author | wenzelm |
Sat, 20 Jan 2007 14:09:27 +0100 | |
changeset 22139 | 539a63b98f76 |
parent 21539 | c5cf9243ad62 |
child 23154 | 5126551e378b |
permissions | -rw-r--r-- |
9487 | 1 |
(* Title: FOL/FOL.thy |
2 |
ID: $Id$ |
|
3 |
Author: Lawrence C Paulson and Markus Wenzel |
|
11678 | 4 |
*) |
9487 | 5 |
|
11678 | 6 |
header {* Classical first-order logic *} |
4093 | 7 |
|
18456 | 8 |
theory FOL |
15481 | 9 |
imports IFOL |
21539 | 10 |
uses ("cladata.ML") ("blastdata.ML") ("simpdata.ML") |
18456 | 11 |
begin |
9487 | 12 |
|
13 |
||
14 |
subsection {* The classical axiom *} |
|
4093 | 15 |
|
7355
4c43090659ca
proper bootstrap of IFOL/FOL theories and packages;
wenzelm
parents:
5887
diff
changeset
|
16 |
axioms |
4c43090659ca
proper bootstrap of IFOL/FOL theories and packages;
wenzelm
parents:
5887
diff
changeset
|
17 |
classical: "(~P ==> P) ==> P" |
4093 | 18 |
|
9487 | 19 |
|
11678 | 20 |
subsection {* Lemmas and proof tools *} |
9487 | 21 |
|
21539 | 22 |
lemma ccontr: "(\<not> P \<Longrightarrow> False) \<Longrightarrow> P" |
23 |
by (erule FalseE [THEN classical]) |
|
24 |
||
25 |
(*** Classical introduction rules for | and EX ***) |
|
26 |
||
27 |
lemma disjCI: "(~Q ==> P) ==> P|Q" |
|
28 |
apply (rule classical) |
|
29 |
apply (assumption | erule meta_mp | rule disjI1 notI)+ |
|
30 |
apply (erule notE disjI2)+ |
|
31 |
done |
|
32 |
||
33 |
(*introduction rule involving only EX*) |
|
34 |
lemma ex_classical: |
|
35 |
assumes r: "~(EX x. P(x)) ==> P(a)" |
|
36 |
shows "EX x. P(x)" |
|
37 |
apply (rule classical) |
|
38 |
apply (rule exI, erule r) |
|
39 |
done |
|
40 |
||
41 |
(*version of above, simplifying ~EX to ALL~ *) |
|
42 |
lemma exCI: |
|
43 |
assumes r: "ALL x. ~P(x) ==> P(a)" |
|
44 |
shows "EX x. P(x)" |
|
45 |
apply (rule ex_classical) |
|
46 |
apply (rule notI [THEN allI, THEN r]) |
|
47 |
apply (erule notE) |
|
48 |
apply (erule exI) |
|
49 |
done |
|
50 |
||
51 |
lemma excluded_middle: "~P | P" |
|
52 |
apply (rule disjCI) |
|
53 |
apply assumption |
|
54 |
done |
|
55 |
||
56 |
(*For disjunctive case analysis*) |
|
57 |
ML {* |
|
22139 | 58 |
fun excluded_middle_tac sP = |
59 |
res_inst_tac [("Q",sP)] (@{thm excluded_middle} RS @{thm disjE}) |
|
21539 | 60 |
*} |
61 |
||
62 |
lemma case_split_thm: |
|
63 |
assumes r1: "P ==> Q" |
|
64 |
and r2: "~P ==> Q" |
|
65 |
shows Q |
|
66 |
apply (rule excluded_middle [THEN disjE]) |
|
67 |
apply (erule r2) |
|
68 |
apply (erule r1) |
|
69 |
done |
|
70 |
||
71 |
lemmas case_split = case_split_thm [case_names True False, cases type: o] |
|
72 |
||
73 |
(*HOL's more natural case analysis tactic*) |
|
74 |
ML {* |
|
22139 | 75 |
fun case_tac a = res_inst_tac [("P",a)] @{thm case_split_thm} |
21539 | 76 |
*} |
77 |
||
78 |
||
79 |
(*** Special elimination rules *) |
|
80 |
||
81 |
||
82 |
(*Classical implies (-->) elimination. *) |
|
83 |
lemma impCE: |
|
84 |
assumes major: "P-->Q" |
|
85 |
and r1: "~P ==> R" |
|
86 |
and r2: "Q ==> R" |
|
87 |
shows R |
|
88 |
apply (rule excluded_middle [THEN disjE]) |
|
89 |
apply (erule r1) |
|
90 |
apply (rule r2) |
|
91 |
apply (erule major [THEN mp]) |
|
92 |
done |
|
93 |
||
94 |
(*This version of --> elimination works on Q before P. It works best for |
|
95 |
those cases in which P holds "almost everywhere". Can't install as |
|
96 |
default: would break old proofs.*) |
|
97 |
lemma impCE': |
|
98 |
assumes major: "P-->Q" |
|
99 |
and r1: "Q ==> R" |
|
100 |
and r2: "~P ==> R" |
|
101 |
shows R |
|
102 |
apply (rule excluded_middle [THEN disjE]) |
|
103 |
apply (erule r2) |
|
104 |
apply (rule r1) |
|
105 |
apply (erule major [THEN mp]) |
|
106 |
done |
|
107 |
||
108 |
(*Double negation law*) |
|
109 |
lemma notnotD: "~~P ==> P" |
|
110 |
apply (rule classical) |
|
111 |
apply (erule notE) |
|
112 |
apply assumption |
|
113 |
done |
|
114 |
||
115 |
lemma contrapos2: "[| Q; ~ P ==> ~ Q |] ==> P" |
|
116 |
apply (rule classical) |
|
117 |
apply (drule (1) meta_mp) |
|
118 |
apply (erule (1) notE) |
|
119 |
done |
|
120 |
||
121 |
(*** Tactics for implication and contradiction ***) |
|
122 |
||
123 |
(*Classical <-> elimination. Proof substitutes P=Q in |
|
124 |
~P ==> ~Q and P ==> Q *) |
|
125 |
lemma iffCE: |
|
126 |
assumes major: "P<->Q" |
|
127 |
and r1: "[| P; Q |] ==> R" |
|
128 |
and r2: "[| ~P; ~Q |] ==> R" |
|
129 |
shows R |
|
130 |
apply (rule major [unfolded iff_def, THEN conjE]) |
|
131 |
apply (elim impCE) |
|
132 |
apply (erule (1) r2) |
|
133 |
apply (erule (1) notE)+ |
|
134 |
apply (erule (1) r1) |
|
135 |
done |
|
136 |
||
137 |
||
138 |
(*Better for fast_tac: needs no quantifier duplication!*) |
|
139 |
lemma alt_ex1E: |
|
140 |
assumes major: "EX! x. P(x)" |
|
141 |
and r: "!!x. [| P(x); ALL y y'. P(y) & P(y') --> y=y' |] ==> R" |
|
142 |
shows R |
|
143 |
using major |
|
144 |
proof (rule ex1E) |
|
145 |
fix x |
|
146 |
assume * : "\<forall>y. P(y) \<longrightarrow> y = x" |
|
147 |
assume "P(x)" |
|
148 |
then show R |
|
149 |
proof (rule r) |
|
150 |
{ fix y y' |
|
151 |
assume "P(y)" and "P(y')" |
|
152 |
with * have "x = y" and "x = y'" by - (tactic "IntPr.fast_tac 1")+ |
|
153 |
then have "y = y'" by (rule subst) |
|
154 |
} note r' = this |
|
155 |
show "\<forall>y y'. P(y) \<and> P(y') \<longrightarrow> y = y'" by (intro strip, elim conjE) (rule r') |
|
156 |
qed |
|
157 |
qed |
|
9525 | 158 |
|
10383 | 159 |
use "cladata.ML" |
160 |
setup Cla.setup |
|
14156 | 161 |
setup cla_setup |
162 |
setup case_setup |
|
10383 | 163 |
|
9487 | 164 |
use "blastdata.ML" |
165 |
setup Blast.setup |
|
13550 | 166 |
|
167 |
||
168 |
lemma ex1_functional: "[| EX! z. P(a,z); P(a,b); P(a,c) |] ==> b = c" |
|
21539 | 169 |
by blast |
20223 | 170 |
|
171 |
(* Elimination of True from asumptions: *) |
|
172 |
lemma True_implies_equals: "(True ==> PROP P) == PROP P" |
|
173 |
proof |
|
174 |
assume "True \<Longrightarrow> PROP P" |
|
175 |
from this and TrueI show "PROP P" . |
|
176 |
next |
|
177 |
assume "PROP P" |
|
178 |
then show "PROP P" . |
|
179 |
qed |
|
9487 | 180 |
|
21539 | 181 |
lemma uncurry: "P --> Q --> R ==> P & Q --> R" |
182 |
by blast |
|
183 |
||
184 |
lemma iff_allI: "(!!x. P(x) <-> Q(x)) ==> (ALL x. P(x)) <-> (ALL x. Q(x))" |
|
185 |
by blast |
|
186 |
||
187 |
lemma iff_exI: "(!!x. P(x) <-> Q(x)) ==> (EX x. P(x)) <-> (EX x. Q(x))" |
|
188 |
by blast |
|
189 |
||
190 |
lemma all_comm: "(ALL x y. P(x,y)) <-> (ALL y x. P(x,y))" by blast |
|
191 |
||
192 |
lemma ex_comm: "(EX x y. P(x,y)) <-> (EX y x. P(x,y))" by blast |
|
193 |
||
9487 | 194 |
use "simpdata.ML" |
195 |
setup simpsetup |
|
196 |
setup "Simplifier.method_setup Splitter.split_modifiers" |
|
197 |
setup Splitter.setup |
|
198 |
setup Clasimp.setup |
|
18591 | 199 |
setup EqSubst.setup |
15481 | 200 |
|
201 |
||
14085 | 202 |
subsection {* Other simple lemmas *} |
203 |
||
204 |
lemma [simp]: "((P-->R) <-> (Q-->R)) <-> ((P<->Q) | R)" |
|
205 |
by blast |
|
206 |
||
207 |
lemma [simp]: "((P-->Q) <-> (P-->R)) <-> (P --> (Q<->R))" |
|
208 |
by blast |
|
209 |
||
210 |
lemma not_disj_iff_imp: "~P | Q <-> (P-->Q)" |
|
211 |
by blast |
|
212 |
||
213 |
(** Monotonicity of implications **) |
|
214 |
||
215 |
lemma conj_mono: "[| P1-->Q1; P2-->Q2 |] ==> (P1&P2) --> (Q1&Q2)" |
|
216 |
by fast (*or (IntPr.fast_tac 1)*) |
|
217 |
||
218 |
lemma disj_mono: "[| P1-->Q1; P2-->Q2 |] ==> (P1|P2) --> (Q1|Q2)" |
|
219 |
by fast (*or (IntPr.fast_tac 1)*) |
|
220 |
||
221 |
lemma imp_mono: "[| Q1-->P1; P2-->Q2 |] ==> (P1-->P2)-->(Q1-->Q2)" |
|
222 |
by fast (*or (IntPr.fast_tac 1)*) |
|
223 |
||
224 |
lemma imp_refl: "P-->P" |
|
225 |
by (rule impI, assumption) |
|
226 |
||
227 |
(*The quantifier monotonicity rules are also intuitionistically valid*) |
|
228 |
lemma ex_mono: "(!!x. P(x) --> Q(x)) ==> (EX x. P(x)) --> (EX x. Q(x))" |
|
229 |
by blast |
|
230 |
||
231 |
lemma all_mono: "(!!x. P(x) --> Q(x)) ==> (ALL x. P(x)) --> (ALL x. Q(x))" |
|
232 |
by blast |
|
233 |
||
11678 | 234 |
|
235 |
subsection {* Proof by cases and induction *} |
|
236 |
||
237 |
text {* Proper handling of non-atomic rule statements. *} |
|
238 |
||
239 |
constdefs |
|
18456 | 240 |
induct_forall where "induct_forall(P) == \<forall>x. P(x)" |
241 |
induct_implies where "induct_implies(A, B) == A \<longrightarrow> B" |
|
242 |
induct_equal where "induct_equal(x, y) == x = y" |
|
243 |
induct_conj where "induct_conj(A, B) == A \<and> B" |
|
11678 | 244 |
|
245 |
lemma induct_forall_eq: "(!!x. P(x)) == Trueprop(induct_forall(\<lambda>x. P(x)))" |
|
18816 | 246 |
unfolding atomize_all induct_forall_def . |
11678 | 247 |
|
248 |
lemma induct_implies_eq: "(A ==> B) == Trueprop(induct_implies(A, B))" |
|
18816 | 249 |
unfolding atomize_imp induct_implies_def . |
11678 | 250 |
|
251 |
lemma induct_equal_eq: "(x == y) == Trueprop(induct_equal(x, y))" |
|
18816 | 252 |
unfolding atomize_eq induct_equal_def . |
11678 | 253 |
|
18456 | 254 |
lemma induct_conj_eq: |
255 |
includes meta_conjunction_syntax |
|
256 |
shows "(A && B) == Trueprop(induct_conj(A, B))" |
|
18816 | 257 |
unfolding atomize_conj induct_conj_def . |
11988 | 258 |
|
18456 | 259 |
lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq |
260 |
lemmas induct_rulify [symmetric, standard] = induct_atomize |
|
261 |
lemmas induct_rulify_fallback = |
|
262 |
induct_forall_def induct_implies_def induct_equal_def induct_conj_def |
|
11678 | 263 |
|
18456 | 264 |
hide const induct_forall induct_implies induct_equal induct_conj |
11678 | 265 |
|
266 |
||
267 |
text {* Method setup. *} |
|
268 |
||
269 |
ML {* |
|
270 |
structure InductMethod = InductMethodFun |
|
271 |
(struct |
|
22139 | 272 |
val cases_default = @{thm case_split} |
273 |
val atomize = @{thms induct_atomize} |
|
274 |
val rulify = @{thms induct_rulify} |
|
275 |
val rulify_fallback = @{thms induct_rulify_fallback} |
|
11678 | 276 |
end); |
277 |
*} |
|
278 |
||
279 |
setup InductMethod.setup |
|
280 |
||
4854 | 281 |
end |