src/HOL/Nat.thy
author wenzelm
Tue, 02 Aug 2016 21:05:34 +0200
changeset 63588 d0e2bad67bd4
parent 63561 fba08009ff3e
child 63648 f9f3006a5579
permissions -rw-r--r--
misc tuning and modernization;
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
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
     2
    Author:     Tobias Nipkow
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
     3
    Author:     Lawrence C Paulson
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
     4
    Author:     Markus Wenzel
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     5
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
     6
Type "nat" is a linear order, and a datatype; arithmetic operators + -
30496
7cdcc9dd95cb vague cleanup in arith proof tools setup: deleted dead code, more proper structures, clearer arrangement
haftmann
parents: 30242
diff changeset
     7
and * (for div and mod, see theory Divides).
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     8
*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     9
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
    10
section \<open>Natural numbers\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    11
15131
c69542757a4d New theory header syntax.
nipkow
parents: 14740
diff changeset
    12
theory Nat
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    13
  imports Inductive Typedef Fun Rings
15131
c69542757a4d New theory header syntax.
nipkow
parents: 14740
diff changeset
    14
begin
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    15
57952
1a9a6dfc255f updated to named_theorems;
wenzelm
parents: 57514
diff changeset
    16
named_theorems arith "arith facts -- only ground formulas"
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 48560
diff changeset
    17
ML_file "Tools/arith_data.ML"
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 48560
diff changeset
    18
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 48560
diff changeset
    19
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
    20
subsection \<open>Type \<open>ind\<close>\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    21
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    22
typedecl ind
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    23
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
    24
axiomatization Zero_Rep :: ind and Suc_Rep :: "ind \<Rightarrow> ind"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
    25
  \<comment> \<open>The axiom of infinity in 2 parts:\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    26
  where Suc_Rep_inject: "Suc_Rep x = Suc_Rep y \<Longrightarrow> x = y"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    27
    and Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    28
19573
340c466c9605 axiomatization;
wenzelm
parents: 18702
diff changeset
    29
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
    30
subsection \<open>Type nat\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
    31
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
    32
text \<open>Type definition\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    33
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    34
inductive Nat :: "ind \<Rightarrow> bool"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    35
  where
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    36
    Zero_RepI: "Nat Zero_Rep"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    37
  | Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    38
49834
b27bbb021df1 discontinued obsolete typedef (open) syntax;
wenzelm
parents: 49723
diff changeset
    39
typedef nat = "{n. Nat n}"
45696
476ad865f125 prefer typedef without alternative name;
wenzelm
parents: 45231
diff changeset
    40
  morphisms Rep_Nat Abs_Nat
44278
1220ecb81e8f observe distinction between sets and predicates more properly
haftmann
parents: 43595
diff changeset
    41
  using Nat.Zero_RepI by auto
1220ecb81e8f observe distinction between sets and predicates more properly
haftmann
parents: 43595
diff changeset
    42
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    43
lemma Nat_Rep_Nat: "Nat (Rep_Nat n)"
44278
1220ecb81e8f observe distinction between sets and predicates more properly
haftmann
parents: 43595
diff changeset
    44
  using Rep_Nat by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    45
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    46
lemma Nat_Abs_Nat_inverse: "Nat n \<Longrightarrow> Rep_Nat (Abs_Nat n) = n"
44278
1220ecb81e8f observe distinction between sets and predicates more properly
haftmann
parents: 43595
diff changeset
    47
  using Abs_Nat_inverse by simp
1220ecb81e8f observe distinction between sets and predicates more properly
haftmann
parents: 43595
diff changeset
    48
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    49
lemma Nat_Abs_Nat_inject: "Nat n \<Longrightarrow> Nat m \<Longrightarrow> Abs_Nat n = Abs_Nat m \<longleftrightarrow> n = m"
44278
1220ecb81e8f observe distinction between sets and predicates more properly
haftmann
parents: 43595
diff changeset
    50
  using Abs_Nat_inject by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    51
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
    52
instantiation nat :: zero
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
    53
begin
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
    54
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    55
definition Zero_nat_def: "0 = Abs_Nat Zero_Rep"
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
    56
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
    57
instance ..
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
    58
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
    59
end
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
    60
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    61
definition Suc :: "nat \<Rightarrow> nat"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    62
  where "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"
44278
1220ecb81e8f observe distinction between sets and predicates more properly
haftmann
parents: 43595
diff changeset
    63
27104
791607529f6d rep_datatype command now takes list of constructors as input arguments
haftmann
parents: 26748
diff changeset
    64
lemma Suc_not_Zero: "Suc m \<noteq> 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    65
  by (simp add: Zero_nat_def Suc_def Suc_RepI Zero_RepI
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    66
      Nat_Abs_Nat_inject Suc_Rep_not_Zero_Rep Nat_Rep_Nat)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    67
27104
791607529f6d rep_datatype command now takes list of constructors as input arguments
haftmann
parents: 26748
diff changeset
    68
lemma Zero_not_Suc: "0 \<noteq> Suc m"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    69
  by (rule not_sym) (rule Suc_not_Zero)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
    70
34208
a7acd6c68d9b more regular axiom of infinity, with no (indirect) reference to overloaded constants
krauss
parents: 33657
diff changeset
    71
lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y \<longleftrightarrow> x = y"
a7acd6c68d9b more regular axiom of infinity, with no (indirect) reference to overloaded constants
krauss
parents: 33657
diff changeset
    72
  by (rule iffI, rule Suc_Rep_inject) simp_all
a7acd6c68d9b more regular axiom of infinity, with no (indirect) reference to overloaded constants
krauss
parents: 33657
diff changeset
    73
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
    74
lemma nat_induct0:
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    75
  assumes "P 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    76
    and "\<And>n. P n \<Longrightarrow> P (Suc n)"
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
    77
  shows "P n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    78
  using assms
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    79
  apply (unfold Zero_nat_def Suc_def)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    80
  apply (rule Rep_Nat_inverse [THEN subst]) \<comment> \<open>types force good instantiation\<close>
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    81
  apply (erule Nat_Rep_Nat [THEN Nat.induct])
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    82
  apply (iprover elim: Nat_Abs_Nat_inverse [THEN subst])
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    83
  done
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    84
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    85
free_constructors case_nat for "0 :: nat" | Suc pred
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    86
  where "pred (0 :: nat) = (0 :: nat)"
58189
9d714be4f028 added 'plugins' option to control which hooks are enabled
blanchet
parents: 57983
diff changeset
    87
    apply atomize_elim
9d714be4f028 added 'plugins' option to control which hooks are enabled
blanchet
parents: 57983
diff changeset
    88
    apply (rename_tac n, induct_tac n rule: nat_induct0, auto)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    89
   apply (simp add: Suc_def Nat_Abs_Nat_inject Nat_Rep_Nat Suc_RepI Suc_Rep_inject' Rep_Nat_inject)
58189
9d714be4f028 added 'plugins' option to control which hooks are enabled
blanchet
parents: 57983
diff changeset
    90
  apply (simp only: Suc_not_Zero)
9d714be4f028 added 'plugins' option to control which hooks are enabled
blanchet
parents: 57983
diff changeset
    91
  done
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
    92
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
    93
\<comment> \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close>
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
    94
setup \<open>Sign.mandatory_path "old"\<close>
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
    95
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
    96
old_rep_datatype "0 :: nat" Suc
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    97
    apply (erule nat_induct0)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    98
    apply assumption
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
    99
   apply (rule nat.inject)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   100
  apply (rule nat.distinct(1))
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   101
  done
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   102
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   103
setup \<open>Sign.parent_path\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   104
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   105
\<comment> \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close>
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   106
setup \<open>Sign.mandatory_path "nat"\<close>
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   107
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   108
declare old.nat.inject[iff del]
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   109
  and old.nat.distinct(1)[simp del, induct_simp del]
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   110
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   111
lemmas induct = old.nat.induct
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   112
lemmas inducts = old.nat.inducts
55642
63beb38e9258 adapted to renaming of datatype 'cases' and 'recs' to 'case' and 'rec'
blanchet
parents: 55575
diff changeset
   113
lemmas rec = old.nat.rec
63beb38e9258 adapted to renaming of datatype 'cases' and 'recs' to 'case' and 'rec'
blanchet
parents: 55575
diff changeset
   114
lemmas simps = nat.inject nat.distinct nat.case nat.rec
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   115
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   116
setup \<open>Sign.parent_path\<close>
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   117
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   118
abbreviation rec_nat :: "'a \<Rightarrow> (nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   119
  where "rec_nat \<equiv> old.rec_nat"
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   120
55424
9ab4129a76a3 remove hidden fact about hidden constant from code generator setup
blanchet
parents: 55423
diff changeset
   121
declare nat.sel[code del]
9ab4129a76a3 remove hidden fact about hidden constant from code generator setup
blanchet
parents: 55423
diff changeset
   122
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   123
hide_const (open) Nat.pred \<comment> \<open>hide everything related to the selector\<close>
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   124
hide_fact
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   125
  nat.case_eq_if
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   126
  nat.collapse
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   127
  nat.expand
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   128
  nat.sel
57983
6edc3529bb4e reordered some (co)datatype property names for more consistency
blanchet
parents: 57952
diff changeset
   129
  nat.exhaust_sel
6edc3529bb4e reordered some (co)datatype property names for more consistency
blanchet
parents: 57952
diff changeset
   130
  nat.split_sel
6edc3529bb4e reordered some (co)datatype property names for more consistency
blanchet
parents: 57952
diff changeset
   131
  nat.split_sel_asm
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   132
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   133
lemma nat_exhaust [case_names 0 Suc, cases type: nat]:
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   134
  "(y = 0 \<Longrightarrow> P) \<Longrightarrow> (\<And>nat. y = Suc nat \<Longrightarrow> P) \<Longrightarrow> P"
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   135
  \<comment> \<open>for backward compatibility -- names of variables differ\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   136
  by (rule old.nat.exhaust)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   137
27104
791607529f6d rep_datatype command now takes list of constructors as input arguments
haftmann
parents: 26748
diff changeset
   138
lemma nat_induct [case_names 0 Suc, induct type: nat]:
791607529f6d rep_datatype command now takes list of constructors as input arguments
haftmann
parents: 26748
diff changeset
   139
  fixes n
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   140
  assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)"
27104
791607529f6d rep_datatype command now takes list of constructors as input arguments
haftmann
parents: 26748
diff changeset
   141
  shows "P n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   142
  \<comment> \<open>for backward compatibility -- names of variables differ\<close>
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   143
  using assms by (rule nat.induct)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   144
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   145
hide_fact
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   146
  nat_exhaust
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55415
diff changeset
   147
  nat_induct0
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   148
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   149
ML \<open>
58389
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   150
val nat_basic_lfp_sugar =
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   151
  let
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   152
    val ctr_sugar = the (Ctr_Sugar.ctr_sugar_of_global @{theory} @{type_name nat});
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   153
    val recx = Logic.varify_types_global @{term rec_nat};
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   154
    val C = body_type (fastype_of recx);
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   155
  in
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   156
    {T = HOLogic.natT, fp_res_index = 0, C = C, fun_arg_Tsss = [[], [[HOLogic.natT, C]]],
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   157
     ctr_sugar = ctr_sugar, recx = recx, rec_thms = @{thms nat.rec}}
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   158
  end;
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   159
\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   160
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   161
setup \<open>
58389
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   162
let
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   163
  fun basic_lfp_sugars_of _ [@{typ nat}] _ _ ctxt =
62326
3cf7a067599c allow predicator instead of map function in 'primrec'
blanchet
parents: 62217
diff changeset
   164
      ([], [0], [nat_basic_lfp_sugar], [], [], [], TrueI (*dummy*), [], false, ctxt)
58389
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   165
    | basic_lfp_sugars_of bs arg_Ts callers callssss ctxt =
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   166
      BNF_LFP_Rec_Sugar.default_basic_lfp_sugars_of bs arg_Ts callers callssss ctxt;
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   167
in
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   168
  BNF_LFP_Rec_Sugar.register_lfp_rec_extension
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   169
    {nested_simps = [], is_new_datatype = K (K true), basic_lfp_sugars_of = basic_lfp_sugars_of,
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   170
     rewrite_nested_rec_call = NONE}
ee1f45ca0d73 made new 'primrec' bootstrapping-capable
blanchet
parents: 58377
diff changeset
   171
end
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   172
\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   173
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   174
text \<open>Injectiveness and distinctness lemmas\<close>
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   175
27104
791607529f6d rep_datatype command now takes list of constructors as input arguments
haftmann
parents: 26748
diff changeset
   176
lemma inj_Suc[simp]: "inj_on Suc N"
791607529f6d rep_datatype command now takes list of constructors as input arguments
haftmann
parents: 26748
diff changeset
   177
  by (simp add: inj_on_def)
791607529f6d rep_datatype command now takes list of constructors as input arguments
haftmann
parents: 26748
diff changeset
   178
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   179
lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   180
  by (rule notE) (rule Suc_not_Zero)
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   181
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   182
lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   183
  by (rule Suc_neq_Zero) (erule sym)
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   184
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   185
lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   186
  by (rule inj_Suc [THEN injD])
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   187
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   188
lemma n_not_Suc_n: "n \<noteq> Suc n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   189
  by (induct n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   190
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   191
lemma Suc_n_not_n: "Suc n \<noteq> n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   192
  by (rule not_sym) (rule n_not_Suc_n)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   193
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   194
text \<open>A special form of induction for reasoning about @{term "m < n"} and @{term "m - n"}.\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   195
lemma diff_induct:
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   196
  assumes "\<And>x. P x 0"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   197
    and "\<And>y. P 0 (Suc y)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   198
    and "\<And>x y. P x y \<Longrightarrow> P (Suc x) (Suc y)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   199
  shows "P m n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   200
proof (induct n arbitrary: m)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   201
  case 0
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   202
  show ?case by (rule assms(1))
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   203
next
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   204
  case (Suc n)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   205
  show ?case
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   206
  proof (induct m)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   207
    case 0
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   208
    show ?case by (rule assms(2))
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   209
  next
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   210
    case (Suc m)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   211
    from \<open>P m n\<close> show ?case by (rule assms(3))
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   212
  qed
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   213
qed
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   214
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   215
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   216
subsection \<open>Arithmetic operators\<close>
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   217
49388
1ffd5a055acf typeclass formalising bounded subtraction
haftmann
parents: 48891
diff changeset
   218
instantiation nat :: comm_monoid_diff
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25563
diff changeset
   219
begin
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   220
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   221
primrec plus_nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   222
  where
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   223
    add_0: "0 + n = (n::nat)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   224
  | add_Suc: "Suc m + n = Suc (m + n)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   225
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   226
lemma add_0_right [simp]: "m + 0 = m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   227
  for m :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   228
  by (induct m) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   229
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   230
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   231
  by (induct m) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   232
28514
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
   233
declare add_0 [code]
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
   234
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   235
lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   236
  by simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   237
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   238
primrec minus_nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   239
  where
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   240
    diff_0 [code]: "m - 0 = (m::nat)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   241
  | diff_Suc: "m - Suc n = (case m - n of 0 \<Rightarrow> 0 | Suc k \<Rightarrow> k)"
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   242
28514
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
   243
declare diff_Suc [simp del]
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   244
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   245
lemma diff_0_eq_0 [simp, code]: "0 - n = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   246
  for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   247
  by (induct n) (simp_all add: diff_Suc)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   248
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   249
lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   250
  by (induct n) (simp_all add: diff_Suc)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   251
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   252
instance
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   253
proof
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   254
  fix n m q :: nat
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   255
  show "(n + m) + q = n + (m + q)" by (induct n) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   256
  show "n + m = m + n" by (induct n) simp_all
59815
cce82e360c2f explicit commutative additive inverse operation;
haftmann
parents: 59582
diff changeset
   257
  show "m + n - m = n" by (induct m) simp_all
cce82e360c2f explicit commutative additive inverse operation;
haftmann
parents: 59582
diff changeset
   258
  show "n - m - q = n - (m + q)" by (induct q) (simp_all add: diff_Suc)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   259
  show "0 + n = n" by simp
49388
1ffd5a055acf typeclass formalising bounded subtraction
haftmann
parents: 48891
diff changeset
   260
  show "0 - n = 0" by simp
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   261
qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   262
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   263
end
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   264
36176
3fe7e97ccca8 replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents: 35828
diff changeset
   265
hide_fact (open) add_0 add_0_right diff_0
35047
1b2bae06c796 hide fact Nat.add_0_right; make add_0_right from Groups priority
haftmann
parents: 35028
diff changeset
   266
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   267
instantiation nat :: comm_semiring_1_cancel
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   268
begin
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   269
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   270
definition One_nat_def [simp]: "1 = Suc 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   271
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   272
primrec times_nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   273
  where
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   274
    mult_0: "0 * n = (0::nat)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   275
  | mult_Suc: "Suc m * n = n + (m * n)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   276
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   277
lemma mult_0_right [simp]: "m * 0 = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   278
  for m :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   279
  by (induct m) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   280
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   281
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   282
  by (induct m) (simp_all add: add.left_commute)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   283
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   284
lemma add_mult_distrib: "(m + n) * k = (m * k) + (n * k)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   285
  for m n k :: nat
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   286
  by (induct m) (simp_all add: add.assoc)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   287
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   288
instance
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   289
proof
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   290
  fix k n m q :: nat
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   291
  show "0 \<noteq> (1::nat)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   292
    by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   293
  show "1 * n = n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   294
    by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   295
  show "n * m = m * n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   296
    by (induct n) simp_all
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   297
  show "(n * m) * q = n * (m * q)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   298
    by (induct n) (simp_all add: add_mult_distrib)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   299
  show "(n + m) * q = n * q + m * q"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   300
    by (rule add_mult_distrib)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   301
  show "k * (m - n) = (k * m) - (k * n)"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   302
    by (induct m n rule: diff_induct) simp_all
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   303
qed
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25563
diff changeset
   304
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25563
diff changeset
   305
end
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   306
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   307
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   308
subsubsection \<open>Addition\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   309
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   310
text \<open>Reasoning about \<open>m + 0 = 0\<close>, etc.\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   311
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   312
lemma add_is_0 [iff]: "m + n = 0 \<longleftrightarrow> m = 0 \<and> n = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   313
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   314
  by (cases m) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   315
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   316
lemma add_is_1: "m + n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = 0 | m = 0 \<and> n = Suc 0"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   317
  by (cases m) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   318
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   319
lemma one_is_add: "Suc 0 = m + n \<longleftrightarrow> m = Suc 0 \<and> n = 0 | m = 0 \<and> n = Suc 0"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   320
  by (rule trans, rule eq_commute, rule add_is_1)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   321
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   322
lemma add_eq_self_zero: "m + n = m \<Longrightarrow> n = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   323
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   324
  by (induct m) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   325
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   326
lemma inj_on_add_nat [simp]: "inj_on (\<lambda>n. n + k) N"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   327
  for k :: nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   328
proof (induct k)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   329
  case 0
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   330
  then show ?case by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   331
next
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   332
  case (Suc k)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   333
  show ?case
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   334
    using comp_inj_on [OF Suc inj_Suc] by (simp add: o_def)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   335
qed
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   336
47208
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   337
lemma Suc_eq_plus1: "Suc n = n + 1"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   338
  by simp
47208
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   339
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   340
lemma Suc_eq_plus1_left: "Suc n = 1 + n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   341
  by simp
47208
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   342
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   343
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   344
subsubsection \<open>Difference\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   345
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   346
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
62365
8a105c235b1f sorted out some duplicate fact bindings
haftmann
parents: 62344
diff changeset
   347
  by (simp add: diff_diff_add)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   348
30093
ecb557b021b2 add lemma diff_Suc_1
huffman
parents: 30079
diff changeset
   349
lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   350
  by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   351
30093
ecb557b021b2 add lemma diff_Suc_1
huffman
parents: 30079
diff changeset
   352
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   353
subsubsection \<open>Multiplication\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   354
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   355
lemma mult_is_0 [simp]: "m * n = 0 \<longleftrightarrow> m = 0 \<or> n = 0" for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   356
  by (induct m) auto
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   357
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   358
lemma mult_eq_1_iff [simp]: "m * n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   359
proof (induct m)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   360
  case 0
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   361
  then show ?case by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   362
next
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   363
  case (Suc m)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   364
  then show ?case by (induct n) auto
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   365
qed
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   366
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   367
lemma one_eq_mult_iff [simp]: "Suc 0 = m * n \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   368
  apply (rule trans)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   369
   apply (rule_tac [2] mult_eq_1_iff)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   370
  apply fastforce
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   371
  done
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   372
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   373
lemma nat_mult_eq_1_iff [simp]: "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   374
  for m n :: nat
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   375
  unfolding One_nat_def by (rule mult_eq_1_iff)
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   376
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   377
lemma nat_1_eq_mult_iff [simp]: "1 = m * n \<longleftrightarrow> m = 1 \<and> n = 1"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   378
  for m n :: nat
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   379
  unfolding One_nat_def by (rule one_eq_mult_iff)
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   380
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   381
lemma mult_cancel1 [simp]: "k * m = k * n \<longleftrightarrow> m = n \<or> k = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   382
  for k m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   383
proof -
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   384
  have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   385
  proof (induct n arbitrary: m)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   386
    case 0
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   387
    then show "m = 0" by simp
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   388
  next
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   389
    case (Suc n)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   390
    then show "m = Suc n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   391
      by (cases m) (simp_all add: eq_commute [of 0])
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   392
  qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   393
  then show ?thesis by auto
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   394
qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   395
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   396
lemma mult_cancel2 [simp]: "m * k = n * k \<longleftrightarrow> m = n \<or> k = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   397
  for k m n :: nat
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   398
  by (simp add: mult.commute)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   399
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   400
lemma Suc_mult_cancel1: "Suc k * m = Suc k * n \<longleftrightarrow> m = n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   401
  by (subst mult_cancel1) simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   402
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   403
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   404
subsection \<open>Orders on @{typ nat}\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   405
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   406
subsubsection \<open>Operation definition\<close>
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   407
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   408
instantiation nat :: linorder
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   409
begin
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   410
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   411
primrec less_eq_nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   412
  where
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   413
    "(0::nat) \<le> n \<longleftrightarrow> True"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   414
  | "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   415
28514
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
   416
declare less_eq_nat.simps [simp del]
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   417
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   418
lemma le0 [iff]: "0 \<le> n" for
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   419
  n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   420
  by (simp add: less_eq_nat.simps)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   421
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   422
lemma [code]: "0 \<le> n \<longleftrightarrow> True"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   423
  for n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   424
  by simp
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   425
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   426
definition less_nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   427
  where less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   428
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   429
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   430
  by (simp add: less_eq_nat.simps(2))
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   431
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   432
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   433
  unfolding less_eq_Suc_le ..
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   434
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   435
lemma le_0_eq [iff]: "n \<le> 0 \<longleftrightarrow> n = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   436
  for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   437
  by (induct n) (simp_all add: less_eq_nat.simps(2))
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   438
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   439
lemma not_less0 [iff]: "\<not> n < 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   440
  for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   441
  by (simp add: less_eq_Suc_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   442
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   443
lemma less_nat_zero_code [code]: "n < 0 \<longleftrightarrow> False"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   444
  for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   445
  by simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   446
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   447
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   448
  by (simp add: less_eq_Suc_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   449
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   450
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   451
  by (simp add: less_eq_Suc_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   452
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   453
lemma Suc_less_eq2: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')"
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   454
  by (cases m) auto
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   455
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   456
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   457
  by (induct m arbitrary: n) (simp_all add: less_eq_nat.simps(2) split: nat.splits)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   458
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   459
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   460
  by (cases n) (auto intro: le_SucI)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   461
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   462
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   463
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   464
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   465
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   466
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   467
26315
cb3badaa192e removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents: 26300
diff changeset
   468
instance
cb3badaa192e removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents: 26300
diff changeset
   469
proof
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   470
  fix n m q :: nat
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   471
  show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   472
  proof (induct n arbitrary: m)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   473
    case 0
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   474
    then show ?case
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   475
      by (cases m) (simp_all add: less_eq_Suc_le)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   476
  next
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   477
    case (Suc n)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   478
    then show ?case
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   479
      by (cases m) (simp_all add: less_eq_Suc_le)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   480
  qed
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   481
  show "n \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   482
    by (induct n) simp_all
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   483
  then show "n = m" if "n \<le> m" and "m \<le> n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   484
    using that by (induct n arbitrary: m)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   485
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   486
  show "n \<le> q" if "n \<le> m" and "m \<le> q"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   487
    using that
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   488
  proof (induct n arbitrary: m q)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   489
    case 0
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   490
    show ?case by simp
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   491
  next
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   492
    case (Suc n)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   493
    then show ?case
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   494
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   495
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   496
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   497
  qed
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   498
  show "n \<le> m \<or> m \<le> n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   499
    by (induct n arbitrary: m)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   500
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   501
qed
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   502
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   503
end
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   504
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52435
diff changeset
   505
instantiation nat :: order_bot
29652
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   506
begin
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   507
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   508
definition bot_nat :: nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   509
  where "bot_nat = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   510
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   511
instance
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   512
  by standard (simp add: bot_nat_def)
29652
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   513
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   514
end
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   515
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
   516
instance nat :: no_top
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 61144
diff changeset
   517
  by standard (auto intro: less_Suc_eq_le [THEN iffD2])
52289
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 51329
diff changeset
   518
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
   519
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   520
subsubsection \<open>Introduction properties\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   521
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   522
lemma lessI [iff]: "n < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   523
  by (simp add: less_Suc_eq_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   524
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   525
lemma zero_less_Suc [iff]: "0 < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   526
  by (simp add: less_Suc_eq_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   527
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   528
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   529
subsubsection \<open>Elimination properties\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   530
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   531
lemma less_not_refl: "\<not> n < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   532
  for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   533
  by (rule order_less_irrefl)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   534
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   535
lemma less_not_refl2: "n < m \<Longrightarrow> m \<noteq> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   536
  for m n :: nat
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   537
  by (rule not_sym) (rule less_imp_neq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   538
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   539
lemma less_not_refl3: "s < t \<Longrightarrow> s \<noteq> t"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   540
  for s t :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   541
  by (rule less_imp_neq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   542
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   543
lemma less_irrefl_nat: "n < n \<Longrightarrow> R"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   544
  for n :: nat
26335
961bbcc9d85b removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents: 26315
diff changeset
   545
  by (rule notE, rule less_not_refl)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   546
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   547
lemma less_zeroE: "n < 0 \<Longrightarrow> R"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   548
  for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   549
  by (rule notE) (rule not_less0)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   550
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   551
lemma less_Suc_eq: "m < Suc n \<longleftrightarrow> m < n \<or> m = n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   552
  unfolding less_Suc_eq_le le_less ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   553
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   554
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   555
  by (simp add: less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   556
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   557
lemma less_one [iff]: "n < 1 \<longleftrightarrow> n = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   558
  for n :: nat
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   559
  unfolding One_nat_def by (rule less_Suc0)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   560
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   561
lemma Suc_mono: "m < n \<Longrightarrow> Suc m < Suc n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   562
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   563
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   564
text \<open>"Less than" is antisymmetric, sort of.\<close>
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   565
lemma less_antisym: "\<not> n < m \<Longrightarrow> n < Suc m \<Longrightarrow> m = n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   566
  unfolding not_less less_Suc_eq_le by (rule antisym)
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   567
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   568
lemma nat_neq_iff: "m \<noteq> n \<longleftrightarrow> m < n \<or> n < m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   569
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   570
  by (rule linorder_neq_iff)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   571
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   572
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   573
subsubsection \<open>Inductive (?) properties\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   574
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   575
lemma Suc_lessI: "m < n \<Longrightarrow> Suc m \<noteq> n \<Longrightarrow> Suc m < n"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   576
  unfolding less_eq_Suc_le [of m] le_less by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   577
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   578
lemma lessE:
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   579
  assumes major: "i < k"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   580
    and 1: "k = Suc i \<Longrightarrow> P"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   581
    and 2: "\<And>j. i < j \<Longrightarrow> k = Suc j \<Longrightarrow> P"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   582
  shows P
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   583
proof -
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   584
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   585
    unfolding less_eq_Suc_le by (induct k) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   586
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   587
    by (auto simp add: less_le)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   588
  with 1 2 show P by auto
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   589
qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   590
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   591
lemma less_SucE:
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   592
  assumes major: "m < Suc n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   593
    and less: "m < n \<Longrightarrow> P"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   594
    and eq: "m = n \<Longrightarrow> P"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   595
  shows P
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   596
  apply (rule major [THEN lessE])
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   597
   apply (rule eq)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   598
   apply blast
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   599
  apply (rule less)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   600
  apply blast
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   601
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   602
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   603
lemma Suc_lessE:
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   604
  assumes major: "Suc i < k"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   605
    and minor: "\<And>j. i < j \<Longrightarrow> k = Suc j \<Longrightarrow> P"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   606
  shows P
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   607
  apply (rule major [THEN lessE])
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   608
   apply (erule lessI [THEN minor])
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   609
  apply (erule Suc_lessD [THEN minor])
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   610
  apply assumption
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   611
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   612
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   613
lemma Suc_less_SucD: "Suc m < Suc n \<Longrightarrow> m < n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   614
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   615
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   616
lemma less_trans_Suc:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   617
  assumes le: "i < j"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   618
  shows "j < k \<Longrightarrow> Suc i < k"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   619
proof (induct k)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   620
  case 0
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   621
  then show ?case by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   622
next
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   623
  case (Suc k)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   624
  with le show ?case
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   625
    by simp (auto simp add: less_Suc_eq dest: Suc_lessD)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   626
qed
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   627
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   628
text \<open>Can be used with \<open>less_Suc_eq\<close> to get @{prop "n = m \<or> n < m"}.\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   629
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   630
  by (simp only: not_less less_Suc_eq_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   631
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   632
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   633
  by (simp only: not_le Suc_le_eq)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   634
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   635
text \<open>Properties of "less than or equal".\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   636
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   637
lemma le_imp_less_Suc: "m \<le> n \<Longrightarrow> m < Suc n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   638
  by (simp only: less_Suc_eq_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   639
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   640
lemma Suc_n_not_le_n: "\<not> Suc n \<le> n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   641
  by (simp add: not_le less_Suc_eq_le)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   642
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   643
lemma le_Suc_eq: "m \<le> Suc n \<longleftrightarrow> m \<le> n \<or> m = Suc n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   644
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   645
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   646
lemma le_SucE: "m \<le> Suc n \<Longrightarrow> (m \<le> n \<Longrightarrow> R) \<Longrightarrow> (m = Suc n \<Longrightarrow> R) \<Longrightarrow> R"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   647
  by (drule le_Suc_eq [THEN iffD1], iprover+)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   648
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   649
lemma Suc_leI: "m < n \<Longrightarrow> Suc m \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   650
  by (simp only: Suc_le_eq)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   651
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   652
text \<open>Stronger version of \<open>Suc_leD\<close>.\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   653
lemma Suc_le_lessD: "Suc m \<le> n \<Longrightarrow> m < n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   654
  by (simp only: Suc_le_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   655
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   656
lemma less_imp_le_nat: "m < n \<Longrightarrow> m \<le> n" for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   657
  unfolding less_eq_Suc_le by (rule Suc_leD)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   658
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   659
text \<open>For instance, \<open>(Suc m < Suc n) = (Suc m \<le> n) = (m < n)\<close>\<close>
26315
cb3badaa192e removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents: 26300
diff changeset
   660
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   661
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   662
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   663
text \<open>Equivalence of \<open>m \<le> n\<close> and \<open>m < n \<or> m = n\<close>\<close>
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   664
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   665
lemma less_or_eq_imp_le: "m < n \<or> m = n \<Longrightarrow> m \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   666
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   667
  unfolding le_less .
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   668
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   669
lemma le_eq_less_or_eq: "m \<le> n \<longleftrightarrow> m < n \<or> m = n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   670
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   671
  by (rule le_less)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   672
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   673
text \<open>Useful with \<open>blast\<close>.\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   674
lemma eq_imp_le: "m = n \<Longrightarrow> m \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   675
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   676
  by auto
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   677
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   678
lemma le_refl: "n \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   679
  for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   680
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   681
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   682
lemma le_trans: "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   683
  for i j k :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   684
  by (rule order_trans)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   685
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   686
lemma le_antisym: "m \<le> n \<Longrightarrow> n \<le> m \<Longrightarrow> m = n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   687
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   688
  by (rule antisym)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   689
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   690
lemma nat_less_le: "m < n \<longleftrightarrow> m \<le> n \<and> m \<noteq> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   691
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   692
  by (rule less_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   693
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   694
lemma le_neq_implies_less: "m \<le> n \<Longrightarrow> m \<noteq> n \<Longrightarrow> m < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   695
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   696
  unfolding less_le ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   697
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   698
lemma nat_le_linear: "m \<le> n | n \<le> m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   699
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   700
  by (rule linear)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   701
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   702
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
15921
b6e345548913 Fixing a problem with lin.arith.
nipkow
parents: 15539
diff changeset
   703
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   704
lemma le_less_Suc_eq: "m \<le> n \<Longrightarrow> n < Suc m \<longleftrightarrow> n = m"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   705
  unfolding less_Suc_eq_le by auto
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   706
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   707
lemma not_less_less_Suc_eq: "\<not> n < m \<Longrightarrow> n < Suc m \<longleftrightarrow> n = m"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   708
  unfolding not_less by (rule le_less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   709
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   710
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   711
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   712
lemma not0_implies_Suc: "n \<noteq> 0 \<Longrightarrow> \<exists>m. n = Suc m"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   713
  by (cases n) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   714
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   715
lemma gr0_implies_Suc: "n > 0 \<Longrightarrow> \<exists>m. n = Suc m"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   716
  by (cases n) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   717
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   718
lemma gr_implies_not0: "m < n \<Longrightarrow> n \<noteq> 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   719
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   720
  by (cases n) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   721
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   722
lemma neq0_conv[iff]: "n \<noteq> 0 \<longleftrightarrow> 0 < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   723
  for n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   724
  by (cases n) simp_all
25140
273772abbea2 More changes from >0 to ~=0::nat
nipkow
parents: 25134
diff changeset
   725
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   726
text \<open>This theorem is useful with \<open>blast\<close>\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   727
lemma gr0I: "(n = 0 \<Longrightarrow> False) \<Longrightarrow> 0 < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   728
  for n :: nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   729
  by (rule neq0_conv[THEN iffD1]) iprover
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   730
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   731
lemma gr0_conv_Suc: "0 < n \<longleftrightarrow> (\<exists>m. n = Suc m)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   732
  by (fast intro: not0_implies_Suc)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   733
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   734
lemma not_gr0 [iff]: "\<not> 0 < n \<longleftrightarrow> n = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   735
  for n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   736
  using neq0_conv by blast
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   737
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   738
lemma Suc_le_D: "Suc n \<le> m' \<Longrightarrow> \<exists>m. m' = Suc m"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   739
  by (induct m') simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   740
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   741
text \<open>Useful in certain inductive arguments\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   742
lemma less_Suc_eq_0_disj: "m < Suc n \<longleftrightarrow> m = 0 \<or> (\<exists>j. m = Suc j \<and> j < n)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   743
  by (cases m) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   744
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   745
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   746
subsubsection \<open>Monotonicity of Addition\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   747
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   748
lemma Suc_pred [simp]: "n > 0 \<Longrightarrow> Suc (n - Suc 0) = n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   749
  by (simp add: diff_Suc split: nat.split)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   750
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   751
lemma Suc_diff_1 [simp]: "0 < n \<Longrightarrow> Suc (n - 1) = n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   752
  unfolding One_nat_def by (rule Suc_pred)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   753
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   754
lemma nat_add_left_cancel_le [simp]: "k + m \<le> k + n \<longleftrightarrow> m \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   755
  for k m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   756
  by (induct k) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   757
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   758
lemma nat_add_left_cancel_less [simp]: "k + m < k + n \<longleftrightarrow> m < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   759
  for k m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   760
  by (induct k) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   761
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   762
lemma add_gr_0 [iff]: "m + n > 0 \<longleftrightarrow> m > 0 \<or> n > 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   763
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   764
  by (auto dest: gr0_implies_Suc)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   765
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   766
text \<open>strict, in 1st argument\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   767
lemma add_less_mono1: "i < j \<Longrightarrow> i + k < j + k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   768
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   769
  by (induct k) simp_all
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   770
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   771
text \<open>strict, in both arguments\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   772
lemma add_less_mono: "i < j \<Longrightarrow> k < l \<Longrightarrow> i + k < j + l"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   773
  for i j k l :: nat
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   774
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   775
  apply (induct j)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   776
   apply simp_all
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   777
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   778
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   779
text \<open>Deleted \<open>less_natE\<close>; use \<open>less_imp_Suc_add RS exE\<close>\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   780
lemma less_imp_Suc_add: "m < n \<Longrightarrow> \<exists>k. n = Suc (m + k)"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   781
proof (induct n)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   782
  case 0
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   783
  then show ?case by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   784
next
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   785
  case Suc
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   786
  then show ?case
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   787
    by (simp add: order_le_less)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   788
      (blast elim!: less_SucE intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   789
qed
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   790
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   791
lemma le_Suc_ex: "k \<le> l \<Longrightarrow> (\<exists>n. l = k + n)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   792
  for k l :: nat
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   793
  by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add)
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   794
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   795
text \<open>strict, in 1st argument; proof is by induction on \<open>k > 0\<close>\<close>
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   796
lemma mult_less_mono2:
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   797
  fixes i j :: nat
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   798
  assumes "i < j" and "0 < k"
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   799
  shows "k * i < k * j"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   800
  using \<open>0 < k\<close>
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   801
proof (induct k)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   802
  case 0
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   803
  then show ?case by simp
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   804
next
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   805
  case (Suc k)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   806
  with \<open>i < j\<close> show ?case
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   807
    by (cases k) (simp_all add: add_less_mono)
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   808
qed
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   809
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   810
text \<open>Addition is the inverse of subtraction:
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   811
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}.\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   812
lemma add_diff_inverse_nat: "\<not> m < n \<Longrightarrow> n + (m - n) = m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   813
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   814
  by (induct m n rule: diff_induct) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   815
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   816
lemma nat_le_iff_add: "m \<le> n \<longleftrightarrow> (\<exists>k. n = m + k)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   817
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   818
  using nat_add_left_cancel_le[of m 0] by (auto dest: le_Suc_ex)
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62365
diff changeset
   819
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   820
text \<open>The naturals form an ordered \<open>semidom\<close> and a \<open>dioid\<close>.\<close>
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62365
diff changeset
   821
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34208
diff changeset
   822
instance nat :: linordered_semidom
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   823
proof
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   824
  fix m n q :: nat
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   825
  show "0 < (1::nat)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   826
    by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   827
  show "m \<le> n \<Longrightarrow> q + m \<le> q + n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   828
    by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   829
  show "m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   830
    by (simp add: mult_less_mono2)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   831
  show "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   832
    by simp
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   833
  show "n \<le> m \<Longrightarrow> (m - n) + n = m"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   834
    by (simp add: add_diff_inverse_nat add.commute linorder_not_less)
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62365
diff changeset
   835
qed
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62365
diff changeset
   836
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62365
diff changeset
   837
instance nat :: dioid
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   838
  by standard (rule nat_le_iff_add)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   839
63145
703edebd1d92 isabelle update_cartouches -c -t;
wenzelm
parents: 63113
diff changeset
   840
declare le0[simp del] \<comment> \<open>This is now @{thm zero_le}\<close>
703edebd1d92 isabelle update_cartouches -c -t;
wenzelm
parents: 63113
diff changeset
   841
declare le_0_eq[simp del] \<comment> \<open>This is now @{thm le_zero_eq}\<close>
703edebd1d92 isabelle update_cartouches -c -t;
wenzelm
parents: 63113
diff changeset
   842
declare not_less0[simp del] \<comment> \<open>This is now @{thm not_less_zero}\<close>
703edebd1d92 isabelle update_cartouches -c -t;
wenzelm
parents: 63113
diff changeset
   843
declare not_gr0[simp del] \<comment> \<open>This is now @{thm not_gr_zero}\<close>
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62365
diff changeset
   844
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   845
instance nat :: ordered_cancel_comm_monoid_add ..
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   846
instance nat :: ordered_cancel_comm_monoid_diff ..
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   847
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   848
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   849
subsubsection \<open>@{term min} and @{term max}\<close>
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   850
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   851
lemma mono_Suc: "mono Suc"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   852
  by (rule monoI) simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   853
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   854
lemma min_0L [simp]: "min 0 n = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   855
  for n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   856
  by (rule min_absorb1) simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   857
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   858
lemma min_0R [simp]: "min n 0 = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   859
  for n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   860
  by (rule min_absorb2) simp
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   861
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   862
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   863
  by (simp add: mono_Suc min_of_mono)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   864
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   865
lemma min_Suc1: "min (Suc n) m = (case m of 0 \<Rightarrow> 0 | Suc m' \<Rightarrow> Suc(min n m'))"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   866
  by (simp split: nat.split)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   867
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   868
lemma min_Suc2: "min m (Suc n) = (case m of 0 \<Rightarrow> 0 | Suc m' \<Rightarrow> Suc(min m' n))"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   869
  by (simp split: nat.split)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   870
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   871
lemma max_0L [simp]: "max 0 n = n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   872
  for n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   873
  by (rule max_absorb2) simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   874
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   875
lemma max_0R [simp]: "max n 0 = n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   876
  for n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   877
  by (rule max_absorb1) simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   878
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   879
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc (max m n)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   880
  by (simp add: mono_Suc max_of_mono)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   881
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   882
lemma max_Suc1: "max (Suc n) m = (case m of 0 \<Rightarrow> Suc n | Suc m' \<Rightarrow> Suc (max n m'))"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   883
  by (simp split: nat.split)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   884
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   885
lemma max_Suc2: "max m (Suc n) = (case m of 0 \<Rightarrow> Suc n | Suc m' \<Rightarrow> Suc (max m' n))"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   886
  by (simp split: nat.split)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   887
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   888
lemma nat_mult_min_left: "min m n * q = min (m * q) (n * q)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   889
  for m n q :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   890
  by (simp add: min_def not_le)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   891
    (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   892
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   893
lemma nat_mult_min_right: "m * min n q = min (m * n) (m * q)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   894
  for m n q :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   895
  by (simp add: min_def not_le)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   896
    (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   897
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   898
lemma nat_add_max_left: "max m n + q = max (m + q) (n + q)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   899
  for m n q :: nat
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   900
  by (simp add: max_def)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   901
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   902
lemma nat_add_max_right: "m + max n q = max (m + n) (m + q)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   903
  for m n q :: nat
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   904
  by (simp add: max_def)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   905
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   906
lemma nat_mult_max_left: "max m n * q = max (m * q) (n * q)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   907
  for m n q :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   908
  by (simp add: max_def not_le)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   909
    (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   910
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   911
lemma nat_mult_max_right: "m * max n q = max (m * n) (m * q)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   912
  for m n q :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   913
  by (simp add: max_def not_le)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   914
    (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   915
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   916
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   917
subsubsection \<open>Additional theorems about @{term "op \<le>"}\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   918
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   919
text \<open>Complete induction, aka course-of-values induction\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   920
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   921
instance nat :: wellorder
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   922
proof
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   923
  fix P and n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   924
  assume step: "(\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n" for n :: nat
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   925
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   926
  proof (induct n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   927
    case (0 n)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   928
    have "P 0" by (rule step) auto
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   929
    with 0 show ?case by auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   930
  next
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   931
    case (Suc m n)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   932
    then have "n \<le> m \<or> n = Suc m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   933
      by (simp add: le_Suc_eq)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   934
    then show ?case
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   935
    proof
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   936
      assume "n \<le> m"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   937
      then show "P n" by (rule Suc(1))
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   938
    next
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   939
      assume n: "n = Suc m"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   940
      show "P n" by (rule step) (rule Suc(1), simp add: n le_simps)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   941
    qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   942
  qed
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   943
  then show "P n" by auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   944
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   945
57015
842bb6d36263 added lemma
nipkow
parents: 56194
diff changeset
   946
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   947
lemma Least_eq_0[simp]: "P 0 \<Longrightarrow> Least P = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   948
  for P :: "nat \<Rightarrow> bool"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   949
  by (rule Least_equality[OF _ le0])
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   950
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   951
lemma Least_Suc: "P n \<Longrightarrow> \<not> P 0 \<Longrightarrow> (LEAST n. P n) = Suc (LEAST m. P (Suc m))"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   952
  apply (cases n)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   953
   apply auto
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   954
  apply (frule LeastI)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   955
  apply (drule_tac P = "\<lambda>x. P (Suc x)" in LeastI)
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   956
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   957
   apply (erule_tac [2] Least_le)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   958
  apply (cases "LEAST x. P x")
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   959
   apply auto
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   960
  apply (drule_tac P = "\<lambda>x. P (Suc x)" in Least_le)
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   961
  apply (blast intro: order_antisym)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   962
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   963
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   964
lemma Least_Suc2: "P n \<Longrightarrow> Q m \<Longrightarrow> \<not> P 0 \<Longrightarrow> \<forall>k. P (Suc k) = Q k \<Longrightarrow> Least P = Suc (Least Q)"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   965
  by (erule (1) Least_Suc [THEN ssubst]) simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   966
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   967
lemma ex_least_nat_le: "\<not> P 0 \<Longrightarrow> P n \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not> P i) \<and> P k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   968
  for P :: "nat \<Rightarrow> bool"
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   969
  apply (cases n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   970
   apply blast
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   971
  apply (rule_tac x="LEAST k. P k" in exI)
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   972
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   973
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   974
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   975
lemma ex_least_nat_less: "\<not> P 0 \<Longrightarrow> P n \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not> P i) \<and> P (k + 1)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   976
  for P :: "nat \<Rightarrow> bool"
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   977
  apply (cases n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   978
   apply blast
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   979
  apply (frule (1) ex_least_nat_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   980
  apply (erule exE)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   981
  apply (case_tac k)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   982
   apply simp
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   983
  apply (rename_tac k1)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   984
  apply (rule_tac x=k1 in exI)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   985
  apply (auto simp add: less_eq_Suc_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   986
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   987
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   988
lemma nat_less_induct:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   989
  fixes P :: "nat \<Rightarrow> bool"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   990
  assumes "\<And>n. \<forall>m. m < n \<longrightarrow> P m \<Longrightarrow> P n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   991
  shows "P n"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   992
  using assms less_induct by blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   993
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   994
lemma measure_induct_rule [case_names less]:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   995
  fixes f :: "'a \<Rightarrow> nat"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   996
  assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   997
  shows "P a"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   998
  by (induct m \<equiv> "f a" arbitrary: a rule: less_induct) (auto intro: step)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   999
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1000
text \<open>old style induction rules:\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1001
lemma measure_induct:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1002
  fixes f :: "'a \<Rightarrow> nat"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1003
  shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1004
  by (rule measure_induct_rule [of f P a]) iprover
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1005
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1006
lemma full_nat_induct:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1007
  assumes step: "\<And>n. (\<forall>m. Suc m \<le> n \<longrightarrow> P m) \<Longrightarrow> P n"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1008
  shows "P n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1009
  by (rule less_induct) (auto intro: step simp:le_simps)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1010
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1011
text\<open>An induction rule for establishing binary relations\<close>
62683
ddd1c864408b clarified rule structure;
wenzelm
parents: 62608
diff changeset
  1012
lemma less_Suc_induct [consumes 1]:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1013
  assumes less: "i < j"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1014
    and step: "\<And>i. P i (Suc i)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1015
    and trans: "\<And>i j k. i < j \<Longrightarrow> j < k \<Longrightarrow> P i j \<Longrightarrow> P j k \<Longrightarrow> P i k"
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1016
  shows "P i j"
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1017
proof -
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1018
  from less obtain k where j: "j = Suc (i + k)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1019
    by (auto dest: less_imp_Suc_add)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1020
  have "P i (Suc (i + k))"
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1021
  proof (induct k)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1022
    case 0
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1023
    show ?case by (simp add: step)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1024
  next
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1025
    case (Suc k)
31714
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
  1026
    have "0 + i < Suc k + i" by (rule add_less_mono1) simp
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1027
    then have "i < Suc (i + k)" by (simp add: add.commute)
31714
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
  1028
    from trans[OF this lessI Suc step]
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
  1029
    show ?case by simp
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1030
  qed
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1031
  then show "P i j" by (simp add: j)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1032
qed
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1033
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1034
text \<open>
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1035
  The method of infinite descent, frequently used in number theory.
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1036
  Provided by Roelof Oosterhuis.
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1037
  \<open>P n\<close> is true for all natural numbers if
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1038
  \<^item> case ``0'': given \<open>n = 0\<close> prove \<open>P n\<close>
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1039
  \<^item> case ``smaller'': given \<open>n > 0\<close> and \<open>\<not> P n\<close> prove there exists
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1040
    a smaller natural number \<open>m\<close> such that \<open>\<not> P m\<close>.
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1041
\<close>
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1042
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1043
lemma infinite_descent: "(\<And>n. \<not> P n \<Longrightarrow> \<exists>m<n. \<not> P m) \<Longrightarrow> P n" for P :: "nat \<Rightarrow> bool"
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1044
  \<comment> \<open>compact version without explicit base case\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1045
  by (induct n rule: less_induct) auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1046
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1047
lemma infinite_descent0 [case_names 0 smaller]:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1048
  fixes P :: "nat \<Rightarrow> bool"
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1049
  assumes "P 0"
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1050
    and "\<And>n. n > 0 \<Longrightarrow> \<not> P n \<Longrightarrow> \<exists>m. m < n \<and> \<not> P m"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1051
  shows "P n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1052
  apply (rule infinite_descent)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1053
  using assms
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1054
  apply (case_tac "n > 0")
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1055
   apply auto
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1056
  done
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1057
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1058
text \<open>
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1059
  Infinite descent using a mapping to \<open>nat\<close>:
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1060
  \<open>P x\<close> is true for all \<open>x \<in> D\<close> if there exists a \<open>V \<in> D \<Rightarrow> nat\<close> and
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1061
  \<^item> case ``0'': given \<open>V x = 0\<close> prove \<open>P x\<close>
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1062
  \<^item> ``smaller'': given \<open>V x > 0\<close> and \<open>\<not> P x\<close> prove
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1063
  there exists a \<open>y \<in> D\<close> such that \<open>V y < V x\<close> and \<open>\<not> P y\<close>.
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1064
\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1065
corollary infinite_descent0_measure [case_names 0 smaller]:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1066
  fixes V :: "'a \<Rightarrow> nat"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1067
  assumes 1: "\<And>x. V x = 0 \<Longrightarrow> P x"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1068
    and 2: "\<And>x. V x > 0 \<Longrightarrow> \<not> P x \<Longrightarrow> \<exists>y. V y < V x \<and> \<not> P y"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1069
  shows "P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1070
proof -
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1071
  obtain n where "n = V x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1072
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1073
  proof (induct n rule: infinite_descent0)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1074
    case 0
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1075
    with 1 show "P x" by auto
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1076
  next
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1077
    case (smaller n)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1078
    then obtain x where *: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1079
    with 2 obtain y where "V y < V x \<and> \<not> P y" by auto
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1080
    with * obtain m where "m = V y \<and> m < n \<and> \<not> P y" by auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1081
    then show ?case by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1082
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1083
  ultimately show "P x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1084
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1085
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1086
text \<open>Again, without explicit base case:\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1087
lemma infinite_descent_measure:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1088
  fixes V :: "'a \<Rightarrow> nat"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1089
  assumes "\<And>x. \<not> P x \<Longrightarrow> \<exists>y. V y < V x \<and> \<not> P y"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1090
  shows "P x"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1091
proof -
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1092
  from assms obtain n where "n = V x" by auto
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1093
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1094
  proof (induct n rule: infinite_descent, auto)
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1095
    show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" if "\<not> P x" for x
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1096
      using assms and that by auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1097
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1098
  ultimately show "P x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1099
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1100
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1101
text \<open>A (clumsy) way of lifting \<open><\<close> monotonicity to \<open>\<le>\<close> monotonicity\<close>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1102
lemma less_mono_imp_le_mono:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1103
  fixes f :: "nat \<Rightarrow> nat"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1104
    and i j :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1105
  assumes "\<And>i j::nat. i < j \<Longrightarrow> f i < f j"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1106
    and "i \<le> j"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1107
  shows "f i \<le> f j"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1108
  using assms by (auto simp add: order_le_less)
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1109
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1110
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1111
text \<open>non-strict, in 1st argument\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1112
lemma add_le_mono1: "i \<le> j \<Longrightarrow> i + k \<le> j + k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1113
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1114
  by (rule add_right_mono)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1115
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1116
text \<open>non-strict, in both arguments\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1117
lemma add_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i + k \<le> j + l"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1118
  for i j k l :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1119
  by (rule add_mono)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1120
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1121
lemma le_add2: "n \<le> m + n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1122
  for m n :: nat
62608
19f87fa0cfcb more theorems on orderings
haftmann
parents: 62502
diff changeset
  1123
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1124
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1125
lemma le_add1: "n \<le> n + m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1126
  for m n :: nat
62608
19f87fa0cfcb more theorems on orderings
haftmann
parents: 62502
diff changeset
  1127
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1128
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1129
lemma less_add_Suc1: "i < Suc (i + m)"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1130
  by (rule le_less_trans, rule le_add1, rule lessI)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1131
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1132
lemma less_add_Suc2: "i < Suc (m + i)"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1133
  by (rule le_less_trans, rule le_add2, rule lessI)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1134
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1135
lemma less_iff_Suc_add: "m < n \<longleftrightarrow> (\<exists>k. n = Suc (m + k))"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1136
  by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1137
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1138
lemma trans_le_add1: "i \<le> j \<Longrightarrow> i \<le> j + m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1139
  for i j m :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1140
  by (rule le_trans, assumption, rule le_add1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1141
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1142
lemma trans_le_add2: "i \<le> j \<Longrightarrow> i \<le> m + j"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1143
  for i j m :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1144
  by (rule le_trans, assumption, rule le_add2)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1145
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1146
lemma trans_less_add1: "i < j \<Longrightarrow> i < j + m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1147
  for i j m :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1148
  by (rule less_le_trans, assumption, rule le_add1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1149
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1150
lemma trans_less_add2: "i < j \<Longrightarrow> i < m + j"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1151
  for i j m :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1152
  by (rule less_le_trans, assumption, rule le_add2)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1153
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1154
lemma add_lessD1: "i + j < k \<Longrightarrow> i < k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1155
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1156
  by (rule le_less_trans [of _ "i+j"]) (simp_all add: le_add1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1157
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1158
lemma not_add_less1 [iff]: "\<not> i + j < i"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1159
  for i j :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1160
  apply (rule notI)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1161
  apply (drule add_lessD1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1162
  apply (erule less_irrefl [THEN notE])
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1163
  done
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1164
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1165
lemma not_add_less2 [iff]: "\<not> j + i < i"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1166
  for i j :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1167
  by (simp add: add.commute)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1168
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1169
lemma add_leD1: "m + k \<le> n \<Longrightarrow> m \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1170
  for k m n :: nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1171
  by (rule order_trans [of _ "m + k"]) (simp_all add: le_add1)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1172
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1173
lemma add_leD2: "m + k \<le> n \<Longrightarrow> k \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1174
  for k m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1175
  apply (simp add: add.commute)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1176
  apply (erule add_leD1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1177
  done
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1178
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1179
lemma add_leE: "m + k \<le> n \<Longrightarrow> (m \<le> n \<Longrightarrow> k \<le> n \<Longrightarrow> R) \<Longrightarrow> R"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1180
  for k m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1181
  by (blast dest: add_leD1 add_leD2)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1182
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1183
text \<open>needs \<open>\<And>k\<close> for \<open>ac_simps\<close> to work\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1184
lemma less_add_eq_less: "\<And>k. k < l \<Longrightarrow> m + l = k + n \<Longrightarrow> m < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1185
  for l m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1186
  by (force simp del: add_Suc_right simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1187
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1188
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1189
subsubsection \<open>More results about difference\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1190
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1191
lemma Suc_diff_le: "n \<le> m \<Longrightarrow> Suc m - n = Suc (m - n)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1192
  by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1193
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1194
lemma diff_less_Suc: "m - n < Suc m"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1195
  apply (induct m n rule: diff_induct)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1196
    apply (erule_tac [3] less_SucE)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1197
     apply (simp_all add: less_Suc_eq)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1198
  done
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1199
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1200
lemma diff_le_self [simp]: "m - n \<le> m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1201
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1202
  by (induct m n rule: diff_induct) (simp_all add: le_SucI)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1203
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1204
lemma less_imp_diff_less: "j < k \<Longrightarrow> j - n < k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1205
  for j k n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1206
  by (rule le_less_trans, rule diff_le_self)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1207
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1208
lemma diff_Suc_less [simp]: "0 < n \<Longrightarrow> n - Suc i < n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1209
  by (cases n) (auto simp add: le_simps)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1210
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1211
lemma diff_add_assoc: "k \<le> j \<Longrightarrow> (i + j) - k = i + (j - k)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1212
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1213
  by (induct j k rule: diff_induct) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1214
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1215
lemma add_diff_assoc [simp]: "k \<le> j \<Longrightarrow> i + (j - k) = i + j - k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1216
  for i j k :: nat
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
  1217
  by (fact diff_add_assoc [symmetric])
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
  1218
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1219
lemma diff_add_assoc2: "k \<le> j \<Longrightarrow> (j + i) - k = (j - k) + i"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1220
  for i j k :: nat
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
  1221
  by (simp add: ac_simps)
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
  1222
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1223
lemma add_diff_assoc2 [simp]: "k \<le> j \<Longrightarrow> j - k + i = j + i - k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1224
  for i j k :: nat
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
  1225
  by (fact diff_add_assoc2 [symmetric])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1226
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1227
lemma le_imp_diff_is_add: "i \<le> j \<Longrightarrow> (j - i = k) = (j = k + i)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1228
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1229
  by auto
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1230
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1231
lemma diff_is_0_eq [simp]: "m - n = 0 \<longleftrightarrow> m \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1232
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1233
  by (induct m n rule: diff_induct) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1234
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1235
lemma diff_is_0_eq' [simp]: "m \<le> n \<Longrightarrow> m - n = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1236
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1237
  by (rule iffD2, rule diff_is_0_eq)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1238
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1239
lemma zero_less_diff [simp]: "0 < n - m \<longleftrightarrow> m < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1240
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1241
  by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1242
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1243
lemma less_imp_add_positive:
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1244
  assumes "i < j"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1245
  shows "\<exists>k::nat. 0 < k \<and> i + k = j"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1246
proof
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1247
  from assms show "0 < j - i \<and> i + (j - i) = j"
23476
839db6346cc8 fix looping simp rule
huffman
parents: 23438
diff changeset
  1248
    by (simp add: order_less_imp_le)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1249
qed
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
  1250
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1251
text \<open>a nice rewrite for bounded subtraction\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1252
lemma nat_minus_add_max: "n - m + m = max n m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1253
  for m n :: nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1254
  by (simp add: max_def not_le order_less_imp_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1255
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1256
lemma nat_diff_split: "P (a - b) \<longleftrightarrow> (a < b \<longrightarrow> P 0) \<and> (\<forall>d. a = b + d \<longrightarrow> P d)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1257
  for a b :: nat
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1258
  \<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close>\<close>
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1259
  by (cases "a < b") (auto simp add: not_less le_less dest!: add_eq_self_zero [OF sym])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1260
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1261
lemma nat_diff_split_asm: "P (a - b) \<longleftrightarrow> \<not> (a < b \<and> \<not> P 0 \<or> (\<exists>d. a = b + d \<and> \<not> P d))"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1262
  for a b :: nat
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1263
  \<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close> in assumptions\<close>
62365
8a105c235b1f sorted out some duplicate fact bindings
haftmann
parents: 62344
diff changeset
  1264
  by (auto split: nat_diff_split)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1265
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1266
lemma Suc_pred': "0 < n \<Longrightarrow> n = Suc(n - 1)"
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1267
  by simp
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1268
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1269
lemma add_eq_if: "m + n = (if m = 0 then n else Suc ((m - 1) + n))"
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1270
  unfolding One_nat_def by (cases m) simp_all
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1271
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1272
lemma mult_eq_if: "m * n = (if m = 0 then 0 else n + ((m - 1) * n))"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1273
  for m n :: nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1274
  by (cases m) simp_all
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1275
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1276
lemma Suc_diff_eq_diff_pred: "0 < n \<Longrightarrow> Suc m - n = m - (n - 1)"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1277
  by (cases n) simp_all
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1278
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1279
lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1280
  by (cases m) simp_all
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1281
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1282
lemma Let_Suc [simp]: "Let (Suc n) f \<equiv> f (Suc n)"
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1283
  by (fact Let_def)
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1284
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1285
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1286
subsubsection \<open>Monotonicity of multiplication\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1287
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1288
lemma mult_le_mono1: "i \<le> j \<Longrightarrow> i * k \<le> j * k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1289
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1290
  by (simp add: mult_right_mono)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1291
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1292
lemma mult_le_mono2: "i \<le> j \<Longrightarrow> k * i \<le> k * j"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1293
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1294
  by (simp add: mult_left_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1295
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1296
text \<open>\<open>\<le>\<close> monotonicity, BOTH arguments\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1297
lemma mult_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i * k \<le> j * l"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1298
  for i j k l :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1299
  by (simp add: mult_mono)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1300
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1301
lemma mult_less_mono1: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> i * k < j * k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1302
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1303
  by (simp add: mult_strict_right_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1304
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1305
text \<open>Differs from the standard \<open>zero_less_mult_iff\<close> in that there are no negative numbers.\<close>
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1306
lemma nat_0_less_mult_iff [simp]: "0 < m * n \<longleftrightarrow> 0 < m \<and> 0 < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1307
  for m n :: nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1308
proof (induct m)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1309
  case 0
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1310
  then show ?case by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1311
next
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1312
  case (Suc m)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: