src/HOL/Nat.thy
author nipkow
Mon, 30 Sep 2002 15:44:21 +0200
changeset 13596 ee5f79b210c1
parent 13585 db4005b40cc6
child 14131 a4fc8b1af5e7
permissions -rw-r--r--
modified induct method
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     1
(*  Title:      HOL/Nat.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     2
    ID:         $Id$
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
     3
    Author:     Tobias Nipkow and Lawrence C Paulson
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     4
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
     5
Type "nat" is a linear order, and a datatype; arithmetic operators + -
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
     6
and * (for div, mod and dvd, see theory Divides).
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     7
*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     8
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
     9
header {* Natural numbers *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    10
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    11
theory Nat = Wellfounded_Recursion:
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    12
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    13
subsection {* Type @{text ind} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    14
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    15
typedecl ind
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    16
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    17
consts
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    18
  Zero_Rep      :: ind
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    19
  Suc_Rep       :: "ind => ind"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    20
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    21
axioms
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    22
  -- {* the axiom of infinity in 2 parts *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    23
  inj_Suc_Rep:          "inj Suc_Rep"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    24
  Suc_Rep_not_Zero_Rep: "Suc_Rep x ~= Zero_Rep"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    25
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    26
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    27
subsection {* Type nat *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    28
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    29
text {* Type definition *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    30
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    31
consts
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    32
  Nat :: "ind set"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    33
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    34
inductive Nat
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    35
intros
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    36
  Zero_RepI: "Zero_Rep : Nat"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    37
  Suc_RepI: "i : Nat ==> Suc_Rep i : Nat"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    38
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    39
global
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    40
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    41
typedef (open Nat)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    42
  nat = "Nat" by (rule exI, rule Nat.Zero_RepI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    43
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    44
instance nat :: ord ..
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    45
instance nat :: zero ..
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    46
instance nat :: one ..
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    47
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    48
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    49
text {* Abstract constants and syntax *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    50
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    51
consts
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    52
  Suc :: "nat => nat"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    53
  pred_nat :: "(nat * nat) set"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    54
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    55
local
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    56
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    57
defs
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    58
  Zero_nat_def: "0 == Abs_Nat Zero_Rep"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    59
  Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    60
  One_nat_def [simp]: "1 == Suc 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    61
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    62
  -- {* nat operations *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    63
  pred_nat_def: "pred_nat == {(m, n). n = Suc m}"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    64
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    65
  less_def: "m < n == (m, n) : trancl pred_nat"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    66
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    67
  le_def: "m <= (n::nat) == ~ (n < m)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    68
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    69
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    70
text {* Induction *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    71
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    72
theorem nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    73
  apply (unfold Zero_nat_def Suc_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    74
  apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    75
  apply (erule Rep_Nat [THEN Nat.induct])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    76
  apply (rules elim: Abs_Nat_inverse [THEN subst])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    77
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    78
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    79
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    80
text {* Isomorphisms: @{text Abs_Nat} and @{text Rep_Nat} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    81
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    82
lemma inj_Rep_Nat: "inj Rep_Nat"
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 13449
diff changeset
    83
  apply (rule inj_on_inverseI)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    84
  apply (rule Rep_Nat_inverse)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    85
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    86
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    87
lemma inj_on_Abs_Nat: "inj_on Abs_Nat Nat"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    88
  apply (rule inj_on_inverseI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    89
  apply (erule Abs_Nat_inverse)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    90
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    91
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    92
text {* Distinctness of constructors *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    93
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    94
lemma Suc_not_Zero [iff]: "Suc m ~= 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    95
  apply (unfold Zero_nat_def Suc_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    96
  apply (rule inj_on_Abs_Nat [THEN inj_on_contraD])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    97
  apply (rule Suc_Rep_not_Zero_Rep)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    98
  apply (rule Rep_Nat Nat.Suc_RepI Nat.Zero_RepI)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    99
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   100
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   101
lemma Zero_not_Suc [iff]: "0 ~= Suc m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   102
  by (rule not_sym, rule Suc_not_Zero not_sym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   103
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   104
lemma Suc_neq_Zero: "Suc m = 0 ==> R"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   105
  by (rule notE, rule Suc_not_Zero)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   106
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   107
lemma Zero_neq_Suc: "0 = Suc m ==> R"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   108
  by (rule Suc_neq_Zero, erule sym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   109
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   110
text {* Injectiveness of @{term Suc} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   111
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   112
lemma inj_Suc: "inj Suc"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   113
  apply (unfold Suc_def)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 13449
diff changeset
   114
  apply (rule inj_onI)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   115
  apply (drule inj_on_Abs_Nat [THEN inj_onD])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   116
  apply (rule Rep_Nat Nat.Suc_RepI)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   117
  apply (drule inj_Suc_Rep [THEN injD])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   118
  apply (erule inj_Rep_Nat [THEN injD])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   119
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   120
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   121
lemma Suc_inject: "Suc x = Suc y ==> x = y"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   122
  by (rule inj_Suc [THEN injD])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   123
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   124
lemma Suc_Suc_eq [iff]: "(Suc m = Suc n) = (m = n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   125
  apply (rule iffI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   126
  apply (erule Suc_inject)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   127
  apply (erule arg_cong)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   128
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   129
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   130
lemma nat_not_singleton: "(ALL x. x = (0::nat)) = False"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   131
  by auto
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   132
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   133
text {* @{typ nat} is a datatype *}
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   134
5188
633ec5f6c155 Declaration of type 'nat' as a datatype (this allows usage of
berghofe
parents: 4640
diff changeset
   135
rep_datatype nat
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   136
  distinct  Suc_not_Zero Zero_not_Suc
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   137
  inject    Suc_Suc_eq
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   138
  induction nat_induct
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   139
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   140
lemma n_not_Suc_n: "n ~= Suc n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   141
  by (induct n) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   142
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   143
lemma Suc_n_not_n: "Suc t ~= t"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   144
  by (rule not_sym, rule n_not_Suc_n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   145
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   146
text {* A special form of induction for reasoning
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   147
  about @{term "m < n"} and @{term "m - n"} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   148
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   149
theorem diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   150
    (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   151
  apply (rule_tac x = "m" in spec)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   152
  apply (induct_tac n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   153
  prefer 2
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   154
  apply (rule allI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   155
  apply (induct_tac x)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   156
  apply rules+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   157
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   158
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   159
subsection {* Basic properties of "less than" *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   160
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   161
lemma wf_pred_nat: "wf pred_nat"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   162
  apply (unfold wf_def pred_nat_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   163
  apply clarify
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   164
  apply (induct_tac x)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   165
  apply blast+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   166
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   167
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   168
lemma wf_less: "wf {(x, y::nat). x < y}"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   169
  apply (unfold less_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   170
  apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_subset])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   171
  apply blast
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   172
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   173
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   174
lemma less_eq: "((m, n) : pred_nat^+) = (m < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   175
  apply (unfold less_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   176
  apply (rule refl)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   177
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   178
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   179
subsubsection {* Introduction properties *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   180
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   181
lemma less_trans: "i < j ==> j < k ==> i < (k::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   182
  apply (unfold less_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   183
  apply (rule trans_trancl [THEN transD])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   184
  apply assumption+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   185
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   186
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   187
lemma lessI [iff]: "n < Suc n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   188
  apply (unfold less_def pred_nat_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   189
  apply (simp add: r_into_trancl)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   190
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   191
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   192
lemma less_SucI: "i < j ==> i < Suc j"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   193
  apply (rule less_trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   194
  apply assumption
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   195
  apply (rule lessI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   196
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   197
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   198
lemma zero_less_Suc [iff]: "0 < Suc n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   199
  apply (induct n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   200
  apply (rule lessI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   201
  apply (erule less_trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   202
  apply (rule lessI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   203
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   204
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   205
subsubsection {* Elimination properties *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   206
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   207
lemma less_not_sym: "n < m ==> ~ m < (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   208
  apply (unfold less_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   209
  apply (blast intro: wf_pred_nat wf_trancl [THEN wf_asym])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   210
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   211
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   212
lemma less_asym:
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   213
  assumes h1: "(n::nat) < m" and h2: "~ P ==> m < n" shows P
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   214
  apply (rule contrapos_np)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   215
  apply (rule less_not_sym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   216
  apply (rule h1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   217
  apply (erule h2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   218
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   219
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   220
lemma less_not_refl: "~ n < (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   221
  apply (unfold less_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   222
  apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_not_refl])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   223
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   224
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   225
lemma less_irrefl [elim!]: "(n::nat) < n ==> R"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   226
  by (rule notE, rule less_not_refl)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   227
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   228
lemma less_not_refl2: "n < m ==> m ~= (n::nat)" by blast
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   229
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   230
lemma less_not_refl3: "(s::nat) < t ==> s ~= t"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   231
  by (rule not_sym, rule less_not_refl2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   232
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   233
lemma lessE:
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   234
  assumes major: "i < k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   235
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   236
  shows P
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   237
  apply (rule major [unfolded less_def pred_nat_def, THEN tranclE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   238
  apply simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   239
  apply (erule p1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   240
  apply (rule p2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   241
  apply (simp add: less_def pred_nat_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   242
  apply assumption
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   243
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   244
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   245
lemma not_less0 [iff]: "~ n < (0::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   246
  by (blast elim: lessE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   247
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   248
lemma less_zeroE: "(n::nat) < 0 ==> R"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   249
  by (rule notE, rule not_less0)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   250
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   251
lemma less_SucE: assumes major: "m < Suc n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   252
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   253
  apply (rule major [THEN lessE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   254
  apply (rule eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   255
  apply blast
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   256
  apply (rule less)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   257
  apply blast
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   258
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   259
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   260
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   261
  by (blast elim!: less_SucE intro: less_trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   262
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   263
lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   264
  by (simp add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   265
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   266
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   267
  by (simp add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   268
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   269
lemma Suc_mono: "m < n ==> Suc m < Suc n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   270
  by (induct n) (fast elim: less_trans lessE)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   271
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   272
text {* "Less than" is a linear ordering *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   273
lemma less_linear: "m < n | m = n | n < (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   274
  apply (induct_tac m)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   275
  apply (induct_tac n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   276
  apply (rule refl [THEN disjI1, THEN disjI2])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   277
  apply (rule zero_less_Suc [THEN disjI1])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   278
  apply (blast intro: Suc_mono less_SucI elim: lessE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   279
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   280
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   281
lemma nat_neq_iff: "((m::nat) ~= n) = (m < n | n < m)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   282
  using less_linear by blast
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   283
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   284
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   285
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   286
  shows "P n m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   287
  apply (rule less_linear [THEN disjE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   288
  apply (erule_tac [2] disjE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   289
  apply (erule lessCase)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   290
  apply (erule sym [THEN eqCase])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   291
  apply (erule major)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   292
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   293
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   294
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   295
subsubsection {* Inductive (?) properties *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   296
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   297
lemma Suc_lessI: "m < n ==> Suc m ~= n ==> Suc m < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   298
  apply (simp add: nat_neq_iff)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   299
  apply (blast elim!: less_irrefl less_SucE elim: less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   300
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   301
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   302
lemma Suc_lessD: "Suc m < n ==> m < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   303
  apply (induct n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   304
  apply (fast intro!: lessI [THEN less_SucI] elim: less_trans lessE)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   305
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   306
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   307
lemma Suc_lessE: assumes major: "Suc i < k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   308
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   309
  apply (rule major [THEN lessE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   310
  apply (erule lessI [THEN minor])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   311
  apply (erule Suc_lessD [THEN minor])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   312
  apply assumption
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   313
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   314
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   315
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   316
  by (blast elim: lessE dest: Suc_lessD)
4104
84433b1ab826 nat datatype_info moved to Nat.thy;
wenzelm
parents: 3370
diff changeset
   317
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   318
lemma Suc_less_eq [iff]: "(Suc m < Suc n) = (m < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   319
  apply (rule iffI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   320
  apply (erule Suc_less_SucD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   321
  apply (erule Suc_mono)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   322
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   323
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   324
lemma less_trans_Suc:
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   325
  assumes le: "i < j" shows "j < k ==> Suc i < k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   326
  apply (induct k)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   327
  apply simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   328
  apply (insert le)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   329
  apply (simp add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   330
  apply (blast dest: Suc_lessD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   331
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   332
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   333
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   334
lemma not_less_eq: "(~ m < n) = (n < Suc m)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   335
  apply (rule_tac m = "m" and n = "n" in diff_induct)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   336
  apply simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   337
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   338
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   339
text {* Complete induction, aka course-of-values induction *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   340
lemma nat_less_induct:
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   341
  assumes prem: "!!n. ALL m::nat. m < n --> P m ==> P n" shows "P n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   342
  apply (rule_tac a=n in wf_induct)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   343
  apply (rule wf_pred_nat [THEN wf_trancl])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   344
  apply (rule prem)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   345
  apply (unfold less_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   346
  apply assumption
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   347
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   348
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   349
subsection {* Properties of "less or equal than" *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   350
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   351
text {* Was @{text le_eq_less_Suc}, but this orientation is more useful *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   352
lemma less_Suc_eq_le: "(m < Suc n) = (m <= n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   353
  by (unfold le_def, rule not_less_eq [symmetric])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   354
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   355
lemma le_imp_less_Suc: "m <= n ==> m < Suc n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   356
  by (rule less_Suc_eq_le [THEN iffD2])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   357
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   358
lemma le0 [iff]: "(0::nat) <= n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   359
  by (unfold le_def, rule not_less0)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   360
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   361
lemma Suc_n_not_le_n: "~ Suc n <= n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   362
  by (simp add: le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   363
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   364
lemma le_0_eq [iff]: "((i::nat) <= 0) = (i = 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   365
  by (induct i) (simp_all add: le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   366
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   367
lemma le_Suc_eq: "(m <= Suc n) = (m <= n | m = Suc n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   368
  by (simp del: less_Suc_eq_le add: less_Suc_eq_le [symmetric] less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   369
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   370
lemma le_SucE: "m <= Suc n ==> (m <= n ==> R) ==> (m = Suc n ==> R) ==> R"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   371
  by (drule le_Suc_eq [THEN iffD1], rules+)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   372
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   373
lemma leI: "~ n < m ==> m <= (n::nat)" by (simp add: le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   374
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   375
lemma leD: "m <= n ==> ~ n < (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   376
  by (simp add: le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   377
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   378
lemmas leE = leD [elim_format]
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   379
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   380
lemma not_less_iff_le: "(~ n < m) = (m <= (n::nat))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   381
  by (blast intro: leI elim: leE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   382
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   383
lemma not_leE: "~ m <= n ==> n<(m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   384
  by (simp add: le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   385
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   386
lemma not_le_iff_less: "(~ n <= m) = (m < (n::nat))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   387
  by (simp add: le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   388
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   389
lemma Suc_leI: "m < n ==> Suc(m) <= n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   390
  apply (simp add: le_def less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   391
  apply (blast elim!: less_irrefl less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   392
  done -- {* formerly called lessD *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   393
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   394
lemma Suc_leD: "Suc(m) <= n ==> m <= n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   395
  by (simp add: le_def less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   396
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   397
text {* Stronger version of @{text Suc_leD} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   398
lemma Suc_le_lessD: "Suc m <= n ==> m < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   399
  apply (simp add: le_def less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   400
  using less_linear
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   401
  apply blast
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   402
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   403
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   404
lemma Suc_le_eq: "(Suc m <= n) = (m < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   405
  by (blast intro: Suc_leI Suc_le_lessD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   406
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   407
lemma le_SucI: "m <= n ==> m <= Suc n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   408
  by (unfold le_def) (blast dest: Suc_lessD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   409
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   410
lemma less_imp_le: "m < n ==> m <= (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   411
  by (unfold le_def) (blast elim: less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   412
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   413
text {* For instance, @{text "(Suc m < Suc n) = (Suc m <= n) = (m < n)"} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   414
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   415
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   416
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   417
text {* Equivalence of @{term "m <= n"} and @{term "m < n | m = n"} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   418
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   419
lemma le_imp_less_or_eq: "m <= n ==> m < n | m = (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   420
  apply (unfold le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   421
  using less_linear
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   422
  apply (blast elim: less_irrefl less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   423
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   424
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   425
lemma less_or_eq_imp_le: "m < n | m = n ==> m <= (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   426
  apply (unfold le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   427
  using less_linear
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   428
  apply (blast elim!: less_irrefl elim: less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   429
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   430
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   431
lemma le_eq_less_or_eq: "(m <= (n::nat)) = (m < n | m=n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   432
  by (rules intro: less_or_eq_imp_le le_imp_less_or_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   433
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   434
text {* Useful with @{text Blast}. *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   435
lemma eq_imp_le: "(m::nat) = n ==> m <= n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   436
  by (rule less_or_eq_imp_le, rule disjI2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   437
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   438
lemma le_refl: "n <= (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   439
  by (simp add: le_eq_less_or_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   440
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   441
lemma le_less_trans: "[| i <= j; j < k |] ==> i < (k::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   442
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   443
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   444
lemma less_le_trans: "[| i < j; j <= k |] ==> i < (k::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   445
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   446
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   447
lemma le_trans: "[| i <= j; j <= k |] ==> i <= (k::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   448
  by (blast dest!: le_imp_less_or_eq intro: less_or_eq_imp_le less_trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   450
lemma le_anti_sym: "[| m <= n; n <= m |] ==> m = (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   451
  by (blast dest!: le_imp_less_or_eq elim!: less_irrefl elim: less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   452
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   453
lemma Suc_le_mono [iff]: "(Suc n <= Suc m) = (n <= m)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   454
  by (simp add: le_simps)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   455
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   456
text {* Axiom @{text order_less_le} of class @{text order}: *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   457
lemma nat_less_le: "((m::nat) < n) = (m <= n & m ~= n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   458
  by (simp add: le_def nat_neq_iff) (blast elim!: less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   459
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   460
lemma le_neq_implies_less: "(m::nat) <= n ==> m ~= n ==> m < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   461
  by (rule iffD2, rule nat_less_le, rule conjI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   462
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   463
text {* Axiom @{text linorder_linear} of class @{text linorder}: *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   464
lemma nat_le_linear: "(m::nat) <= n | n <= m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   465
  apply (simp add: le_eq_less_or_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   466
  using less_linear
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   467
  apply blast
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   468
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   469
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   470
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   471
  by (blast elim!: less_SucE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   472
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   473
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   474
text {*
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   475
  Rewrite @{term "n < Suc m"} to @{term "n = m"}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   476
  if @{term "~ n < m"} or @{term "m <= n"} hold.
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   477
  Not suitable as default simprules because they often lead to looping
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   478
*}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   479
lemma le_less_Suc_eq: "m <= n ==> (n < Suc m) = (n = m)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   480
  by (rule not_less_less_Suc_eq, rule leD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   481
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   482
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   483
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   484
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   485
text {*
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   486
  Re-orientation of the equations @{text "0 = x"} and @{text "1 = x"}. 
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   487
  No longer added as simprules (they loop) 
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   488
  but via @{text reorient_simproc} in Bin
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   489
*}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   490
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   491
text {* Polymorphic, not just for @{typ nat} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   492
lemma zero_reorient: "(0 = x) = (x = 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   493
  by auto
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   494
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   495
lemma one_reorient: "(1 = x) = (x = 1)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   496
  by auto
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   497
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   498
text {* Type {@typ nat} is a wellfounded linear order *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   499
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   500
instance nat :: order by (intro_classes,
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   501
  (assumption | rule le_refl le_trans le_anti_sym nat_less_le)+)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   502
instance nat :: linorder by (intro_classes, rule nat_le_linear)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   503
instance nat :: wellorder by (intro_classes, rule wf_less)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   504
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   505
subsection {* Arithmetic operators *}
1660
8cb42cd97579 *** empty log message ***
oheimb
parents: 1625
diff changeset
   506
12338
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 11451
diff changeset
   507
axclass power < type
10435
b100e8d2c355 added axclass power (from HOL.thy);
wenzelm
parents: 9436
diff changeset
   508
3370
5c5fdce3a4e4 Overloading of "^" requires new type class "power", with types "nat" and
paulson
parents: 2608
diff changeset
   509
consts
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   510
  power :: "('a::power) => nat => 'a"            (infixr "^" 80)
3370
5c5fdce3a4e4 Overloading of "^" requires new type class "power", with types "nat" and
paulson
parents: 2608
diff changeset
   511
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   512
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   513
text {* arithmetic operators @{text "+ -"} and @{text "*"} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   514
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   515
instance nat :: plus ..
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   516
instance nat :: minus ..
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   517
instance nat :: times ..
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   518
instance nat :: power ..
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   519
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   520
text {* size of a datatype value; overloaded *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   521
consts size :: "'a => nat"
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   522
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   523
primrec
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   524
  add_0:    "0 + n = n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   525
  add_Suc:  "Suc m + n = Suc (m + n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   526
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   527
primrec
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   528
  diff_0:   "m - 0 = m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   529
  diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   530
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   531
primrec
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   532
  mult_0:   "0 * n = 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   533
  mult_Suc: "Suc m * n = n + (m * n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   534
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   535
text {* These 2 rules ease the use of primitive recursion. NOTE USE OF @{text "=="} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   536
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   537
  by simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   538
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   539
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   540
  by simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   541
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   542
lemma not0_implies_Suc: "n ~= 0 ==> EX m. n = Suc m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   543
  by (case_tac n) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   544
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   545
lemma gr_implies_not0: "!!n::nat. m<n ==> n ~= 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   546
  by (case_tac n) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   547
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   548
lemma neq0_conv [iff]: "!!n::nat. (n ~= 0) = (0 < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   549
  by (case_tac n) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   550
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   551
text {* This theorem is useful with @{text blast} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   552
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   553
  by (rule iffD1, rule neq0_conv, rules)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   554
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   555
lemma gr0_conv_Suc: "(0 < n) = (EX m. n = Suc m)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   556
  by (fast intro: not0_implies_Suc)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   557
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   558
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   559
  apply (rule iffI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   560
  apply (rule ccontr)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   561
  apply simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   562
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   563
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   564
lemma Suc_le_D: "(Suc n <= m') ==> (? m. m' = Suc m)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   565
  by (induct m') simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   566
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   567
text {* Useful in certain inductive arguments *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   568
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (EX j. m = Suc j & j < n))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   569
  by (case_tac m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   570
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   571
lemma nat_induct2: "P 0 ==> P (Suc 0) ==> (!!k. P k ==> P (Suc (Suc k))) ==> P n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   572
  apply (rule nat_less_induct)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   573
  apply (case_tac n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   574
  apply (case_tac [2] nat)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   575
  apply (blast intro: less_trans)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   576
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   577
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   578
subsection {* @{text LEAST} theorems for type @{typ nat} by specialization *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   579
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   580
lemmas LeastI = wellorder_LeastI
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   581
lemmas Least_le = wellorder_Least_le
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   582
lemmas not_less_Least = wellorder_not_less_Least
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   583
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   584
lemma Least_Suc: "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   585
  apply (case_tac "n")
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   586
  apply auto
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   587
  apply (frule LeastI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   588
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   589
  apply (subgoal_tac " (LEAST x. P x) <= Suc (LEAST x. P (Suc x))")
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   590
  apply (erule_tac [2] Least_le)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   591
  apply (case_tac "LEAST x. P x")
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   592
  apply auto
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   593
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   594
  apply (blast intro: order_antisym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   595
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   596
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   597
lemma Least_Suc2: "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   598
  apply (erule (1) Least_Suc [THEN ssubst])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   599
  apply simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   600
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   601
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   602
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   603
subsection {* @{term min} and @{term max} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   604
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   605
lemma min_0L [simp]: "min 0 n = (0::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   606
  by (rule min_leastL) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   607
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   608
lemma min_0R [simp]: "min n 0 = (0::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   609
  by (rule min_leastR) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   610
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   611
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   612
  by (simp add: min_of_mono)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   613
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   614
lemma max_0L [simp]: "max 0 n = (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   615
  by (rule max_leastL) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   616
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   617
lemma max_0R [simp]: "max n 0 = (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   618
  by (rule max_leastR) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   619
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   620
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   621
  by (simp add: max_of_mono)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   622
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   623
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   624
subsection {* Basic rewrite rules for the arithmetic operators *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   625
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   626
text {* Difference *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   627
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   628
lemma diff_0_eq_0 [simp]: "0 - n = (0::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   629
  by (induct_tac n) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   630
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   631
lemma diff_Suc_Suc [simp]: "Suc(m) - Suc(n) = m - n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   632
  by (induct_tac n) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   633
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   634
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   635
text {*
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   636
  Could be (and is, below) generalized in various ways
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   637
  However, none of the generalizations are currently in the simpset,
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   638
  and I dread to think what happens if I put them in
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   639
*}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   640
lemma Suc_pred [simp]: "0 < n ==> Suc (n - Suc 0) = n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   641
  by (simp split add: nat.split)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   642
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   643
declare diff_Suc [simp del]
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   644
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   645
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   646
subsection {* Addition *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   647
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   648
lemma add_0_right [simp]: "m + 0 = (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   649
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   650
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   651
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   652
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   653
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   654
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   655
text {* Associative law for addition *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   656
lemma add_assoc: "(m + n) + k = m + ((n + k)::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   657
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   658
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   659
text {* Commutative law for addition *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   660
lemma add_commute: "m + n = n + (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   661
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   662
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   663
lemma add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   664
  apply (rule mk_left_commute [of "op +"])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   665
  apply (rule add_assoc)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   666
  apply (rule add_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   667
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   668
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   669
text {* Addition is an AC-operator *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   670
lemmas add_ac = add_assoc add_commute add_left_commute
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   671
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   672
lemma add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   673
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   674
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   675
lemma add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   676
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   677
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   678
lemma add_left_cancel_le [simp]: "(k + m <= k + n) = (m<=(n::nat))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   679
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   680
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   681
lemma add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   682
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   683
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   684
text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   685
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   686
lemma add_is_0 [iff]: "!!m::nat. (m + n = 0) = (m = 0 & n = 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   687
  by (case_tac m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   688
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   689
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   690
  by (case_tac m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   691
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   692
lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   693
  by (rule trans, rule eq_commute, rule add_is_1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   694
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   695
lemma add_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m | 0 < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   696
  by (simp del: neq0_conv add: neq0_conv [symmetric])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   697
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   698
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   699
  apply (drule add_0_right [THEN ssubst])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   700
  apply (simp add: add_assoc del: add_0_right)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   701
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   702
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   703
subsection {* Additional theorems about "less than" *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   704
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   705
text {* Deleted @{text less_natE}; instead use @{text "less_imp_Suc_add RS exE"} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   706
lemma less_imp_Suc_add: "m < n ==> (EX k. n = Suc (m + k))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   707
  apply (induct n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   708
  apply (simp_all add: order_le_less)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   709
  apply (blast elim!: less_SucE intro!: add_0_right [symmetric] add_Suc_right [symmetric])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   710
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   711
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   712
lemma le_add2: "n <= ((m + n)::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   713
  apply (induct m)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   714
  apply simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   715
  apply (erule le_SucI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   716
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   717
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   718
lemma le_add1: "n <= ((n + m)::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   719
  apply (simp add: add_ac)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   720
  apply (rule le_add2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   721
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   722
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   723
lemma less_add_Suc1: "i < Suc (i + m)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   724
  by (rule le_less_trans, rule le_add1, rule lessI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   725
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   726
lemma less_add_Suc2: "i < Suc (m + i)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   727
  by (rule le_less_trans, rule le_add2, rule lessI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   728
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   729
lemma less_iff_Suc_add: "(m < n) = (EX k. n = Suc (m + k))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   730
  by (rules intro!: less_add_Suc1 less_imp_Suc_add)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   731
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   732
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   733
lemma trans_le_add1: "(i::nat) <= j ==> i <= j + m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   734
  by (rule le_trans, assumption, rule le_add1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   735
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   736
lemma trans_le_add2: "(i::nat) <= j ==> i <= m + j"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   737
  by (rule le_trans, assumption, rule le_add2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   738
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   739
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   740
  by (rule less_le_trans, assumption, rule le_add1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   741
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   742
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   743
  by (rule less_le_trans, assumption, rule le_add2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   744
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   745
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   746
  apply (induct j)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   747
  apply simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   748
  apply (blast dest: Suc_lessD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   749
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   750
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   751
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   752
  apply (rule notI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   753
  apply (erule add_lessD1 [THEN less_irrefl])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   754
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   755
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   756
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   757
  by (simp add: add_commute not_add_less1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   758
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   759
lemma add_leD1: "m + k <= n ==> m <= (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   760
  by (induct k) (simp_all add: le_simps)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   761
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   762
lemma add_leD2: "m + k <= n ==> k <= (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   763
  apply (simp add: add_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   764
  apply (erule add_leD1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   765
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   766
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   767
lemma add_leE: "(m::nat) + k <= n ==> (m <= n ==> k <= n ==> R) ==> R"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   768
  by (blast dest: add_leD1 add_leD2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   769
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   770
text {* needs @{text "!!k"} for @{text add_ac} to work *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   771
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   772
  by (force simp del: add_Suc_right
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   773
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   774
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   775
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   776
subsection {* Monotonicity of Addition *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   777
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   778
text {* strict, in 1st argument *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   779
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   780
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   781
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   782
text {* strict, in both arguments *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   783
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   784
  apply (rule add_less_mono1 [THEN less_trans])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   785
  apply assumption+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   786
  apply (induct_tac j)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   787
  apply simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   788
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   789
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   790
text {* A [clumsy] way of lifting @{text "<"}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   791
  monotonicity to @{text "<="} monotonicity *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   792
lemma less_mono_imp_le_mono:
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   793
  assumes lt_mono: "!!i j::nat. i < j ==> f i < f j"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   794
  and le: "i <= j" shows "f i <= ((f j)::nat)" using le
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   795
  apply (simp add: order_le_less)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   796
  apply (blast intro!: lt_mono)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   797
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   798
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   799
text {* non-strict, in 1st argument *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   800
lemma add_le_mono1: "i <= j ==> i + k <= j + (k::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   801
  apply (rule_tac f = "%j. j + k" in less_mono_imp_le_mono)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   802
  apply (erule add_less_mono1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   803
  apply assumption
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   804
  done
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   805
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   806
text {* non-strict, in both arguments *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   807
lemma add_le_mono: "[| i <= j;  k <= l |] ==> i + k <= j + (l::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   808
  apply (erule add_le_mono1 [THEN le_trans])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   809
  apply (simp add: add_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   810
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   811
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   812
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   813
subsection {* Multiplication *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   814
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   815
text {* right annihilation in product *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   816
lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   817
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   818
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   819
text {* right successor law for multiplication *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   820
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   821
  by (induct m) (simp_all add: add_ac)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   822
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   823
lemma mult_1: "(1::nat) * n = n" by simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   824
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   825
lemma mult_1_right: "n * (1::nat) = n" by simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   826
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   827
text {* Commutative law for multiplication *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   828
lemma mult_commute: "m * n = n * (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   829
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   830
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   831
text {* addition distributes over multiplication *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   832
lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   833
  by (induct m) (simp_all add: add_ac)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   834
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   835
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   836
  by (induct m) (simp_all add: add_ac)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   837
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   838
text {* Associative law for multiplication *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   839
lemma mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   840
  by (induct m) (simp_all add: add_mult_distrib)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   841
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   842
lemma mult_left_commute: "x * (y * z) = y * ((x * z)::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   843
  apply (rule mk_left_commute [of "op *"])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   844
  apply (rule mult_assoc)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   845
  apply (rule mult_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   846
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   847
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   848
lemmas mult_ac = mult_assoc mult_commute mult_left_commute
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   849
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   850
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   851
  apply (induct_tac m)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   852
  apply (induct_tac [2] n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   853
  apply simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   854
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   855
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   856
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   857
subsection {* Difference *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   858
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   859
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   860
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   861
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   862
text {* Addition is the inverse of subtraction:
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   863
  if @{term "n <= m"} then @{term "n + (m - n) = m"}. *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   864
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   865
  by (induct m n rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   866
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   867
lemma le_add_diff_inverse [simp]: "n <= m ==> n + (m - n) = (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   868
  by (simp add: add_diff_inverse not_less_iff_le)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   869
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   870
lemma le_add_diff_inverse2 [simp]: "n <= m ==> (m - n) + n = (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   871
  by (simp add: le_add_diff_inverse add_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   872
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   873
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   874
subsection {* More results about difference *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   875
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   876
lemma Suc_diff_le: "n <= m ==> Suc m - n = Suc (m - n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   877
  by (induct m n rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   878
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   879
lemma diff_less_Suc: "m - n < Suc m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   880
  apply (induct m n rule: diff_induct)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   881
  apply (erule_tac [3] less_SucE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   882
  apply (simp_all add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   883
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   884
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   885
lemma diff_le_self [simp]: "m - n <= (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   886
  by (induct m n rule: diff_induct) (simp_all add: le_SucI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   887
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   888
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   889
  by (rule le_less_trans, rule diff_le_self)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   890
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   891
lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   892
  by (induct i j rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   893
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   894
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   895
  by (simp add: diff_diff_left)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   896
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   897
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   898
  apply (case_tac "n")
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   899
  apply safe
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   900
  apply (simp add: le_simps)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   901
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   902
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   903
text {* This and the next few suggested by Florian Kammueller *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   904
lemma diff_commute: "(i::nat) - j - k = i - k - j"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   905
  by (simp add: diff_diff_left add_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   906
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   907
lemma diff_add_assoc: "k <= (j::nat) ==> (i + j) - k = i + (j - k)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   908
  by (induct j k rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   909
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   910
lemma diff_add_assoc2: "k <= (j::nat) ==> (j + i) - k = (j - k) + i"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   911
  by (simp add: add_commute diff_add_assoc)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   912
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   913
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   914
  by (induct n) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   915
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   916
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   917
  by (simp add: diff_add_assoc)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   918
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   919
lemma le_imp_diff_is_add: "i <= (j::nat) ==> (j - i = k) = (j = k + i)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   920
  apply safe
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   921
  apply (simp_all add: diff_add_inverse2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   922
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   923
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   924
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m <= n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   925
  by (induct m n rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   926
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   927
lemma diff_is_0_eq' [simp]: "m <= n ==> (m::nat) - n = 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   928
  by (rule iffD2, rule diff_is_0_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   929
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   930
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   931
  by (induct m n rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   932
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   933
lemma less_imp_add_positive: "i < j  ==> EX k::nat. 0 < k & i + k = j"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   934
  apply (rule_tac x = "j - i" in exI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   935
  apply (simp (no_asm_simp) add: add_diff_inverse less_not_sym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   936
  done
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   937
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   938
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   939
  apply (induct k i rule: diff_induct)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   940
  apply (simp_all (no_asm))
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   941
  apply rules
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   942
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   943
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   944
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   945
  apply (rule diff_self_eq_0 [THEN subst])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   946
  apply (rule zero_induct_lemma)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   947
  apply rules+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   948
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   949
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   950
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   951
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   952
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   953
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   954
  by (simp add: diff_cancel add_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   955
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   956
lemma diff_add_0: "n - (n + m) = (0::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   957
  by (induct n) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   958
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   959
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   960
text {* Difference distributes over multiplication *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   961
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   962
lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   963
  by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   964
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   965
lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   966
  by (simp add: diff_mult_distrib mult_commute [of k])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   967
  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   968
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   969
lemmas nat_distrib =
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   970
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   971
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   972
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   973
subsection {* Monotonicity of Multiplication *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   974
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   975
lemma mult_le_mono1: "i <= (j::nat) ==> i * k <= j * k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   976
  by (induct k) (simp_all add: add_le_mono)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   977
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   978
lemma mult_le_mono2: "i <= (j::nat) ==> k * i <= k * j"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   979
  apply (drule mult_le_mono1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   980
  apply (simp add: mult_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   981
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   982
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   983
text {* @{text "<="} monotonicity, BOTH arguments *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   984
lemma mult_le_mono: "i <= (j::nat) ==> k <= l ==> i * k <= j * l"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   985
  apply (erule mult_le_mono1 [THEN le_trans])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   986
  apply (erule mult_le_mono2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   987
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   988
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   989
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   990
lemma mult_less_mono2: "(i::nat) < j ==> 0 < k ==> k * i < k * j"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   991
  apply (erule_tac m1 = "0" in less_imp_Suc_add [THEN exE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   992
  apply simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   993
  apply (induct_tac x)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   994
  apply (simp_all add: add_less_mono)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   995
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   996
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   997
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   998
  by (drule mult_less_mono2) (simp_all add: mult_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   999
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1000
lemma zero_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1001
  apply (induct m)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1002
  apply (case_tac [2] n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1003
  apply simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1004
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1005
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1006
lemma one_le_mult_iff [simp]: "(Suc 0 <= m * n) = (1 <= m & 1 <= n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1007
  apply (induct m)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1008
  apply (case_tac [2] n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1009
  apply simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1010
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1011
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1012
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1013
  apply (induct_tac m)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1014
  apply simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1015
  apply (induct_tac n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1016
  apply simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1017
  apply fastsimp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1018
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1019
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1020
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1021
  apply (rule trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1022
  apply (rule_tac [2] mult_eq_1_iff)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1023
  apply fastsimp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1024
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1025
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1026
lemma mult_less_cancel2: "((m::nat) * k < n * k) = (0 < k & m < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1027
  apply (safe intro!: mult_less_mono1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1028
  apply (case_tac k)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1029
  apply auto
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1030
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1031
  apply (blast intro: mult_le_mono1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1032
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1033
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1034
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1035
  by (simp add: mult_less_cancel2 mult_commute [of k])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1036
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1037
declare mult_less_cancel2 [simp]
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1038
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1039
lemma mult_le_cancel1 [simp]: "(k * (m::nat) <= k * n) = (0 < k --> m <= n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1040
  apply (simp add: linorder_not_less [symmetric])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1041
  apply auto
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1042
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1043
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1044
lemma mult_le_cancel2 [simp]: "((m::nat) * k <= n * k) = (0 < k --> m <= n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1045
  apply (simp add: linorder_not_less [symmetric])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1046
  apply auto
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1047
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1048
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1049
lemma mult_cancel2: "(m * k = n * k) = (m = n | (k = (0::nat)))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1050
  apply (cut_tac less_linear)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1051
  apply safe
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1052
  apply auto
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1053
  apply (drule mult_less_mono1, assumption, simp)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1054
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1055
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1056
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1057
  by (simp add: mult_cancel2 mult_commute [of k])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1058
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1059
declare mult_cancel2 [simp]
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1060
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1061
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1062
  by (subst mult_less_cancel1) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1063
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1064
lemma Suc_mult_le_cancel1: "(Suc k * m <= Suc k * n) = (m <= n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1065
  by (subst mult_le_cancel1) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1066
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1067
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1068
  by (subst mult_cancel1) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1069
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1070
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1071
text {* Lemma for @{text gcd} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1072
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1073
  apply (drule sym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1074
  apply (rule disjCI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1075
  apply (rule nat_less_cases, erule_tac [2] _)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1076
  apply (fastsimp elim!: less_SucE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1077
  apply (fastsimp dest: mult_less_mono2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1078
  done
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
  1079
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
  1080
end