src/HOL/Orderings.thy
author haftmann
Tue, 05 Jun 2007 19:23:09 +0200
changeset 23263 0c227412b285
parent 23247 b99dce43d252
child 23417 42c1a89b45c1
permissions -rw-r--r--
tuned boostrap
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
     1
(*  Title:      HOL/Orderings.thy
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
     2
    ID:         $Id$
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
     3
    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
     4
*)
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
     5
21329
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
     6
header {* Syntactic and abstract orders *}
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
     7
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
     8
theory Orderings
23247
b99dce43d252 merged Code_Generator.thy into HOL.thy
haftmann
parents: 23212
diff changeset
     9
imports HOL
23263
0c227412b285 tuned boostrap
haftmann
parents: 23247
diff changeset
    10
uses
0c227412b285 tuned boostrap
haftmann
parents: 23247
diff changeset
    11
  (*"~~/src/Provers/quasi.ML"*)
0c227412b285 tuned boostrap
haftmann
parents: 23247
diff changeset
    12
  "~~/src/Provers/order.ML"
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
    13
begin
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
    14
21329
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
    15
subsection {* Order syntax *}
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
    16
22473
753123c89d72 explizit "type" superclass
haftmann
parents: 22424
diff changeset
    17
class ord = type +
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    18
  fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubseteq>" 50)
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    19
    and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubset>" 50)
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    20
begin
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    21
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    22
notation
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    23
  less_eq  ("op \<^loc><=") and
21620
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    24
  less_eq  ("(_/ \<^loc><= _)" [51, 51] 50) and
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    25
  less  ("op \<^loc><") and
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    26
  less  ("(_/ \<^loc>< _)"  [51, 51] 50)
21620
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    27
  
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    28
notation (xsymbols)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21383
diff changeset
    29
  less_eq  ("op \<^loc>\<le>") and
21259
63ab016c99ca modified less/less_eq syntax to avoid "x < y < z" etc.;
wenzelm
parents: 21248
diff changeset
    30
  less_eq  ("(_/ \<^loc>\<le> _)"  [51, 51] 50)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
    31
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    32
notation (HTML output)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21383
diff changeset
    33
  less_eq  ("op \<^loc>\<le>") and
21259
63ab016c99ca modified less/less_eq syntax to avoid "x < y < z" etc.;
wenzelm
parents: 21248
diff changeset
    34
  less_eq  ("(_/ \<^loc>\<le> _)"  [51, 51] 50)
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    35
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    36
abbreviation (input)
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    37
  greater  (infix "\<^loc>>" 50) where
21620
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    38
  "x \<^loc>> y \<equiv> y \<^loc>< x"
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    39
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    40
abbreviation (input)
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    41
  greater_eq  (infix "\<^loc>>=" 50) where
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    42
  "x \<^loc>>= y \<equiv> y \<^loc><= x"
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    43
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    44
notation (input)
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    45
  greater_eq  (infix "\<^loc>\<ge>" 50)
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    46
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    47
text {*
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    48
  syntactic min/max -- these definitions reach
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    49
  their usual semantics in class linorder ahead.
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    50
*}
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    51
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    52
definition
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    53
  min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
    54
  "min a b = (if a \<^loc>\<le> b then a else b)"
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    55
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    56
definition
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    57
  max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
    58
  "max a b = (if a \<^loc>\<le> b then b else a)"
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    59
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    60
end
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    61
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    62
notation
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    63
  less_eq  ("op <=") and
21620
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    64
  less_eq  ("(_/ <= _)" [51, 51] 50) and
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    65
  less  ("op <") and
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    66
  less  ("(_/ < _)"  [51, 51] 50)
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    67
  
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    68
notation (xsymbols)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21383
diff changeset
    69
  less_eq  ("op \<le>") and
21259
63ab016c99ca modified less/less_eq syntax to avoid "x < y < z" etc.;
wenzelm
parents: 21248
diff changeset
    70
  less_eq  ("(_/ \<le> _)"  [51, 51] 50)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
    71
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    72
notation (HTML output)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21383
diff changeset
    73
  less_eq  ("op \<le>") and
21259
63ab016c99ca modified less/less_eq syntax to avoid "x < y < z" etc.;
wenzelm
parents: 21248
diff changeset
    74
  less_eq  ("(_/ \<le> _)"  [51, 51] 50)
20714
6a122dba034c tuned syntax for <= <
haftmann
parents: 20588
diff changeset
    75
19536
1a3a3cf8b4fa replaced syntax/translations by abbreviation;
wenzelm
parents: 19527
diff changeset
    76
abbreviation (input)
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    77
  greater  (infix ">" 50) where
21620
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    78
  "x > y \<equiv> y < x"
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    79
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    80
abbreviation (input)
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    81
  greater_eq  (infix ">=" 50) where
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    82
  "x >= y \<equiv> y <= x"
21620
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    83
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    84
notation (input)
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    85
  greater_eq  (infix "\<ge>" 50)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
    86
23087
ad7244663431 rudimentary class target implementation
haftmann
parents: 23032
diff changeset
    87
lemmas min_def [code func, code unfold, code inline del] = min_def [folded ord_class.min]
ad7244663431 rudimentary class target implementation
haftmann
parents: 23032
diff changeset
    88
lemmas max_def [code func, code unfold, code inline del] = max_def [folded ord_class.max]
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    89
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
    90
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
    91
subsection {* Partial orders *}
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
    92
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
    93
class order = ord +
22316
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
    94
  assumes less_le: "x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y"
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
    95
  and order_refl [iff]: "x \<sqsubseteq> x"
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
    96
  and order_trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
    97
  assumes antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
    98
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
    99
begin
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   100
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   101
text {* Reflexivity. *}
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   102
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   103
lemma eq_refl: "x = y \<Longrightarrow> x \<^loc>\<le> y"
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   104
    -- {* This form is useful with the classical reasoner. *}
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   105
by (erule ssubst) (rule order_refl)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   106
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   107
lemma less_irrefl [iff]: "\<not> x \<^loc>< x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   108
by (simp add: less_le)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   109
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   110
lemma le_less: "x \<^loc>\<le> y \<longleftrightarrow> x \<^loc>< y \<or> x = y"
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   111
    -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   112
by (simp add: less_le) blast
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   113
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   114
lemma le_imp_less_or_eq: "x \<^loc>\<le> y \<Longrightarrow> x \<^loc>< y \<or> x = y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   115
unfolding less_le by blast
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   116
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   117
lemma less_imp_le: "x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   118
unfolding less_le by blast
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   119
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   120
lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   121
by (erule contrapos_pn, erule subst, rule less_irrefl)
21329
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   122
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   123
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   124
text {* Useful for simplification, but too risky to include by default. *}
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   125
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   126
lemma less_imp_not_eq: "x \<^loc>< y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   127
by auto
21329
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   128
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   129
lemma less_imp_not_eq2: "x \<^loc>< y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   130
by auto
21329
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   131
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   132
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   133
text {* Transitivity rules for calculational reasoning *}
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   134
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   135
lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<^loc>\<le> b \<Longrightarrow> a \<^loc>< b"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   136
by (simp add: less_le)
21329
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   137
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   138
lemma le_neq_trans: "a \<^loc>\<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<^loc>< b"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   139
by (simp add: less_le)
21329
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   140
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   141
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   142
text {* Asymmetry. *}
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   143
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   144
lemma less_not_sym: "x \<^loc>< y \<Longrightarrow> \<not> (y \<^loc>< x)"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   145
by (simp add: less_le antisym)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   146
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   147
lemma less_asym: "x \<^loc>< y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<^loc>< x) \<Longrightarrow> P"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   148
by (drule less_not_sym, erule contrapos_np) simp
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   149
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   150
lemma eq_iff: "x = y \<longleftrightarrow> x \<^loc>\<le> y \<and> y \<^loc>\<le> x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   151
by (blast intro: antisym)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   152
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   153
lemma antisym_conv: "y \<^loc>\<le> x \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   154
by (blast intro: antisym)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   155
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   156
lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   157
by (erule contrapos_pn, erule subst, rule less_irrefl)
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   158
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   159
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   160
text {* Transitivity. *}
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   161
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   162
lemma less_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   163
by (simp add: less_le) (blast intro: order_trans antisym)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   164
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   165
lemma le_less_trans: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   166
by (simp add: less_le) (blast intro: order_trans antisym)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   167
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   168
lemma less_le_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<^loc>< z"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   169
by (simp add: less_le) (blast intro: order_trans antisym)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   170
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   171
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   172
text {* Useful for simplification, but too risky to include by default. *}
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   173
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   174
lemma less_imp_not_less: "x \<^loc>< y \<Longrightarrow> (\<not> y \<^loc>< x) \<longleftrightarrow> True"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   175
by (blast elim: less_asym)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   176
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   177
lemma less_imp_triv: "x \<^loc>< y \<Longrightarrow> (y \<^loc>< x \<longrightarrow> P) \<longleftrightarrow> True"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   178
by (blast elim: less_asym)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   179
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   180
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   181
text {* Transitivity rules for calculational reasoning *}
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   182
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   183
lemma less_asym': "a \<^loc>< b \<Longrightarrow> b \<^loc>< a \<Longrightarrow> P"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   184
by (rule less_asym)
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   185
22916
haftmann
parents: 22886
diff changeset
   186
haftmann
parents: 22886
diff changeset
   187
text {* Reverse order *}
haftmann
parents: 22886
diff changeset
   188
haftmann
parents: 22886
diff changeset
   189
lemma order_reverse:
23018
1d29bc31b0cb no special treatment in naming of locale predicates stemming form classes
haftmann
parents: 22997
diff changeset
   190
  "order (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   191
by unfold_locales
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   192
   (simp add: less_le, auto intro: antisym order_trans)
22916
haftmann
parents: 22886
diff changeset
   193
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   194
end
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   195
21329
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   196
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   197
subsection {* Linear (total) orders *}
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   198
22316
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   199
class linorder = order +
21216
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   200
  assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   201
begin
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   202
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   203
lemma less_linear: "x \<^loc>< y \<or> x = y \<or> y \<^loc>< x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   204
unfolding less_le using less_le linear by blast
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   205
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   206
lemma le_less_linear: "x \<^loc>\<le> y \<or> y \<^loc>< x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   207
by (simp add: le_less less_linear)
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   208
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   209
lemma le_cases [case_names le ge]:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   210
  "(x \<^loc>\<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>\<le> x \<Longrightarrow> P) \<Longrightarrow> P"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   211
using linear by blast
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   212
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   213
lemma linorder_cases [case_names less equal greater]:
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   214
  "(x \<^loc>< y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> P) \<Longrightarrow> P"
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   215
using less_linear by blast
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   216
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   217
lemma not_less: "\<not> x \<^loc>< y \<longleftrightarrow> y \<^loc>\<le> x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   218
apply (simp add: less_le)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   219
using linear apply (blast intro: antisym)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   220
done
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   221
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   222
lemma not_less_iff_gr_or_eq:
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   223
 "\<not>(x \<^loc>< y) \<longleftrightarrow> (x \<^loc>> y | x = y)"
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   224
apply(simp add:not_less le_less)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   225
apply blast
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   226
done
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   227
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   228
lemma not_le: "\<not> x \<^loc>\<le> y \<longleftrightarrow> y \<^loc>< x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   229
apply (simp add: less_le)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   230
using linear apply (blast intro: antisym)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   231
done
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   232
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   233
lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x \<^loc>< y \<or> y \<^loc>< x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   234
by (cut_tac x = x and y = y in less_linear, auto)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   235
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   236
lemma neqE: "x \<noteq> y \<Longrightarrow> (x \<^loc>< y \<Longrightarrow> R) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> R) \<Longrightarrow> R"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   237
by (simp add: neq_iff) blast
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   238
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   239
lemma antisym_conv1: "\<not> x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   240
by (blast intro: antisym dest: not_less [THEN iffD1])
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   241
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   242
lemma antisym_conv2: "x \<^loc>\<le> y \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   243
by (blast intro: antisym dest: not_less [THEN iffD1])
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   244
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   245
lemma antisym_conv3: "\<not> y \<^loc>< x \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   246
by (blast intro: antisym dest: not_less [THEN iffD1])
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   247
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   248
text{*Replacing the old Nat.leI*}
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   249
lemma leI: "\<not> x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   250
unfolding not_less .
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   251
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   252
lemma leD: "y \<^loc>\<le> x \<Longrightarrow> \<not> x \<^loc>< y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   253
unfolding not_less .
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   254
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   255
(*FIXME inappropriate name (or delete altogether)*)
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   256
lemma not_leE: "\<not> y \<^loc>\<le> x \<Longrightarrow> x \<^loc>< y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   257
unfolding not_le .
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   258
22916
haftmann
parents: 22886
diff changeset
   259
haftmann
parents: 22886
diff changeset
   260
text {* Reverse order *}
haftmann
parents: 22886
diff changeset
   261
haftmann
parents: 22886
diff changeset
   262
lemma linorder_reverse:
23018
1d29bc31b0cb no special treatment in naming of locale predicates stemming form classes
haftmann
parents: 22997
diff changeset
   263
  "linorder (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   264
by unfold_locales
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   265
  (simp add: less_le, auto intro: antisym order_trans simp add: linear)
22916
haftmann
parents: 22886
diff changeset
   266
haftmann
parents: 22886
diff changeset
   267
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
   268
text {* min/max properties *}
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   269
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   270
lemma min_le_iff_disj:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   271
  "min x y \<^loc>\<le> z \<longleftrightarrow> x \<^loc>\<le> z \<or> y \<^loc>\<le> z"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   272
unfolding min_def using linear by (auto intro: order_trans)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   273
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   274
lemma le_max_iff_disj:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   275
  "z \<^loc>\<le> max x y \<longleftrightarrow> z \<^loc>\<le> x \<or> z \<^loc>\<le> y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   276
unfolding max_def using linear by (auto intro: order_trans)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   277
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   278
lemma min_less_iff_disj:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   279
  "min x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<or> y \<^loc>< z"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   280
unfolding min_def le_less using less_linear by (auto intro: less_trans)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   281
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   282
lemma less_max_iff_disj:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   283
  "z \<^loc>< max x y \<longleftrightarrow> z \<^loc>< x \<or> z \<^loc>< y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   284
unfolding max_def le_less using less_linear by (auto intro: less_trans)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   285
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   286
lemma min_less_iff_conj [simp]:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   287
  "z \<^loc>< min x y \<longleftrightarrow> z \<^loc>< x \<and> z \<^loc>< y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   288
unfolding min_def le_less using less_linear by (auto intro: less_trans)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   289
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   290
lemma max_less_iff_conj [simp]:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   291
  "max x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<and> y \<^loc>< z"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   292
unfolding max_def le_less using less_linear by (auto intro: less_trans)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   293
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   294
lemma split_min:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   295
  "P (min i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P i) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P j)"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   296
by (simp add: min_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   297
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   298
lemma split_max:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   299
  "P (max i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P j) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P i)"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   300
by (simp add: max_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   301
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   302
end
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   303
22916
haftmann
parents: 22886
diff changeset
   304
subsection {* Name duplicates -- including min/max interpretation *}
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   305
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   306
lemmas order_less_le = less_le
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   307
lemmas order_eq_refl = order_class.eq_refl
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   308
lemmas order_less_irrefl = order_class.less_irrefl
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   309
lemmas order_le_less = order_class.le_less
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   310
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   311
lemmas order_less_imp_le = order_class.less_imp_le
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   312
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   313
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   314
lemmas order_neq_le_trans = order_class.neq_le_trans
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   315
lemmas order_le_neq_trans = order_class.le_neq_trans
22316
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   316
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   317
lemmas order_antisym = antisym
22316
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   318
lemmas order_less_not_sym = order_class.less_not_sym
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   319
lemmas order_less_asym = order_class.less_asym
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   320
lemmas order_eq_iff = order_class.eq_iff
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   321
lemmas order_antisym_conv = order_class.antisym_conv
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   322
lemmas order_less_trans = order_class.less_trans
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   323
lemmas order_le_less_trans = order_class.le_less_trans
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   324
lemmas order_less_le_trans = order_class.less_le_trans
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   325
lemmas order_less_imp_not_less = order_class.less_imp_not_less
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   326
lemmas order_less_imp_triv = order_class.less_imp_triv
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   327
lemmas order_less_asym' = order_class.less_asym'
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   328
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   329
lemmas linorder_linear = linear
22316
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   330
lemmas linorder_less_linear = linorder_class.less_linear
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   331
lemmas linorder_le_less_linear = linorder_class.le_less_linear
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   332
lemmas linorder_le_cases = linorder_class.le_cases
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   333
lemmas linorder_not_less = linorder_class.not_less
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   334
lemmas linorder_not_le = linorder_class.not_le
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   335
lemmas linorder_neq_iff = linorder_class.neq_iff
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   336
lemmas linorder_neqE = linorder_class.neqE
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   337
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   338
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   339
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   340
23087
ad7244663431 rudimentary class target implementation
haftmann
parents: 23032
diff changeset
   341
lemmas min_le_iff_disj = linorder_class.min_le_iff_disj [folded ord_class.min]
ad7244663431 rudimentary class target implementation
haftmann
parents: 23032
diff changeset
   342
lemmas le_max_iff_disj = linorder_class.le_max_iff_disj [folded ord_class.max]
ad7244663431 rudimentary class target implementation
haftmann
parents: 23032
diff changeset
   343
lemmas min_less_iff_disj = linorder_class.min_less_iff_disj [folded ord_class.min]
ad7244663431 rudimentary class target implementation
haftmann
parents: 23032
diff changeset
   344
lemmas less_max_iff_disj = linorder_class.less_max_iff_disj [folded ord_class.max]
ad7244663431 rudimentary class target implementation
haftmann
parents: 23032
diff changeset
   345
lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj [folded ord_class.min]
ad7244663431 rudimentary class target implementation
haftmann
parents: 23032
diff changeset
   346
lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj [folded ord_class.max]
ad7244663431 rudimentary class target implementation
haftmann
parents: 23032
diff changeset
   347
lemmas split_min = linorder_class.split_min [folded ord_class.min]
ad7244663431 rudimentary class target implementation
haftmann
parents: 23032
diff changeset
   348
lemmas split_max = linorder_class.split_max [folded ord_class.max]
22916
haftmann
parents: 22886
diff changeset
   349
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   350
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   351
subsection {* Reasoning tools setup *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   352
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   353
ML {*
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   354
local
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   355
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   356
fun decomp_gen sort thy (Trueprop $ t) =
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   357
  let
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   358
    fun of_sort t =
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   359
      let
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   360
        val T = type_of t
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   361
      in
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   362
        (* exclude numeric types: linear arithmetic subsumes transitivity *)
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   363
        T <> HOLogic.natT andalso T <> HOLogic.intT
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   364
          andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   365
      end;
22916
haftmann
parents: 22886
diff changeset
   366
    fun dec (Const (@{const_name Not}, _) $ t) = (case dec t
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   367
          of NONE => NONE
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   368
           | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
22916
haftmann
parents: 22886
diff changeset
   369
      | dec (Const (@{const_name "op ="},  _) $ t1 $ t2) =
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   370
          if of_sort t1
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   371
          then SOME (t1, "=", t2)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   372
          else NONE
22997
d4f3b015b50b canonical prefixing of class constants
haftmann
parents: 22916
diff changeset
   373
      | dec (Const (@{const_name Orderings.less_eq},  _) $ t1 $ t2) =
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   374
          if of_sort t1
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   375
          then SOME (t1, "<=", t2)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   376
          else NONE
22997
d4f3b015b50b canonical prefixing of class constants
haftmann
parents: 22916
diff changeset
   377
      | dec (Const (@{const_name Orderings.less},  _) $ t1 $ t2) =
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   378
          if of_sort t1
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   379
          then SOME (t1, "<", t2)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   380
          else NONE
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   381
      | dec _ = NONE;
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   382
  in dec t end;
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   383
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   384
in
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   385
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   386
(* sorry - there is no preorder class
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   387
structure Quasi_Tac = Quasi_Tac_Fun (
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   388
struct
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   389
  val le_trans = thm "order_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   390
  val le_refl = thm "order_refl";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   391
  val eqD1 = thm "order_eq_refl";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   392
  val eqD2 = thm "sym" RS thm "order_eq_refl";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   393
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   394
  val less_imp_le = thm "order_less_imp_le";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   395
  val le_neq_trans = thm "order_le_neq_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   396
  val neq_le_trans = thm "order_neq_le_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   397
  val less_imp_neq = thm "less_imp_neq";
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
   398
  val decomp_trans = decomp_gen ["Orderings.preorder"];
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
   399
  val decomp_quasi = decomp_gen ["Orderings.preorder"];
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   400
end);*)
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   401
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   402
structure Order_Tac = Order_Tac_Fun (
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   403
struct
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   404
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   405
  val le_refl = thm "order_refl";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   406
  val less_imp_le = thm "order_less_imp_le";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   407
  val not_lessI = thm "linorder_not_less" RS thm "iffD2";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   408
  val not_leI = thm "linorder_not_le" RS thm "iffD2";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   409
  val not_lessD = thm "linorder_not_less" RS thm "iffD1";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   410
  val not_leD = thm "linorder_not_le" RS thm "iffD1";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   411
  val eqI = thm "order_antisym";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   412
  val eqD1 = thm "order_eq_refl";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   413
  val eqD2 = thm "sym" RS thm "order_eq_refl";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   414
  val less_trans = thm "order_less_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   415
  val less_le_trans = thm "order_less_le_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   416
  val le_less_trans = thm "order_le_less_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   417
  val le_trans = thm "order_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   418
  val le_neq_trans = thm "order_le_neq_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   419
  val neq_le_trans = thm "order_neq_le_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   420
  val less_imp_neq = thm "less_imp_neq";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   421
  val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   422
  val not_sym = thm "not_sym";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   423
  val decomp_part = decomp_gen ["Orderings.order"];
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   424
  val decomp_lin = decomp_gen ["Orderings.linorder"];
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   425
end);
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   426
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   427
end;
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   428
*}
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   429
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   430
setup {*
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   431
let
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   432
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   433
fun prp t thm = (#prop (rep_thm thm) = t);
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   434
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   435
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   436
  let val prems = prems_of_ss ss;
22916
haftmann
parents: 22886
diff changeset
   437
      val less = Const (@{const_name less}, T);
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   438
      val t = HOLogic.mk_Trueprop(le $ s $ r);
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   439
  in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   440
       NONE =>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   441
         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   442
         in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   443
              NONE => NONE
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
   444
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv1}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   445
         end
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
   446
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_antisym_conv}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   447
  end
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   448
  handle THM _ => NONE;
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   449
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   450
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   451
  let val prems = prems_of_ss ss;
22916
haftmann
parents: 22886
diff changeset
   452
      val le = Const (@{const_name less_eq}, T);
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   453
      val t = HOLogic.mk_Trueprop(le $ r $ s);
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   454
  in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   455
       NONE =>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   456
         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   457
         in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   458
              NONE => NONE
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
   459
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv3}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   460
         end
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
   461
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv2}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   462
  end
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   463
  handle THM _ => NONE;
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   464
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   465
fun add_simprocs procs thy =
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   466
  (Simplifier.change_simpset_of thy (fn ss => ss
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   467
    addsimprocs (map (fn (name, raw_ts, proc) =>
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   468
      Simplifier.simproc thy name raw_ts proc)) procs); thy);
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   469
fun add_solver name tac thy =
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   470
  (Simplifier.change_simpset_of thy (fn ss => ss addSolver
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   471
    (mk_solver name (K tac))); thy);
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   472
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   473
in
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   474
  add_simprocs [
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   475
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   476
       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   477
     ]
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   478
  #> add_solver "Trans_linear" Order_Tac.linear_tac
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   479
  #> add_solver "Trans_partial" Order_Tac.partial_tac
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   480
  (* Adding the transitivity reasoners also as safe solvers showed a slight
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   481
     speed up, but the reasoning strength appears to be not higher (at least
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   482
     no breaking of additional proofs in the entire HOL distribution, as
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   483
     of 5 March 2004, was observed). *)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   484
end
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   485
*}
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   486
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   487
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   488
subsection {* Bounded quantifiers *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   489
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   490
syntax
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   491
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   492
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   493
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   494
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   495
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   496
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   497
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   498
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   499
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   500
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   501
syntax (xsymbols)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   502
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   503
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   504
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   505
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   506
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   507
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   508
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   509
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   510
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   511
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   512
syntax (HOL)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   513
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   514
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   515
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   516
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   517
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   518
syntax (HTML output)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   519
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   520
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   521
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   522
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   523
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   524
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   525
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   526
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   527
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   528
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   529
translations
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   530
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   531
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   532
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   533
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   534
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   535
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   536
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   537
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   538
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   539
print_translation {*
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   540
let
22916
haftmann
parents: 22886
diff changeset
   541
  val All_binder = Syntax.binder_name @{const_syntax All};
haftmann
parents: 22886
diff changeset
   542
  val Ex_binder = Syntax.binder_name @{const_syntax Ex};
22377
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   543
  val impl = @{const_syntax "op -->"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   544
  val conj = @{const_syntax "op &"};
22916
haftmann
parents: 22886
diff changeset
   545
  val less = @{const_syntax less};
haftmann
parents: 22886
diff changeset
   546
  val less_eq = @{const_syntax less_eq};
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   547
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   548
  val trans =
21524
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   549
   [((All_binder, impl, less), ("_All_less", "_All_greater")),
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   550
    ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   551
    ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   552
    ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   553
22344
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   554
  fun matches_bound v t = 
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   555
     case t of (Const ("_bound", _) $ Free (v', _)) => (v = v')
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   556
              | _ => false
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   557
  fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false)
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   558
  fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   559
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   560
  fun tr' q = (q,
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   561
    fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   562
      (case AList.lookup (op =) trans (q, c, d) of
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   563
        NONE => raise Match
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   564
      | SOME (l, g) =>
22344
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   565
          if matches_bound v t andalso not (contains_var v u) then mk v l u P
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   566
          else if matches_bound v u andalso not (contains_var v t) then mk v g t P
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   567
          else raise Match)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   568
     | _ => raise Match);
21524
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   569
in [tr' All_binder, tr' Ex_binder] end
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   570
*}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   571
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   572
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   573
subsection {* Transitivity reasoning *}
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   574
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   575
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   576
by (rule subst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   577
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   578
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   579
by (rule ssubst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   580
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   581
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   582
by (rule subst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   583
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   584
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   585
by (rule ssubst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   586
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   587
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   588
  (!!x y. x < y ==> f x < f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   589
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   590
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   591
  assume "a < b" hence "f a < f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   592
  also assume "f b < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   593
  finally (order_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   594
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   595
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   596
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   597
  (!!x y. x < y ==> f x < f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   598
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   599
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   600
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   601
  also assume "b < c" hence "f b < f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   602
  finally (order_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   603
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   604
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   605
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   606
  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   607
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   608
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   609
  assume "a <= b" hence "f a <= f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   610
  also assume "f b < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   611
  finally (order_le_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   612
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   613
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   614
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   615
  (!!x y. x < y ==> f x < f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   616
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   617
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   618
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   619
  also assume "b < c" hence "f b < f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   620
  finally (order_le_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   621
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   622
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   623
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   624
  (!!x y. x < y ==> f x < f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   625
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   626
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   627
  assume "a < b" hence "f a < f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   628
  also assume "f b <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   629
  finally (order_less_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   630
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   631
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   632
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   633
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   634
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   635
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   636
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   637
  also assume "b <= c" hence "f b <= f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   638
  finally (order_less_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   639
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   640
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   641
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   642
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   643
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   644
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   645
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   646
  also assume "b <= c" hence "f b <= f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   647
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   648
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   649
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   650
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   651
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   652
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   653
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   654
  assume "a <= b" hence "f a <= f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   655
  also assume "f b <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   656
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   657
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   658
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   659
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   660
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   661
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   662
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   663
  assume "a <= b" hence "f a <= f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   664
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   665
  finally (ord_le_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   666
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   667
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   668
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   669
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   670
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   671
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   672
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   673
  also assume "b <= c" hence "f b <= f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   674
  finally (ord_eq_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   675
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   676
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   677
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   678
  (!!x y. x < y ==> f x < f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   679
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   680
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   681
  assume "a < b" hence "f a < f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   682
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   683
  finally (ord_less_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   684
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   685
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   686
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   687
  (!!x y. x < y ==> f x < f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   688
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   689
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   690
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   691
  also assume "b < c" hence "f b < f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   692
  finally (ord_eq_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   693
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   694
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   695
text {*
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   696
  Note that this list of rules is in reverse order of priorities.
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   697
*}
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   698
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   699
lemmas order_trans_rules [trans] =
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   700
  order_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   701
  order_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   702
  order_le_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   703
  order_le_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   704
  order_less_le_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   705
  order_less_le_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   706
  order_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   707
  order_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   708
  ord_le_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   709
  ord_eq_le_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   710
  ord_less_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   711
  ord_eq_less_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   712
  forw_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   713
  back_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   714
  rev_mp
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   715
  mp
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   716
  order_neq_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   717
  order_le_neq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   718
  order_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   719
  order_less_asym'
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   720
  order_le_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   721
  order_less_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   722
  order_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   723
  order_antisym
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   724
  ord_le_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   725
  ord_eq_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   726
  ord_less_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   727
  ord_eq_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   728
  trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   729
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   730
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   731
(* FIXME cleanup *)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   732
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   733
text {* These support proving chains of decreasing inequalities
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   734
    a >= b >= c ... in Isar proofs. *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   735
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   736
lemma xt1:
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   737
  "a = b ==> b > c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   738
  "a > b ==> b = c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   739
  "a = b ==> b >= c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   740
  "a >= b ==> b = c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   741
  "(x::'a::order) >= y ==> y >= x ==> x = y"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   742
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   743
  "(x::'a::order) > y ==> y >= z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   744
  "(x::'a::order) >= y ==> y > z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   745
  "(a::'a::order) > b ==> b > a ==> ?P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   746
  "(x::'a::order) > y ==> y > z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   747
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   748
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   749
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   750
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   751
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   752
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   753
by auto
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   754
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   755
lemma xt2:
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   756
  "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   757
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   758
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   759
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   760
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   761
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   762
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   763
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   764
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   765
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   766
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   767
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   768
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   769
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   770
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   771
lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   772
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   773
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   774
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   775
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   776
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   777
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   778
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   779
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   780
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   781
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   782
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   783
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   784
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   785
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   786
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   787
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   788
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   789
(* 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   790
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   791
  for the wrong thing in an Isar proof.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   792
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   793
  The extra transitivity rules can be used as follows: 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   794
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   795
lemma "(a::'a::order) > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   796
proof -
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   797
  have "a >= b" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   798
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   799
  also have "?rhs >= c" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   800
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   801
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   802
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   803
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   804
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   805
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   806
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   807
  also (xtrans) have "?rhs > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   808
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   809
  finally (xtrans) show ?thesis .
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   810
qed
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   811
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   812
  Alternatively, one can use "declare xtrans [trans]" and then
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   813
  leave out the "(xtrans)" above.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   814
*)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   815
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   816
subsection {* Order on bool *}
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   817
22886
cdff6ef76009 moved recfun_codegen.ML to Code_Generator.thy
haftmann
parents: 22841
diff changeset
   818
instance bool :: order 
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   819
  le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   820
  less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
22916
haftmann
parents: 22886
diff changeset
   821
  by intro_classes (auto simp add: le_bool_def less_bool_def)
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   822
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   823
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   824
by (simp add: le_bool_def)
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   825
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   826
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   827
by (simp add: le_bool_def)
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   828
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   829
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   830
by (simp add: le_bool_def)
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   831
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   832
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   833
by (simp add: le_bool_def)
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   834
22348
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   835
lemma [code func]:
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   836
  "False \<le> b \<longleftrightarrow> True"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   837
  "True \<le> b \<longleftrightarrow> b"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   838
  "False < b \<longleftrightarrow> b"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   839
  "True < b \<longleftrightarrow> False"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   840
  unfolding le_bool_def less_bool_def by simp_all
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   841
22424
8a5412121687 *** empty log message ***
haftmann
parents: 22384
diff changeset
   842
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   843
subsection {* Monotonicity, syntactic least value operator and min/max *}
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   844
21216
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   845
locale mono =
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   846
  fixes f
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   847
  assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   848
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   849
lemmas monoI [intro?] = mono.intro
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   850
  and monoD [dest?] = mono.mono
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   851
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   852
constdefs
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   853
  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   854
  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   855
    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   856
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   857
lemma LeastI2_order:
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   858
  "[| P (x::'a::order);
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   859
      !!y. P y ==> x <= y;
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   860
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   861
   ==> Q (Least P)"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   862
apply (unfold Least_def)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   863
apply (rule theI2)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   864
  apply (blast intro: order_antisym)+
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   865
done
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   866
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   867
lemma Least_equality:
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   868
  "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   869
apply (simp add: Least_def)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   870
apply (rule the_equality)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   871
apply (auto intro!: order_antisym)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   872
done
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   873
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   874
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   875
by (simp add: min_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   876
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   877
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   878
by (simp add: max_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   879
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   880
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   881
apply (simp add: min_def)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   882
apply (blast intro: order_antisym)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   883
done
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   884
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   885
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   886
apply (simp add: max_def)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   887
apply (blast intro: order_antisym)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   888
done
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   889
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   890
lemma min_of_mono:
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   891
  "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   892
by (simp add: min_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   893
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   894
lemma max_of_mono:
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   895
  "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   896
by (simp add: max_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   897
22548
6ce4bddf3bcb dropped legacy ML bindings
haftmann
parents: 22473
diff changeset
   898
6ce4bddf3bcb dropped legacy ML bindings
haftmann
parents: 22473
diff changeset
   899
subsection {* legacy ML bindings *}
21673
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   900
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   901
ML {*
22548
6ce4bddf3bcb dropped legacy ML bindings
haftmann
parents: 22473
diff changeset
   902
val monoI = @{thm monoI};
22886
cdff6ef76009 moved recfun_codegen.ML to Code_Generator.thy
haftmann
parents: 22841
diff changeset
   903
*}
21673
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   904
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   905
end