src/HOL/Orderings.thy
author haftmann
Wed, 09 May 2007 07:53:06 +0200
changeset 22886 cdff6ef76009
parent 22841 83b9f2d3fb3c
child 22916 8caf6da610e2
permissions -rw-r--r--
moved recfun_codegen.ML to Code_Generator.thy
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
22886
cdff6ef76009 moved recfun_codegen.ML to Code_Generator.thy
haftmann
parents: 22841
diff changeset
     9
imports Code_Generator
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
    10
begin
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
    11
21329
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
    12
subsection {* Order syntax *}
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
    13
22473
753123c89d72 explizit "type" superclass
haftmann
parents: 22424
diff changeset
    14
class ord = type +
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    15
  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
    16
    and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubset>" 50)
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    17
begin
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    18
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    19
notation
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    20
  less_eq  ("op \<^loc><=") and
21620
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    21
  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
    22
  less  ("op \<^loc><") and
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    23
  less  ("(_/ \<^loc>< _)"  [51, 51] 50)
21620
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    24
  
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    25
notation (xsymbols)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21383
diff changeset
    26
  less_eq  ("op \<^loc>\<le>") and
21259
63ab016c99ca modified less/less_eq syntax to avoid "x < y < z" etc.;
wenzelm
parents: 21248
diff changeset
    27
  less_eq  ("(_/ \<^loc>\<le> _)"  [51, 51] 50)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
    28
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    29
notation (HTML output)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21383
diff changeset
    30
  less_eq  ("op \<^loc>\<le>") and
21259
63ab016c99ca modified less/less_eq syntax to avoid "x < y < z" etc.;
wenzelm
parents: 21248
diff changeset
    31
  less_eq  ("(_/ \<^loc>\<le> _)"  [51, 51] 50)
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    32
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    33
abbreviation (input)
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    34
  greater  (infix "\<^loc>>" 50) where
21620
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    35
  "x \<^loc>> y \<equiv> y \<^loc>< x"
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    36
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    37
abbreviation (input)
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    38
  greater_eq  (infix "\<^loc>>=" 50) where
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    39
  "x \<^loc>>= y \<equiv> y \<^loc><= x"
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    40
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    41
notation (input)
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    42
  greater_eq  (infix "\<^loc>\<ge>" 50)
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    43
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    44
text {*
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    45
  syntactic min/max -- these definitions reach
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    46
  their usual semantics in class linorder ahead.
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    47
*}
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    48
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    49
definition
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    50
  min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
    51
  "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
    52
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    53
definition
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    54
  max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
    55
  "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
    56
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    57
end
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    58
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    59
notation
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    60
  less_eq  ("op <=") and
21620
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    61
  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
    62
  less  ("op <") and
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    63
  less  ("(_/ < _)"  [51, 51] 50)
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    64
  
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    65
notation (xsymbols)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21383
diff changeset
    66
  less_eq  ("op \<le>") and
21259
63ab016c99ca modified less/less_eq syntax to avoid "x < y < z" etc.;
wenzelm
parents: 21248
diff changeset
    67
  less_eq  ("(_/ \<le> _)"  [51, 51] 50)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
    68
21204
1e96553668c6 renamed 'const_syntax' to 'notation';
wenzelm
parents: 21194
diff changeset
    69
notation (HTML output)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21383
diff changeset
    70
  less_eq  ("op \<le>") and
21259
63ab016c99ca modified less/less_eq syntax to avoid "x < y < z" etc.;
wenzelm
parents: 21248
diff changeset
    71
  less_eq  ("(_/ \<le> _)"  [51, 51] 50)
20714
6a122dba034c tuned syntax for <= <
haftmann
parents: 20588
diff changeset
    72
19536
1a3a3cf8b4fa replaced syntax/translations by abbreviation;
wenzelm
parents: 19527
diff changeset
    73
abbreviation (input)
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    74
  greater  (infix ">" 50) where
21620
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    75
  "x > y \<equiv> y < x"
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    76
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    77
abbreviation (input)
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    78
  greater_eq  (infix ">=" 50) where
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    79
  "x >= y \<equiv> y <= x"
21620
3d4bfc7f6363 some syntax cleanup
haftmann
parents: 21546
diff changeset
    80
21656
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    81
notation (input)
43d709faa9dc restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents: 21620
diff changeset
    82
  greater_eq  (infix "\<ge>" 50)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
    83
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    84
definition
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    85
  min :: "'a\<Colon>ord \<Rightarrow> 'a \<Rightarrow> 'a" where
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    86
  "min a b = (if a \<le> b then a else b)"
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    87
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    88
definition
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    89
  max :: "'a\<Colon>ord \<Rightarrow> 'a \<Rightarrow> 'a" where
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    90
  "max a b = (if a \<le> b then b else a)"
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    91
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    92
lemma min_linorder:
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    93
  "ord.min (op \<le> \<Colon> 'a\<Colon>ord \<Rightarrow> 'a \<Rightarrow> bool) = min"
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    94
  by rule+ (simp add: min_def ord_class.min_def)
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    95
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    96
lemma max_linorder:
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    97
  "ord.max (op \<le> \<Colon> 'a\<Colon>ord \<Rightarrow> 'a \<Rightarrow> bool) = max"
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    98
  by rule+ (simp add: max_def ord_class.max_def)
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
    99
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   100
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   101
subsection {* Partial orders *}
15524
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
class order = ord +
22316
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   104
  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
   105
  and order_refl [iff]: "x \<sqsubseteq> x"
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   106
  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
   107
  assumes antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   108
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   109
begin
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   110
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   111
text {* Reflexivity. *}
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   112
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   113
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
   114
    -- {* This form is useful with the classical reasoner. *}
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   115
  by (erule ssubst) (rule order_refl)
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_irrefl [iff]: "\<not> x \<^loc>< x"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   118
  by (simp add: less_le)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   119
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   120
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
   121
    -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   122
  by (simp add: less_le) blast
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   123
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   124
lemma le_imp_less_or_eq: "x \<^loc>\<le> y \<Longrightarrow> x \<^loc>< y \<or> x = y"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   125
  unfolding less_le by blast
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   126
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   127
lemma less_imp_le: "x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   128
  unfolding less_le by blast
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   129
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   130
lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"
21329
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   131
  by (erule contrapos_pn, erule subst, rule less_irrefl)
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   132
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   133
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   134
text {* Useful for simplification, but too risky to include by default. *}
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   135
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   136
lemma less_imp_not_eq: "x \<^loc>< y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
21329
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   137
  by auto
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   138
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   139
lemma less_imp_not_eq2: "x \<^loc>< y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
21329
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   140
  by auto
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   141
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   142
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   143
text {* Transitivity rules for calculational reasoning *}
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   144
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   145
lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<^loc>\<le> b \<Longrightarrow> a \<^loc>< b"
21329
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   146
  by (simp add: less_le)
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   147
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   148
lemma le_neq_trans: "a \<^loc>\<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<^loc>< b"
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   149
  by (simp add: less_le)
21329
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   150
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   151
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   152
text {* Asymmetry. *}
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   153
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   154
lemma less_not_sym: "x \<^loc>< y \<Longrightarrow> \<not> (y \<^loc>< x)"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   155
  by (simp add: less_le antisym)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   156
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   157
lemma less_asym: "x \<^loc>< y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<^loc>< x) \<Longrightarrow> P"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   158
  by (drule less_not_sym, erule contrapos_np) simp
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   159
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   160
lemma eq_iff: "x = y \<longleftrightarrow> x \<^loc>\<le> y \<and> y \<^loc>\<le> x"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   161
  by (blast intro: antisym)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   162
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   163
lemma antisym_conv: "y \<^loc>\<le> x \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   164
  by (blast intro: antisym)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   165
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   166
lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   167
  by (erule contrapos_pn, erule subst, rule less_irrefl)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   168
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   169
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   170
text {* Transitivity. *}
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   171
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   172
lemma less_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   173
  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
   174
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   175
lemma le_less_trans: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   176
  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
   177
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   178
lemma less_le_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<^loc>< z"
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   179
  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
   180
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   181
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   182
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
   183
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   184
lemma less_imp_not_less: "x \<^loc>< y \<Longrightarrow> (\<not> y \<^loc>< x) \<longleftrightarrow> True"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   185
  by (blast elim: less_asym)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   186
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   187
lemma less_imp_triv: "x \<^loc>< y \<Longrightarrow> (y \<^loc>< x \<longrightarrow> P) \<longleftrightarrow> True"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   188
  by (blast elim: less_asym)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   189
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   190
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   191
text {* Transitivity rules for calculational reasoning *}
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   192
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   193
lemma less_asym': "a \<^loc>< b \<Longrightarrow> b \<^loc>< a \<Longrightarrow> P"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   194
  by (rule less_asym)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   195
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   196
end
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   197
21329
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   198
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   199
subsection {* Linear (total) orders *}
7338206d75f1 introduces preorders
haftmann
parents: 21259
diff changeset
   200
22316
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   201
class linorder = order +
21216
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   202
  assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   203
begin
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   204
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   205
lemma less_linear: "x \<^loc>< y \<or> x = y \<or> y \<^loc>< x"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   206
  unfolding less_le using less_le linear by blast 
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   207
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   208
lemma le_less_linear: "x \<^loc>\<le> y \<or> y \<^loc>< x"
21412
a259061ad0b0 re-eliminated thm trichotomy
haftmann
parents: 21404
diff changeset
   209
  by (simp add: le_less less_linear)
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   210
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   211
lemma le_cases [case_names le ge]:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   212
  "(x \<^loc>\<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>\<le> x \<Longrightarrow> P) \<Longrightarrow> P"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   213
  using linear by blast
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   214
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   215
lemma linorder_cases [case_names less equal greater]:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   216
    "(x \<^loc>< y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> P) \<Longrightarrow> P"
21412
a259061ad0b0 re-eliminated thm trichotomy
haftmann
parents: 21404
diff changeset
   217
  using less_linear by blast
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   218
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   219
lemma not_less: "\<not> x \<^loc>< y \<longleftrightarrow> y \<^loc>\<le> x"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   220
  apply (simp add: less_le)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   221
  using linear apply (blast intro: antisym)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   222
  done
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   223
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   224
lemma not_le: "\<not> x \<^loc>\<le> y \<longleftrightarrow> y \<^loc>< x"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   225
  apply (simp add: less_le)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   226
  using linear apply (blast intro: antisym)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   227
  done
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   228
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   229
lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x \<^loc>< y \<or> y \<^loc>< x"
21412
a259061ad0b0 re-eliminated thm trichotomy
haftmann
parents: 21404
diff changeset
   230
  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
   231
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   232
lemma neqE: "x \<noteq> y \<Longrightarrow> (x \<^loc>< y \<Longrightarrow> R) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> R) \<Longrightarrow> R"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   233
  by (simp add: neq_iff) blast
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   234
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   235
lemma antisym_conv1: "\<not> x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   236
  by (blast intro: antisym dest: not_less [THEN iffD1])
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   237
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   238
lemma antisym_conv2: "x \<^loc>\<le> y \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   239
  by (blast intro: antisym dest: not_less [THEN iffD1])
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   240
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   241
lemma antisym_conv3: "\<not> y \<^loc>< x \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   242
  by (blast intro: antisym dest: not_less [THEN iffD1])
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   243
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   244
text{*Replacing the old Nat.leI*}
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   245
lemma leI: "\<not> x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> x"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   246
  unfolding not_less .
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   247
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   248
lemma leD: "y \<^loc>\<le> x \<Longrightarrow> \<not> x \<^loc>< y"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   249
  unfolding not_less .
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   250
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   251
(*FIXME inappropriate name (or delete altogether)*)
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   252
lemma not_leE: "\<not> y \<^loc>\<le> x \<Longrightarrow> x \<^loc>< y"
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   253
  unfolding not_le .
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   254
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
   255
text {* min/max properties *}
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   256
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   257
lemma min_le_iff_disj:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   258
  "min x y \<^loc>\<le> z \<longleftrightarrow> x \<^loc>\<le> z \<or> y \<^loc>\<le> z"
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   259
  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
   260
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   261
lemma le_max_iff_disj:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   262
  "z \<^loc>\<le> max x y \<longleftrightarrow> z \<^loc>\<le> x \<or> z \<^loc>\<le> y"
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   263
  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
   264
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   265
lemma min_less_iff_disj:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   266
  "min x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<or> y \<^loc>< z"
21412
a259061ad0b0 re-eliminated thm trichotomy
haftmann
parents: 21404
diff changeset
   267
  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
   268
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   269
lemma less_max_iff_disj:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   270
  "z \<^loc>< max x y \<longleftrightarrow> z \<^loc>< x \<or> z \<^loc>< y"
21412
a259061ad0b0 re-eliminated thm trichotomy
haftmann
parents: 21404
diff changeset
   271
  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
   272
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   273
lemma min_less_iff_conj [simp]:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   274
  "z \<^loc>< min x y \<longleftrightarrow> z \<^loc>< x \<and> z \<^loc>< y"
21412
a259061ad0b0 re-eliminated thm trichotomy
haftmann
parents: 21404
diff changeset
   275
  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
   276
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   277
lemma max_less_iff_conj [simp]:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   278
  "max x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<and> y \<^loc>< z"
21412
a259061ad0b0 re-eliminated thm trichotomy
haftmann
parents: 21404
diff changeset
   279
  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
   280
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   281
lemma split_min:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   282
  "P (min i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P i) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P j)"
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   283
  by (simp add: min_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   284
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   285
lemma split_max:
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   286
  "P (max i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P j) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P i)"
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   287
  by (simp add: max_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   288
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   289
end
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   290
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   291
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   292
subsection {* Name duplicates *}
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   293
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   294
lemmas order_less_le = less_le
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   295
lemmas order_eq_refl = order_class.eq_refl
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   296
lemmas order_less_irrefl = order_class.less_irrefl
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   297
lemmas order_le_less = order_class.le_less
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   298
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   299
lemmas order_less_imp_le = order_class.less_imp_le
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   300
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   301
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   302
lemmas order_neq_le_trans = order_class.neq_le_trans
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   303
lemmas order_le_neq_trans = order_class.le_neq_trans
22316
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   304
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   305
lemmas order_antisym = antisym
22316
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   306
lemmas order_less_not_sym = order_class.less_not_sym
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   307
lemmas order_less_asym = order_class.less_asym
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   308
lemmas order_eq_iff = order_class.eq_iff
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   309
lemmas order_antisym_conv = order_class.antisym_conv
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   310
lemmas less_imp_neq = order_class.less_imp_neq
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   311
lemmas order_less_trans = order_class.less_trans
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   312
lemmas order_le_less_trans = order_class.le_less_trans
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   313
lemmas order_less_le_trans = order_class.less_le_trans
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   314
lemmas order_less_imp_not_less = order_class.less_imp_not_less
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   315
lemmas order_less_imp_triv = order_class.less_imp_triv
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   316
lemmas order_less_asym' = order_class.less_asym'
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   317
22384
33a46e6c7f04 prefix of class interpretation not mandatory any longer
haftmann
parents: 22377
diff changeset
   318
lemmas linorder_linear = linear
22316
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   319
lemmas linorder_less_linear = linorder_class.less_linear
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   320
lemmas linorder_le_less_linear = linorder_class.le_less_linear
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   321
lemmas linorder_le_cases = linorder_class.le_cases
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   322
lemmas linorder_not_less = linorder_class.not_less
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   323
lemmas linorder_not_le = linorder_class.not_le
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   324
lemmas linorder_neq_iff = linorder_class.neq_iff
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   325
lemmas linorder_neqE = linorder_class.neqE
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   326
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   327
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   328
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   329
lemmas leI = linorder_class.leI
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   330
lemmas leD = linorder_class.leD
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   331
lemmas not_leE = linorder_class.not_leE
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   332
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   333
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   334
subsection {* Reasoning tools setup *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   335
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   336
ML {*
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   337
local
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   338
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   339
fun decomp_gen sort thy (Trueprop $ t) =
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   340
  let
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   341
    fun of_sort t =
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   342
      let
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   343
        val T = type_of t
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   344
      in
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   345
        (* exclude numeric types: linear arithmetic subsumes transitivity *)
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   346
        T <> HOLogic.natT andalso T <> HOLogic.intT
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   347
          andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   348
      end;
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   349
    fun dec (Const ("Not", _) $ t) = (case dec t
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   350
          of NONE => NONE
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   351
           | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   352
      | dec (Const ("op =",  _) $ t1 $ t2) =
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   353
          if of_sort t1
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   354
          then SOME (t1, "=", t2)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   355
          else NONE
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   356
      | dec (Const ("Orderings.less_eq",  _) $ t1 $ t2) =
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   357
          if of_sort t1
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   358
          then SOME (t1, "<=", t2)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   359
          else NONE
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   360
      | dec (Const ("Orderings.less",  _) $ t1 $ t2) =
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   361
          if of_sort t1
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   362
          then SOME (t1, "<", t2)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   363
          else NONE
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   364
      | dec _ = NONE;
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   365
  in dec t end;
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   366
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   367
in
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   368
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   369
(* sorry - there is no preorder class
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   370
structure Quasi_Tac = Quasi_Tac_Fun (
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   371
struct
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   372
  val le_trans = thm "order_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   373
  val le_refl = thm "order_refl";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   374
  val eqD1 = thm "order_eq_refl";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   375
  val eqD2 = thm "sym" RS thm "order_eq_refl";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   376
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   377
  val less_imp_le = thm "order_less_imp_le";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   378
  val le_neq_trans = thm "order_le_neq_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   379
  val neq_le_trans = thm "order_neq_le_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   380
  val less_imp_neq = thm "less_imp_neq";
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
   381
  val decomp_trans = decomp_gen ["Orderings.preorder"];
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
   382
  val decomp_quasi = decomp_gen ["Orderings.preorder"];
22841
83b9f2d3fb3c dropped preorders, unified syntax
haftmann
parents: 22738
diff changeset
   383
end);*)
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   384
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   385
structure Order_Tac = Order_Tac_Fun (
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   386
struct
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   387
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   388
  val le_refl = thm "order_refl";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   389
  val less_imp_le = thm "order_less_imp_le";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   390
  val not_lessI = thm "linorder_not_less" RS thm "iffD2";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   391
  val not_leI = thm "linorder_not_le" RS thm "iffD2";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   392
  val not_lessD = thm "linorder_not_less" RS thm "iffD1";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   393
  val not_leD = thm "linorder_not_le" RS thm "iffD1";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   394
  val eqI = thm "order_antisym";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   395
  val eqD1 = thm "order_eq_refl";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   396
  val eqD2 = thm "sym" RS thm "order_eq_refl";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   397
  val less_trans = thm "order_less_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   398
  val less_le_trans = thm "order_less_le_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   399
  val le_less_trans = thm "order_le_less_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   400
  val le_trans = thm "order_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   401
  val le_neq_trans = thm "order_le_neq_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   402
  val neq_le_trans = thm "order_neq_le_trans";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   403
  val less_imp_neq = thm "less_imp_neq";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   404
  val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   405
  val not_sym = thm "not_sym";
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   406
  val decomp_part = decomp_gen ["Orderings.order"];
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   407
  val decomp_lin = decomp_gen ["Orderings.linorder"];
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   408
end);
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   409
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   410
end;
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   411
*}
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   412
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   413
setup {*
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   414
let
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   415
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   416
fun prp t thm = (#prop (rep_thm thm) = t);
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   417
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   418
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   419
  let val prems = prems_of_ss ss;
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   420
      val less = Const("Orderings.less",T);
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   421
      val t = HOLogic.mk_Trueprop(le $ s $ r);
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   422
  in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   423
       NONE =>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   424
         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   425
         in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   426
              NONE => NONE
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
   427
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv1}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   428
         end
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
   429
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_antisym_conv}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   430
  end
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   431
  handle THM _ => NONE;
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   432
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   433
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   434
  let val prems = prems_of_ss ss;
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   435
      val le = Const("Orderings.less_eq",T);
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   436
      val t = HOLogic.mk_Trueprop(le $ r $ s);
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   437
  in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   438
       NONE =>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   439
         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   440
         in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   441
              NONE => NONE
22738
4899f06effc6 shifted min/max to class order
haftmann
parents: 22548
diff changeset
   442
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv3}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   443
         end
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_conv2}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   445
  end
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   446
  handle THM _ => NONE;
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   447
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   448
fun add_simprocs procs thy =
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   449
  (Simplifier.change_simpset_of thy (fn ss => ss
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   450
    addsimprocs (map (fn (name, raw_ts, proc) =>
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   451
      Simplifier.simproc thy name raw_ts proc)) procs); thy);
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   452
fun add_solver name tac thy =
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   453
  (Simplifier.change_simpset_of thy (fn ss => ss addSolver
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   454
    (mk_solver name (K tac))); thy);
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   455
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   456
in
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   457
  add_simprocs [
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   458
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   459
       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   460
     ]
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   461
  #> add_solver "Trans_linear" Order_Tac.linear_tac
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   462
  #> add_solver "Trans_partial" Order_Tac.partial_tac
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   463
  (* Adding the transitivity reasoners also as safe solvers showed a slight
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   464
     speed up, but the reasoning strength appears to be not higher (at least
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   465
     no breaking of additional proofs in the entire HOL distribution, as
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   466
     of 5 March 2004, was observed). *)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   467
end
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   468
*}
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   469
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   470
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   471
subsection {* Bounded quantifiers *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   472
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   473
syntax
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   474
  "_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
   475
  "_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
   476
  "_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
   477
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   478
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   479
  "_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
   480
  "_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
   481
  "_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
   482
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   483
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   484
syntax (xsymbols)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   485
  "_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
   486
  "_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
   487
  "_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
   488
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   489
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   490
  "_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
   491
  "_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
   492
  "_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
   493
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   494
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   495
syntax (HOL)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   496
  "_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
   497
  "_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
   498
  "_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
   499
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   500
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   501
syntax (HTML output)
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
translations
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   513
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   514
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   515
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   516
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   517
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   518
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   519
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   520
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   521
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   522
print_translation {*
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   523
let
22377
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   524
  val All_binder = Syntax.binder_name @{const_syntax "All"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   525
  val Ex_binder = Syntax.binder_name @{const_syntax "Ex"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   526
  val impl = @{const_syntax "op -->"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   527
  val conj = @{const_syntax "op &"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   528
  val less = @{const_syntax "less"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   529
  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
   530
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   531
  val trans =
21524
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   532
   [((All_binder, impl, less), ("_All_less", "_All_greater")),
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   533
    ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   534
    ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   535
    ((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
   536
22344
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   537
  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
   538
     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
   539
              | _ => false
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   540
  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
   541
  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
   542
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   543
  fun tr' q = (q,
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   544
    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
   545
      (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
   546
        NONE => raise Match
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   547
      | SOME (l, g) =>
22344
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   548
          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
   549
          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
   550
          else raise Match)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   551
     | _ => raise Match);
21524
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   552
in [tr' All_binder, tr' Ex_binder] end
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   553
*}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   554
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   555
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   556
subsection {* Transitivity reasoning *}
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   557
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   558
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   559
  by (rule subst)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   560
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   561
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   562
  by (rule ssubst)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   563
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   564
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   565
  by (rule subst)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   566
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   567
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   568
  by (rule ssubst)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   569
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   570
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
   571
  (!!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
   572
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   573
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   574
  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
   575
  also assume "f b < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   576
  finally (order_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   577
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   578
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   579
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
   580
  (!!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
   581
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   582
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   583
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   584
  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
   585
  finally (order_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   586
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   587
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   588
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
   589
  (!!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
   590
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   591
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   592
  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
   593
  also assume "f b < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   594
  finally (order_le_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   595
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   596
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   597
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
   598
  (!!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
   599
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   600
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   601
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   602
  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
   603
  finally (order_le_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   604
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   605
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   606
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
   607
  (!!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
   608
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   609
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   610
  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
   611
  also assume "f b <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   612
  finally (order_less_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   613
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   614
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   615
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
   616
  (!!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
   617
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   618
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   619
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   620
  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
   621
  finally (order_less_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   622
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   623
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   624
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
   625
  (!!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
   626
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   627
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   628
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   629
  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
   630
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   631
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   632
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   633
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
   634
  (!!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
   635
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   636
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   637
  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
   638
  also assume "f b <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   639
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   640
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   641
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   642
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   643
  (!!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
   644
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   645
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   646
  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
   647
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   648
  finally (ord_le_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   649
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   650
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   651
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   652
  (!!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
   653
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   654
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   655
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   656
  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
   657
  finally (ord_eq_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   658
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   659
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   660
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   661
  (!!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
   662
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   663
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   664
  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
   665
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   666
  finally (ord_less_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   667
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   668
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   669
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   670
  (!!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
   671
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   672
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   673
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   674
  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
   675
  finally (ord_eq_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   676
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   677
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   678
text {*
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   679
  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
   680
*}
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   681
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   682
lemmas order_trans_rules [trans] =
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   683
  order_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   684
  order_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   685
  order_le_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   686
  order_le_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   687
  order_less_le_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   688
  order_less_le_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   689
  order_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   690
  order_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   691
  ord_le_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   692
  ord_eq_le_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   693
  ord_less_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   694
  ord_eq_less_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   695
  forw_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   696
  back_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   697
  rev_mp
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   698
  mp
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   699
  order_neq_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   700
  order_le_neq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   701
  order_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   702
  order_less_asym'
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   703
  order_le_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   704
  order_less_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   705
  order_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   706
  order_antisym
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   707
  ord_le_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   708
  ord_eq_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   709
  ord_less_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   710
  ord_eq_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   711
  trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   712
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   713
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   714
(* FIXME cleanup *)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   715
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   716
text {* These support proving chains of decreasing inequalities
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   717
    a >= b >= c ... in Isar proofs. *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   718
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   719
lemma xt1:
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   720
  "a = b ==> b > c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   721
  "a > b ==> b = c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   722
  "a = b ==> b >= c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   723
  "a >= b ==> b = c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   724
  "(x::'a::order) >= y ==> y >= x ==> x = y"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   725
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   726
  "(x::'a::order) > y ==> y >= z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   727
  "(x::'a::order) >= y ==> y > z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   728
  "(a::'a::order) > b ==> b > a ==> ?P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   729
  "(x::'a::order) > y ==> y > z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   730
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   731
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   732
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   733
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   734
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   735
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   736
by auto
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   737
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   738
lemma xt2:
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   739
  "(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
   740
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   741
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   742
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   743
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   744
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   745
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   746
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   747
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   748
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   749
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   750
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   751
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   752
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   753
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   754
lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   755
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   756
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   757
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   758
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   759
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   760
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   761
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   762
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   763
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   764
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   765
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   766
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   767
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   768
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   769
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   770
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   771
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   772
(* 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   773
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   774
  for the wrong thing in an Isar proof.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   775
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   776
  The extra transitivity rules can be used as follows: 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   777
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   778
lemma "(a::'a::order) > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   779
proof -
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   780
  have "a >= b" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   781
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   782
  also have "?rhs >= c" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   783
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   784
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   785
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   786
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   787
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   788
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   789
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   790
  also (xtrans) have "?rhs > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   791
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   792
  finally (xtrans) show ?thesis .
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   793
qed
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   794
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   795
  Alternatively, one can use "declare xtrans [trans]" and then
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   796
  leave out the "(xtrans)" above.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   797
*)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   798
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   799
subsection {* Order on bool *}
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   800
22886
cdff6ef76009 moved recfun_codegen.ML to Code_Generator.thy
haftmann
parents: 22841
diff changeset
   801
instance bool :: order 
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   802
  le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   803
  less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   804
  by default (auto simp add: le_bool_def less_bool_def)
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   805
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   806
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   807
  by (simp add: le_bool_def)
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   808
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   809
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   810
  by (simp add: le_bool_def)
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   811
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   812
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   813
  by (simp add: le_bool_def)
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   814
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   815
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   816
  by (simp add: le_bool_def)
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   817
22348
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   818
lemma [code func]:
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   819
  "False \<le> b \<longleftrightarrow> True"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   820
  "True \<le> b \<longleftrightarrow> b"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   821
  "False < b \<longleftrightarrow> b"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   822
  "True < b \<longleftrightarrow> False"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   823
  unfolding le_bool_def less_bool_def by simp_all
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   824
22424
8a5412121687 *** empty log message ***
haftmann
parents: 22384
diff changeset
   825
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   826
subsection {* Monotonicity, syntactic least value operator and min/max *}
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   827
21216
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   828
locale mono =
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   829
  fixes f
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   830
  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
   831
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   832
lemmas monoI [intro?] = mono.intro
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   833
  and monoD [dest?] = mono.mono
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   834
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   835
constdefs
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   836
  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   837
  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   838
    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   839
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   840
lemma LeastI2_order:
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   841
  "[| P (x::'a::order);
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   842
      !!y. P y ==> x <= y;
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   843
      !!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
   844
   ==> Q (Least P)"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   845
  apply (unfold Least_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   846
  apply (rule theI2)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   847
    apply (blast intro: order_antisym)+
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   848
  done
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   849
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   850
lemma Least_equality:
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   851
    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   852
  apply (simp add: Least_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   853
  apply (rule the_equality)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   854
  apply (auto intro!: order_antisym)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   855
  done
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   856
22316
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   857
lemmas min_le_iff_disj = linorder_class.min_le_iff_disj [unfolded min_linorder]
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   858
lemmas le_max_iff_disj = linorder_class.le_max_iff_disj [unfolded max_linorder]
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   859
lemmas min_less_iff_disj = linorder_class.min_less_iff_disj [unfolded min_linorder]
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   860
lemmas less_max_iff_disj = linorder_class.less_max_iff_disj [unfolded max_linorder]
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   861
lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj [unfolded min_linorder]
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   862
lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj [unfolded max_linorder]
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   863
lemmas split_min = linorder_class.split_min [unfolded min_linorder]
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   864
lemmas split_max = linorder_class.split_max [unfolded max_linorder]
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   865
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   866
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   867
  by (simp add: min_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   868
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   869
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   870
  by (simp add: max_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   871
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   872
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   873
  apply (simp add: min_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   874
  apply (blast intro: order_antisym)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   875
  done
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_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   878
  apply (simp add: max_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   879
  apply (blast intro: order_antisym)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   880
  done
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   881
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   882
lemma min_of_mono:
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   883
    "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   884
  by (simp add: min_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   885
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   886
lemma max_of_mono:
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   887
    "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   888
  by (simp add: max_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   889
22548
6ce4bddf3bcb dropped legacy ML bindings
haftmann
parents: 22473
diff changeset
   890
6ce4bddf3bcb dropped legacy ML bindings
haftmann
parents: 22473
diff changeset
   891
subsection {* legacy ML bindings *}
21673
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   892
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   893
ML {*
22548
6ce4bddf3bcb dropped legacy ML bindings
haftmann
parents: 22473
diff changeset
   894
val monoI = @{thm monoI};
22886
cdff6ef76009 moved recfun_codegen.ML to Code_Generator.thy
haftmann
parents: 22841
diff changeset
   895
*}
21673
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   896
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   897
end