author | huffman |
Thu, 04 Sep 2008 17:19:57 +0200 | |
changeset 28131 | 3130d7b3149d |
parent 27823 | 52971512d1a2 |
child 28516 | e6fdcaaadbd3 |
permissions | -rw-r--r-- |
15524 | 1 |
(* Title: HOL/Orderings.thy |
2 |
ID: $Id$ |
|
3 |
Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson |
|
4 |
*) |
|
5 |
||
25614 | 6 |
header {* Abstract orderings *} |
15524 | 7 |
|
8 |
theory Orderings |
|
26796
c554b77061e5
- Now imports Code_Setup, rather than Set and Fun, since the theorems
berghofe
parents:
26496
diff
changeset
|
9 |
imports Code_Setup |
23263 | 10 |
uses |
11 |
"~~/src/Provers/order.ML" |
|
15524 | 12 |
begin |
13 |
||
27682 | 14 |
subsection {* Quasi orders *} |
15524 | 15 |
|
27682 | 16 |
class preorder = ord + |
17 |
assumes less_le_not_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> (y \<le> x)" |
|
25062 | 18 |
and order_refl [iff]: "x \<le> x" |
19 |
and order_trans: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z" |
|
21248 | 20 |
begin |
21 |
||
15524 | 22 |
text {* Reflexivity. *} |
23 |
||
25062 | 24 |
lemma eq_refl: "x = y \<Longrightarrow> x \<le> y" |
15524 | 25 |
-- {* This form is useful with the classical reasoner. *} |
23212 | 26 |
by (erule ssubst) (rule order_refl) |
15524 | 27 |
|
25062 | 28 |
lemma less_irrefl [iff]: "\<not> x < x" |
27682 | 29 |
by (simp add: less_le_not_le) |
30 |
||
31 |
lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y" |
|
32 |
unfolding less_le_not_le by blast |
|
33 |
||
34 |
||
35 |
text {* Asymmetry. *} |
|
36 |
||
37 |
lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)" |
|
38 |
by (simp add: less_le_not_le) |
|
39 |
||
40 |
lemma less_asym: "x < y \<Longrightarrow> (\<not> P \<Longrightarrow> y < x) \<Longrightarrow> P" |
|
41 |
by (drule less_not_sym, erule contrapos_np) simp |
|
42 |
||
43 |
||
44 |
text {* Transitivity. *} |
|
45 |
||
46 |
lemma less_trans: "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" |
|
47 |
by (auto simp add: less_le_not_le intro: order_trans) |
|
48 |
||
49 |
lemma le_less_trans: "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z" |
|
50 |
by (auto simp add: less_le_not_le intro: order_trans) |
|
51 |
||
52 |
lemma less_le_trans: "x < y \<Longrightarrow> y \<le> z \<Longrightarrow> x < z" |
|
53 |
by (auto simp add: less_le_not_le intro: order_trans) |
|
54 |
||
55 |
||
56 |
text {* Useful for simplification, but too risky to include by default. *} |
|
57 |
||
58 |
lemma less_imp_not_less: "x < y \<Longrightarrow> (\<not> y < x) \<longleftrightarrow> True" |
|
59 |
by (blast elim: less_asym) |
|
60 |
||
61 |
lemma less_imp_triv: "x < y \<Longrightarrow> (y < x \<longrightarrow> P) \<longleftrightarrow> True" |
|
62 |
by (blast elim: less_asym) |
|
63 |
||
64 |
||
65 |
text {* Transitivity rules for calculational reasoning *} |
|
66 |
||
67 |
lemma less_asym': "a < b \<Longrightarrow> b < a \<Longrightarrow> P" |
|
68 |
by (rule less_asym) |
|
69 |
||
70 |
||
71 |
text {* Dual order *} |
|
72 |
||
73 |
lemma dual_preorder: |
|
74 |
"preorder (op \<ge>) (op >)" |
|
75 |
by unfold_locales (auto simp add: less_le_not_le intro: order_trans) |
|
76 |
||
77 |
end |
|
78 |
||
79 |
||
80 |
subsection {* Partial orders *} |
|
81 |
||
82 |
class order = preorder + |
|
83 |
assumes antisym: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y" |
|
84 |
begin |
|
85 |
||
86 |
text {* Reflexivity. *} |
|
87 |
||
88 |
lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> y" |
|
89 |
by (auto simp add: less_le_not_le intro: antisym) |
|
15524 | 90 |
|
25062 | 91 |
lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y" |
15524 | 92 |
-- {* NOT suitable for iff, since it can cause PROOF FAILED. *} |
23212 | 93 |
by (simp add: less_le) blast |
15524 | 94 |
|
25062 | 95 |
lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x = y" |
23212 | 96 |
unfolding less_le by blast |
15524 | 97 |
|
21329 | 98 |
|
99 |
text {* Useful for simplification, but too risky to include by default. *} |
|
100 |
||
25062 | 101 |
lemma less_imp_not_eq: "x < y \<Longrightarrow> (x = y) \<longleftrightarrow> False" |
23212 | 102 |
by auto |
21329 | 103 |
|
25062 | 104 |
lemma less_imp_not_eq2: "x < y \<Longrightarrow> (y = x) \<longleftrightarrow> False" |
23212 | 105 |
by auto |
21329 | 106 |
|
107 |
||
108 |
text {* Transitivity rules for calculational reasoning *} |
|
109 |
||
25062 | 110 |
lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b" |
23212 | 111 |
by (simp add: less_le) |
21329 | 112 |
|
25062 | 113 |
lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b" |
23212 | 114 |
by (simp add: less_le) |
21329 | 115 |
|
15524 | 116 |
|
117 |
text {* Asymmetry. *} |
|
118 |
||
25062 | 119 |
lemma eq_iff: "x = y \<longleftrightarrow> x \<le> y \<and> y \<le> x" |
23212 | 120 |
by (blast intro: antisym) |
15524 | 121 |
|
25062 | 122 |
lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y" |
23212 | 123 |
by (blast intro: antisym) |
15524 | 124 |
|
25062 | 125 |
lemma less_imp_neq: "x < y \<Longrightarrow> x \<noteq> y" |
23212 | 126 |
by (erule contrapos_pn, erule subst, rule less_irrefl) |
21248 | 127 |
|
21083 | 128 |
|
27107 | 129 |
text {* Least value operator *} |
130 |
||
27299 | 131 |
definition (in ord) |
27107 | 132 |
Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "LEAST " 10) where |
133 |
"Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y))" |
|
134 |
||
135 |
lemma Least_equality: |
|
136 |
assumes "P x" |
|
137 |
and "\<And>y. P y \<Longrightarrow> x \<le> y" |
|
138 |
shows "Least P = x" |
|
139 |
unfolding Least_def by (rule the_equality) |
|
140 |
(blast intro: assms antisym)+ |
|
141 |
||
142 |
lemma LeastI2_order: |
|
143 |
assumes "P x" |
|
144 |
and "\<And>y. P y \<Longrightarrow> x \<le> y" |
|
145 |
and "\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> x \<le> y \<Longrightarrow> Q x" |
|
146 |
shows "Q (Least P)" |
|
147 |
unfolding Least_def by (rule theI2) |
|
148 |
(blast intro: assms antisym)+ |
|
149 |
||
150 |
||
26014 | 151 |
text {* Dual order *} |
22916 | 152 |
|
26014 | 153 |
lemma dual_order: |
25103 | 154 |
"order (op \<ge>) (op >)" |
27682 | 155 |
by (intro_locales, rule dual_preorder) (unfold_locales, rule antisym) |
22916 | 156 |
|
21248 | 157 |
end |
15524 | 158 |
|
21329 | 159 |
|
160 |
subsection {* Linear (total) orders *} |
|
161 |
||
22316 | 162 |
class linorder = order + |
25207 | 163 |
assumes linear: "x \<le> y \<or> y \<le> x" |
21248 | 164 |
begin |
165 |
||
25062 | 166 |
lemma less_linear: "x < y \<or> x = y \<or> y < x" |
23212 | 167 |
unfolding less_le using less_le linear by blast |
21248 | 168 |
|
25062 | 169 |
lemma le_less_linear: "x \<le> y \<or> y < x" |
23212 | 170 |
by (simp add: le_less less_linear) |
21248 | 171 |
|
172 |
lemma le_cases [case_names le ge]: |
|
25062 | 173 |
"(x \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<le> x \<Longrightarrow> P) \<Longrightarrow> P" |
23212 | 174 |
using linear by blast |
21248 | 175 |
|
22384
33a46e6c7f04
prefix of class interpretation not mandatory any longer
haftmann
parents:
22377
diff
changeset
|
176 |
lemma linorder_cases [case_names less equal greater]: |
25062 | 177 |
"(x < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P" |
23212 | 178 |
using less_linear by blast |
21248 | 179 |
|
25062 | 180 |
lemma not_less: "\<not> x < y \<longleftrightarrow> y \<le> x" |
23212 | 181 |
apply (simp add: less_le) |
182 |
using linear apply (blast intro: antisym) |
|
183 |
done |
|
184 |
||
185 |
lemma not_less_iff_gr_or_eq: |
|
25062 | 186 |
"\<not>(x < y) \<longleftrightarrow> (x > y | x = y)" |
23212 | 187 |
apply(simp add:not_less le_less) |
188 |
apply blast |
|
189 |
done |
|
15524 | 190 |
|
25062 | 191 |
lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x" |
23212 | 192 |
apply (simp add: less_le) |
193 |
using linear apply (blast intro: antisym) |
|
194 |
done |
|
15524 | 195 |
|
25062 | 196 |
lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x" |
23212 | 197 |
by (cut_tac x = x and y = y in less_linear, auto) |
15524 | 198 |
|
25062 | 199 |
lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R" |
23212 | 200 |
by (simp add: neq_iff) blast |
15524 | 201 |
|
25062 | 202 |
lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y" |
23212 | 203 |
by (blast intro: antisym dest: not_less [THEN iffD1]) |
15524 | 204 |
|
25062 | 205 |
lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y" |
23212 | 206 |
by (blast intro: antisym dest: not_less [THEN iffD1]) |
15524 | 207 |
|
25062 | 208 |
lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y" |
23212 | 209 |
by (blast intro: antisym dest: not_less [THEN iffD1]) |
15524 | 210 |
|
25062 | 211 |
lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x" |
23212 | 212 |
unfolding not_less . |
16796 | 213 |
|
25062 | 214 |
lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y" |
23212 | 215 |
unfolding not_less . |
16796 | 216 |
|
217 |
(*FIXME inappropriate name (or delete altogether)*) |
|
25062 | 218 |
lemma not_leE: "\<not> y \<le> x \<Longrightarrow> x < y" |
23212 | 219 |
unfolding not_le . |
21248 | 220 |
|
22916 | 221 |
|
26014 | 222 |
text {* Dual order *} |
22916 | 223 |
|
26014 | 224 |
lemma dual_linorder: |
25103 | 225 |
"linorder (op \<ge>) (op >)" |
27682 | 226 |
by (rule linorder.intro, rule dual_order) (unfold_locales, rule linear) |
22916 | 227 |
|
228 |
||
23881 | 229 |
text {* min/max *} |
230 |
||
27299 | 231 |
definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
25062 | 232 |
[code unfold, code inline del]: "min a b = (if a \<le> b then a else b)" |
23881 | 233 |
|
27299 | 234 |
definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
25062 | 235 |
[code unfold, code inline del]: "max a b = (if a \<le> b then b else a)" |
22384
33a46e6c7f04
prefix of class interpretation not mandatory any longer
haftmann
parents:
22377
diff
changeset
|
236 |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
237 |
lemma min_le_iff_disj: |
25062 | 238 |
"min x y \<le> z \<longleftrightarrow> x \<le> z \<or> y \<le> z" |
23212 | 239 |
unfolding min_def using linear by (auto intro: order_trans) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
240 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
241 |
lemma le_max_iff_disj: |
25062 | 242 |
"z \<le> max x y \<longleftrightarrow> z \<le> x \<or> z \<le> y" |
23212 | 243 |
unfolding max_def using linear by (auto intro: order_trans) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
244 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
245 |
lemma min_less_iff_disj: |
25062 | 246 |
"min x y < z \<longleftrightarrow> x < z \<or> y < z" |
23212 | 247 |
unfolding min_def le_less using less_linear by (auto intro: less_trans) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
248 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
249 |
lemma less_max_iff_disj: |
25062 | 250 |
"z < max x y \<longleftrightarrow> z < x \<or> z < y" |
23212 | 251 |
unfolding max_def le_less using less_linear by (auto intro: less_trans) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
252 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
253 |
lemma min_less_iff_conj [simp]: |
25062 | 254 |
"z < min x y \<longleftrightarrow> z < x \<and> z < y" |
23212 | 255 |
unfolding min_def le_less using less_linear by (auto intro: less_trans) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
256 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
257 |
lemma max_less_iff_conj [simp]: |
25062 | 258 |
"max x y < z \<longleftrightarrow> x < z \<and> y < z" |
23212 | 259 |
unfolding max_def le_less using less_linear by (auto intro: less_trans) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
260 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23948
diff
changeset
|
261 |
lemma split_min [noatp]: |
25062 | 262 |
"P (min i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P i) \<and> (\<not> i \<le> j \<longrightarrow> P j)" |
23212 | 263 |
by (simp add: min_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
264 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23948
diff
changeset
|
265 |
lemma split_max [noatp]: |
25062 | 266 |
"P (max i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P j) \<and> (\<not> i \<le> j \<longrightarrow> P i)" |
23212 | 267 |
by (simp add: max_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
268 |
|
21248 | 269 |
end |
270 |
||
23948 | 271 |
|
21083 | 272 |
subsection {* Reasoning tools setup *} |
273 |
||
21091 | 274 |
ML {* |
275 |
||
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
276 |
signature ORDERS = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
277 |
sig |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
278 |
val print_structures: Proof.context -> unit |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
279 |
val setup: theory -> theory |
24704
9a95634ab135
Transitivity reasoner gets additional argument of premises to improve integration with simplifier.
ballarin
parents:
24641
diff
changeset
|
280 |
val order_tac: thm list -> Proof.context -> int -> tactic |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
281 |
end; |
21091 | 282 |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
283 |
structure Orders: ORDERS = |
21248 | 284 |
struct |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
285 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
286 |
(** Theory and context data **) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
287 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
288 |
fun struct_eq ((s1: string, ts1), (s2, ts2)) = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
289 |
(s1 = s2) andalso eq_list (op aconv) (ts1, ts2); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
290 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
291 |
structure Data = GenericDataFun |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
292 |
( |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
293 |
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
|
294 |
(* Order structures: |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
295 |
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
|
296 |
needed to set up the transitivity reasoner, |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
297 |
identifier and operations identify the structure uniquely. *) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
298 |
val empty = []; |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
299 |
val extend = I; |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
300 |
fun merge _ = AList.join struct_eq (K fst); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
301 |
); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
302 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
303 |
fun print_structures ctxt = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
304 |
let |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
305 |
val structs = Data.get (Context.Proof ctxt); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
306 |
fun pretty_term t = Pretty.block |
24920 | 307 |
[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
|
308 |
Pretty.str "::", Pretty.brk 1, |
24920 | 309 |
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
|
310 |
fun pretty_struct ((s, ts), _) = Pretty.block |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
311 |
[Pretty.str s, Pretty.str ":", Pretty.brk 1, |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
312 |
Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))]; |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
313 |
in |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
314 |
Pretty.writeln (Pretty.big_list "Order structures:" (map pretty_struct structs)) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
315 |
end; |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
316 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
317 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
318 |
(** Method **) |
21091 | 319 |
|
24704
9a95634ab135
Transitivity reasoner gets additional argument of premises to improve integration with simplifier.
ballarin
parents:
24641
diff
changeset
|
320 |
fun struct_tac ((s, [eq, le, less]), thms) prems = |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
321 |
let |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
322 |
fun decomp thy (Trueprop $ t) = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
323 |
let |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
324 |
fun excluded t = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
325 |
(* exclude numeric types: linear arithmetic subsumes transitivity *) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
326 |
let val T = type_of t |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
327 |
in |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
328 |
T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
329 |
end; |
24741
a53f5db5acbb
Fixed setup of transitivity reasoner (function decomp).
ballarin
parents:
24704
diff
changeset
|
330 |
fun rel (bin_op $ t1 $ t2) = |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
331 |
if excluded t1 then NONE |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
332 |
else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
333 |
else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
334 |
else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
335 |
else NONE |
24741
a53f5db5acbb
Fixed setup of transitivity reasoner (function decomp).
ballarin
parents:
24704
diff
changeset
|
336 |
| rel _ = NONE; |
a53f5db5acbb
Fixed setup of transitivity reasoner (function decomp).
ballarin
parents:
24704
diff
changeset
|
337 |
fun dec (Const (@{const_name Not}, _) $ t) = (case rel t |
a53f5db5acbb
Fixed setup of transitivity reasoner (function decomp).
ballarin
parents:
24704
diff
changeset
|
338 |
of NONE => NONE |
a53f5db5acbb
Fixed setup of transitivity reasoner (function decomp).
ballarin
parents:
24704
diff
changeset
|
339 |
| SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2)) |
a53f5db5acbb
Fixed setup of transitivity reasoner (function decomp).
ballarin
parents:
24704
diff
changeset
|
340 |
| dec x = rel x; |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
341 |
in dec t end; |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
342 |
in |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
343 |
case s of |
24704
9a95634ab135
Transitivity reasoner gets additional argument of premises to improve integration with simplifier.
ballarin
parents:
24641
diff
changeset
|
344 |
"order" => Order_Tac.partial_tac decomp thms prems |
9a95634ab135
Transitivity reasoner gets additional argument of premises to improve integration with simplifier.
ballarin
parents:
24641
diff
changeset
|
345 |
| "linorder" => Order_Tac.linear_tac decomp thms prems |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
346 |
| _ => error ("Unknown kind of order `" ^ s ^ "' encountered in transitivity reasoner.") |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
347 |
end |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
348 |
|
24704
9a95634ab135
Transitivity reasoner gets additional argument of premises to improve integration with simplifier.
ballarin
parents:
24641
diff
changeset
|
349 |
fun order_tac prems ctxt = |
9a95634ab135
Transitivity reasoner gets additional argument of premises to improve integration with simplifier.
ballarin
parents:
24641
diff
changeset
|
350 |
FIRST' (map (fn s => CHANGED o struct_tac s prems) (Data.get (Context.Proof ctxt))); |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
351 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
352 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
353 |
(** Attribute **) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
354 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
355 |
fun add_struct_thm s tag = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
356 |
Thm.declaration_attribute |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
357 |
(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
|
358 |
fun del_struct s = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
359 |
Thm.declaration_attribute |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
360 |
(fn _ => Data.map (AList.delete struct_eq s)); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
361 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
362 |
val attribute = Attrib.syntax |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
363 |
(Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
364 |
Args.del >> K NONE) --| Args.colon (* FIXME || |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
365 |
Scan.succeed true *) ) -- Scan.lift Args.name -- |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
366 |
Scan.repeat Args.term |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
367 |
>> (fn ((SOME tag, n), ts) => add_struct_thm (n, ts) tag |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
368 |
| ((NONE, n), ts) => del_struct (n, ts))); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
369 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
370 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
371 |
(** Diagnostic command **) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
372 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
373 |
val print = Toplevel.unknown_context o |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
374 |
Toplevel.keep (Toplevel.node_case |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
375 |
(Context.cases (print_structures o ProofContext.init) print_structures) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
376 |
(print_structures o Proof.context_of)); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
377 |
|
24867 | 378 |
val _ = |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
379 |
OuterSyntax.improper_command "print_orders" |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
380 |
"print order structures available to transitivity reasoner" OuterKeyword.diag |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
381 |
(Scan.succeed (Toplevel.no_timing o print)); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
382 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
383 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
384 |
(** Setup **) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
385 |
|
24867 | 386 |
val setup = |
387 |
Method.add_methods |
|
388 |
[("order", Method.ctxt_args (Method.SIMPLE_METHOD' o order_tac []), "transitivity reasoner")] #> |
|
389 |
Attrib.add_attributes [("order", attribute, "theorems controlling transitivity reasoner")]; |
|
21091 | 390 |
|
391 |
end; |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
392 |
|
21091 | 393 |
*} |
394 |
||
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
395 |
setup Orders.setup |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
396 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
397 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
398 |
text {* Declarations to set up transitivity reasoner of partial and linear orders. *} |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
399 |
|
25076 | 400 |
context order |
401 |
begin |
|
402 |
||
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
403 |
(* The type constraint on @{term op =} below is necessary since the operation |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
404 |
is not a parameter of the locale. *) |
25076 | 405 |
|
27689 | 406 |
declare less_irrefl [THEN notE, order add less_reflE: order "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool" "op <=" "op <"] |
407 |
||
408 |
declare order_refl [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
409 |
||
410 |
declare less_imp_le [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
411 |
||
412 |
declare antisym [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
413 |
||
414 |
declare eq_refl [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
415 |
||
416 |
declare sym [THEN eq_refl, order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
417 |
||
418 |
declare less_trans [order add less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
419 |
||
420 |
declare less_le_trans [order add less_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
421 |
||
422 |
declare le_less_trans [order add le_less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
423 |
||
424 |
declare order_trans [order add le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
425 |
||
426 |
declare le_neq_trans [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
427 |
||
428 |
declare neq_le_trans [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
429 |
||
430 |
declare less_imp_neq [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
431 |
||
432 |
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
433 |
||
434 |
declare not_sym [order add not_sym: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
435 |
|
25076 | 436 |
end |
437 |
||
438 |
context linorder |
|
439 |
begin |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
440 |
|
27689 | 441 |
declare [[order del: order "op = :: 'a => 'a => bool" "op <=" "op <"]] |
442 |
||
443 |
declare less_irrefl [THEN notE, order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
444 |
||
445 |
declare order_refl [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
446 |
||
447 |
declare less_imp_le [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
448 |
||
449 |
declare not_less [THEN iffD2, order add not_lessI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
450 |
||
451 |
declare not_le [THEN iffD2, order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
452 |
||
453 |
declare not_less [THEN iffD1, order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
454 |
||
455 |
declare not_le [THEN iffD1, order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
456 |
||
457 |
declare antisym [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
458 |
||
459 |
declare eq_refl [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
25076 | 460 |
|
27689 | 461 |
declare sym [THEN eq_refl, order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
462 |
||
463 |
declare less_trans [order add less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
464 |
||
465 |
declare less_le_trans [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
466 |
||
467 |
declare le_less_trans [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
468 |
||
469 |
declare order_trans [order add le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
470 |
||
471 |
declare le_neq_trans [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
472 |
||
473 |
declare neq_le_trans [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
474 |
||
475 |
declare less_imp_neq [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
476 |
||
477 |
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
478 |
||
479 |
declare not_sym [order add not_sym: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
480 |
|
25076 | 481 |
end |
482 |
||
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
483 |
|
21083 | 484 |
setup {* |
485 |
let |
|
486 |
||
487 |
fun prp t thm = (#prop (rep_thm thm) = t); |
|
15524 | 488 |
|
21083 | 489 |
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) = |
490 |
let val prems = prems_of_ss ss; |
|
22916 | 491 |
val less = Const (@{const_name less}, T); |
21083 | 492 |
val t = HOLogic.mk_Trueprop(le $ s $ r); |
493 |
in case find_first (prp t) prems of |
|
494 |
NONE => |
|
495 |
let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) |
|
496 |
in case find_first (prp t) prems of |
|
497 |
NONE => NONE |
|
24422 | 498 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1})) |
21083 | 499 |
end |
24422 | 500 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv})) |
21083 | 501 |
end |
502 |
handle THM _ => NONE; |
|
15524 | 503 |
|
21083 | 504 |
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) = |
505 |
let val prems = prems_of_ss ss; |
|
22916 | 506 |
val le = Const (@{const_name less_eq}, T); |
21083 | 507 |
val t = HOLogic.mk_Trueprop(le $ r $ s); |
508 |
in case find_first (prp t) prems of |
|
509 |
NONE => |
|
510 |
let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r)) |
|
511 |
in case find_first (prp t) prems of |
|
512 |
NONE => NONE |
|
24422 | 513 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3})) |
21083 | 514 |
end |
24422 | 515 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv2})) |
21083 | 516 |
end |
517 |
handle THM _ => NONE; |
|
15524 | 518 |
|
21248 | 519 |
fun add_simprocs procs thy = |
26496
49ae9456eba9
purely functional setup of claset/simpset/clasimpset;
wenzelm
parents:
26324
diff
changeset
|
520 |
Simplifier.map_simpset (fn ss => ss |
21248 | 521 |
addsimprocs (map (fn (name, raw_ts, proc) => |
26496
49ae9456eba9
purely functional setup of claset/simpset/clasimpset;
wenzelm
parents:
26324
diff
changeset
|
522 |
Simplifier.simproc thy name raw_ts proc) procs)) thy; |
49ae9456eba9
purely functional setup of claset/simpset/clasimpset;
wenzelm
parents:
26324
diff
changeset
|
523 |
fun add_solver name tac = |
49ae9456eba9
purely functional setup of claset/simpset/clasimpset;
wenzelm
parents:
26324
diff
changeset
|
524 |
Simplifier.map_simpset (fn ss => ss addSolver |
49ae9456eba9
purely functional setup of claset/simpset/clasimpset;
wenzelm
parents:
26324
diff
changeset
|
525 |
mk_solver' name (fn ss => tac (Simplifier.prems_of_ss ss) (Simplifier.the_context ss))); |
21083 | 526 |
|
527 |
in |
|
21248 | 528 |
add_simprocs [ |
529 |
("antisym le", ["(x::'a::order) <= y"], prove_antisym_le), |
|
530 |
("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less) |
|
531 |
] |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
532 |
#> add_solver "Transitivity" Orders.order_tac |
21248 | 533 |
(* Adding the transitivity reasoners also as safe solvers showed a slight |
534 |
speed up, but the reasoning strength appears to be not higher (at least |
|
535 |
no breaking of additional proofs in the entire HOL distribution, as |
|
536 |
of 5 March 2004, was observed). *) |
|
21083 | 537 |
end |
538 |
*} |
|
15524 | 539 |
|
540 |
||
24422 | 541 |
subsection {* Name duplicates *} |
542 |
||
543 |
lemmas order_less_le = less_le |
|
27682 | 544 |
lemmas order_eq_refl = preorder_class.eq_refl |
545 |
lemmas order_less_irrefl = preorder_class.less_irrefl |
|
24422 | 546 |
lemmas order_le_less = order_class.le_less |
547 |
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq |
|
27682 | 548 |
lemmas order_less_imp_le = preorder_class.less_imp_le |
24422 | 549 |
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq |
550 |
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2 |
|
551 |
lemmas order_neq_le_trans = order_class.neq_le_trans |
|
552 |
lemmas order_le_neq_trans = order_class.le_neq_trans |
|
553 |
||
554 |
lemmas order_antisym = antisym |
|
27682 | 555 |
lemmas order_less_not_sym = preorder_class.less_not_sym |
556 |
lemmas order_less_asym = preorder_class.less_asym |
|
24422 | 557 |
lemmas order_eq_iff = order_class.eq_iff |
558 |
lemmas order_antisym_conv = order_class.antisym_conv |
|
27682 | 559 |
lemmas order_less_trans = preorder_class.less_trans |
560 |
lemmas order_le_less_trans = preorder_class.le_less_trans |
|
561 |
lemmas order_less_le_trans = preorder_class.less_le_trans |
|
562 |
lemmas order_less_imp_not_less = preorder_class.less_imp_not_less |
|
563 |
lemmas order_less_imp_triv = preorder_class.less_imp_triv |
|
564 |
lemmas order_less_asym' = preorder_class.less_asym' |
|
24422 | 565 |
|
566 |
lemmas linorder_linear = linear |
|
567 |
lemmas linorder_less_linear = linorder_class.less_linear |
|
568 |
lemmas linorder_le_less_linear = linorder_class.le_less_linear |
|
569 |
lemmas linorder_le_cases = linorder_class.le_cases |
|
570 |
lemmas linorder_not_less = linorder_class.not_less |
|
571 |
lemmas linorder_not_le = linorder_class.not_le |
|
572 |
lemmas linorder_neq_iff = linorder_class.neq_iff |
|
573 |
lemmas linorder_neqE = linorder_class.neqE |
|
574 |
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1 |
|
575 |
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2 |
|
576 |
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3 |
|
577 |
||
578 |
||
21083 | 579 |
subsection {* Bounded quantifiers *} |
580 |
||
581 |
syntax |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
582 |
"_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
|
583 |
"_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
|
584 |
"_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
|
585 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) |
21083 | 586 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
587 |
"_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
|
588 |
"_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
|
589 |
"_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
|
590 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10) |
21083 | 591 |
|
592 |
syntax (xsymbols) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
593 |
"_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
|
594 |
"_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
|
595 |
"_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
|
596 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 597 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
598 |
"_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
|
599 |
"_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
|
600 |
"_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
|
601 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 602 |
|
603 |
syntax (HOL) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
604 |
"_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
|
605 |
"_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
|
606 |
"_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
|
607 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) |
21083 | 608 |
|
609 |
syntax (HTML output) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
610 |
"_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
|
611 |
"_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
|
612 |
"_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
|
613 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 614 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
615 |
"_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
|
616 |
"_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
|
617 |
"_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
|
618 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 619 |
|
620 |
translations |
|
621 |
"ALL x<y. P" => "ALL x. x < y \<longrightarrow> P" |
|
622 |
"EX x<y. P" => "EX x. x < y \<and> P" |
|
623 |
"ALL x<=y. P" => "ALL x. x <= y \<longrightarrow> P" |
|
624 |
"EX x<=y. P" => "EX x. x <= y \<and> P" |
|
625 |
"ALL x>y. P" => "ALL x. x > y \<longrightarrow> P" |
|
626 |
"EX x>y. P" => "EX x. x > y \<and> P" |
|
627 |
"ALL x>=y. P" => "ALL x. x >= y \<longrightarrow> P" |
|
628 |
"EX x>=y. P" => "EX x. x >= y \<and> P" |
|
629 |
||
630 |
print_translation {* |
|
631 |
let |
|
22916 | 632 |
val All_binder = Syntax.binder_name @{const_syntax All}; |
633 |
val Ex_binder = Syntax.binder_name @{const_syntax Ex}; |
|
22377 | 634 |
val impl = @{const_syntax "op -->"}; |
635 |
val conj = @{const_syntax "op &"}; |
|
22916 | 636 |
val less = @{const_syntax less}; |
637 |
val less_eq = @{const_syntax less_eq}; |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
638 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
639 |
val trans = |
21524 | 640 |
[((All_binder, impl, less), ("_All_less", "_All_greater")), |
641 |
((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")), |
|
642 |
((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")), |
|
643 |
((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))]; |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
644 |
|
22344
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
645 |
fun matches_bound v t = |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
646 |
case t of (Const ("_bound", _) $ Free (v', _)) => (v = v') |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
647 |
| _ => false |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
648 |
fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false) |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
649 |
fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
650 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
651 |
fun tr' q = (q, |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
652 |
fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] => |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
653 |
(case AList.lookup (op =) trans (q, c, d) of |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
654 |
NONE => raise Match |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
655 |
| SOME (l, g) => |
22344
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
656 |
if matches_bound v t andalso not (contains_var v u) then mk v l u P |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
657 |
else if matches_bound v u andalso not (contains_var v t) then mk v g t P |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
658 |
else raise Match) |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
659 |
| _ => raise Match); |
21524 | 660 |
in [tr' All_binder, tr' Ex_binder] end |
21083 | 661 |
*} |
662 |
||
663 |
||
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
664 |
subsection {* Transitivity reasoning *} |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
665 |
|
25193 | 666 |
context ord |
667 |
begin |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
668 |
|
25193 | 669 |
lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c" |
670 |
by (rule subst) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
671 |
|
25193 | 672 |
lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c" |
673 |
by (rule ssubst) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
674 |
|
25193 | 675 |
lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c" |
676 |
by (rule subst) |
|
677 |
||
678 |
lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c" |
|
679 |
by (rule ssubst) |
|
680 |
||
681 |
end |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
682 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
683 |
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
|
684 |
(!!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
|
685 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
686 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
687 |
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
|
688 |
also assume "f b < c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
689 |
finally (order_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
690 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
691 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
692 |
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
|
693 |
(!!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
|
694 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
695 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
696 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
697 |
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
|
698 |
finally (order_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
699 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
700 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
701 |
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
|
702 |
(!!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
|
703 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
704 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
705 |
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
|
706 |
also assume "f b < c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
707 |
finally (order_le_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
708 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
709 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
710 |
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
|
711 |
(!!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
|
712 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
713 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
714 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
715 |
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
|
716 |
finally (order_le_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
717 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
718 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
719 |
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
|
720 |
(!!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
|
721 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
722 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
723 |
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
|
724 |
also assume "f b <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
725 |
finally (order_less_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
726 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
727 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
728 |
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
|
729 |
(!!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
|
730 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
731 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
732 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
733 |
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
|
734 |
finally (order_less_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
735 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
736 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
737 |
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
|
738 |
(!!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
|
739 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
740 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
741 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
742 |
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
|
743 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
744 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
745 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
746 |
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
|
747 |
(!!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
|
748 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
749 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
750 |
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
|
751 |
also assume "f b <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
752 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
753 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
754 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
755 |
lemma ord_le_eq_subst: "a <= b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
756 |
(!!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
|
757 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
758 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
759 |
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
|
760 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
761 |
finally (ord_le_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
762 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
763 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
764 |
lemma ord_eq_le_subst: "a = f b ==> b <= c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
765 |
(!!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
|
766 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
767 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
768 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
769 |
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
|
770 |
finally (ord_eq_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
771 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
772 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
773 |
lemma ord_less_eq_subst: "a < b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
774 |
(!!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
|
775 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
776 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
777 |
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
|
778 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
779 |
finally (ord_less_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
780 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
781 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
782 |
lemma ord_eq_less_subst: "a = f b ==> b < c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
783 |
(!!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
|
784 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
785 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
786 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
787 |
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
|
788 |
finally (ord_eq_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
789 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
790 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
791 |
text {* |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
792 |
Note that this list of rules is in reverse order of priorities. |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
793 |
*} |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
794 |
|
27682 | 795 |
lemmas [trans] = |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
796 |
order_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
797 |
order_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
798 |
order_le_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
799 |
order_le_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
800 |
order_less_le_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
801 |
order_less_le_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
802 |
order_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
803 |
order_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
804 |
ord_le_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
805 |
ord_eq_le_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
806 |
ord_less_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
807 |
ord_eq_less_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
808 |
forw_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
809 |
back_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
810 |
rev_mp |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
811 |
mp |
27682 | 812 |
|
813 |
lemmas (in order) [trans] = |
|
814 |
neq_le_trans |
|
815 |
le_neq_trans |
|
816 |
||
817 |
lemmas (in preorder) [trans] = |
|
818 |
less_trans |
|
819 |
less_asym' |
|
820 |
le_less_trans |
|
821 |
less_le_trans |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
822 |
order_trans |
27682 | 823 |
|
824 |
lemmas (in order) [trans] = |
|
825 |
antisym |
|
826 |
||
827 |
lemmas (in ord) [trans] = |
|
828 |
ord_le_eq_trans |
|
829 |
ord_eq_le_trans |
|
830 |
ord_less_eq_trans |
|
831 |
ord_eq_less_trans |
|
832 |
||
833 |
lemmas [trans] = |
|
834 |
trans |
|
835 |
||
836 |
lemmas order_trans_rules = |
|
837 |
order_less_subst2 |
|
838 |
order_less_subst1 |
|
839 |
order_le_less_subst2 |
|
840 |
order_le_less_subst1 |
|
841 |
order_less_le_subst2 |
|
842 |
order_less_le_subst1 |
|
843 |
order_subst2 |
|
844 |
order_subst1 |
|
845 |
ord_le_eq_subst |
|
846 |
ord_eq_le_subst |
|
847 |
ord_less_eq_subst |
|
848 |
ord_eq_less_subst |
|
849 |
forw_subst |
|
850 |
back_subst |
|
851 |
rev_mp |
|
852 |
mp |
|
853 |
neq_le_trans |
|
854 |
le_neq_trans |
|
855 |
less_trans |
|
856 |
less_asym' |
|
857 |
le_less_trans |
|
858 |
less_le_trans |
|
859 |
order_trans |
|
860 |
antisym |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
861 |
ord_le_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
862 |
ord_eq_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
863 |
ord_less_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
864 |
ord_eq_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
865 |
trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
866 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
867 |
(* FIXME cleanup *) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
868 |
|
21083 | 869 |
text {* These support proving chains of decreasing inequalities |
870 |
a >= b >= c ... in Isar proofs. *} |
|
871 |
||
872 |
lemma xt1: |
|
873 |
"a = b ==> b > c ==> a > c" |
|
874 |
"a > b ==> b = c ==> a > c" |
|
875 |
"a = b ==> b >= c ==> a >= c" |
|
876 |
"a >= b ==> b = c ==> a >= c" |
|
877 |
"(x::'a::order) >= y ==> y >= x ==> x = y" |
|
878 |
"(x::'a::order) >= y ==> y >= z ==> x >= z" |
|
879 |
"(x::'a::order) > y ==> y >= z ==> x > z" |
|
880 |
"(x::'a::order) >= y ==> y > z ==> x > z" |
|
23417 | 881 |
"(a::'a::order) > b ==> b > a ==> P" |
21083 | 882 |
"(x::'a::order) > y ==> y > z ==> x > z" |
883 |
"(a::'a::order) >= b ==> a ~= b ==> a > b" |
|
884 |
"(a::'a::order) ~= b ==> a >= b ==> a > b" |
|
885 |
"a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" |
|
886 |
"a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c" |
|
887 |
"a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
|
888 |
"a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c" |
|
25076 | 889 |
by auto |
21083 | 890 |
|
891 |
lemma xt2: |
|
892 |
"(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
|
893 |
by (subgoal_tac "f b >= f c", force, force) |
|
894 |
||
895 |
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> |
|
896 |
(!!x y. x >= y ==> f x >= f y) ==> f a >= c" |
|
897 |
by (subgoal_tac "f a >= f b", force, force) |
|
898 |
||
899 |
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==> |
|
900 |
(!!x y. x >= y ==> f x >= f y) ==> a > f c" |
|
901 |
by (subgoal_tac "f b >= f c", force, force) |
|
902 |
||
903 |
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==> |
|
904 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
|
905 |
by (subgoal_tac "f a > f b", force, force) |
|
906 |
||
907 |
lemma xt6: "(a::'a::order) >= f b ==> b > c ==> |
|
908 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
|
909 |
by (subgoal_tac "f b > f c", force, force) |
|
910 |
||
911 |
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==> |
|
912 |
(!!x y. x >= y ==> f x >= f y) ==> f a > c" |
|
913 |
by (subgoal_tac "f a >= f b", force, force) |
|
914 |
||
915 |
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==> |
|
916 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
|
917 |
by (subgoal_tac "f b > f c", force, force) |
|
918 |
||
919 |
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==> |
|
920 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
|
921 |
by (subgoal_tac "f a > f b", force, force) |
|
922 |
||
923 |
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9 |
|
924 |
||
925 |
(* |
|
926 |
Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands |
|
927 |
for the wrong thing in an Isar proof. |
|
928 |
||
929 |
The extra transitivity rules can be used as follows: |
|
930 |
||
931 |
lemma "(a::'a::order) > z" |
|
932 |
proof - |
|
933 |
have "a >= b" (is "_ >= ?rhs") |
|
934 |
sorry |
|
935 |
also have "?rhs >= c" (is "_ >= ?rhs") |
|
936 |
sorry |
|
937 |
also (xtrans) have "?rhs = d" (is "_ = ?rhs") |
|
938 |
sorry |
|
939 |
also (xtrans) have "?rhs >= e" (is "_ >= ?rhs") |
|
940 |
sorry |
|
941 |
also (xtrans) have "?rhs > f" (is "_ > ?rhs") |
|
942 |
sorry |
|
943 |
also (xtrans) have "?rhs > z" |
|
944 |
sorry |
|
945 |
finally (xtrans) show ?thesis . |
|
946 |
qed |
|
947 |
||
948 |
Alternatively, one can use "declare xtrans [trans]" and then |
|
949 |
leave out the "(xtrans)" above. |
|
950 |
*) |
|
951 |
||
21546 | 952 |
subsection {* Order on bool *} |
953 |
||
26324 | 954 |
instantiation bool :: order |
25510 | 955 |
begin |
956 |
||
957 |
definition |
|
958 |
le_bool_def [code func del]: "P \<le> Q \<longleftrightarrow> P \<longrightarrow> Q" |
|
959 |
||
960 |
definition |
|
961 |
less_bool_def [code func del]: "(P\<Colon>bool) < Q \<longleftrightarrow> P \<le> Q \<and> P \<noteq> Q" |
|
962 |
||
963 |
instance |
|
22916 | 964 |
by intro_classes (auto simp add: le_bool_def less_bool_def) |
25510 | 965 |
|
966 |
end |
|
21546 | 967 |
|
968 |
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q" |
|
23212 | 969 |
by (simp add: le_bool_def) |
21546 | 970 |
|
971 |
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q" |
|
23212 | 972 |
by (simp add: le_bool_def) |
21546 | 973 |
|
974 |
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R" |
|
23212 | 975 |
by (simp add: le_bool_def) |
21546 | 976 |
|
977 |
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q" |
|
23212 | 978 |
by (simp add: le_bool_def) |
21546 | 979 |
|
22348 | 980 |
lemma [code func]: |
981 |
"False \<le> b \<longleftrightarrow> True" |
|
982 |
"True \<le> b \<longleftrightarrow> b" |
|
983 |
"False < b \<longleftrightarrow> b" |
|
984 |
"True < b \<longleftrightarrow> False" |
|
985 |
unfolding le_bool_def less_bool_def by simp_all |
|
986 |
||
22424 | 987 |
|
23881 | 988 |
subsection {* Order on functions *} |
989 |
||
25510 | 990 |
instantiation "fun" :: (type, ord) ord |
991 |
begin |
|
992 |
||
993 |
definition |
|
994 |
le_fun_def [code func del]: "f \<le> g \<longleftrightarrow> (\<forall>x. f x \<le> g x)" |
|
23881 | 995 |
|
25510 | 996 |
definition |
997 |
less_fun_def [code func del]: "(f\<Colon>'a \<Rightarrow> 'b) < g \<longleftrightarrow> f \<le> g \<and> f \<noteq> g" |
|
998 |
||
999 |
instance .. |
|
1000 |
||
1001 |
end |
|
23881 | 1002 |
|
1003 |
instance "fun" :: (type, order) order |
|
1004 |
by default |
|
26796
c554b77061e5
- Now imports Code_Setup, rather than Set and Fun, since the theorems
berghofe
parents:
26496
diff
changeset
|
1005 |
(auto simp add: le_fun_def less_fun_def |
c554b77061e5
- Now imports Code_Setup, rather than Set and Fun, since the theorems
berghofe
parents:
26496
diff
changeset
|
1006 |
intro: order_trans order_antisym intro!: ext) |
23881 | 1007 |
|
1008 |
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g" |
|
1009 |
unfolding le_fun_def by simp |
|
1010 |
||
1011 |
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P" |
|
1012 |
unfolding le_fun_def by simp |
|
1013 |
||
1014 |
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x" |
|
1015 |
unfolding le_fun_def by simp |
|
1016 |
||
1017 |
text {* |
|
1018 |
Handy introduction and elimination rules for @{text "\<le>"} |
|
1019 |
on unary and binary predicates |
|
1020 |
*} |
|
1021 |
||
26796
c554b77061e5
- Now imports Code_Setup, rather than Set and Fun, since the theorems
berghofe
parents:
26496
diff
changeset
|
1022 |
lemma predicate1I: |
23881 | 1023 |
assumes PQ: "\<And>x. P x \<Longrightarrow> Q x" |
1024 |
shows "P \<le> Q" |
|
1025 |
apply (rule le_funI) |
|
1026 |
apply (rule le_boolI) |
|
1027 |
apply (rule PQ) |
|
1028 |
apply assumption |
|
1029 |
done |
|
1030 |
||
1031 |
lemma predicate1D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x" |
|
1032 |
apply (erule le_funE) |
|
1033 |
apply (erule le_boolE) |
|
1034 |
apply assumption+ |
|
1035 |
done |
|
1036 |
||
1037 |
lemma predicate2I [Pure.intro!, intro!]: |
|
1038 |
assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y" |
|
1039 |
shows "P \<le> Q" |
|
1040 |
apply (rule le_funI)+ |
|
1041 |
apply (rule le_boolI) |
|
1042 |
apply (rule PQ) |
|
1043 |
apply assumption |
|
1044 |
done |
|
1045 |
||
1046 |
lemma predicate2D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y" |
|
1047 |
apply (erule le_funE)+ |
|
1048 |
apply (erule le_boolE) |
|
1049 |
apply assumption+ |
|
1050 |
done |
|
1051 |
||
1052 |
lemma rev_predicate1D: "P x ==> P <= Q ==> Q x" |
|
1053 |
by (rule predicate1D) |
|
1054 |
||
1055 |
lemma rev_predicate2D: "P x y ==> P <= Q ==> Q x y" |
|
1056 |
by (rule predicate2D) |
|
1057 |
||
1058 |
||
1059 |
subsection {* Monotonicity, least value operator and min/max *} |
|
21083 | 1060 |
|
25076 | 1061 |
context order |
1062 |
begin |
|
1063 |
||
1064 |
definition |
|
1065 |
mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" |
|
1066 |
where |
|
1067 |
"mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)" |
|
1068 |
||
1069 |
lemma monoI [intro?]: |
|
1070 |
fixes f :: "'a \<Rightarrow> 'b\<Colon>order" |
|
1071 |
shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y) \<Longrightarrow> mono f" |
|
1072 |
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
|
1073 |
|
25076 | 1074 |
lemma monoD [dest?]: |
1075 |
fixes f :: "'a \<Rightarrow> 'b\<Colon>order" |
|
1076 |
shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" |
|
1077 |
unfolding mono_def by iprover |
|
1078 |
||
1079 |
end |
|
1080 |
||
1081 |
context linorder |
|
1082 |
begin |
|
1083 |
||
1084 |
lemma min_of_mono: |
|
1085 |
fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder" |
|
25377 | 1086 |
shows "mono f \<Longrightarrow> min (f m) (f n) = f (min m n)" |
25076 | 1087 |
by (auto simp: mono_def Orderings.min_def min_def intro: Orderings.antisym) |
1088 |
||
1089 |
lemma max_of_mono: |
|
1090 |
fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder" |
|
25377 | 1091 |
shows "mono f \<Longrightarrow> max (f m) (f n) = f (max m n)" |
25076 | 1092 |
by (auto simp: mono_def Orderings.max_def max_def intro: Orderings.antisym) |
1093 |
||
1094 |
end |
|
21083 | 1095 |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1096 |
lemma min_leastL: "(!!x. least <= x) ==> min least x = least" |
23212 | 1097 |
by (simp add: min_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1098 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1099 |
lemma max_leastL: "(!!x. least <= x) ==> max least x = x" |
23212 | 1100 |
by (simp add: max_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1101 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1102 |
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least" |
23212 | 1103 |
apply (simp add: min_def) |
1104 |
apply (blast intro: order_antisym) |
|
1105 |
done |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1106 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1107 |
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x" |
23212 | 1108 |
apply (simp add: max_def) |
1109 |
apply (blast intro: order_antisym) |
|
1110 |
done |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1111 |
|
27823 | 1112 |
|
1113 |
subsection {* Dense orders *} |
|
1114 |
||
1115 |
class dense_linear_order = linorder + |
|
1116 |
assumes gt_ex: "\<exists>y. x < y" |
|
1117 |
and lt_ex: "\<exists>y. y < x" |
|
1118 |
and dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)" |
|
1119 |
(*see further theory Dense_Linear_Order*) |
|
1120 |
||
1121 |
||
1122 |
subsection {* Wellorders *} |
|
1123 |
||
1124 |
class wellorder = linorder + |
|
1125 |
assumes less_induct [case_names less]: "(\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P a" |
|
1126 |
begin |
|
1127 |
||
1128 |
lemma wellorder_Least_lemma: |
|
1129 |
fixes k :: 'a |
|
1130 |
assumes "P k" |
|
1131 |
shows "P (LEAST x. P x)" and "(LEAST x. P x) \<le> k" |
|
1132 |
proof - |
|
1133 |
have "P (LEAST x. P x) \<and> (LEAST x. P x) \<le> k" |
|
1134 |
using assms proof (induct k rule: less_induct) |
|
1135 |
case (less x) then have "P x" by simp |
|
1136 |
show ?case proof (rule classical) |
|
1137 |
assume assm: "\<not> (P (LEAST a. P a) \<and> (LEAST a. P a) \<le> x)" |
|
1138 |
have "\<And>y. P y \<Longrightarrow> x \<le> y" |
|
1139 |
proof (rule classical) |
|
1140 |
fix y |
|
1141 |
assume "P y" and "\<not> x \<le> y" |
|
1142 |
with less have "P (LEAST a. P a)" and "(LEAST a. P a) \<le> y" |
|
1143 |
by (auto simp add: not_le) |
|
1144 |
with assm have "x < (LEAST a. P a)" and "(LEAST a. P a) \<le> y" |
|
1145 |
by auto |
|
1146 |
then show "x \<le> y" by auto |
|
1147 |
qed |
|
1148 |
with `P x` have Least: "(LEAST a. P a) = x" |
|
1149 |
by (rule Least_equality) |
|
1150 |
with `P x` show ?thesis by simp |
|
1151 |
qed |
|
1152 |
qed |
|
1153 |
then show "P (LEAST x. P x)" and "(LEAST x. P x) \<le> k" by auto |
|
1154 |
qed |
|
1155 |
||
1156 |
lemmas LeastI = wellorder_Least_lemma(1) |
|
1157 |
lemmas Least_le = wellorder_Least_lemma(2) |
|
1158 |
||
1159 |
-- "The following 3 lemmas are due to Brian Huffman" |
|
1160 |
lemma LeastI_ex: "\<exists>x. P x \<Longrightarrow> P (Least P)" |
|
1161 |
by (erule exE) (erule LeastI) |
|
1162 |
||
1163 |
lemma LeastI2: |
|
1164 |
"P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)" |
|
1165 |
by (blast intro: LeastI) |
|
1166 |
||
1167 |
lemma LeastI2_ex: |
|
1168 |
"\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)" |
|
1169 |
by (blast intro: LeastI_ex) |
|
1170 |
||
1171 |
lemma not_less_Least: "k < (LEAST x. P x) \<Longrightarrow> \<not> P k" |
|
1172 |
apply (simp (no_asm_use) add: not_le [symmetric]) |
|
1173 |
apply (erule contrapos_nn) |
|
1174 |
apply (erule Least_le) |
|
1175 |
done |
|
1176 |
||
1177 |
end |
|
1178 |
||
15524 | 1179 |
end |