author | wenzelm |
Tue, 16 Jan 2018 09:30:00 +0100 | |
changeset 67443 | 3abf6a722518 |
parent 62020 | 5d208fd2507d |
child 69590 | e65314985426 |
permissions | -rw-r--r-- |
31974 | 1 |
(* Title: FOL/ex/Intuitionistic.thy |
14239 | 2 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
3 |
Copyright 1991 University of Cambridge |
|
4 |
*) |
|
5 |
||
60770 | 6 |
section \<open>Intuitionistic First-Order Logic\<close> |
14239 | 7 |
|
31974 | 8 |
theory Intuitionistic |
9 |
imports IFOL |
|
10 |
begin |
|
14239 | 11 |
|
12 |
(* |
|
13 |
Single-step ML commands: |
|
14 |
by (IntPr.step_tac 1) |
|
15 |
by (biresolve_tac safe_brls 1); |
|
16 |
by (biresolve_tac haz_brls 1); |
|
17 |
by (assume_tac 1); |
|
18 |
by (IntPr.safe_tac 1); |
|
19 |
by (IntPr.mp_tac 1); |
|
51798 | 20 |
by (IntPr.fast_tac @{context} 1); |
14239 | 21 |
*) |
22 |
||
23 |
||
60770 | 24 |
text\<open>Metatheorem (for \emph{propositional} formulae): |
14239 | 25 |
$P$ is classically provable iff $\neg\neg P$ is intuitionistically provable. |
26 |
Therefore $\neg P$ is classically provable iff it is intuitionistically |
|
27 |
provable. |
|
28 |
||
29 |
Proof: Let $Q$ be the conjuction of the propositions $A\vee\neg A$, one for |
|
30 |
each atom $A$ in $P$. Now $\neg\neg Q$ is intuitionistically provable because |
|
31 |
$\neg\neg(A\vee\neg A)$ is and because double-negation distributes over |
|
32 |
conjunction. If $P$ is provable classically, then clearly $Q\rightarrow P$ is |
|
33 |
provable intuitionistically, so $\neg\neg(Q\rightarrow P)$ is also provable |
|
34 |
intuitionistically. The latter is intuitionistically equivalent to $\neg\neg |
|
35 |
Q\rightarrow\neg\neg P$, hence to $\neg\neg P$, since $\neg\neg Q$ is |
|
36 |
intuitionistically provable. Finally, if $P$ is a negation then $\neg\neg P$ |
|
60770 | 37 |
is intuitionstically equivalent to $P$. [Andy Pitts]\<close> |
14239 | 38 |
|
61489 | 39 |
lemma "\<not> \<not> (P \<and> Q) \<longleftrightarrow> \<not> \<not> P \<and> \<not> \<not> Q" |
40 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 41 |
|
61489 | 42 |
lemma "\<not> \<not> ((\<not> P \<longrightarrow> Q) \<longrightarrow> (\<not> P \<longrightarrow> \<not> Q) \<longrightarrow> P)" |
43 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 44 |
|
61489 | 45 |
text \<open>Double-negation does NOT distribute over disjunction.\<close> |
14239 | 46 |
|
61489 | 47 |
lemma "\<not> \<not> (P \<longrightarrow> Q) \<longleftrightarrow> (\<not> \<not> P \<longrightarrow> \<not> \<not> Q)" |
48 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 49 |
|
61489 | 50 |
lemma "\<not> \<not> \<not> P \<longleftrightarrow> \<not> P" |
51 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 52 |
|
61489 | 53 |
lemma "\<not> \<not> ((P \<longrightarrow> Q \<or> R) \<longrightarrow> (P \<longrightarrow> Q) \<or> (P \<longrightarrow> R))" |
54 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 55 |
|
61489 | 56 |
lemma "(P \<longleftrightarrow> Q) \<longleftrightarrow> (Q \<longleftrightarrow> P)" |
57 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 58 |
|
61489 | 59 |
lemma "((P \<longrightarrow> (Q \<or> (Q \<longrightarrow> R))) \<longrightarrow> R) \<longrightarrow> R" |
60 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 61 |
|
61489 | 62 |
lemma |
63 |
"(((G \<longrightarrow> A) \<longrightarrow> J) \<longrightarrow> D \<longrightarrow> E) \<longrightarrow> (((H \<longrightarrow> B) \<longrightarrow> I) \<longrightarrow> C \<longrightarrow> J) |
|
64 |
\<longrightarrow> (A \<longrightarrow> H) \<longrightarrow> F \<longrightarrow> G \<longrightarrow> (((C \<longrightarrow> B) \<longrightarrow> I) \<longrightarrow> D) \<longrightarrow> (A \<longrightarrow> C) |
|
65 |
\<longrightarrow> (((F \<longrightarrow> A) \<longrightarrow> B) \<longrightarrow> I) \<longrightarrow> E" |
|
66 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 67 |
|
68 |
||
61489 | 69 |
subsection \<open>Lemmas for the propositional double-negation translation\<close> |
14239 | 70 |
|
61489 | 71 |
lemma "P \<longrightarrow> \<not> \<not> P" |
72 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 73 |
|
61489 | 74 |
lemma "\<not> \<not> (\<not> \<not> P \<longrightarrow> P)" |
75 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 76 |
|
61489 | 77 |
lemma "\<not> \<not> P \<and> \<not> \<not> (P \<longrightarrow> Q) \<longrightarrow> \<not> \<not> Q" |
78 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 79 |
|
80 |
||
61489 | 81 |
text \<open>The following are classically but not constructively valid. |
82 |
The attempt to prove them terminates quickly!\<close> |
|
83 |
lemma "((P \<longrightarrow> Q) \<longrightarrow> P) \<longrightarrow> P" |
|
84 |
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)? |
|
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
62020
diff
changeset
|
85 |
apply (rule asm_rl) \<comment> \<open>Checks that subgoals remain: proof failed.\<close> |
14239 | 86 |
oops |
87 |
||
61489 | 88 |
lemma "(P \<and> Q \<longrightarrow> R) \<longrightarrow> (P \<longrightarrow> R) \<or> (Q \<longrightarrow> R)" |
89 |
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)? |
|
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
62020
diff
changeset
|
90 |
apply (rule asm_rl) \<comment> \<open>Checks that subgoals remain: proof failed.\<close> |
14239 | 91 |
oops |
92 |
||
93 |
||
61489 | 94 |
subsection \<open>de Bruijn formulae\<close> |
14239 | 95 |
|
61489 | 96 |
text \<open>de Bruijn formula with three predicates\<close> |
97 |
lemma |
|
98 |
"((P \<longleftrightarrow> Q) \<longrightarrow> P \<and> Q \<and> R) \<and> |
|
99 |
((Q \<longleftrightarrow> R) \<longrightarrow> P \<and> Q \<and> R) \<and> |
|
100 |
((R \<longleftrightarrow> P) \<longrightarrow> P \<and> Q \<and> R) \<longrightarrow> P \<and> Q \<and> R" |
|
101 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 102 |
|
103 |
||
61489 | 104 |
text \<open>de Bruijn formula with five predicates\<close> |
105 |
lemma |
|
106 |
"((P \<longleftrightarrow> Q) \<longrightarrow> P \<and> Q \<and> R \<and> S \<and> T) \<and> |
|
107 |
((Q \<longleftrightarrow> R) \<longrightarrow> P \<and> Q \<and> R \<and> S \<and> T) \<and> |
|
108 |
((R \<longleftrightarrow> S) \<longrightarrow> P \<and> Q \<and> R \<and> S \<and> T) \<and> |
|
109 |
((S \<longleftrightarrow> T) \<longrightarrow> P \<and> Q \<and> R \<and> S \<and> T) \<and> |
|
110 |
((T \<longleftrightarrow> P) \<longrightarrow> P \<and> Q \<and> R \<and> S \<and> T) \<longrightarrow> P \<and> Q \<and> R \<and> S \<and> T" |
|
111 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 112 |
|
113 |
||
61489 | 114 |
text \<open> |
115 |
Problems from of Sahlin, Franzen and Haridi, |
|
116 |
An Intuitionistic Predicate Logic Theorem Prover. |
|
117 |
J. Logic and Comp. 2 (5), October 1992, 619-656. |
|
118 |
\<close> |
|
14239 | 119 |
|
60770 | 120 |
text\<open>Problem 1.1\<close> |
61489 | 121 |
lemma |
122 |
"(\<forall>x. \<exists>y. \<forall>z. p(x) \<and> q(y) \<and> r(z)) \<longleftrightarrow> |
|
123 |
(\<forall>z. \<exists>y. \<forall>x. p(x) \<and> q(y) \<and> r(z))" |
|
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
62020
diff
changeset
|
124 |
by (tactic \<open>IntPr.best_dup_tac @{context} 1\<close>) \<comment> \<open>SLOW\<close> |
14239 | 125 |
|
60770 | 126 |
text\<open>Problem 3.1\<close> |
61489 | 127 |
lemma "\<not> (\<exists>x. \<forall>y. mem(y,x) \<longleftrightarrow> \<not> mem(x,x))" |
128 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 129 |
|
60770 | 130 |
text\<open>Problem 4.1: hopeless!\<close> |
61489 | 131 |
lemma |
132 |
"(\<forall>x. p(x) \<longrightarrow> p(h(x)) \<or> p(g(x))) \<and> (\<exists>x. p(x)) \<and> (\<forall>x. \<not> p(h(x))) |
|
133 |
\<longrightarrow> (\<exists>x. p(g(g(g(g(g(x)))))))" |
|
134 |
oops |
|
14239 | 135 |
|
136 |
||
61489 | 137 |
subsection \<open>Intuitionistic FOL: propositional problems based on Pelletier.\<close> |
14239 | 138 |
|
62020 | 139 |
text\<open>\<open>\<not>\<not>\<close>1\<close> |
61489 | 140 |
lemma "\<not> \<not> ((P \<longrightarrow> Q) \<longleftrightarrow> (\<not> Q \<longrightarrow> \<not> P))" |
141 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 142 |
|
62020 | 143 |
text\<open>\<open>\<not>\<not>\<close>2\<close> |
61489 | 144 |
lemma "\<not> \<not> (\<not> \<not> P \<longleftrightarrow> P)" |
145 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 146 |
|
60770 | 147 |
text\<open>3\<close> |
61489 | 148 |
lemma "\<not> (P \<longrightarrow> Q) \<longrightarrow> (Q \<longrightarrow> P)" |
149 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 150 |
|
62020 | 151 |
text\<open>\<open>\<not>\<not>\<close>4\<close> |
61489 | 152 |
lemma "\<not> \<not> ((\<not> P \<longrightarrow> Q) \<longleftrightarrow> (\<not> Q \<longrightarrow> P))" |
153 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 154 |
|
62020 | 155 |
text\<open>\<open>\<not>\<not>\<close>5\<close> |
61490 | 156 |
lemma "\<not> \<not> ((P \<or> Q \<longrightarrow> P \<or> R) \<longrightarrow> P \<or> (Q \<longrightarrow> R))" |
61489 | 157 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
14239 | 158 |
|
62020 | 159 |
text\<open>\<open>\<not>\<not>\<close>6\<close> |
61489 | 160 |
lemma "\<not> \<not> (P \<or> \<not> P)" |
161 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 162 |
|
62020 | 163 |
text\<open>\<open>\<not>\<not>\<close>7\<close> |
61489 | 164 |
lemma "\<not> \<not> (P \<or> \<not> \<not> \<not> P)" |
165 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 166 |
|
62020 | 167 |
text\<open>\<open>\<not>\<not>\<close>8. Peirce's law\<close> |
61489 | 168 |
lemma "\<not> \<not> (((P \<longrightarrow> Q) \<longrightarrow> P) \<longrightarrow> P)" |
169 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 170 |
|
60770 | 171 |
text\<open>9\<close> |
61489 | 172 |
lemma "((P \<or> Q) \<and> (\<not> P \<or> Q) \<and> (P \<or> \<not> Q)) \<longrightarrow> \<not> (\<not> P \<or> \<not> Q)" |
173 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 174 |
|
60770 | 175 |
text\<open>10\<close> |
61489 | 176 |
lemma "(Q \<longrightarrow> R) \<longrightarrow> (R \<longrightarrow> P \<and> Q) \<longrightarrow> (P \<longrightarrow> (Q \<or> R)) \<longrightarrow> (P \<longleftrightarrow> Q)" |
177 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 178 |
|
179 |
||
61489 | 180 |
subsection\<open>11. Proved in each direction (incorrectly, says Pelletier!!)\<close> |
181 |
||
182 |
lemma "P \<longleftrightarrow> P" |
|
183 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
184 |
||
62020 | 185 |
text\<open>\<open>\<not>\<not>\<close>12. Dijkstra's law\<close> |
61489 | 186 |
lemma "\<not> \<not> (((P \<longleftrightarrow> Q) \<longleftrightarrow> R) \<longleftrightarrow> (P \<longleftrightarrow> (Q \<longleftrightarrow> R)))" |
187 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
188 |
||
189 |
lemma "((P \<longleftrightarrow> Q) \<longleftrightarrow> R) \<longrightarrow> \<not> \<not> (P \<longleftrightarrow> (Q \<longleftrightarrow> R))" |
|
190 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
191 |
||
192 |
text\<open>13. Distributive law\<close> |
|
193 |
lemma "P \<or> (Q \<and> R) \<longleftrightarrow> (P \<or> Q) \<and> (P \<or> R)" |
|
194 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
195 |
||
62020 | 196 |
text\<open>\<open>\<not>\<not>\<close>14\<close> |
61489 | 197 |
lemma "\<not> \<not> ((P \<longleftrightarrow> Q) \<longleftrightarrow> ((Q \<or> \<not> P) \<and> (\<not> Q \<or> P)))" |
198 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
199 |
||
62020 | 200 |
text\<open>\<open>\<not>\<not>\<close>15\<close> |
61489 | 201 |
lemma "\<not> \<not> ((P \<longrightarrow> Q) \<longleftrightarrow> (\<not> P \<or> Q))" |
202 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
203 |
||
62020 | 204 |
text\<open>\<open>\<not>\<not>\<close>16\<close> |
61489 | 205 |
lemma "\<not> \<not> ((P \<longrightarrow> Q) \<or> (Q \<longrightarrow> P))" |
206 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
207 |
||
62020 | 208 |
text\<open>\<open>\<not>\<not>\<close>17\<close> |
61489 | 209 |
lemma "\<not> \<not> (((P \<and> (Q \<longrightarrow> R)) \<longrightarrow> S) \<longleftrightarrow> ((\<not> P \<or> Q \<or> S) \<and> (\<not> P \<or> \<not> R \<or> S)))" |
210 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
211 |
||
212 |
text \<open>Dijkstra's ``Golden Rule''\<close> |
|
213 |
lemma "(P \<and> Q) \<longleftrightarrow> P \<longleftrightarrow> Q \<longleftrightarrow> (P \<or> Q)" |
|
214 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
215 |
||
216 |
||
217 |
section \<open>Examples with quantifiers\<close> |
|
218 |
||
219 |
subsection \<open>The converse is classical in the following implications \dots\<close> |
|
220 |
||
221 |
lemma "(\<exists>x. P(x) \<longrightarrow> Q) \<longrightarrow> (\<forall>x. P(x)) \<longrightarrow> Q" |
|
222 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
223 |
||
224 |
lemma "((\<forall>x. P(x)) \<longrightarrow> Q) \<longrightarrow> \<not> (\<forall>x. P(x) \<and> \<not> Q)" |
|
225 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
226 |
||
227 |
lemma "((\<forall>x. \<not> P(x)) \<longrightarrow> Q) \<longrightarrow> \<not> (\<forall>x. \<not> (P(x) \<or> Q))" |
|
228 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
229 |
||
230 |
lemma "(\<forall>x. P(x)) \<or> Q \<longrightarrow> (\<forall>x. P(x) \<or> Q)" |
|
231 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
232 |
||
233 |
lemma "(\<exists>x. P \<longrightarrow> Q(x)) \<longrightarrow> (P \<longrightarrow> (\<exists>x. Q(x)))" |
|
234 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 235 |
|
236 |
||
61489 | 237 |
subsection \<open>The following are not constructively valid!\<close> |
238 |
text \<open>The attempt to prove them terminates quickly!\<close> |
|
14239 | 239 |
|
61489 | 240 |
lemma "((\<forall>x. P(x)) \<longrightarrow> Q) \<longrightarrow> (\<exists>x. P(x) \<longrightarrow> Q)" |
241 |
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)? |
|
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
62020
diff
changeset
|
242 |
apply (rule asm_rl) \<comment> \<open>Checks that subgoals remain: proof failed.\<close> |
61489 | 243 |
oops |
14239 | 244 |
|
61489 | 245 |
lemma "(P \<longrightarrow> (\<exists>x. Q(x))) \<longrightarrow> (\<exists>x. P \<longrightarrow> Q(x))" |
246 |
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)? |
|
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
62020
diff
changeset
|
247 |
apply (rule asm_rl) \<comment> \<open>Checks that subgoals remain: proof failed.\<close> |
61489 | 248 |
oops |
14239 | 249 |
|
61489 | 250 |
lemma "(\<forall>x. P(x) \<or> Q) \<longrightarrow> ((\<forall>x. P(x)) \<or> Q)" |
251 |
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)? |
|
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
62020
diff
changeset
|
252 |
apply (rule asm_rl) \<comment> \<open>Checks that subgoals remain: proof failed.\<close> |
61489 | 253 |
oops |
14239 | 254 |
|
61489 | 255 |
lemma "(\<forall>x. \<not> \<not> P(x)) \<longrightarrow> \<not> \<not> (\<forall>x. P(x))" |
256 |
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)? |
|
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
62020
diff
changeset
|
257 |
apply (rule asm_rl) \<comment> \<open>Checks that subgoals remain: proof failed.\<close> |
61489 | 258 |
oops |
14239 | 259 |
|
61489 | 260 |
text \<open>Classically but not intuitionistically valid. Proved by a bug in 1986!\<close> |
261 |
lemma "\<exists>x. Q(x) \<longrightarrow> (\<forall>x. Q(x))" |
|
262 |
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)? |
|
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
62020
diff
changeset
|
263 |
apply (rule asm_rl) \<comment> \<open>Checks that subgoals remain: proof failed.\<close> |
61489 | 264 |
oops |
14239 | 265 |
|
266 |
||
61489 | 267 |
subsection \<open>Hard examples with quantifiers\<close> |
14239 | 268 |
|
61489 | 269 |
text \<open> |
270 |
The ones that have not been proved are not known to be valid! Some will |
|
271 |
require quantifier duplication -- not currently available. |
|
272 |
\<close> |
|
14239 | 273 |
|
62020 | 274 |
text\<open>\<open>\<not>\<not>\<close>18\<close> |
61489 | 275 |
lemma "\<not> \<not> (\<exists>y. \<forall>x. P(y) \<longrightarrow> P(x))" |
62020 | 276 |
oops \<comment> \<open>NOT PROVED\<close> |
14239 | 277 |
|
62020 | 278 |
text\<open>\<open>\<not>\<not>\<close>19\<close> |
61489 | 279 |
lemma "\<not> \<not> (\<exists>x. \<forall>y z. (P(y) \<longrightarrow> Q(z)) \<longrightarrow> (P(x) \<longrightarrow> Q(x)))" |
62020 | 280 |
oops \<comment> \<open>NOT PROVED\<close> |
14239 | 281 |
|
60770 | 282 |
text\<open>20\<close> |
61489 | 283 |
lemma |
284 |
"(\<forall>x y. \<exists>z. \<forall>w. (P(x) \<and> Q(y) \<longrightarrow> R(z) \<and> S(w))) |
|
285 |
\<longrightarrow> (\<exists>x y. P(x) \<and> Q(y)) \<longrightarrow> (\<exists>z. R(z))" |
|
286 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 287 |
|
60770 | 288 |
text\<open>21\<close> |
61489 | 289 |
lemma "(\<exists>x. P \<longrightarrow> Q(x)) \<and> (\<exists>x. Q(x) \<longrightarrow> P) \<longrightarrow> \<not> \<not> (\<exists>x. P \<longleftrightarrow> Q(x))" |
62020 | 290 |
oops \<comment> \<open>NOT PROVED; needs quantifier duplication\<close> |
14239 | 291 |
|
60770 | 292 |
text\<open>22\<close> |
61489 | 293 |
lemma "(\<forall>x. P \<longleftrightarrow> Q(x)) \<longrightarrow> (P \<longleftrightarrow> (\<forall>x. Q(x)))" |
294 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 295 |
|
62020 | 296 |
text\<open>\<open>\<not>\<not>\<close>23\<close> |
61489 | 297 |
lemma "\<not> \<not> ((\<forall>x. P \<or> Q(x)) \<longleftrightarrow> (P \<or> (\<forall>x. Q(x))))" |
298 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 299 |
|
60770 | 300 |
text\<open>24\<close> |
61489 | 301 |
lemma |
302 |
"\<not> (\<exists>x. S(x) \<and> Q(x)) \<and> (\<forall>x. P(x) \<longrightarrow> Q(x) \<or> R(x)) \<and> |
|
303 |
(\<not> (\<exists>x. P(x)) \<longrightarrow> (\<exists>x. Q(x))) \<and> (\<forall>x. Q(x) \<or> R(x) \<longrightarrow> S(x)) |
|
304 |
\<longrightarrow> \<not> \<not> (\<exists>x. P(x) \<and> R(x))" |
|
305 |
text \<open> |
|
62020 | 306 |
Not clear why \<open>fast_tac\<close>, \<open>best_tac\<close>, \<open>ASTAR\<close> and |
307 |
\<open>ITER_DEEPEN\<close> all take forever. |
|
61489 | 308 |
\<close> |
309 |
apply (tactic \<open>IntPr.safe_tac @{context}\<close>) |
|
310 |
apply (erule impE) |
|
311 |
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
312 |
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
313 |
done |
|
14239 | 314 |
|
60770 | 315 |
text\<open>25\<close> |
61489 | 316 |
lemma |
317 |
"(\<exists>x. P(x)) \<and> |
|
318 |
(\<forall>x. L(x) \<longrightarrow> \<not> (M(x) \<and> R(x))) \<and> |
|
319 |
(\<forall>x. P(x) \<longrightarrow> (M(x) \<and> L(x))) \<and> |
|
320 |
((\<forall>x. P(x) \<longrightarrow> Q(x)) \<or> (\<exists>x. P(x) \<and> R(x))) |
|
321 |
\<longrightarrow> (\<exists>x. Q(x) \<and> P(x))" |
|
322 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 323 |
|
62020 | 324 |
text\<open>\<open>\<not>\<not>\<close>26\<close> |
61489 | 325 |
lemma |
326 |
"(\<not> \<not> (\<exists>x. p(x)) \<longleftrightarrow> \<not> \<not> (\<exists>x. q(x))) \<and> |
|
327 |
(\<forall>x. \<forall>y. p(x) \<and> q(y) \<longrightarrow> (r(x) \<longleftrightarrow> s(y))) |
|
328 |
\<longrightarrow> ((\<forall>x. p(x) \<longrightarrow> r(x)) \<longleftrightarrow> (\<forall>x. q(x) \<longrightarrow> s(x)))" |
|
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
62020
diff
changeset
|
329 |
oops \<comment> \<open>NOT PROVED\<close> |
14239 | 330 |
|
60770 | 331 |
text\<open>27\<close> |
61489 | 332 |
lemma |
333 |
"(\<exists>x. P(x) \<and> \<not> Q(x)) \<and> |
|
334 |
(\<forall>x. P(x) \<longrightarrow> R(x)) \<and> |
|
335 |
(\<forall>x. M(x) \<and> L(x) \<longrightarrow> P(x)) \<and> |
|
336 |
((\<exists>x. R(x) \<and> \<not> Q(x)) \<longrightarrow> (\<forall>x. L(x) \<longrightarrow> \<not> R(x))) |
|
337 |
\<longrightarrow> (\<forall>x. M(x) \<longrightarrow> \<not> L(x))" |
|
338 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 339 |
|
62020 | 340 |
text\<open>\<open>\<not>\<not>\<close>28. AMENDED\<close> |
61489 | 341 |
lemma |
342 |
"(\<forall>x. P(x) \<longrightarrow> (\<forall>x. Q(x))) \<and> |
|
343 |
(\<not> \<not> (\<forall>x. Q(x) \<or> R(x)) \<longrightarrow> (\<exists>x. Q(x) \<and> S(x))) \<and> |
|
344 |
(\<not> \<not> (\<exists>x. S(x)) \<longrightarrow> (\<forall>x. L(x) \<longrightarrow> M(x))) |
|
345 |
\<longrightarrow> (\<forall>x. P(x) \<and> L(x) \<longrightarrow> M(x))" |
|
346 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 347 |
|
61489 | 348 |
text\<open>29. Essentially the same as Principia Mathematica *11.71\<close> |
349 |
lemma |
|
350 |
"(\<exists>x. P(x)) \<and> (\<exists>y. Q(y)) |
|
351 |
\<longrightarrow> ((\<forall>x. P(x) \<longrightarrow> R(x)) \<and> (\<forall>y. Q(y) \<longrightarrow> S(y)) \<longleftrightarrow> |
|
352 |
(\<forall>x y. P(x) \<and> Q(y) \<longrightarrow> R(x) \<and> S(y)))" |
|
353 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 354 |
|
62020 | 355 |
text\<open>\<open>\<not>\<not>\<close>30\<close> |
61489 | 356 |
lemma |
357 |
"(\<forall>x. (P(x) \<or> Q(x)) \<longrightarrow> \<not> R(x)) \<and> |
|
358 |
(\<forall>x. (Q(x) \<longrightarrow> \<not> S(x)) \<longrightarrow> P(x) \<and> R(x)) |
|
359 |
\<longrightarrow> (\<forall>x. \<not> \<not> S(x))" |
|
360 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 361 |
|
60770 | 362 |
text\<open>31\<close> |
61489 | 363 |
lemma |
364 |
"\<not> (\<exists>x. P(x) \<and> (Q(x) \<or> R(x))) \<and> |
|
365 |
(\<exists>x. L(x) \<and> P(x)) \<and> |
|
366 |
(\<forall>x. \<not> R(x) \<longrightarrow> M(x)) |
|
367 |
\<longrightarrow> (\<exists>x. L(x) \<and> M(x))" |
|
368 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 369 |
|
60770 | 370 |
text\<open>32\<close> |
61489 | 371 |
lemma |
372 |
"(\<forall>x. P(x) \<and> (Q(x) \<or> R(x)) \<longrightarrow> S(x)) \<and> |
|
373 |
(\<forall>x. S(x) \<and> R(x) \<longrightarrow> L(x)) \<and> |
|
374 |
(\<forall>x. M(x) \<longrightarrow> R(x)) |
|
375 |
\<longrightarrow> (\<forall>x. P(x) \<and> M(x) \<longrightarrow> L(x))" |
|
376 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 377 |
|
62020 | 378 |
text\<open>\<open>\<not>\<not>\<close>33\<close> |
61489 | 379 |
lemma |
380 |
"(\<forall>x. \<not> \<not> (P(a) \<and> (P(x) \<longrightarrow> P(b)) \<longrightarrow> P(c))) \<longleftrightarrow> |
|
381 |
(\<forall>x. \<not> \<not> ((\<not> P(a) \<or> P(x) \<or> P(c)) \<and> (\<not> P(a) \<or> \<not> P(b) \<or> P(c))))" |
|
382 |
apply (tactic \<open>IntPr.best_tac @{context} 1\<close>) |
|
383 |
done |
|
14239 | 384 |
|
385 |
||
60770 | 386 |
text\<open>36\<close> |
61489 | 387 |
lemma |
388 |
"(\<forall>x. \<exists>y. J(x,y)) \<and> |
|
389 |
(\<forall>x. \<exists>y. G(x,y)) \<and> |
|
390 |
(\<forall>x y. J(x,y) \<or> G(x,y) \<longrightarrow> (\<forall>z. J(y,z) \<or> G(y,z) \<longrightarrow> H(x,z))) |
|
391 |
\<longrightarrow> (\<forall>x. \<exists>y. H(x,y))" |
|
392 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 393 |
|
60770 | 394 |
text\<open>37\<close> |
61489 | 395 |
lemma |
396 |
"(\<forall>z. \<exists>w. \<forall>x. \<exists>y. |
|
397 |
\<not> \<not> (P(x,z) \<longrightarrow> P(y,w)) \<and> P(y,z) \<and> (P(y,w) \<longrightarrow> (\<exists>u. Q(u,w)))) \<and> |
|
398 |
(\<forall>x z. \<not> P(x,z) \<longrightarrow> (\<exists>y. Q(y,z))) \<and> |
|
399 |
(\<not> \<not> (\<exists>x y. Q(x,y)) \<longrightarrow> (\<forall>x. R(x,x))) |
|
400 |
\<longrightarrow> \<not> \<not> (\<forall>x. \<exists>y. R(x,y))" |
|
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
62020
diff
changeset
|
401 |
oops \<comment> \<open>NOT PROVED\<close> |
14239 | 402 |
|
60770 | 403 |
text\<open>39\<close> |
61489 | 404 |
lemma "\<not> (\<exists>x. \<forall>y. F(y,x) \<longleftrightarrow> \<not> F(y,y))" |
405 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 406 |
|
61489 | 407 |
text\<open>40. AMENDED\<close> |
408 |
lemma |
|
409 |
"(\<exists>y. \<forall>x. F(x,y) \<longleftrightarrow> F(x,x)) \<longrightarrow> |
|
410 |
\<not> (\<forall>x. \<exists>y. \<forall>z. F(z,y) \<longleftrightarrow> \<not> F(z,x))" |
|
411 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 412 |
|
60770 | 413 |
text\<open>44\<close> |
61489 | 414 |
lemma |
415 |
"(\<forall>x. f(x) \<longrightarrow> |
|
416 |
(\<exists>y. g(y) \<and> h(x,y) \<and> (\<exists>y. g(y) \<and> \<not> h(x,y)))) \<and> |
|
417 |
(\<exists>x. j(x) \<and> (\<forall>y. g(y) \<longrightarrow> h(x,y))) |
|
418 |
\<longrightarrow> (\<exists>x. j(x) \<and> \<not> f(x))" |
|
419 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 420 |
|
60770 | 421 |
text\<open>48\<close> |
61489 | 422 |
lemma "(a = b \<or> c = d) \<and> (a = c \<or> b = d) \<longrightarrow> a = d \<or> b = c" |
423 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 424 |
|
60770 | 425 |
text\<open>51\<close> |
61489 | 426 |
lemma |
427 |
"(\<exists>z w. \<forall>x y. P(x,y) \<longleftrightarrow> (x = z \<and> y = w)) \<longrightarrow> |
|
428 |
(\<exists>z. \<forall>x. \<exists>w. (\<forall>y. P(x,y) \<longleftrightarrow> y = w) \<longleftrightarrow> x = z)" |
|
429 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 430 |
|
60770 | 431 |
text\<open>52\<close> |
61489 | 432 |
text \<open>Almost the same as 51.\<close> |
433 |
lemma |
|
434 |
"(\<exists>z w. \<forall>x y. P(x,y) \<longleftrightarrow> (x = z \<and> y = w)) \<longrightarrow> |
|
435 |
(\<exists>w. \<forall>y. \<exists>z. (\<forall>x. P(x,y) \<longleftrightarrow> x = z) \<longleftrightarrow> y = w)" |
|
436 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 437 |
|
60770 | 438 |
text\<open>56\<close> |
61489 | 439 |
lemma "(\<forall>x. (\<exists>y. P(y) \<and> x = f(y)) \<longrightarrow> P(x)) \<longleftrightarrow> (\<forall>x. P(x) \<longrightarrow> P(f(x)))" |
440 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 441 |
|
60770 | 442 |
text\<open>57\<close> |
61489 | 443 |
lemma |
444 |
"P(f(a,b), f(b,c)) \<and> P(f(b,c), f(a,c)) \<and> |
|
445 |
(\<forall>x y z. P(x,y) \<and> P(y,z) \<longrightarrow> P(x,z)) \<longrightarrow> P(f(a,b), f(a,c))" |
|
446 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 447 |
|
60770 | 448 |
text\<open>60\<close> |
61489 | 449 |
lemma "\<forall>x. P(x,f(x)) \<longleftrightarrow> (\<exists>y. (\<forall>z. P(z,y) \<longrightarrow> P(z,f(x))) \<and> P(x,y))" |
450 |
by (tactic \<open>IntPr.fast_tac @{context} 1\<close>) |
|
14239 | 451 |
|
452 |
end |