author | haftmann |
Thu, 02 Jul 2020 12:10:58 +0000 | |
changeset 71989 | bad75618fb82 |
parent 71851 | 34ecb540a079 |
child 73271 | 05a873f90655 |
permissions | -rw-r--r-- |
28685 | 1 |
(* Title: HOL/Orderings.thy |
15524 | 2 |
Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson |
3 |
*) |
|
4 |
||
60758 | 5 |
section \<open>Abstract orderings\<close> |
15524 | 6 |
|
7 |
theory Orderings |
|
35301
90e42f9ba4d1
distributed theory Algebras to theories Groups and Lattices
haftmann
parents:
35115
diff
changeset
|
8 |
imports HOL |
46950
d0181abdbdac
declare command keywords via theory header, including strict checking outside Pure;
wenzelm
parents:
46884
diff
changeset
|
9 |
keywords "print_orders" :: diag |
15524 | 10 |
begin |
11 |
||
69605 | 12 |
ML_file \<open>~~/src/Provers/order.ML\<close> |
48891 | 13 |
|
60758 | 14 |
subsection \<open>Abstract ordering\<close> |
51487 | 15 |
|
16 |
locale ordering = |
|
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
17 |
fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<^bold>\<le>" 50) |
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
18 |
and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<^bold><" 50) |
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
19 |
assumes strict_iff_order: "a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b" |
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
20 |
assumes refl: "a \<^bold>\<le> a" \<comment> \<open>not \<open>iff\<close>: makes problems due to multiple (dual) interpretations\<close> |
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
21 |
and antisym: "a \<^bold>\<le> b \<Longrightarrow> b \<^bold>\<le> a \<Longrightarrow> a = b" |
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
22 |
and trans: "a \<^bold>\<le> b \<Longrightarrow> b \<^bold>\<le> c \<Longrightarrow> a \<^bold>\<le> c" |
51487 | 23 |
begin |
24 |
||
25 |
lemma strict_implies_order: |
|
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
26 |
"a \<^bold>< b \<Longrightarrow> a \<^bold>\<le> b" |
51487 | 27 |
by (simp add: strict_iff_order) |
28 |
||
29 |
lemma strict_implies_not_eq: |
|
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
30 |
"a \<^bold>< b \<Longrightarrow> a \<noteq> b" |
51487 | 31 |
by (simp add: strict_iff_order) |
32 |
||
33 |
lemma not_eq_order_implies_strict: |
|
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
34 |
"a \<noteq> b \<Longrightarrow> a \<^bold>\<le> b \<Longrightarrow> a \<^bold>< b" |
51487 | 35 |
by (simp add: strict_iff_order) |
36 |
||
37 |
lemma order_iff_strict: |
|
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
38 |
"a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b" |
51487 | 39 |
by (auto simp add: strict_iff_order refl) |
40 |
||
61799 | 41 |
lemma irrefl: \<comment> \<open>not \<open>iff\<close>: makes problems due to multiple (dual) interpretations\<close> |
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
42 |
"\<not> a \<^bold>< a" |
51487 | 43 |
by (simp add: strict_iff_order) |
44 |
||
45 |
lemma asym: |
|
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
46 |
"a \<^bold>< b \<Longrightarrow> b \<^bold>< a \<Longrightarrow> False" |
51487 | 47 |
by (auto simp add: strict_iff_order intro: antisym) |
48 |
||
49 |
lemma strict_trans1: |
|
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
50 |
"a \<^bold>\<le> b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c" |
51487 | 51 |
by (auto simp add: strict_iff_order intro: trans antisym) |
52 |
||
53 |
lemma strict_trans2: |
|
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
54 |
"a \<^bold>< b \<Longrightarrow> b \<^bold>\<le> c \<Longrightarrow> a \<^bold>< c" |
51487 | 55 |
by (auto simp add: strict_iff_order intro: trans antisym) |
56 |
||
57 |
lemma strict_trans: |
|
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
58 |
"a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c" |
51487 | 59 |
by (auto intro: strict_trans1 strict_implies_order) |
60 |
||
71851 | 61 |
lemma eq_iff: "a = b \<longleftrightarrow> a \<^bold>\<le> b \<and> b \<^bold>\<le> a" |
62 |
by (auto simp add: refl intro: antisym) |
|
63 |
||
51487 | 64 |
end |
65 |
||
63819 | 66 |
text \<open>Alternative introduction rule with bias towards strict order\<close> |
67 |
||
68 |
lemma ordering_strictI: |
|
69 |
fixes less_eq (infix "\<^bold>\<le>" 50) |
|
70 |
and less (infix "\<^bold><" 50) |
|
71 |
assumes less_eq_less: "\<And>a b. a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b" |
|
72 |
assumes asym: "\<And>a b. a \<^bold>< b \<Longrightarrow> \<not> b \<^bold>< a" |
|
73 |
assumes irrefl: "\<And>a. \<not> a \<^bold>< a" |
|
74 |
assumes trans: "\<And>a b c. a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c" |
|
75 |
shows "ordering less_eq less" |
|
76 |
proof |
|
77 |
fix a b |
|
78 |
show "a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b" |
|
79 |
by (auto simp add: less_eq_less asym irrefl) |
|
80 |
next |
|
81 |
fix a |
|
82 |
show "a \<^bold>\<le> a" |
|
83 |
by (auto simp add: less_eq_less) |
|
84 |
next |
|
85 |
fix a b c |
|
86 |
assume "a \<^bold>\<le> b" and "b \<^bold>\<le> c" then show "a \<^bold>\<le> c" |
|
87 |
by (auto simp add: less_eq_less intro: trans) |
|
88 |
next |
|
89 |
fix a b |
|
90 |
assume "a \<^bold>\<le> b" and "b \<^bold>\<le> a" then show "a = b" |
|
91 |
by (auto simp add: less_eq_less asym) |
|
92 |
qed |
|
93 |
||
94 |
lemma ordering_dualI: |
|
95 |
fixes less_eq (infix "\<^bold>\<le>" 50) |
|
96 |
and less (infix "\<^bold><" 50) |
|
97 |
assumes "ordering (\<lambda>a b. b \<^bold>\<le> a) (\<lambda>a b. b \<^bold>< a)" |
|
98 |
shows "ordering less_eq less" |
|
99 |
proof - |
|
100 |
from assms interpret ordering "\<lambda>a b. b \<^bold>\<le> a" "\<lambda>a b. b \<^bold>< a" . |
|
101 |
show ?thesis |
|
102 |
by standard (auto simp: strict_iff_order refl intro: antisym trans) |
|
103 |
qed |
|
104 |
||
51487 | 105 |
locale ordering_top = ordering + |
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
106 |
fixes top :: "'a" ("\<^bold>\<top>") |
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
107 |
assumes extremum [simp]: "a \<^bold>\<le> \<^bold>\<top>" |
51487 | 108 |
begin |
109 |
||
110 |
lemma extremum_uniqueI: |
|
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
111 |
"\<^bold>\<top> \<^bold>\<le> a \<Longrightarrow> a = \<^bold>\<top>" |
51487 | 112 |
by (rule antisym) auto |
113 |
||
114 |
lemma extremum_unique: |
|
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
115 |
"\<^bold>\<top> \<^bold>\<le> a \<longleftrightarrow> a = \<^bold>\<top>" |
51487 | 116 |
by (auto intro: antisym) |
117 |
||
118 |
lemma extremum_strict [simp]: |
|
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
119 |
"\<not> (\<^bold>\<top> \<^bold>< a)" |
51487 | 120 |
using extremum [of a] by (auto simp add: order_iff_strict intro: asym irrefl) |
121 |
||
122 |
lemma not_eq_extremum: |
|
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63172
diff
changeset
|
123 |
"a \<noteq> \<^bold>\<top> \<longleftrightarrow> a \<^bold>< \<^bold>\<top>" |
51487 | 124 |
by (auto simp add: order_iff_strict intro: not_eq_order_implies_strict extremum) |
125 |
||
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
126 |
end |
51487 | 127 |
|
128 |
||
60758 | 129 |
subsection \<open>Syntactic orders\<close> |
35092
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35028
diff
changeset
|
130 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35028
diff
changeset
|
131 |
class ord = |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35028
diff
changeset
|
132 |
fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35028
diff
changeset
|
133 |
and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35028
diff
changeset
|
134 |
begin |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35028
diff
changeset
|
135 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35028
diff
changeset
|
136 |
notation |
67403 | 137 |
less_eq ("'(\<le>')") and |
138 |
less_eq ("(_/ \<le> _)" [51, 51] 50) and |
|
139 |
less ("'(<')") and |
|
140 |
less ("(_/ < _)" [51, 51] 50) |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
141 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61824
diff
changeset
|
142 |
abbreviation (input) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61824
diff
changeset
|
143 |
greater_eq (infix "\<ge>" 50) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61824
diff
changeset
|
144 |
where "x \<ge> y \<equiv> y \<le> x" |
35092
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35028
diff
changeset
|
145 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35028
diff
changeset
|
146 |
abbreviation (input) |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61824
diff
changeset
|
147 |
greater (infix ">" 50) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61824
diff
changeset
|
148 |
where "x > y \<equiv> y < x" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61824
diff
changeset
|
149 |
|
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61824
diff
changeset
|
150 |
notation (ASCII) |
67403 | 151 |
less_eq ("'(<=')") and |
152 |
less_eq ("(_/ <= _)" [51, 51] 50) |
|
35092
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35028
diff
changeset
|
153 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35028
diff
changeset
|
154 |
notation (input) |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61824
diff
changeset
|
155 |
greater_eq (infix ">=" 50) |
35092
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35028
diff
changeset
|
156 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35028
diff
changeset
|
157 |
end |
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35028
diff
changeset
|
158 |
|
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35028
diff
changeset
|
159 |
|
60758 | 160 |
subsection \<open>Quasi orders\<close> |
15524 | 161 |
|
27682 | 162 |
class preorder = ord + |
163 |
assumes less_le_not_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> (y \<le> x)" |
|
25062 | 164 |
and order_refl [iff]: "x \<le> x" |
165 |
and order_trans: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z" |
|
21248 | 166 |
begin |
167 |
||
60758 | 168 |
text \<open>Reflexivity.\<close> |
15524 | 169 |
|
25062 | 170 |
lemma eq_refl: "x = y \<Longrightarrow> x \<le> y" |
61799 | 171 |
\<comment> \<open>This form is useful with the classical reasoner.\<close> |
23212 | 172 |
by (erule ssubst) (rule order_refl) |
15524 | 173 |
|
25062 | 174 |
lemma less_irrefl [iff]: "\<not> x < x" |
27682 | 175 |
by (simp add: less_le_not_le) |
176 |
||
177 |
lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y" |
|
63172 | 178 |
by (simp add: less_le_not_le) |
27682 | 179 |
|
180 |
||
60758 | 181 |
text \<open>Asymmetry.\<close> |
27682 | 182 |
|
183 |
lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)" |
|
184 |
by (simp add: less_le_not_le) |
|
185 |
||
186 |
lemma less_asym: "x < y \<Longrightarrow> (\<not> P \<Longrightarrow> y < x) \<Longrightarrow> P" |
|
187 |
by (drule less_not_sym, erule contrapos_np) simp |
|
188 |
||
189 |
||
60758 | 190 |
text \<open>Transitivity.\<close> |
27682 | 191 |
|
192 |
lemma less_trans: "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
193 |
by (auto simp add: less_le_not_le intro: order_trans) |
27682 | 194 |
|
195 |
lemma le_less_trans: "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z" |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
196 |
by (auto simp add: less_le_not_le intro: order_trans) |
27682 | 197 |
|
198 |
lemma less_le_trans: "x < y \<Longrightarrow> y \<le> z \<Longrightarrow> x < z" |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
199 |
by (auto simp add: less_le_not_le intro: order_trans) |
27682 | 200 |
|
201 |
||
60758 | 202 |
text \<open>Useful for simplification, but too risky to include by default.\<close> |
27682 | 203 |
|
204 |
lemma less_imp_not_less: "x < y \<Longrightarrow> (\<not> y < x) \<longleftrightarrow> True" |
|
205 |
by (blast elim: less_asym) |
|
206 |
||
207 |
lemma less_imp_triv: "x < y \<Longrightarrow> (y < x \<longrightarrow> P) \<longleftrightarrow> True" |
|
208 |
by (blast elim: less_asym) |
|
209 |
||
210 |
||
60758 | 211 |
text \<open>Transitivity rules for calculational reasoning\<close> |
27682 | 212 |
|
213 |
lemma less_asym': "a < b \<Longrightarrow> b < a \<Longrightarrow> P" |
|
214 |
by (rule less_asym) |
|
215 |
||
216 |
||
60758 | 217 |
text \<open>Dual order\<close> |
27682 | 218 |
|
219 |
lemma dual_preorder: |
|
67398 | 220 |
"class.preorder (\<ge>) (>)" |
63819 | 221 |
by standard (auto simp add: less_le_not_le intro: order_trans) |
27682 | 222 |
|
223 |
end |
|
224 |
||
225 |
||
60758 | 226 |
subsection \<open>Partial orders\<close> |
27682 | 227 |
|
228 |
class order = preorder + |
|
229 |
assumes antisym: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y" |
|
230 |
begin |
|
231 |
||
51487 | 232 |
lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> y" |
233 |
by (auto simp add: less_le_not_le intro: antisym) |
|
234 |
||
63819 | 235 |
sublocale order: ordering less_eq less + dual_order: ordering greater_eq greater |
236 |
proof - |
|
237 |
interpret ordering less_eq less |
|
238 |
by standard (auto intro: antisym order_trans simp add: less_le) |
|
239 |
show "ordering less_eq less" |
|
240 |
by (fact ordering_axioms) |
|
241 |
then show "ordering greater_eq greater" |
|
242 |
by (rule ordering_dualI) |
|
243 |
qed |
|
51487 | 244 |
|
60758 | 245 |
text \<open>Reflexivity.\<close> |
15524 | 246 |
|
25062 | 247 |
lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y" |
61799 | 248 |
\<comment> \<open>NOT suitable for iff, since it can cause PROOF FAILED.\<close> |
51546
2e26df807dc7
more uniform style for interpretation and sublocale declarations
haftmann
parents:
51487
diff
changeset
|
249 |
by (fact order.order_iff_strict) |
15524 | 250 |
|
25062 | 251 |
lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x = y" |
63172 | 252 |
by (simp add: less_le) |
15524 | 253 |
|
21329 | 254 |
|
60758 | 255 |
text \<open>Useful for simplification, but too risky to include by default.\<close> |
21329 | 256 |
|
25062 | 257 |
lemma less_imp_not_eq: "x < y \<Longrightarrow> (x = y) \<longleftrightarrow> False" |
23212 | 258 |
by auto |
21329 | 259 |
|
25062 | 260 |
lemma less_imp_not_eq2: "x < y \<Longrightarrow> (y = x) \<longleftrightarrow> False" |
23212 | 261 |
by auto |
21329 | 262 |
|
263 |
||
60758 | 264 |
text \<open>Transitivity rules for calculational reasoning\<close> |
21329 | 265 |
|
25062 | 266 |
lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b" |
51546
2e26df807dc7
more uniform style for interpretation and sublocale declarations
haftmann
parents:
51487
diff
changeset
|
267 |
by (fact order.not_eq_order_implies_strict) |
21329 | 268 |
|
25062 | 269 |
lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b" |
51546
2e26df807dc7
more uniform style for interpretation and sublocale declarations
haftmann
parents:
51487
diff
changeset
|
270 |
by (rule order.not_eq_order_implies_strict) |
21329 | 271 |
|
15524 | 272 |
|
60758 | 273 |
text \<open>Asymmetry.\<close> |
15524 | 274 |
|
25062 | 275 |
lemma eq_iff: "x = y \<longleftrightarrow> x \<le> y \<and> y \<le> x" |
71851 | 276 |
by (fact order.eq_iff) |
15524 | 277 |
|
25062 | 278 |
lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y" |
71851 | 279 |
by (simp add: eq_iff) |
15524 | 280 |
|
25062 | 281 |
lemma less_imp_neq: "x < y \<Longrightarrow> x \<noteq> y" |
71851 | 282 |
by (fact order.strict_implies_not_eq) |
21248 | 283 |
|
70749
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
284 |
lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y" |
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
285 |
by (simp add: local.le_less) |
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
286 |
|
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
287 |
lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y" |
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
288 |
by (simp add: local.less_le) |
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
289 |
|
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
290 |
lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y" |
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
291 |
by (auto simp: less_le antisym) |
21083 | 292 |
|
60758 | 293 |
text \<open>Least value operator\<close> |
27107 | 294 |
|
27299 | 295 |
definition (in ord) |
27107 | 296 |
Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "LEAST " 10) where |
297 |
"Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y))" |
|
298 |
||
299 |
lemma Least_equality: |
|
300 |
assumes "P x" |
|
301 |
and "\<And>y. P y \<Longrightarrow> x \<le> y" |
|
302 |
shows "Least P = x" |
|
303 |
unfolding Least_def by (rule the_equality) |
|
304 |
(blast intro: assms antisym)+ |
|
305 |
||
306 |
lemma LeastI2_order: |
|
307 |
assumes "P x" |
|
308 |
and "\<And>y. P y \<Longrightarrow> x \<le> y" |
|
309 |
and "\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> x \<le> y \<Longrightarrow> Q x" |
|
310 |
shows "Q (Least P)" |
|
311 |
unfolding Least_def by (rule theI2) |
|
312 |
(blast intro: assms antisym)+ |
|
313 |
||
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69605
diff
changeset
|
314 |
lemma Least_ex1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69605
diff
changeset
|
315 |
assumes "\<exists>!x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69605
diff
changeset
|
316 |
shows Least1I: "P (Least P)" and Least1_le: "P z \<Longrightarrow> Least P \<le> z" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69605
diff
changeset
|
317 |
using theI'[OF assms] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69605
diff
changeset
|
318 |
unfolding Least_def |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69605
diff
changeset
|
319 |
by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69605
diff
changeset
|
320 |
|
65963 | 321 |
text \<open>Greatest value operator\<close> |
322 |
||
323 |
definition Greatest :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "GREATEST " 10) where |
|
324 |
"Greatest P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<ge> y))" |
|
325 |
||
326 |
lemma GreatestI2_order: |
|
327 |
"\<lbrakk> P x; |
|
328 |
\<And>y. P y \<Longrightarrow> x \<ge> y; |
|
329 |
\<And>x. \<lbrakk> P x; \<forall>y. P y \<longrightarrow> x \<ge> y \<rbrakk> \<Longrightarrow> Q x \<rbrakk> |
|
330 |
\<Longrightarrow> Q (Greatest P)" |
|
331 |
unfolding Greatest_def |
|
332 |
by (rule theI2) (blast intro: antisym)+ |
|
333 |
||
334 |
lemma Greatest_equality: |
|
335 |
"\<lbrakk> P x; \<And>y. P y \<Longrightarrow> x \<ge> y \<rbrakk> \<Longrightarrow> Greatest P = x" |
|
336 |
unfolding Greatest_def |
|
337 |
by (rule the_equality) (blast intro: antisym)+ |
|
338 |
||
21248 | 339 |
end |
15524 | 340 |
|
63819 | 341 |
lemma ordering_orderI: |
342 |
fixes less_eq (infix "\<^bold>\<le>" 50) |
|
343 |
and less (infix "\<^bold><" 50) |
|
344 |
assumes "ordering less_eq less" |
|
345 |
shows "class.order less_eq less" |
|
346 |
proof - |
|
347 |
from assms interpret ordering less_eq less . |
|
348 |
show ?thesis |
|
349 |
by standard (auto intro: antisym trans simp add: refl strict_iff_order) |
|
350 |
qed |
|
56545 | 351 |
|
352 |
lemma order_strictI: |
|
353 |
fixes less (infix "\<sqsubset>" 50) |
|
354 |
and less_eq (infix "\<sqsubseteq>" 50) |
|
63819 | 355 |
assumes "\<And>a b. a \<sqsubseteq> b \<longleftrightarrow> a \<sqsubset> b \<or> a = b" |
356 |
assumes "\<And>a b. a \<sqsubset> b \<Longrightarrow> \<not> b \<sqsubset> a" |
|
357 |
assumes "\<And>a. \<not> a \<sqsubset> a" |
|
358 |
assumes "\<And>a b c. a \<sqsubset> b \<Longrightarrow> b \<sqsubset> c \<Longrightarrow> a \<sqsubset> c" |
|
56545 | 359 |
shows "class.order less_eq less" |
63819 | 360 |
by (rule ordering_orderI) (rule ordering_strictI, (fact assms)+) |
361 |
||
362 |
context order |
|
363 |
begin |
|
364 |
||
365 |
text \<open>Dual order\<close> |
|
366 |
||
367 |
lemma dual_order: |
|
67398 | 368 |
"class.order (\<ge>) (>)" |
63819 | 369 |
using dual_order.ordering_axioms by (rule ordering_orderI) |
370 |
||
371 |
end |
|
56545 | 372 |
|
373 |
||
60758 | 374 |
subsection \<open>Linear (total) orders\<close> |
21329 | 375 |
|
22316 | 376 |
class linorder = order + |
25207 | 377 |
assumes linear: "x \<le> y \<or> y \<le> x" |
21248 | 378 |
begin |
379 |
||
25062 | 380 |
lemma less_linear: "x < y \<or> x = y \<or> y < x" |
23212 | 381 |
unfolding less_le using less_le linear by blast |
21248 | 382 |
|
25062 | 383 |
lemma le_less_linear: "x \<le> y \<or> y < x" |
23212 | 384 |
by (simp add: le_less less_linear) |
21248 | 385 |
|
386 |
lemma le_cases [case_names le ge]: |
|
25062 | 387 |
"(x \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<le> x \<Longrightarrow> P) \<Longrightarrow> P" |
23212 | 388 |
using linear by blast |
21248 | 389 |
|
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
390 |
lemma (in linorder) le_cases3: |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
391 |
"\<lbrakk>\<lbrakk>x \<le> y; y \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> x; x \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>x \<le> z; z \<le> y\<rbrakk> \<Longrightarrow> P; |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
392 |
\<lbrakk>z \<le> y; y \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> z; z \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>z \<le> x; x \<le> y\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
393 |
by (blast intro: le_cases) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
394 |
|
22384
33a46e6c7f04
prefix of class interpretation not mandatory any longer
haftmann
parents:
22377
diff
changeset
|
395 |
lemma linorder_cases [case_names less equal greater]: |
25062 | 396 |
"(x < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P" |
23212 | 397 |
using less_linear by blast |
21248 | 398 |
|
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56545
diff
changeset
|
399 |
lemma linorder_wlog[case_names le sym]: |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56545
diff
changeset
|
400 |
"(\<And>a b. a \<le> b \<Longrightarrow> P a b) \<Longrightarrow> (\<And>a b. P b a \<Longrightarrow> P a b) \<Longrightarrow> P a b" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56545
diff
changeset
|
401 |
by (cases rule: le_cases[of a b]) blast+ |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56545
diff
changeset
|
402 |
|
25062 | 403 |
lemma not_less: "\<not> x < y \<longleftrightarrow> y \<le> x" |
70749
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
404 |
unfolding less_le |
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
405 |
using linear by (blast intro: antisym) |
23212 | 406 |
|
70749
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
407 |
lemma not_less_iff_gr_or_eq: "\<not>(x < y) \<longleftrightarrow> (x > y \<or> x = y)" |
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
408 |
by (auto simp add:not_less le_less) |
15524 | 409 |
|
25062 | 410 |
lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x" |
70749
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
411 |
unfolding less_le |
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
412 |
using linear by (blast intro: antisym) |
15524 | 413 |
|
25062 | 414 |
lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x" |
23212 | 415 |
by (cut_tac x = x and y = y in less_linear, auto) |
15524 | 416 |
|
25062 | 417 |
lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R" |
23212 | 418 |
by (simp add: neq_iff) blast |
15524 | 419 |
|
25062 | 420 |
lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y" |
23212 | 421 |
by (blast intro: antisym dest: not_less [THEN iffD1]) |
15524 | 422 |
|
25062 | 423 |
lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x" |
23212 | 424 |
unfolding not_less . |
16796 | 425 |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
426 |
lemma not_le_imp_less: "\<not> y \<le> x \<Longrightarrow> x < y" |
23212 | 427 |
unfolding not_le . |
21248 | 428 |
|
64758
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64287
diff
changeset
|
429 |
lemma linorder_less_wlog[case_names less refl sym]: |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64287
diff
changeset
|
430 |
"\<lbrakk>\<And>a b. a < b \<Longrightarrow> P a b; \<And>a. P a a; \<And>a b. P b a \<Longrightarrow> P a b\<rbrakk> \<Longrightarrow> P a b" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64287
diff
changeset
|
431 |
using antisym_conv3 by blast |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64287
diff
changeset
|
432 |
|
60758 | 433 |
text \<open>Dual order\<close> |
22916 | 434 |
|
26014 | 435 |
lemma dual_linorder: |
67398 | 436 |
"class.linorder (\<ge>) (>)" |
36635
080b755377c0
locale predicates of classes carry a mandatory "class" prefix
haftmann
parents:
35828
diff
changeset
|
437 |
by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear) |
22916 | 438 |
|
21248 | 439 |
end |
440 |
||
23948 | 441 |
|
60758 | 442 |
text \<open>Alternative introduction rule with bias towards strict order\<close> |
56545 | 443 |
|
444 |
lemma linorder_strictI: |
|
63819 | 445 |
fixes less_eq (infix "\<^bold>\<le>" 50) |
446 |
and less (infix "\<^bold><" 50) |
|
56545 | 447 |
assumes "class.order less_eq less" |
63819 | 448 |
assumes trichotomy: "\<And>a b. a \<^bold>< b \<or> a = b \<or> b \<^bold>< a" |
56545 | 449 |
shows "class.linorder less_eq less" |
450 |
proof - |
|
451 |
interpret order less_eq less |
|
60758 | 452 |
by (fact \<open>class.order less_eq less\<close>) |
56545 | 453 |
show ?thesis |
454 |
proof |
|
455 |
fix a b |
|
63819 | 456 |
show "a \<^bold>\<le> b \<or> b \<^bold>\<le> a" |
56545 | 457 |
using trichotomy by (auto simp add: le_less) |
458 |
qed |
|
459 |
qed |
|
460 |
||
461 |
||
60758 | 462 |
subsection \<open>Reasoning tools setup\<close> |
21083 | 463 |
|
60758 | 464 |
ML \<open> |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
465 |
signature ORDERS = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
466 |
sig |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
467 |
val print_structures: Proof.context -> unit |
32215 | 468 |
val order_tac: Proof.context -> thm list -> int -> tactic |
58826 | 469 |
val add_struct: string * term list -> string -> attribute |
470 |
val del_struct: string * term list -> attribute |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
471 |
end; |
21091 | 472 |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
473 |
structure Orders: ORDERS = |
21248 | 474 |
struct |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
475 |
|
56508 | 476 |
(* context data *) |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
477 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
478 |
fun struct_eq ((s1: string, ts1), (s2, ts2)) = |
67405
e9ab4ad7bd15
uniform use of Standard ML op-infix -- eliminated warnings;
wenzelm
parents:
67403
diff
changeset
|
479 |
s1 = s2 andalso eq_list (op aconv) (ts1, ts2); |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
480 |
|
33519 | 481 |
structure Data = Generic_Data |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
482 |
( |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
483 |
type T = ((string * term list) * Order_Tac.less_arith) list; |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
484 |
(* Order structures: |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
485 |
identifier of the structure, list of operations and record of theorems |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
486 |
needed to set up the transitivity reasoner, |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
487 |
identifier and operations identify the structure uniquely. *) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
488 |
val empty = []; |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
489 |
val extend = I; |
33519 | 490 |
fun merge data = AList.join struct_eq (K fst) data; |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
491 |
); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
492 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
493 |
fun print_structures ctxt = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
494 |
let |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
495 |
val structs = Data.get (Context.Proof ctxt); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
496 |
fun pretty_term t = Pretty.block |
24920 | 497 |
[Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1, |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
498 |
Pretty.str "::", Pretty.brk 1, |
24920 | 499 |
Pretty.quote (Syntax.pretty_typ ctxt (type_of t))]; |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
500 |
fun pretty_struct ((s, ts), _) = Pretty.block |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
501 |
[Pretty.str s, Pretty.str ":", Pretty.brk 1, |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
502 |
Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))]; |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
503 |
in |
51579 | 504 |
Pretty.writeln (Pretty.big_list "order structures:" (map pretty_struct structs)) |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
505 |
end; |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
506 |
|
56508 | 507 |
val _ = |
69593 | 508 |
Outer_Syntax.command \<^command_keyword>\<open>print_orders\<close> |
56508 | 509 |
"print order structures available to transitivity reasoner" |
60097
d20ca79d50e4
discontinued pointless warnings: commands are only defined inside a theory context;
wenzelm
parents:
59936
diff
changeset
|
510 |
(Scan.succeed (Toplevel.keep (print_structures o Toplevel.context_of))); |
21091 | 511 |
|
56508 | 512 |
|
513 |
(* tactics *) |
|
514 |
||
515 |
fun struct_tac ((s, ops), thms) ctxt facts = |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
516 |
let |
56508 | 517 |
val [eq, le, less] = ops; |
69597 | 518 |
fun decomp thy (\<^const>\<open>Trueprop\<close> $ t) = |
56508 | 519 |
let |
520 |
fun excluded t = |
|
521 |
(* exclude numeric types: linear arithmetic subsumes transitivity *) |
|
522 |
let val T = type_of t |
|
523 |
in |
|
524 |
T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT |
|
525 |
end; |
|
526 |
fun rel (bin_op $ t1 $ t2) = |
|
527 |
if excluded t1 then NONE |
|
528 |
else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2) |
|
529 |
else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2) |
|
530 |
else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2) |
|
531 |
else NONE |
|
532 |
| rel _ = NONE; |
|
69593 | 533 |
fun dec (Const (\<^const_name>\<open>Not\<close>, _) $ t) = |
56508 | 534 |
(case rel t of NONE => |
535 |
NONE |
|
536 |
| SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2)) |
|
537 |
| dec x = rel x; |
|
538 |
in dec t end |
|
539 |
| decomp _ _ = NONE; |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
540 |
in |
56508 | 541 |
(case s of |
542 |
"order" => Order_Tac.partial_tac decomp thms ctxt facts |
|
543 |
| "linorder" => Order_Tac.linear_tac decomp thms ctxt facts |
|
544 |
| _ => error ("Unknown order kind " ^ quote s ^ " encountered in transitivity reasoner")) |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
545 |
end |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
546 |
|
56508 | 547 |
fun order_tac ctxt facts = |
548 |
FIRST' (map (fn s => CHANGED o struct_tac s ctxt facts) (Data.get (Context.Proof ctxt))); |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
549 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
550 |
|
56508 | 551 |
(* attributes *) |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
552 |
|
58826 | 553 |
fun add_struct s tag = |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
554 |
Thm.declaration_attribute |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
555 |
(fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm))); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
556 |
fun del_struct s = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
557 |
Thm.declaration_attribute |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
558 |
(fn _ => Data.map (AList.delete struct_eq s)); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
559 |
|
21091 | 560 |
end; |
60758 | 561 |
\<close> |
21091 | 562 |
|
60758 | 563 |
attribute_setup order = \<open> |
58826 | 564 |
Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || Args.del >> K NONE) --| |
565 |
Args.colon (* FIXME || Scan.succeed true *) ) -- Scan.lift Args.name -- |
|
566 |
Scan.repeat Args.term |
|
567 |
>> (fn ((SOME tag, n), ts) => Orders.add_struct (n, ts) tag |
|
568 |
| ((NONE, n), ts) => Orders.del_struct (n, ts)) |
|
60758 | 569 |
\<close> "theorems controlling transitivity reasoner" |
58826 | 570 |
|
60758 | 571 |
method_setup order = \<open> |
47432 | 572 |
Scan.succeed (fn ctxt => SIMPLE_METHOD' (Orders.order_tac ctxt [])) |
60758 | 573 |
\<close> "transitivity reasoner" |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
574 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
575 |
|
60758 | 576 |
text \<open>Declarations to set up transitivity reasoner of partial and linear orders.\<close> |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
577 |
|
25076 | 578 |
context order |
579 |
begin |
|
580 |
||
67398 | 581 |
(* The type constraint on @{term (=}) below is necessary since the operation |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
582 |
is not a parameter of the locale. *) |
25076 | 583 |
|
67398 | 584 |
declare less_irrefl [THEN notE, order add less_reflE: order "(=) :: 'a \<Rightarrow> 'a \<Rightarrow> bool" "(<=)" "(<)"] |
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
585 |
|
67398 | 586 |
declare order_refl [order add le_refl: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
587 |
|
67398 | 588 |
declare less_imp_le [order add less_imp_le: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
589 |
|
67398 | 590 |
declare antisym [order add eqI: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 591 |
|
67398 | 592 |
declare eq_refl [order add eqD1: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 593 |
|
67398 | 594 |
declare sym [THEN eq_refl, order add eqD2: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 595 |
|
67398 | 596 |
declare less_trans [order add less_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
597 |
|
67398 | 598 |
declare less_le_trans [order add less_le_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
599 |
|
67398 | 600 |
declare le_less_trans [order add le_less_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 601 |
|
67398 | 602 |
declare order_trans [order add le_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 603 |
|
67398 | 604 |
declare le_neq_trans [order add le_neq_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 605 |
|
67398 | 606 |
declare neq_le_trans [order add neq_le_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 607 |
|
67398 | 608 |
declare less_imp_neq [order add less_imp_neq: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 609 |
|
67398 | 610 |
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 611 |
|
67398 | 612 |
declare not_sym [order add not_sym: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
613 |
|
25076 | 614 |
end |
615 |
||
616 |
context linorder |
|
617 |
begin |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
618 |
|
67398 | 619 |
declare [[order del: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]] |
27689 | 620 |
|
67398 | 621 |
declare less_irrefl [THEN notE, order add less_reflE: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 622 |
|
67398 | 623 |
declare order_refl [order add le_refl: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 624 |
|
67398 | 625 |
declare less_imp_le [order add less_imp_le: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 626 |
|
67398 | 627 |
declare not_less [THEN iffD2, order add not_lessI: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 628 |
|
67398 | 629 |
declare not_le [THEN iffD2, order add not_leI: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 630 |
|
67398 | 631 |
declare not_less [THEN iffD1, order add not_lessD: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 632 |
|
67398 | 633 |
declare not_le [THEN iffD1, order add not_leD: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 634 |
|
67398 | 635 |
declare antisym [order add eqI: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 636 |
|
67398 | 637 |
declare eq_refl [order add eqD1: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
25076 | 638 |
|
67398 | 639 |
declare sym [THEN eq_refl, order add eqD2: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 640 |
|
67398 | 641 |
declare less_trans [order add less_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 642 |
|
67398 | 643 |
declare less_le_trans [order add less_le_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 644 |
|
67398 | 645 |
declare le_less_trans [order add le_less_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 646 |
|
67398 | 647 |
declare order_trans [order add le_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 648 |
|
67398 | 649 |
declare le_neq_trans [order add le_neq_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 650 |
|
67398 | 651 |
declare neq_le_trans [order add neq_le_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 652 |
|
67398 | 653 |
declare less_imp_neq [order add less_imp_neq: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 654 |
|
67398 | 655 |
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
27689 | 656 |
|
67398 | 657 |
declare not_sym [order add not_sym: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"] |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
658 |
|
25076 | 659 |
end |
660 |
||
60758 | 661 |
setup \<open> |
56509 | 662 |
map_theory_simpset (fn ctxt0 => ctxt0 addSolver |
663 |
mk_solver "Transitivity" (fn ctxt => Orders.order_tac ctxt (Simplifier.prems_of ctxt))) |
|
664 |
(*Adding the transitivity reasoners also as safe solvers showed a slight |
|
665 |
speed up, but the reasoning strength appears to be not higher (at least |
|
666 |
no breaking of additional proofs in the entire HOL distribution, as |
|
667 |
of 5 March 2004, was observed).*) |
|
60758 | 668 |
\<close> |
15524 | 669 |
|
60758 | 670 |
ML \<open> |
56509 | 671 |
local |
672 |
fun prp t thm = Thm.prop_of thm = t; (* FIXME proper aconv!? *) |
|
673 |
in |
|
15524 | 674 |
|
56509 | 675 |
fun antisym_le_simproc ctxt ct = |
59582 | 676 |
(case Thm.term_of ct of |
56509 | 677 |
(le as Const (_, T)) $ r $ s => |
678 |
(let |
|
679 |
val prems = Simplifier.prems_of ctxt; |
|
69593 | 680 |
val less = Const (\<^const_name>\<open>less\<close>, T); |
56509 | 681 |
val t = HOLogic.mk_Trueprop(le $ s $ r); |
682 |
in |
|
683 |
(case find_first (prp t) prems of |
|
684 |
NONE => |
|
685 |
let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) in |
|
686 |
(case find_first (prp t) prems of |
|
687 |
NONE => NONE |
|
70749
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
688 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm antisym_conv1}))) |
56509 | 689 |
end |
690 |
| SOME thm => SOME (mk_meta_eq (thm RS @{thm order_class.antisym_conv}))) |
|
691 |
end handle THM _ => NONE) |
|
692 |
| _ => NONE); |
|
15524 | 693 |
|
56509 | 694 |
fun antisym_less_simproc ctxt ct = |
59582 | 695 |
(case Thm.term_of ct of |
56509 | 696 |
NotC $ ((less as Const(_,T)) $ r $ s) => |
697 |
(let |
|
698 |
val prems = Simplifier.prems_of ctxt; |
|
69593 | 699 |
val le = Const (\<^const_name>\<open>less_eq\<close>, T); |
56509 | 700 |
val t = HOLogic.mk_Trueprop(le $ r $ s); |
701 |
in |
|
702 |
(case find_first (prp t) prems of |
|
703 |
NONE => |
|
704 |
let val t = HOLogic.mk_Trueprop (NotC $ (less $ s $ r)) in |
|
705 |
(case find_first (prp t) prems of |
|
706 |
NONE => NONE |
|
707 |
| SOME thm => SOME (mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3}))) |
|
708 |
end |
|
70749
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
709 |
| SOME thm => SOME (mk_meta_eq (thm RS @{thm antisym_conv2}))) |
56509 | 710 |
end handle THM _ => NONE) |
711 |
| _ => NONE); |
|
21083 | 712 |
|
56509 | 713 |
end; |
60758 | 714 |
\<close> |
15524 | 715 |
|
56509 | 716 |
simproc_setup antisym_le ("(x::'a::order) \<le> y") = "K antisym_le_simproc" |
717 |
simproc_setup antisym_less ("\<not> (x::'a::linorder) < y") = "K antisym_less_simproc" |
|
718 |
||
15524 | 719 |
|
60758 | 720 |
subsection \<open>Bounded quantifiers\<close> |
21083 | 721 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61824
diff
changeset
|
722 |
syntax (ASCII) |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
723 |
"_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
|
724 |
"_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
|
725 |
"_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
|
726 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) |
21083 | 727 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
728 |
"_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
|
729 |
"_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
|
730 |
"_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
|
731 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10) |
21083 | 732 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
733 |
"_All_neq" :: "[idt, 'a, bool] => bool" ("(3ALL _~=_./ _)" [0, 0, 10] 10) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
734 |
"_Ex_neq" :: "[idt, 'a, bool] => bool" ("(3EX _~=_./ _)" [0, 0, 10] 10) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
735 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61824
diff
changeset
|
736 |
syntax |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
737 |
"_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
|
738 |
"_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
|
739 |
"_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
|
740 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 741 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
742 |
"_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
|
743 |
"_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
|
744 |
"_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
|
745 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 746 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
747 |
"_All_neq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<noteq>_./ _)" [0, 0, 10] 10) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
748 |
"_Ex_neq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<noteq>_./ _)" [0, 0, 10] 10) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
749 |
|
62521 | 750 |
syntax (input) |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
751 |
"_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
|
752 |
"_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
|
753 |
"_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
|
754 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
755 |
"_All_neq" :: "[idt, 'a, bool] => bool" ("(3! _~=_./ _)" [0, 0, 10] 10) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
756 |
"_Ex_neq" :: "[idt, 'a, bool] => bool" ("(3? _~=_./ _)" [0, 0, 10] 10) |
21083 | 757 |
|
758 |
translations |
|
67091 | 759 |
"\<forall>x<y. P" \<rightharpoonup> "\<forall>x. x < y \<longrightarrow> P" |
760 |
"\<exists>x<y. P" \<rightharpoonup> "\<exists>x. x < y \<and> P" |
|
761 |
"\<forall>x\<le>y. P" \<rightharpoonup> "\<forall>x. x \<le> y \<longrightarrow> P" |
|
762 |
"\<exists>x\<le>y. P" \<rightharpoonup> "\<exists>x. x \<le> y \<and> P" |
|
763 |
"\<forall>x>y. P" \<rightharpoonup> "\<forall>x. x > y \<longrightarrow> P" |
|
764 |
"\<exists>x>y. P" \<rightharpoonup> "\<exists>x. x > y \<and> P" |
|
765 |
"\<forall>x\<ge>y. P" \<rightharpoonup> "\<forall>x. x \<ge> y \<longrightarrow> P" |
|
766 |
"\<exists>x\<ge>y. P" \<rightharpoonup> "\<exists>x. x \<ge> y \<and> P" |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
767 |
"\<forall>x\<noteq>y. P" \<rightharpoonup> "\<forall>x. x \<noteq> y \<longrightarrow> P" |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
768 |
"\<exists>x\<noteq>y. P" \<rightharpoonup> "\<exists>x. x \<noteq> y \<and> P" |
21083 | 769 |
|
60758 | 770 |
print_translation \<open> |
21083 | 771 |
let |
69593 | 772 |
val All_binder = Mixfix.binder_name \<^const_syntax>\<open>All\<close>; |
773 |
val Ex_binder = Mixfix.binder_name \<^const_syntax>\<open>Ex\<close>; |
|
774 |
val impl = \<^const_syntax>\<open>HOL.implies\<close>; |
|
775 |
val conj = \<^const_syntax>\<open>HOL.conj\<close>; |
|
776 |
val less = \<^const_syntax>\<open>less\<close>; |
|
777 |
val less_eq = \<^const_syntax>\<open>less_eq\<close>; |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
778 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
779 |
val trans = |
35115 | 780 |
[((All_binder, impl, less), |
69593 | 781 |
(\<^syntax_const>\<open>_All_less\<close>, \<^syntax_const>\<open>_All_greater\<close>)), |
35115 | 782 |
((All_binder, impl, less_eq), |
69593 | 783 |
(\<^syntax_const>\<open>_All_less_eq\<close>, \<^syntax_const>\<open>_All_greater_eq\<close>)), |
35115 | 784 |
((Ex_binder, conj, less), |
69593 | 785 |
(\<^syntax_const>\<open>_Ex_less\<close>, \<^syntax_const>\<open>_Ex_greater\<close>)), |
35115 | 786 |
((Ex_binder, conj, less_eq), |
69593 | 787 |
(\<^syntax_const>\<open>_Ex_less_eq\<close>, \<^syntax_const>\<open>_Ex_greater_eq\<close>))]; |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
788 |
|
35115 | 789 |
fun matches_bound v t = |
790 |
(case t of |
|
69593 | 791 |
Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v', _) => v = v' |
35115 | 792 |
| _ => false); |
793 |
fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false); |
|
49660
de49d9b4d7bc
more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents:
48891
diff
changeset
|
794 |
fun mk x c n P = Syntax.const c $ Syntax_Trans.mark_bound_body x $ n $ P; |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
795 |
|
52143 | 796 |
fun tr' q = (q, fn _ => |
69593 | 797 |
(fn [Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v, T), |
35364 | 798 |
Const (c, _) $ (Const (d, _) $ t $ u) $ P] => |
67398 | 799 |
(case AList.lookup (=) trans (q, c, d) of |
35115 | 800 |
NONE => raise Match |
801 |
| SOME (l, g) => |
|
49660
de49d9b4d7bc
more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents:
48891
diff
changeset
|
802 |
if matches_bound v t andalso not (contains_var v u) then mk (v, T) l u P |
de49d9b4d7bc
more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents:
48891
diff
changeset
|
803 |
else if matches_bound v u andalso not (contains_var v t) then mk (v, T) g t P |
35115 | 804 |
else raise Match) |
52143 | 805 |
| _ => raise Match)); |
21524 | 806 |
in [tr' All_binder, tr' Ex_binder] end |
60758 | 807 |
\<close> |
21083 | 808 |
|
809 |
||
60758 | 810 |
subsection \<open>Transitivity reasoning\<close> |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
811 |
|
25193 | 812 |
context ord |
813 |
begin |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
814 |
|
25193 | 815 |
lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c" |
816 |
by (rule subst) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
817 |
|
25193 | 818 |
lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c" |
819 |
by (rule ssubst) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
820 |
|
25193 | 821 |
lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c" |
822 |
by (rule subst) |
|
823 |
||
824 |
lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c" |
|
825 |
by (rule ssubst) |
|
826 |
||
827 |
end |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
828 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
829 |
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
|
830 |
(!!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
|
831 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
832 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
833 |
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
|
834 |
also assume "f b < c" |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
835 |
finally (less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
836 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
837 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
838 |
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
|
839 |
(!!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
|
840 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
841 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
842 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
843 |
also assume "b < c" hence "f b < f c" by (rule r) |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
844 |
finally (less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
845 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
846 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
847 |
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
|
848 |
(!!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
|
849 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
850 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
851 |
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
|
852 |
also assume "f b < c" |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
853 |
finally (le_less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
854 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
855 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
856 |
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
|
857 |
(!!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
|
858 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
859 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
860 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
861 |
also assume "b < c" hence "f b < f c" by (rule r) |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
862 |
finally (le_less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
863 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
864 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
865 |
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
|
866 |
(!!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
|
867 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
868 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
869 |
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
|
870 |
also assume "f b <= c" |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
871 |
finally (less_le_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
872 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
873 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
874 |
lemma 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
|
875 |
(!!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
|
876 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
877 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
878 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
879 |
also assume "b <= c" hence "f b <= f c" by (rule r) |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
880 |
finally (less_le_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
881 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
882 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
883 |
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
|
884 |
(!!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
|
885 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
886 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
887 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
888 |
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
|
889 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
890 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
891 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
892 |
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
|
893 |
(!!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
|
894 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
895 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
896 |
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
|
897 |
also assume "f b <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
898 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
899 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
900 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
901 |
lemma ord_le_eq_subst: "a <= b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
902 |
(!!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
|
903 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
904 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
905 |
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
|
906 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
907 |
finally (ord_le_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
908 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
909 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
910 |
lemma ord_eq_le_subst: "a = f b ==> b <= c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
911 |
(!!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
|
912 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
913 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
914 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
915 |
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
|
916 |
finally (ord_eq_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
917 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
918 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
919 |
lemma ord_less_eq_subst: "a < b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
920 |
(!!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
|
921 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
922 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
923 |
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
|
924 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
925 |
finally (ord_less_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
926 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
927 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
928 |
lemma ord_eq_less_subst: "a = f b ==> b < c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
929 |
(!!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
|
930 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
931 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
932 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
933 |
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
|
934 |
finally (ord_eq_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
935 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
936 |
|
60758 | 937 |
text \<open> |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
938 |
Note that this list of rules is in reverse order of priorities. |
60758 | 939 |
\<close> |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
940 |
|
27682 | 941 |
lemmas [trans] = |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
942 |
order_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
943 |
order_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
944 |
order_le_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
945 |
order_le_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
946 |
order_less_le_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
947 |
order_less_le_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
948 |
order_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
949 |
order_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
950 |
ord_le_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
951 |
ord_eq_le_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
952 |
ord_less_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
953 |
ord_eq_less_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
954 |
forw_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
955 |
back_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
956 |
rev_mp |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
957 |
mp |
27682 | 958 |
|
959 |
lemmas (in order) [trans] = |
|
960 |
neq_le_trans |
|
961 |
le_neq_trans |
|
962 |
||
963 |
lemmas (in preorder) [trans] = |
|
964 |
less_trans |
|
965 |
less_asym' |
|
966 |
le_less_trans |
|
967 |
less_le_trans |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
968 |
order_trans |
27682 | 969 |
|
970 |
lemmas (in order) [trans] = |
|
971 |
antisym |
|
972 |
||
973 |
lemmas (in ord) [trans] = |
|
974 |
ord_le_eq_trans |
|
975 |
ord_eq_le_trans |
|
976 |
ord_less_eq_trans |
|
977 |
ord_eq_less_trans |
|
978 |
||
979 |
lemmas [trans] = |
|
980 |
trans |
|
981 |
||
982 |
lemmas order_trans_rules = |
|
983 |
order_less_subst2 |
|
984 |
order_less_subst1 |
|
985 |
order_le_less_subst2 |
|
986 |
order_le_less_subst1 |
|
987 |
order_less_le_subst2 |
|
988 |
order_less_le_subst1 |
|
989 |
order_subst2 |
|
990 |
order_subst1 |
|
991 |
ord_le_eq_subst |
|
992 |
ord_eq_le_subst |
|
993 |
ord_less_eq_subst |
|
994 |
ord_eq_less_subst |
|
995 |
forw_subst |
|
996 |
back_subst |
|
997 |
rev_mp |
|
998 |
mp |
|
999 |
neq_le_trans |
|
1000 |
le_neq_trans |
|
1001 |
less_trans |
|
1002 |
less_asym' |
|
1003 |
le_less_trans |
|
1004 |
less_le_trans |
|
1005 |
order_trans |
|
1006 |
antisym |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1007 |
ord_le_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1008 |
ord_eq_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1009 |
ord_less_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1010 |
ord_eq_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1011 |
trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1012 |
|
60758 | 1013 |
text \<open>These support proving chains of decreasing inequalities |
1014 |
a >= b >= c ... in Isar proofs.\<close> |
|
21083 | 1015 |
|
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
1016 |
lemma xt1 [no_atp]: |
67091 | 1017 |
"a = b \<Longrightarrow> b > c \<Longrightarrow> a > c" |
1018 |
"a > b \<Longrightarrow> b = c \<Longrightarrow> a > c" |
|
1019 |
"a = b \<Longrightarrow> b \<ge> c \<Longrightarrow> a \<ge> c" |
|
1020 |
"a \<ge> b \<Longrightarrow> b = c \<Longrightarrow> a \<ge> c" |
|
1021 |
"(x::'a::order) \<ge> y \<Longrightarrow> y \<ge> x \<Longrightarrow> x = y" |
|
1022 |
"(x::'a::order) \<ge> y \<Longrightarrow> y \<ge> z \<Longrightarrow> x \<ge> z" |
|
1023 |
"(x::'a::order) > y \<Longrightarrow> y \<ge> z \<Longrightarrow> x > z" |
|
1024 |
"(x::'a::order) \<ge> y \<Longrightarrow> y > z \<Longrightarrow> x > z" |
|
1025 |
"(a::'a::order) > b \<Longrightarrow> b > a \<Longrightarrow> P" |
|
1026 |
"(x::'a::order) > y \<Longrightarrow> y > z \<Longrightarrow> x > z" |
|
1027 |
"(a::'a::order) \<ge> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a > b" |
|
1028 |
"(a::'a::order) \<noteq> b \<Longrightarrow> a \<ge> b \<Longrightarrow> a > b" |
|
1029 |
"a = f b \<Longrightarrow> b > c \<Longrightarrow> (\<And>x y. x > y \<Longrightarrow> f x > f y) \<Longrightarrow> a > f c" |
|
1030 |
"a > b \<Longrightarrow> f b = c \<Longrightarrow> (\<And>x y. x > y \<Longrightarrow> f x > f y) \<Longrightarrow> f a > c" |
|
1031 |
"a = f b \<Longrightarrow> b \<ge> c \<Longrightarrow> (\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> a \<ge> f c" |
|
1032 |
"a \<ge> b \<Longrightarrow> f b = c \<Longrightarrow> (\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> f a \<ge> c" |
|
25076 | 1033 |
by auto |
21083 | 1034 |
|
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
1035 |
lemma xt2 [no_atp]: |
21083 | 1036 |
"(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
1037 |
by (subgoal_tac "f b >= f c", force, force) |
|
1038 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
1039 |
lemma xt3 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> |
21083 | 1040 |
(!!x y. x >= y ==> f x >= f y) ==> f a >= c" |
1041 |
by (subgoal_tac "f a >= f b", force, force) |
|
1042 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
1043 |
lemma xt4 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) >= c ==> |
21083 | 1044 |
(!!x y. x >= y ==> f x >= f y) ==> a > f c" |
1045 |
by (subgoal_tac "f b >= f c", force, force) |
|
1046 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
1047 |
lemma xt5 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) >= c==> |
21083 | 1048 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
1049 |
by (subgoal_tac "f a > f b", force, force) |
|
1050 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
1051 |
lemma xt6 [no_atp]: "(a::'a::order) >= f b ==> b > c ==> |
21083 | 1052 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
1053 |
by (subgoal_tac "f b > f c", force, force) |
|
1054 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
1055 |
lemma xt7 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) > c ==> |
21083 | 1056 |
(!!x y. x >= y ==> f x >= f y) ==> f a > c" |
1057 |
by (subgoal_tac "f a >= f b", force, force) |
|
1058 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
1059 |
lemma xt8 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) > c ==> |
21083 | 1060 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
1061 |
by (subgoal_tac "f b > f c", force, force) |
|
1062 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
1063 |
lemma xt9 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) > c ==> |
21083 | 1064 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
1065 |
by (subgoal_tac "f a > f b", force, force) |
|
1066 |
||
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53216
diff
changeset
|
1067 |
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9 |
21083 | 1068 |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
1069 |
(* |
21083 | 1070 |
Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands |
1071 |
for the wrong thing in an Isar proof. |
|
1072 |
||
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
1073 |
The extra transitivity rules can be used as follows: |
21083 | 1074 |
|
1075 |
lemma "(a::'a::order) > z" |
|
1076 |
proof - |
|
1077 |
have "a >= b" (is "_ >= ?rhs") |
|
1078 |
sorry |
|
1079 |
also have "?rhs >= c" (is "_ >= ?rhs") |
|
1080 |
sorry |
|
1081 |
also (xtrans) have "?rhs = d" (is "_ = ?rhs") |
|
1082 |
sorry |
|
1083 |
also (xtrans) have "?rhs >= e" (is "_ >= ?rhs") |
|
1084 |
sorry |
|
1085 |
also (xtrans) have "?rhs > f" (is "_ > ?rhs") |
|
1086 |
sorry |
|
1087 |
also (xtrans) have "?rhs > z" |
|
1088 |
sorry |
|
1089 |
finally (xtrans) show ?thesis . |
|
1090 |
qed |
|
1091 |
||
1092 |
Alternatively, one can use "declare xtrans [trans]" and then |
|
1093 |
leave out the "(xtrans)" above. |
|
1094 |
*) |
|
1095 |
||
23881 | 1096 |
|
60758 | 1097 |
subsection \<open>Monotonicity\<close> |
21083 | 1098 |
|
25076 | 1099 |
context order |
1100 |
begin |
|
1101 |
||
61076 | 1102 |
definition mono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where |
25076 | 1103 |
"mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)" |
1104 |
||
1105 |
lemma monoI [intro?]: |
|
61076 | 1106 |
fixes f :: "'a \<Rightarrow> 'b::order" |
25076 | 1107 |
shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y) \<Longrightarrow> mono f" |
1108 |
unfolding mono_def by iprover |
|
21216
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
1109 |
|
25076 | 1110 |
lemma monoD [dest?]: |
61076 | 1111 |
fixes f :: "'a \<Rightarrow> 'b::order" |
25076 | 1112 |
shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" |
1113 |
unfolding mono_def by iprover |
|
1114 |
||
51263
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1115 |
lemma monoE: |
61076 | 1116 |
fixes f :: "'a \<Rightarrow> 'b::order" |
51263
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1117 |
assumes "mono f" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1118 |
assumes "x \<le> y" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1119 |
obtains "f x \<le> f y" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1120 |
proof |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1121 |
from assms show "f x \<le> f y" by (simp add: mono_def) |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1122 |
qed |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1123 |
|
61076 | 1124 |
definition antimono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where |
56020
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1125 |
"antimono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<ge> f y)" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1126 |
|
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1127 |
lemma antimonoI [intro?]: |
61076 | 1128 |
fixes f :: "'a \<Rightarrow> 'b::order" |
56020
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1129 |
shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> antimono f" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1130 |
unfolding antimono_def by iprover |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1131 |
|
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1132 |
lemma antimonoD [dest?]: |
61076 | 1133 |
fixes f :: "'a \<Rightarrow> 'b::order" |
56020
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1134 |
shows "antimono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<ge> f y" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1135 |
unfolding antimono_def by iprover |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1136 |
|
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1137 |
lemma antimonoE: |
61076 | 1138 |
fixes f :: "'a \<Rightarrow> 'b::order" |
56020
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1139 |
assumes "antimono f" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1140 |
assumes "x \<le> y" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1141 |
obtains "f x \<ge> f y" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1142 |
proof |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1143 |
from assms show "f x \<ge> f y" by (simp add: antimono_def) |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1144 |
qed |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1145 |
|
61076 | 1146 |
definition strict_mono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where |
30298 | 1147 |
"strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)" |
1148 |
||
1149 |
lemma strict_monoI [intro?]: |
|
1150 |
assumes "\<And>x y. x < y \<Longrightarrow> f x < f y" |
|
1151 |
shows "strict_mono f" |
|
1152 |
using assms unfolding strict_mono_def by auto |
|
1153 |
||
1154 |
lemma strict_monoD [dest?]: |
|
1155 |
"strict_mono f \<Longrightarrow> x < y \<Longrightarrow> f x < f y" |
|
1156 |
unfolding strict_mono_def by auto |
|
1157 |
||
1158 |
lemma strict_mono_mono [dest?]: |
|
1159 |
assumes "strict_mono f" |
|
1160 |
shows "mono f" |
|
1161 |
proof (rule monoI) |
|
1162 |
fix x y |
|
1163 |
assume "x \<le> y" |
|
1164 |
show "f x \<le> f y" |
|
1165 |
proof (cases "x = y") |
|
1166 |
case True then show ?thesis by simp |
|
1167 |
next |
|
60758 | 1168 |
case False with \<open>x \<le> y\<close> have "x < y" by simp |
30298 | 1169 |
with assms strict_monoD have "f x < f y" by auto |
1170 |
then show ?thesis by simp |
|
1171 |
qed |
|
1172 |
qed |
|
1173 |
||
25076 | 1174 |
end |
1175 |
||
1176 |
context linorder |
|
1177 |
begin |
|
1178 |
||
51263
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1179 |
lemma mono_invE: |
61076 | 1180 |
fixes f :: "'a \<Rightarrow> 'b::order" |
51263
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1181 |
assumes "mono f" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1182 |
assumes "f x < f y" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1183 |
obtains "x \<le> y" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1184 |
proof |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1185 |
show "x \<le> y" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1186 |
proof (rule ccontr) |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1187 |
assume "\<not> x \<le> y" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1188 |
then have "y \<le> x" by simp |
60758 | 1189 |
with \<open>mono f\<close> obtain "f y \<le> f x" by (rule monoE) |
1190 |
with \<open>f x < f y\<close> show False by simp |
|
51263
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1191 |
qed |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1192 |
qed |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1193 |
|
66936 | 1194 |
lemma mono_strict_invE: |
1195 |
fixes f :: "'a \<Rightarrow> 'b::order" |
|
1196 |
assumes "mono f" |
|
1197 |
assumes "f x < f y" |
|
1198 |
obtains "x < y" |
|
1199 |
proof |
|
1200 |
show "x < y" |
|
1201 |
proof (rule ccontr) |
|
1202 |
assume "\<not> x < y" |
|
1203 |
then have "y \<le> x" by simp |
|
1204 |
with \<open>mono f\<close> obtain "f y \<le> f x" by (rule monoE) |
|
1205 |
with \<open>f x < f y\<close> show False by simp |
|
1206 |
qed |
|
1207 |
qed |
|
1208 |
||
30298 | 1209 |
lemma strict_mono_eq: |
1210 |
assumes "strict_mono f" |
|
1211 |
shows "f x = f y \<longleftrightarrow> x = y" |
|
1212 |
proof |
|
1213 |
assume "f x = f y" |
|
1214 |
show "x = y" proof (cases x y rule: linorder_cases) |
|
1215 |
case less with assms strict_monoD have "f x < f y" by auto |
|
60758 | 1216 |
with \<open>f x = f y\<close> show ?thesis by simp |
30298 | 1217 |
next |
1218 |
case equal then show ?thesis . |
|
1219 |
next |
|
1220 |
case greater with assms strict_monoD have "f y < f x" by auto |
|
60758 | 1221 |
with \<open>f x = f y\<close> show ?thesis by simp |
30298 | 1222 |
qed |
1223 |
qed simp |
|
1224 |
||
1225 |
lemma strict_mono_less_eq: |
|
1226 |
assumes "strict_mono f" |
|
1227 |
shows "f x \<le> f y \<longleftrightarrow> x \<le> y" |
|
1228 |
proof |
|
1229 |
assume "x \<le> y" |
|
1230 |
with assms strict_mono_mono monoD show "f x \<le> f y" by auto |
|
1231 |
next |
|
1232 |
assume "f x \<le> f y" |
|
1233 |
show "x \<le> y" proof (rule ccontr) |
|
1234 |
assume "\<not> x \<le> y" then have "y < x" by simp |
|
1235 |
with assms strict_monoD have "f y < f x" by auto |
|
60758 | 1236 |
with \<open>f x \<le> f y\<close> show False by simp |
30298 | 1237 |
qed |
1238 |
qed |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
1239 |
|
30298 | 1240 |
lemma strict_mono_less: |
1241 |
assumes "strict_mono f" |
|
1242 |
shows "f x < f y \<longleftrightarrow> x < y" |
|
1243 |
using assms |
|
1244 |
by (auto simp add: less_le Orderings.less_le strict_mono_eq strict_mono_less_eq) |
|
1245 |
||
54860 | 1246 |
end |
1247 |
||
1248 |
||
60758 | 1249 |
subsection \<open>min and max -- fundamental\<close> |
54860 | 1250 |
|
1251 |
definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
|
1252 |
"min a b = (if a \<le> b then a else b)" |
|
1253 |
||
1254 |
definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
|
1255 |
"max a b = (if a \<le> b then b else a)" |
|
1256 |
||
45931 | 1257 |
lemma min_absorb1: "x \<le> y \<Longrightarrow> min x y = x" |
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset
|
1258 |
by (simp add: min_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1259 |
|
54857 | 1260 |
lemma max_absorb2: "x \<le> y \<Longrightarrow> max x y = y" |
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset
|
1261 |
by (simp add: max_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1262 |
|
61076 | 1263 |
lemma min_absorb2: "(y::'a::order) \<le> x \<Longrightarrow> min x y = y" |
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset
|
1264 |
by (simp add:min_def) |
45893 | 1265 |
|
61076 | 1266 |
lemma max_absorb1: "(y::'a::order) \<le> x \<Longrightarrow> max x y = x" |
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset
|
1267 |
by (simp add: max_def) |
45893 | 1268 |
|
61630 | 1269 |
lemma max_min_same [simp]: |
1270 |
fixes x y :: "'a :: linorder" |
|
1271 |
shows "max x (min x y) = x" "max (min x y) x = x" "max (min x y) y = y" "max y (min x y) = y" |
|
1272 |
by(auto simp add: max_def min_def) |
|
45893 | 1273 |
|
66936 | 1274 |
|
60758 | 1275 |
subsection \<open>(Unique) top and bottom elements\<close> |
28685 | 1276 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1277 |
class bot = |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1278 |
fixes bot :: 'a ("\<bottom>") |
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1279 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1280 |
class order_bot = order + bot + |
51487 | 1281 |
assumes bot_least: "\<bottom> \<le> a" |
54868 | 1282 |
begin |
51487 | 1283 |
|
61605 | 1284 |
sublocale bot: ordering_top greater_eq greater bot |
61169 | 1285 |
by standard (fact bot_least) |
51487 | 1286 |
|
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1287 |
lemma le_bot: |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1288 |
"a \<le> \<bottom> \<Longrightarrow> a = \<bottom>" |
51487 | 1289 |
by (fact bot.extremum_uniqueI) |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1290 |
|
43816 | 1291 |
lemma bot_unique: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1292 |
"a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>" |
51487 | 1293 |
by (fact bot.extremum_unique) |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1294 |
|
51487 | 1295 |
lemma not_less_bot: |
1296 |
"\<not> a < \<bottom>" |
|
1297 |
by (fact bot.extremum_strict) |
|
43816 | 1298 |
|
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1299 |
lemma bot_less: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1300 |
"a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a" |
51487 | 1301 |
by (fact bot.not_eq_extremum) |
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1302 |
|
67452 | 1303 |
lemma max_bot[simp]: "max bot x = x" |
1304 |
by(simp add: max_def bot_unique) |
|
1305 |
||
1306 |
lemma max_bot2[simp]: "max x bot = x" |
|
1307 |
by(simp add: max_def bot_unique) |
|
1308 |
||
1309 |
lemma min_bot[simp]: "min bot x = bot" |
|
1310 |
by(simp add: min_def bot_unique) |
|
1311 |
||
1312 |
lemma min_bot2[simp]: "min x bot = bot" |
|
1313 |
by(simp add: min_def bot_unique) |
|
1314 |
||
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1315 |
end |
41082 | 1316 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1317 |
class top = |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1318 |
fixes top :: 'a ("\<top>") |
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1319 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1320 |
class order_top = order + top + |
51487 | 1321 |
assumes top_greatest: "a \<le> \<top>" |
54868 | 1322 |
begin |
51487 | 1323 |
|
61605 | 1324 |
sublocale top: ordering_top less_eq less top |
61169 | 1325 |
by standard (fact top_greatest) |
51487 | 1326 |
|
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1327 |
lemma top_le: |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1328 |
"\<top> \<le> a \<Longrightarrow> a = \<top>" |
51487 | 1329 |
by (fact top.extremum_uniqueI) |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1330 |
|
43816 | 1331 |
lemma top_unique: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1332 |
"\<top> \<le> a \<longleftrightarrow> a = \<top>" |
51487 | 1333 |
by (fact top.extremum_unique) |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1334 |
|
51487 | 1335 |
lemma not_top_less: |
1336 |
"\<not> \<top> < a" |
|
1337 |
by (fact top.extremum_strict) |
|
43816 | 1338 |
|
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1339 |
lemma less_top: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1340 |
"a \<noteq> \<top> \<longleftrightarrow> a < \<top>" |
51487 | 1341 |
by (fact top.not_eq_extremum) |
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1342 |
|
67452 | 1343 |
lemma max_top[simp]: "max top x = top" |
1344 |
by(simp add: max_def top_unique) |
|
1345 |
||
1346 |
lemma max_top2[simp]: "max x top = top" |
|
1347 |
by(simp add: max_def top_unique) |
|
1348 |
||
1349 |
lemma min_top[simp]: "min top x = x" |
|
1350 |
by(simp add: min_def top_unique) |
|
1351 |
||
1352 |
lemma min_top2[simp]: "min x top = x" |
|
1353 |
by(simp add: min_def top_unique) |
|
1354 |
||
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1355 |
end |
28685 | 1356 |
|
1357 |
||
60758 | 1358 |
subsection \<open>Dense orders\<close> |
27823 | 1359 |
|
53216 | 1360 |
class dense_order = order + |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1361 |
assumes dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1362 |
|
53216 | 1363 |
class dense_linorder = linorder + dense_order |
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1364 |
begin |
27823 | 1365 |
|
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1366 |
lemma dense_le: |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1367 |
fixes y z :: 'a |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1368 |
assumes "\<And>x. x < y \<Longrightarrow> x \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1369 |
shows "y \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1370 |
proof (rule ccontr) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1371 |
assume "\<not> ?thesis" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1372 |
hence "z < y" by simp |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1373 |
from dense[OF this] |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1374 |
obtain x where "x < y" and "z < x" by safe |
60758 | 1375 |
moreover have "x \<le> z" using assms[OF \<open>x < y\<close>] . |
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1376 |
ultimately show False by auto |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1377 |
qed |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1378 |
|
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1379 |
lemma dense_le_bounded: |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1380 |
fixes x y z :: 'a |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1381 |
assumes "x < y" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1382 |
assumes *: "\<And>w. \<lbrakk> x < w ; w < y \<rbrakk> \<Longrightarrow> w \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1383 |
shows "y \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1384 |
proof (rule dense_le) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1385 |
fix w assume "w < y" |
60758 | 1386 |
from dense[OF \<open>x < y\<close>] obtain u where "x < u" "u < y" by safe |
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1387 |
from linear[of u w] |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1388 |
show "w \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1389 |
proof (rule disjE) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1390 |
assume "u \<le> w" |
60758 | 1391 |
from less_le_trans[OF \<open>x < u\<close> \<open>u \<le> w\<close>] \<open>w < y\<close> |
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1392 |
show "w \<le> z" by (rule *) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1393 |
next |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1394 |
assume "w \<le> u" |
60758 | 1395 |
from \<open>w \<le> u\<close> *[OF \<open>x < u\<close> \<open>u < y\<close>] |
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1396 |
show "w \<le> z" by (rule order_trans) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1397 |
qed |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1398 |
qed |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1399 |
|
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1400 |
lemma dense_ge: |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1401 |
fixes y z :: 'a |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1402 |
assumes "\<And>x. z < x \<Longrightarrow> y \<le> x" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1403 |
shows "y \<le> z" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1404 |
proof (rule ccontr) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1405 |
assume "\<not> ?thesis" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1406 |
hence "z < y" by simp |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1407 |
from dense[OF this] |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1408 |
obtain x where "x < y" and "z < x" by safe |
60758 | 1409 |
moreover have "y \<le> x" using assms[OF \<open>z < x\<close>] . |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1410 |
ultimately show False by auto |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1411 |
qed |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1412 |
|
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1413 |
lemma dense_ge_bounded: |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1414 |
fixes x y z :: 'a |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1415 |
assumes "z < x" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1416 |
assumes *: "\<And>w. \<lbrakk> z < w ; w < x \<rbrakk> \<Longrightarrow> y \<le> w" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1417 |
shows "y \<le> z" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1418 |
proof (rule dense_ge) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1419 |
fix w assume "z < w" |
60758 | 1420 |
from dense[OF \<open>z < x\<close>] obtain u where "z < u" "u < x" by safe |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1421 |
from linear[of u w] |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1422 |
show "y \<le> w" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1423 |
proof (rule disjE) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1424 |
assume "w \<le> u" |
60758 | 1425 |
from \<open>z < w\<close> le_less_trans[OF \<open>w \<le> u\<close> \<open>u < x\<close>] |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1426 |
show "y \<le> w" by (rule *) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1427 |
next |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1428 |
assume "u \<le> w" |
60758 | 1429 |
from *[OF \<open>z < u\<close> \<open>u < x\<close>] \<open>u \<le> w\<close> |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1430 |
show "y \<le> w" by (rule order_trans) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1431 |
qed |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1432 |
qed |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1433 |
|
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1434 |
end |
27823 | 1435 |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
1436 |
class no_top = order + |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1437 |
assumes gt_ex: "\<exists>y. x < y" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1438 |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
1439 |
class no_bot = order + |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1440 |
assumes lt_ex: "\<exists>y. y < x" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1441 |
|
53216 | 1442 |
class unbounded_dense_linorder = dense_linorder + no_top + no_bot |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1443 |
|
51546
2e26df807dc7
more uniform style for interpretation and sublocale declarations
haftmann
parents:
51487
diff
changeset
|
1444 |
|
60758 | 1445 |
subsection \<open>Wellorders\<close> |
27823 | 1446 |
|
1447 |
class wellorder = linorder + |
|
1448 |
assumes less_induct [case_names less]: "(\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P a" |
|
1449 |
begin |
|
1450 |
||
1451 |
lemma wellorder_Least_lemma: |
|
1452 |
fixes k :: 'a |
|
1453 |
assumes "P k" |
|
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1454 |
shows LeastI: "P (LEAST x. P x)" and Least_le: "(LEAST x. P x) \<le> k" |
27823 | 1455 |
proof - |
1456 |
have "P (LEAST x. P x) \<and> (LEAST x. P x) \<le> k" |
|
1457 |
using assms proof (induct k rule: less_induct) |
|
1458 |
case (less x) then have "P x" by simp |
|
1459 |
show ?case proof (rule classical) |
|
1460 |
assume assm: "\<not> (P (LEAST a. P a) \<and> (LEAST a. P a) \<le> x)" |
|
1461 |
have "\<And>y. P y \<Longrightarrow> x \<le> y" |
|
1462 |
proof (rule classical) |
|
1463 |
fix y |
|
38705 | 1464 |
assume "P y" and "\<not> x \<le> y" |
27823 | 1465 |
with less have "P (LEAST a. P a)" and "(LEAST a. P a) \<le> y" |
1466 |
by (auto simp add: not_le) |
|
1467 |
with assm have "x < (LEAST a. P a)" and "(LEAST a. P a) \<le> y" |
|
1468 |
by auto |
|
1469 |
then show "x \<le> y" by auto |
|
1470 |
qed |
|
60758 | 1471 |
with \<open>P x\<close> have Least: "(LEAST a. P a) = x" |
27823 | 1472 |
by (rule Least_equality) |
60758 | 1473 |
with \<open>P x\<close> show ?thesis by simp |
27823 | 1474 |
qed |
1475 |
qed |
|
1476 |
then show "P (LEAST x. P x)" and "(LEAST x. P x) \<le> k" by auto |
|
1477 |
qed |
|
1478 |
||
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
67405
diff
changeset
|
1479 |
\<comment> \<open>The following 3 lemmas are due to Brian Huffman\<close> |
27823 | 1480 |
lemma LeastI_ex: "\<exists>x. P x \<Longrightarrow> P (Least P)" |
1481 |
by (erule exE) (erule LeastI) |
|
1482 |
||
1483 |
lemma LeastI2: |
|
1484 |
"P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)" |
|
1485 |
by (blast intro: LeastI) |
|
1486 |
||
1487 |
lemma LeastI2_ex: |
|
1488 |
"\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)" |
|
1489 |
by (blast intro: LeastI_ex) |
|
1490 |
||
38705 | 1491 |
lemma LeastI2_wellorder: |
1492 |
assumes "P a" |
|
1493 |
and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a" |
|
1494 |
shows "Q (Least P)" |
|
1495 |
proof (rule LeastI2_order) |
|
60758 | 1496 |
show "P (Least P)" using \<open>P a\<close> by (rule LeastI) |
38705 | 1497 |
next |
1498 |
fix y assume "P y" thus "Least P \<le> y" by (rule Least_le) |
|
1499 |
next |
|
1500 |
fix x assume "P x" "\<forall>y. P y \<longrightarrow> x \<le> y" thus "Q x" by (rule assms(2)) |
|
1501 |
qed |
|
1502 |
||
61699
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1503 |
lemma LeastI2_wellorder_ex: |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1504 |
assumes "\<exists>x. P x" |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1505 |
and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a" |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1506 |
shows "Q (Least P)" |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1507 |
using assms by clarify (blast intro!: LeastI2_wellorder) |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1508 |
|
27823 | 1509 |
lemma not_less_Least: "k < (LEAST x. P x) \<Longrightarrow> \<not> P k" |
61699
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1510 |
apply (simp add: not_le [symmetric]) |
27823 | 1511 |
apply (erule contrapos_nn) |
1512 |
apply (erule Least_le) |
|
1513 |
done |
|
1514 |
||
64287 | 1515 |
lemma exists_least_iff: "(\<exists>n. P n) \<longleftrightarrow> (\<exists>n. P n \<and> (\<forall>m < n. \<not> P m))" (is "?lhs \<longleftrightarrow> ?rhs") |
1516 |
proof |
|
1517 |
assume ?rhs thus ?lhs by blast |
|
1518 |
next |
|
1519 |
assume H: ?lhs then obtain n where n: "P n" by blast |
|
1520 |
let ?x = "Least P" |
|
1521 |
{ fix m assume m: "m < ?x" |
|
1522 |
from not_less_Least[OF m] have "\<not> P m" . } |
|
1523 |
with LeastI_ex[OF H] show ?rhs by blast |
|
1524 |
qed |
|
1525 |
||
38705 | 1526 |
end |
27823 | 1527 |
|
28685 | 1528 |
|
69593 | 1529 |
subsection \<open>Order on \<^typ>\<open>bool\<close>\<close> |
28685 | 1530 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1531 |
instantiation bool :: "{order_bot, order_top, linorder}" |
28685 | 1532 |
begin |
1533 |
||
1534 |
definition |
|
41080 | 1535 |
le_bool_def [simp]: "P \<le> Q \<longleftrightarrow> P \<longrightarrow> Q" |
28685 | 1536 |
|
1537 |
definition |
|
61076 | 1538 |
[simp]: "(P::bool) < Q \<longleftrightarrow> \<not> P \<and> Q" |
28685 | 1539 |
|
1540 |
definition |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1541 |
[simp]: "\<bottom> \<longleftrightarrow> False" |
28685 | 1542 |
|
1543 |
definition |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1544 |
[simp]: "\<top> \<longleftrightarrow> True" |
28685 | 1545 |
|
1546 |
instance proof |
|
41080 | 1547 |
qed auto |
28685 | 1548 |
|
15524 | 1549 |
end |
28685 | 1550 |
|
1551 |
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q" |
|
41080 | 1552 |
by simp |
28685 | 1553 |
|
1554 |
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q" |
|
41080 | 1555 |
by simp |
28685 | 1556 |
|
1557 |
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R" |
|
41080 | 1558 |
by simp |
28685 | 1559 |
|
1560 |
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q" |
|
41080 | 1561 |
by simp |
32899 | 1562 |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1563 |
lemma bot_boolE: "\<bottom> \<Longrightarrow> P" |
41080 | 1564 |
by simp |
32899 | 1565 |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1566 |
lemma top_boolI: \<top> |
41080 | 1567 |
by simp |
28685 | 1568 |
|
1569 |
lemma [code]: |
|
1570 |
"False \<le> b \<longleftrightarrow> True" |
|
1571 |
"True \<le> b \<longleftrightarrow> b" |
|
1572 |
"False < b \<longleftrightarrow> b" |
|
1573 |
"True < b \<longleftrightarrow> False" |
|
41080 | 1574 |
by simp_all |
28685 | 1575 |
|
1576 |
||
69593 | 1577 |
subsection \<open>Order on \<^typ>\<open>_ \<Rightarrow> _\<close>\<close> |
28685 | 1578 |
|
1579 |
instantiation "fun" :: (type, ord) ord |
|
1580 |
begin |
|
1581 |
||
1582 |
definition |
|
37767 | 1583 |
le_fun_def: "f \<le> g \<longleftrightarrow> (\<forall>x. f x \<le> g x)" |
28685 | 1584 |
|
1585 |
definition |
|
61076 | 1586 |
"(f::'a \<Rightarrow> 'b) < g \<longleftrightarrow> f \<le> g \<and> \<not> (g \<le> f)" |
28685 | 1587 |
|
1588 |
instance .. |
|
1589 |
||
1590 |
end |
|
1591 |
||
1592 |
instance "fun" :: (type, preorder) preorder proof |
|
1593 |
qed (auto simp add: le_fun_def less_fun_def |
|
44921 | 1594 |
intro: order_trans antisym) |
28685 | 1595 |
|
1596 |
instance "fun" :: (type, order) order proof |
|
44921 | 1597 |
qed (auto simp add: le_fun_def intro: antisym) |
28685 | 1598 |
|
41082 | 1599 |
instantiation "fun" :: (type, bot) bot |
1600 |
begin |
|
1601 |
||
1602 |
definition |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1603 |
"\<bottom> = (\<lambda>x. \<bottom>)" |
41082 | 1604 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1605 |
instance .. |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1606 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1607 |
end |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1608 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1609 |
instantiation "fun" :: (type, order_bot) order_bot |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1610 |
begin |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1611 |
|
49769 | 1612 |
lemma bot_apply [simp, code]: |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1613 |
"\<bottom> x = \<bottom>" |
41082 | 1614 |
by (simp add: bot_fun_def) |
1615 |
||
1616 |
instance proof |
|
46884 | 1617 |
qed (simp add: le_fun_def) |
41082 | 1618 |
|
1619 |
end |
|
1620 |
||
28685 | 1621 |
instantiation "fun" :: (type, top) top |
1622 |
begin |
|
1623 |
||
1624 |
definition |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1625 |
[no_atp]: "\<top> = (\<lambda>x. \<top>)" |
28685 | 1626 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1627 |
instance .. |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1628 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1629 |
end |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1630 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1631 |
instantiation "fun" :: (type, order_top) order_top |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1632 |
begin |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1633 |
|
49769 | 1634 |
lemma top_apply [simp, code]: |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1635 |
"\<top> x = \<top>" |
41080 | 1636 |
by (simp add: top_fun_def) |
1637 |
||
28685 | 1638 |
instance proof |
46884 | 1639 |
qed (simp add: le_fun_def) |
28685 | 1640 |
|
1641 |
end |
|
1642 |
||
1643 |
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g" |
|
1644 |
unfolding le_fun_def by simp |
|
1645 |
||
1646 |
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P" |
|
1647 |
unfolding le_fun_def by simp |
|
1648 |
||
1649 |
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x" |
|
54860 | 1650 |
by (rule le_funE) |
28685 | 1651 |
|
59000 | 1652 |
lemma mono_compose: "mono Q \<Longrightarrow> mono (\<lambda>i x. Q i (f x))" |
1653 |
unfolding mono_def le_fun_def by auto |
|
1654 |
||
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1655 |
|
60758 | 1656 |
subsection \<open>Order on unary and binary predicates\<close> |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1657 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1658 |
lemma predicate1I: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1659 |
assumes PQ: "\<And>x. P x \<Longrightarrow> Q x" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1660 |
shows "P \<le> Q" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1661 |
apply (rule le_funI) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1662 |
apply (rule le_boolI) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1663 |
apply (rule PQ) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1664 |
apply assumption |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1665 |
done |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1666 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1667 |
lemma predicate1D: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1668 |
"P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1669 |
apply (erule le_funE) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1670 |
apply (erule le_boolE) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1671 |
apply assumption+ |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1672 |
done |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1673 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1674 |
lemma rev_predicate1D: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1675 |
"P x \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1676 |
by (rule predicate1D) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1677 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1678 |
lemma predicate2I: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1679 |
assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1680 |
shows "P \<le> Q" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1681 |
apply (rule le_funI)+ |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1682 |
apply (rule le_boolI) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1683 |
apply (rule PQ) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1684 |
apply assumption |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1685 |
done |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1686 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1687 |
lemma predicate2D: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1688 |
"P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1689 |
apply (erule le_funE)+ |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1690 |
apply (erule le_boolE) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1691 |
apply assumption+ |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1692 |
done |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1693 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1694 |
lemma rev_predicate2D: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1695 |
"P x y \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x y" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1696 |
by (rule predicate2D) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1697 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1698 |
lemma bot1E [no_atp]: "\<bottom> x \<Longrightarrow> P" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1699 |
by (simp add: bot_fun_def) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1700 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1701 |
lemma bot2E: "\<bottom> x y \<Longrightarrow> P" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1702 |
by (simp add: bot_fun_def) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1703 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1704 |
lemma top1I: "\<top> x" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1705 |
by (simp add: top_fun_def) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1706 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1707 |
lemma top2I: "\<top> x y" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1708 |
by (simp add: top_fun_def) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1709 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1710 |
|
60758 | 1711 |
subsection \<open>Name duplicates\<close> |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1712 |
|
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1713 |
lemmas order_eq_refl = preorder_class.eq_refl |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1714 |
lemmas order_less_irrefl = preorder_class.less_irrefl |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1715 |
lemmas order_less_imp_le = preorder_class.less_imp_le |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1716 |
lemmas order_less_not_sym = preorder_class.less_not_sym |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1717 |
lemmas order_less_asym = preorder_class.less_asym |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1718 |
lemmas order_less_trans = preorder_class.less_trans |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1719 |
lemmas order_le_less_trans = preorder_class.le_less_trans |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1720 |
lemmas order_less_le_trans = preorder_class.less_le_trans |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1721 |
lemmas order_less_imp_not_less = preorder_class.less_imp_not_less |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1722 |
lemmas order_less_imp_triv = preorder_class.less_imp_triv |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1723 |
lemmas order_less_asym' = preorder_class.less_asym' |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1724 |
|
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1725 |
lemmas order_less_le = order_class.less_le |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1726 |
lemmas order_le_less = order_class.le_less |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1727 |
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1728 |
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1729 |
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2 |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1730 |
lemmas order_neq_le_trans = order_class.neq_le_trans |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1731 |
lemmas order_le_neq_trans = order_class.le_neq_trans |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1732 |
lemmas order_antisym = order_class.antisym |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1733 |
lemmas order_eq_iff = order_class.eq_iff |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1734 |
lemmas order_antisym_conv = order_class.antisym_conv |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1735 |
|
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1736 |
lemmas linorder_linear = linorder_class.linear |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1737 |
lemmas linorder_less_linear = linorder_class.less_linear |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1738 |
lemmas linorder_le_less_linear = linorder_class.le_less_linear |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1739 |
lemmas linorder_le_cases = linorder_class.le_cases |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1740 |
lemmas linorder_not_less = linorder_class.not_less |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1741 |
lemmas linorder_not_le = linorder_class.not_le |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1742 |
lemmas linorder_neq_iff = linorder_class.neq_iff |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1743 |
lemmas linorder_neqE = linorder_class.neqE |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1744 |
|
28685 | 1745 |
end |