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   914
done
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   915
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   916
theorem int_induct [induct type: int, case_names nonneg neg]:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   917
     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   918
  by (cases z) auto
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   919
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   920
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   921
(*Legacy ML bindings, but no longer the structure Int.*)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   922
ML
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   923
{*
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   924
val zabs_def = thm "zabs_def"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   925
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   926
val int_0 = thm "int_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   927
val int_1 = thm "int_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   928
val int_Suc0_eq_1 = thm "int_Suc0_eq_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   929
val neg_eq_less_0 = thm "neg_eq_less_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   930
val not_neg_eq_ge_0 = thm "not_neg_eq_ge_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   931
val not_neg_0 = thm "not_neg_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   932
val not_neg_1 = thm "not_neg_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   933
val iszero_0 = thm "iszero_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   934
val not_iszero_1 = thm "not_iszero_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   935
val int_0_less_1 = thm "int_0_less_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   936
val int_0_neq_1 = thm "int_0_neq_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   937
val negative_zless = thm "negative_zless";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   938
val negative_zle = thm "negative_zle";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   939
val not_zle_0_negative = thm "not_zle_0_negative";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   940
val not_int_zless_negative = thm "not_int_zless_negative";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   941
val negative_eq_positive = thm "negative_eq_positive";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   942
val zle_iff_zadd = thm "zle_iff_zadd";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   943
val abs_int_eq = thm "abs_int_eq";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   944
val abs_split = thm"abs_split";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   945
val nat_int = thm "nat_int";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   946
val nat_zminus_int = thm "nat_zminus_int";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   947
val nat_zero = thm "nat_zero";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   948
val not_neg_nat = thm "not_neg_nat";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   949
val neg_nat = thm "neg_nat";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   950
val zless_nat_eq_int_zless = thm "zless_nat_eq_int_zless";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   951
val nat_0_le = thm "nat_0_le";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   952
val nat_le_0 = thm "nat_le_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   953
val zless_nat_conj = thm "zless_nat_conj";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   954
val int_cases = thm "int_cases";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   955
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   956
val int_def = thm "int_def";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   957
val Zero_int_def = thm "Zero_int_def";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   958
val One_int_def = thm "One_int_def";
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   959
val diff_int_def = thm "diff_int_def";
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   960
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   961
val inj_int = thm "inj_int";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   962
val zminus_zminus = thm "zminus_zminus";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   963
val zminus_0 = thm "zminus_0";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   964
val zminus_zadd_distrib = thm "zminus_zadd_distrib";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   965
val zadd_commute = thm "zadd_commute";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   966
val zadd_assoc = thm "zadd_assoc";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   967
val zadd_left_commute = thm "zadd_left_commute";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   968
val zadd_ac = thms "zadd_ac";
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 14269
diff changeset
   969
val zmult_ac = thms "zmult_ac";
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   970
val zadd_int = thm "zadd_int";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   971
val zadd_int_left = thm "zadd_int_left";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   972
val int_Suc = thm "int_Suc";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   973
val zadd_0 = thm "zadd_0";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   974
val zadd_0_right = thm "zadd_0_right";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   975
val zmult_zminus = thm "zmult_zminus";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   976
val zmult_commute = thm "zmult_commute";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   977
val zmult_assoc = thm "zmult_assoc";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   978
val zadd_zmult_distrib = thm "zadd_zmult_distrib";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   979
val zadd_zmult_distrib2 = thm "zadd_zmult_distrib2";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   980
val zdiff_zmult_distrib = thm "zdiff_zmult_distrib";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   981
val zdiff_zmult_distrib2 = thm "zdiff_zmult_distrib2";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   982
val int_distrib = thms "int_distrib";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   983
val zmult_int = thm "zmult_int";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   984
val zmult_1 = thm "zmult_1";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   985
val zmult_1_right = thm "zmult_1_right";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   986
val int_int_eq = thm "int_int_eq";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   987
val int_eq_0_conv = thm "int_eq_0_conv";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   988
val zless_int = thm "zless_int";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   989
val int_less_0_conv = thm "int_less_0_conv";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   990
val zero_less_int_conv = thm "zero_less_int_conv";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   991
val zle_int = thm "zle_int";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   992
val zero_zle_int = thm "zero_zle_int";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   993
val int_le_0_conv = thm "int_le_0_conv";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   994
val zle_refl = thm "zle_refl";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   995
val zle_linear = thm "zle_linear";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   996
val zle_trans = thm "zle_trans";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   997
val zle_anti_sym = thm "zle_anti_sym";
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   998
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   999
val Ints_def = thm "Ints_def";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1000
val Nats_def = thm "Nats_def";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1001
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1002
val of_nat_0 = thm "of_nat_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1003
val of_nat_Suc = thm "of_nat_Suc";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1004
val of_nat_1 = thm "of_nat_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1005
val of_nat_add = thm "of_nat_add";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1006
val of_nat_mult = thm "of_nat_mult";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1007
val zero_le_imp_of_nat = thm "zero_le_imp_of_nat";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1008
val less_imp_of_nat_less = thm "less_imp_of_nat_less";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1009
val of_nat_less_imp_less = thm "of_nat_less_imp_less";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1010
val of_nat_less_iff = thm "of_nat_less_iff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1011
val of_nat_le_iff = thm "of_nat_le_iff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1012
val of_nat_eq_iff = thm "of_nat_eq_iff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1013
val Nats_0 = thm "Nats_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1014
val Nats_1 = thm "Nats_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1015
val Nats_add = thm "Nats_add";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1016
val Nats_mult = thm "Nats_mult";
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1017
val int_eq_of_nat = thm"int_eq_of_nat";
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1018
val of_int = thm "of_int";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1019
val of_int_0 = thm "of_int_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1020
val of_int_1 = thm "of_int_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1021
val of_int_add = thm "of_int_add";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1022
val of_int_minus = thm "of_int_minus";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1023
val of_int_diff = thm "of_int_diff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1024
val of_int_mult = thm "of_int_mult";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1025
val of_int_le_iff = thm "of_int_le_iff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1026
val of_int_less_iff = thm "of_int_less_iff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1027
val of_int_eq_iff = thm "of_int_eq_iff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1028
val Ints_0 = thm "Ints_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1029
val Ints_1 = thm "Ints_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1030
val Ints_add = thm "Ints_add";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1031
val Ints_minus = thm "Ints_minus";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1032
val Ints_diff = thm "Ints_diff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1033
val Ints_mult = thm "Ints_mult";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1034
val of_int_of_nat_eq = thm"of_int_of_nat_eq";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1035
val Ints_cases = thm "Ints_cases";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1036
val Ints_induct = thm "Ints_induct";
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
  1037
*}
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
  1038
5508
691c70898518 new files in Integ
paulson
parents:
diff changeset
  1039
end