author | Christian Sternagel |
Thu, 30 Aug 2012 15:44:03 +0900 | |
changeset 49093 | fdc301f592c4 |
parent 41460 | ea56b98aee83 |
child 55380 | 4de48353034e |
permissions | -rw-r--r-- |
12360 | 1 |
(* Title: HOL/ex/Higher_Order_Logic.thy |
2 |
Author: Gertrud Bauer and Markus Wenzel, TU Muenchen |
|
3 |
*) |
|
4 |
||
5 |
header {* Foundations of HOL *} |
|
6 |
||
26957 | 7 |
theory Higher_Order_Logic imports Pure begin |
12360 | 8 |
|
9 |
text {* |
|
10 |
The following theory development demonstrates Higher-Order Logic |
|
11 |
itself, represented directly within the Pure framework of Isabelle. |
|
12 |
The ``HOL'' logic given here is essentially that of Gordon |
|
13 |
\cite{Gordon:1985:HOL}, although we prefer to present basic concepts |
|
14 |
in a slightly more conventional manner oriented towards plain |
|
15 |
Natural Deduction. |
|
16 |
*} |
|
17 |
||
18 |
||
19 |
subsection {* Pure Logic *} |
|
20 |
||
14854 | 21 |
classes type |
36452 | 22 |
default_sort type |
12360 | 23 |
|
24 |
typedecl o |
|
25 |
arities |
|
26 |
o :: type |
|
20523
36a59e5d0039
Major update to function package, including new syntax and the (only theoretical)
krauss
parents:
19736
diff
changeset
|
27 |
"fun" :: (type, type) type |
12360 | 28 |
|
29 |
||
30 |
subsubsection {* Basic logical connectives *} |
|
31 |
||
32 |
judgment |
|
33 |
Trueprop :: "o \<Rightarrow> prop" ("_" 5) |
|
34 |
||
23822 | 35 |
axiomatization |
36 |
imp :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<longrightarrow>" 25) and |
|
12360 | 37 |
All :: "('a \<Rightarrow> o) \<Rightarrow> o" (binder "\<forall>" 10) |
23822 | 38 |
where |
39 |
impI [intro]: "(A \<Longrightarrow> B) \<Longrightarrow> A \<longrightarrow> B" and |
|
40 |
impE [dest, trans]: "A \<longrightarrow> B \<Longrightarrow> A \<Longrightarrow> B" and |
|
41 |
allI [intro]: "(\<And>x. P x) \<Longrightarrow> \<forall>x. P x" and |
|
12360 | 42 |
allE [dest]: "\<forall>x. P x \<Longrightarrow> P a" |
43 |
||
44 |
||
45 |
subsubsection {* Extensional equality *} |
|
46 |
||
23822 | 47 |
axiomatization |
12360 | 48 |
equal :: "'a \<Rightarrow> 'a \<Rightarrow> o" (infixl "=" 50) |
23822 | 49 |
where |
50 |
refl [intro]: "x = x" and |
|
51 |
subst: "x = y \<Longrightarrow> P x \<Longrightarrow> P y" |
|
12360 | 52 |
|
23822 | 53 |
axiomatization where |
54 |
ext [intro]: "(\<And>x. f x = g x) \<Longrightarrow> f = g" and |
|
12360 | 55 |
iff [intro]: "(A \<Longrightarrow> B) \<Longrightarrow> (B \<Longrightarrow> A) \<Longrightarrow> A = B" |
56 |
||
12394 | 57 |
theorem sym [sym]: "x = y \<Longrightarrow> y = x" |
12360 | 58 |
proof - |
59 |
assume "x = y" |
|
23373 | 60 |
then show "y = x" by (rule subst) (rule refl) |
12360 | 61 |
qed |
62 |
||
63 |
lemma [trans]: "x = y \<Longrightarrow> P y \<Longrightarrow> P x" |
|
64 |
by (rule subst) (rule sym) |
|
65 |
||
66 |
lemma [trans]: "P x \<Longrightarrow> x = y \<Longrightarrow> P y" |
|
67 |
by (rule subst) |
|
68 |
||
69 |
theorem trans [trans]: "x = y \<Longrightarrow> y = z \<Longrightarrow> x = z" |
|
70 |
by (rule subst) |
|
71 |
||
72 |
theorem iff1 [elim]: "A = B \<Longrightarrow> A \<Longrightarrow> B" |
|
73 |
by (rule subst) |
|
74 |
||
75 |
theorem iff2 [elim]: "A = B \<Longrightarrow> B \<Longrightarrow> A" |
|
76 |
by (rule subst) (rule sym) |
|
77 |
||
78 |
||
79 |
subsubsection {* Derived connectives *} |
|
80 |
||
19736 | 81 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
82 |
false :: o ("\<bottom>") where |
12360 | 83 |
"\<bottom> \<equiv> \<forall>A. A" |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
84 |
|
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
85 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
86 |
true :: o ("\<top>") where |
12360 | 87 |
"\<top> \<equiv> \<bottom> \<longrightarrow> \<bottom>" |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
88 |
|
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
89 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
90 |
not :: "o \<Rightarrow> o" ("\<not> _" [40] 40) where |
12360 | 91 |
"not \<equiv> \<lambda>A. A \<longrightarrow> \<bottom>" |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
92 |
|
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
93 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
94 |
conj :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<and>" 35) where |
12360 | 95 |
"conj \<equiv> \<lambda>A B. \<forall>C. (A \<longrightarrow> B \<longrightarrow> C) \<longrightarrow> C" |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
96 |
|
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
97 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
98 |
disj :: "o \<Rightarrow> o \<Rightarrow> o" (infixr "\<or>" 30) where |
12360 | 99 |
"disj \<equiv> \<lambda>A B. \<forall>C. (A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C" |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
100 |
|
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
101 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
102 |
Ex :: "('a \<Rightarrow> o) \<Rightarrow> o" (binder "\<exists>" 10) where |
23822 | 103 |
"\<exists>x. P x \<equiv> \<forall>C. (\<forall>x. P x \<longrightarrow> C) \<longrightarrow> C" |
12360 | 104 |
|
19380 | 105 |
abbreviation |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20523
diff
changeset
|
106 |
not_equal :: "'a \<Rightarrow> 'a \<Rightarrow> o" (infixl "\<noteq>" 50) where |
19380 | 107 |
"x \<noteq> y \<equiv> \<not> (x = y)" |
12360 | 108 |
|
109 |
theorem falseE [elim]: "\<bottom> \<Longrightarrow> A" |
|
110 |
proof (unfold false_def) |
|
111 |
assume "\<forall>A. A" |
|
23373 | 112 |
then show A .. |
12360 | 113 |
qed |
114 |
||
115 |
theorem trueI [intro]: \<top> |
|
116 |
proof (unfold true_def) |
|
117 |
show "\<bottom> \<longrightarrow> \<bottom>" .. |
|
118 |
qed |
|
119 |
||
120 |
theorem notI [intro]: "(A \<Longrightarrow> \<bottom>) \<Longrightarrow> \<not> A" |
|
121 |
proof (unfold not_def) |
|
122 |
assume "A \<Longrightarrow> \<bottom>" |
|
23373 | 123 |
then show "A \<longrightarrow> \<bottom>" .. |
12360 | 124 |
qed |
125 |
||
126 |
theorem notE [elim]: "\<not> A \<Longrightarrow> A \<Longrightarrow> B" |
|
127 |
proof (unfold not_def) |
|
128 |
assume "A \<longrightarrow> \<bottom>" |
|
129 |
also assume A |
|
130 |
finally have \<bottom> .. |
|
23373 | 131 |
then show B .. |
12360 | 132 |
qed |
133 |
||
134 |
lemma notE': "A \<Longrightarrow> \<not> A \<Longrightarrow> B" |
|
135 |
by (rule notE) |
|
136 |
||
137 |
lemmas contradiction = notE notE' -- {* proof by contradiction in any order *} |
|
138 |
||
139 |
theorem conjI [intro]: "A \<Longrightarrow> B \<Longrightarrow> A \<and> B" |
|
140 |
proof (unfold conj_def) |
|
141 |
assume A and B |
|
142 |
show "\<forall>C. (A \<longrightarrow> B \<longrightarrow> C) \<longrightarrow> C" |
|
143 |
proof |
|
144 |
fix C show "(A \<longrightarrow> B \<longrightarrow> C) \<longrightarrow> C" |
|
145 |
proof |
|
146 |
assume "A \<longrightarrow> B \<longrightarrow> C" |
|
23373 | 147 |
also note `A` |
148 |
also note `B` |
|
12360 | 149 |
finally show C . |
150 |
qed |
|
151 |
qed |
|
152 |
qed |
|
153 |
||
154 |
theorem conjE [elim]: "A \<and> B \<Longrightarrow> (A \<Longrightarrow> B \<Longrightarrow> C) \<Longrightarrow> C" |
|
155 |
proof (unfold conj_def) |
|
156 |
assume c: "\<forall>C. (A \<longrightarrow> B \<longrightarrow> C) \<longrightarrow> C" |
|
157 |
assume "A \<Longrightarrow> B \<Longrightarrow> C" |
|
158 |
moreover { |
|
159 |
from c have "(A \<longrightarrow> B \<longrightarrow> A) \<longrightarrow> A" .. |
|
160 |
also have "A \<longrightarrow> B \<longrightarrow> A" |
|
161 |
proof |
|
162 |
assume A |
|
23373 | 163 |
then show "B \<longrightarrow> A" .. |
12360 | 164 |
qed |
165 |
finally have A . |
|
166 |
} moreover { |
|
167 |
from c have "(A \<longrightarrow> B \<longrightarrow> B) \<longrightarrow> B" .. |
|
168 |
also have "A \<longrightarrow> B \<longrightarrow> B" |
|
169 |
proof |
|
170 |
show "B \<longrightarrow> B" .. |
|
171 |
qed |
|
172 |
finally have B . |
|
173 |
} ultimately show C . |
|
174 |
qed |
|
175 |
||
176 |
theorem disjI1 [intro]: "A \<Longrightarrow> A \<or> B" |
|
177 |
proof (unfold disj_def) |
|
178 |
assume A |
|
179 |
show "\<forall>C. (A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C" |
|
180 |
proof |
|
181 |
fix C show "(A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C" |
|
182 |
proof |
|
183 |
assume "A \<longrightarrow> C" |
|
23373 | 184 |
also note `A` |
12360 | 185 |
finally have C . |
23373 | 186 |
then show "(B \<longrightarrow> C) \<longrightarrow> C" .. |
12360 | 187 |
qed |
188 |
qed |
|
189 |
qed |
|
190 |
||
191 |
theorem disjI2 [intro]: "B \<Longrightarrow> A \<or> B" |
|
192 |
proof (unfold disj_def) |
|
193 |
assume B |
|
194 |
show "\<forall>C. (A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C" |
|
195 |
proof |
|
196 |
fix C show "(A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C" |
|
197 |
proof |
|
198 |
show "(B \<longrightarrow> C) \<longrightarrow> C" |
|
199 |
proof |
|
200 |
assume "B \<longrightarrow> C" |
|
23373 | 201 |
also note `B` |
12360 | 202 |
finally show C . |
203 |
qed |
|
204 |
qed |
|
205 |
qed |
|
206 |
qed |
|
207 |
||
208 |
theorem disjE [elim]: "A \<or> B \<Longrightarrow> (A \<Longrightarrow> C) \<Longrightarrow> (B \<Longrightarrow> C) \<Longrightarrow> C" |
|
209 |
proof (unfold disj_def) |
|
210 |
assume c: "\<forall>C. (A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C" |
|
211 |
assume r1: "A \<Longrightarrow> C" and r2: "B \<Longrightarrow> C" |
|
212 |
from c have "(A \<longrightarrow> C) \<longrightarrow> (B \<longrightarrow> C) \<longrightarrow> C" .. |
|
213 |
also have "A \<longrightarrow> C" |
|
214 |
proof |
|
23373 | 215 |
assume A then show C by (rule r1) |
12360 | 216 |
qed |
217 |
also have "B \<longrightarrow> C" |
|
218 |
proof |
|
23373 | 219 |
assume B then show C by (rule r2) |
12360 | 220 |
qed |
221 |
finally show C . |
|
222 |
qed |
|
223 |
||
224 |
theorem exI [intro]: "P a \<Longrightarrow> \<exists>x. P x" |
|
225 |
proof (unfold Ex_def) |
|
226 |
assume "P a" |
|
227 |
show "\<forall>C. (\<forall>x. P x \<longrightarrow> C) \<longrightarrow> C" |
|
228 |
proof |
|
229 |
fix C show "(\<forall>x. P x \<longrightarrow> C) \<longrightarrow> C" |
|
230 |
proof |
|
231 |
assume "\<forall>x. P x \<longrightarrow> C" |
|
23373 | 232 |
then have "P a \<longrightarrow> C" .. |
233 |
also note `P a` |
|
12360 | 234 |
finally show C . |
235 |
qed |
|
236 |
qed |
|
237 |
qed |
|
238 |
||
239 |
theorem exE [elim]: "\<exists>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> C) \<Longrightarrow> C" |
|
240 |
proof (unfold Ex_def) |
|
241 |
assume c: "\<forall>C. (\<forall>x. P x \<longrightarrow> C) \<longrightarrow> C" |
|
242 |
assume r: "\<And>x. P x \<Longrightarrow> C" |
|
243 |
from c have "(\<forall>x. P x \<longrightarrow> C) \<longrightarrow> C" .. |
|
244 |
also have "\<forall>x. P x \<longrightarrow> C" |
|
245 |
proof |
|
246 |
fix x show "P x \<longrightarrow> C" |
|
247 |
proof |
|
248 |
assume "P x" |
|
23373 | 249 |
then show C by (rule r) |
12360 | 250 |
qed |
251 |
qed |
|
252 |
finally show C . |
|
253 |
qed |
|
254 |
||
255 |
||
256 |
subsection {* Classical logic *} |
|
257 |
||
258 |
locale classical = |
|
259 |
assumes classical: "(\<not> A \<Longrightarrow> A) \<Longrightarrow> A" |
|
260 |
||
261 |
theorem (in classical) |
|
262 |
Peirce's_Law: "((A \<longrightarrow> B) \<longrightarrow> A) \<longrightarrow> A" |
|
263 |
proof |
|
264 |
assume a: "(A \<longrightarrow> B) \<longrightarrow> A" |
|
265 |
show A |
|
266 |
proof (rule classical) |
|
267 |
assume "\<not> A" |
|
268 |
have "A \<longrightarrow> B" |
|
269 |
proof |
|
270 |
assume A |
|
23373 | 271 |
with `\<not> A` show B by (rule contradiction) |
12360 | 272 |
qed |
273 |
with a show A .. |
|
274 |
qed |
|
275 |
qed |
|
276 |
||
277 |
theorem (in classical) |
|
278 |
double_negation: "\<not> \<not> A \<Longrightarrow> A" |
|
279 |
proof - |
|
280 |
assume "\<not> \<not> A" |
|
281 |
show A |
|
282 |
proof (rule classical) |
|
283 |
assume "\<not> A" |
|
23373 | 284 |
with `\<not> \<not> A` show ?thesis by (rule contradiction) |
12360 | 285 |
qed |
286 |
qed |
|
287 |
||
288 |
theorem (in classical) |
|
289 |
tertium_non_datur: "A \<or> \<not> A" |
|
290 |
proof (rule double_negation) |
|
291 |
show "\<not> \<not> (A \<or> \<not> A)" |
|
292 |
proof |
|
293 |
assume "\<not> (A \<or> \<not> A)" |
|
294 |
have "\<not> A" |
|
295 |
proof |
|
23373 | 296 |
assume A then have "A \<or> \<not> A" .. |
297 |
with `\<not> (A \<or> \<not> A)` show \<bottom> by (rule contradiction) |
|
12360 | 298 |
qed |
23373 | 299 |
then have "A \<or> \<not> A" .. |
300 |
with `\<not> (A \<or> \<not> A)` show \<bottom> by (rule contradiction) |
|
12360 | 301 |
qed |
302 |
qed |
|
303 |
||
304 |
theorem (in classical) |
|
305 |
classical_cases: "(A \<Longrightarrow> C) \<Longrightarrow> (\<not> A \<Longrightarrow> C) \<Longrightarrow> C" |
|
306 |
proof - |
|
307 |
assume r1: "A \<Longrightarrow> C" and r2: "\<not> A \<Longrightarrow> C" |
|
308 |
from tertium_non_datur show C |
|
309 |
proof |
|
310 |
assume A |
|
23373 | 311 |
then show ?thesis by (rule r1) |
12360 | 312 |
next |
313 |
assume "\<not> A" |
|
23373 | 314 |
then show ?thesis by (rule r2) |
12360 | 315 |
qed |
316 |
qed |
|
317 |
||
12573 | 318 |
lemma (in classical) "(\<not> A \<Longrightarrow> A) \<Longrightarrow> A" (* FIXME *) |
319 |
proof - |
|
320 |
assume r: "\<not> A \<Longrightarrow> A" |
|
321 |
show A |
|
322 |
proof (rule classical_cases) |
|
23373 | 323 |
assume A then show A . |
12573 | 324 |
next |
23373 | 325 |
assume "\<not> A" then show A by (rule r) |
12573 | 326 |
qed |
327 |
qed |
|
328 |
||
12360 | 329 |
end |