src/FOL/ex/Intuitionistic.thy
 author haftmann Thu Nov 23 17:03:27 2017 +0000 (21 months ago) changeset 67087 733017b19de9 parent 62020 5d208fd2507d child 67443 3abf6a722518 permissions -rw-r--r--
generalized more lemmas
     1 (*  Title:      FOL/ex/Intuitionistic.thy

     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     3     Copyright   1991  University of Cambridge

     4 *)

     5

     6 section \<open>Intuitionistic First-Order Logic\<close>

     7

     8 theory Intuitionistic

     9 imports IFOL

    10 begin

    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);

    20 by (IntPr.fast_tac @{context} 1);

    21 *)

    22

    23

    24 text\<open>Metatheorem (for \emph{propositional} formulae):

    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 $\neg\neg P$

    37 is intuitionstically equivalent to $P$.  [Andy Pitts]\<close>

    38

    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>)

    41

    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>)

    44

    45 text \<open>Double-negation does NOT distribute over disjunction.\<close>

    46

    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>)

    49

    50 lemma "\<not> \<not> \<not> P \<longleftrightarrow> \<not> P"

    51   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)

    52

    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>)

    55

    56 lemma "(P \<longleftrightarrow> Q) \<longleftrightarrow> (Q \<longleftrightarrow> P)"

    57   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)

    58

    59 lemma "((P \<longrightarrow> (Q \<or> (Q \<longrightarrow> R))) \<longrightarrow> R) \<longrightarrow> R"

    60   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)

    61

    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>)

    67

    68

    69 subsection \<open>Lemmas for the propositional double-negation translation\<close>

    70

    71 lemma "P \<longrightarrow> \<not> \<not> P"

    72   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)

    73

    74 lemma "\<not> \<not> (\<not> \<not> P \<longrightarrow> P)"

    75   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)

    76

    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>)

    79

    80

    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>)?

    85 apply (rule asm_rl) \<comment>\<open>Checks that subgoals remain: proof failed.\<close>

    86 oops

    87

    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>)?

    90 apply (rule asm_rl) \<comment>\<open>Checks that subgoals remain: proof failed.\<close>

    91 oops

    92

    93

    94 subsection \<open>de Bruijn formulae\<close>

    95

    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>)

   102

   103

   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>)

   112

   113

   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>

   119

   120 text\<open>Problem 1.1\<close>

   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))"

   124   by (tactic \<open>IntPr.best_dup_tac @{context} 1\<close>)  \<comment>\<open>SLOW\<close>

   125

   126 text\<open>Problem 3.1\<close>

   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>)

   129

   130 text\<open>Problem 4.1: hopeless!\<close>

   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

   135

   136

   137 subsection \<open>Intuitionistic FOL: propositional problems based on Pelletier.\<close>

   138

   139 text\<open>\<open>\<not>\<not>\<close>1\<close>

   140 lemma "\<not> \<not> ((P \<longrightarrow> Q) \<longleftrightarrow> (\<not> Q \<longrightarrow> \<not> P))"

   141   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)

   142

   143 text\<open>\<open>\<not>\<not>\<close>2\<close>

   144 lemma "\<not> \<not> (\<not> \<not> P \<longleftrightarrow> P)"

   145   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)

   146

   147 text\<open>3\<close>

   148 lemma "\<not> (P \<longrightarrow> Q) \<longrightarrow> (Q \<longrightarrow> P)"

   149   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)

   150

   151 text\<open>\<open>\<not>\<not>\<close>4\<close>

   152 lemma "\<not> \<not> ((\<not> P \<longrightarrow> Q) \<longleftrightarrow> (\<not> Q \<longrightarrow> P))"

   153   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)

   154

   155 text\<open>\<open>\<not>\<not>\<close>5\<close>

   156 lemma "\<not> \<not> ((P \<or> Q \<longrightarrow> P \<or> R) \<longrightarrow> P \<or> (Q \<longrightarrow> R))"

   157   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)

   158

   159 text\<open>\<open>\<not>\<not>\<close>6\<close>

   160 lemma "\<not> \<not> (P \<or> \<not> P)"

   161   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)

   162

   163 text\<open>\<open>\<not>\<not>\<close>7\<close>

   164 lemma "\<not> \<not> (P \<or> \<not> \<not> \<not> P)"

   165   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)

   166

   167 text\<open>\<open>\<not>\<not>\<close>8. Peirce's law\<close>

   168 lemma "\<not> \<not> (((P \<longrightarrow> Q) \<longrightarrow> P) \<longrightarrow> P)"

   169   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)

   170

   171 text\<open>9\<close>

   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>)

   174

   175 text\<open>10\<close>

   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>)

   178

   179

   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

   185 text\<open>\<open>\<not>\<not>\<close>12. Dijkstra's law\<close>

   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

   196 text\<open>\<open>\<not>\<not>\<close>14\<close>

   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

   200 text\<open>\<open>\<not>\<not>\<close>15\<close>

   201 lemma "\<not> \<not> ((P \<longrightarrow> Q) \<longleftrightarrow> (\<not> P \<or> Q))"

   202   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)

   203

   204 text\<open>\<open>\<not>\<not>\<close>16\<close>

   205 lemma "\<not> \<not> ((P \<longrightarrow> Q) \<or> (Q \<longrightarrow> P))"

   206   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)

   207

   208 text\<open>\<open>\<not>\<not>\<close>17\<close>

   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>)

   235

   236

   237 subsection \<open>The following are not constructively valid!\<close>

   238 text \<open>The attempt to prove them terminates quickly!\<close>

   239

   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>)?

   242   apply (rule asm_rl) \<comment>\<open>Checks that subgoals remain: proof failed.\<close>

   243   oops

   244

   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>)?

   247   apply (rule asm_rl) \<comment>\<open>Checks that subgoals remain: proof failed.\<close>

   248   oops

   249

   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>)?

   252   apply (rule asm_rl) \<comment>\<open>Checks that subgoals remain: proof failed.\<close>

   253   oops

   254

   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>)?

   257   apply (rule asm_rl) \<comment>\<open>Checks that subgoals remain: proof failed.\<close>

   258   oops

   259

   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>)?

   263   apply (rule asm_rl) \<comment>\<open>Checks that subgoals remain: proof failed.\<close>

   264   oops

   265

   266

   267 subsection \<open>Hard examples with quantifiers\<close>

   268

   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>

   273

   274 text\<open>\<open>\<not>\<not>\<close>18\<close>

   275 lemma "\<not> \<not> (\<exists>y. \<forall>x. P(y) \<longrightarrow> P(x))"

   276   oops  \<comment> \<open>NOT PROVED\<close>

   277

   278 text\<open>\<open>\<not>\<not>\<close>19\<close>

   279 lemma "\<not> \<not> (\<exists>x. \<forall>y z. (P(y) \<longrightarrow> Q(z)) \<longrightarrow> (P(x) \<longrightarrow> Q(x)))"

   280   oops  \<comment> \<open>NOT PROVED\<close>

   281

   282 text\<open>20\<close>

   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>)

   287

   288 text\<open>21\<close>

   289 lemma "(\<exists>x. P \<longrightarrow> Q(x)) \<and> (\<exists>x. Q(x) \<longrightarrow> P) \<longrightarrow> \<not> \<not> (\<exists>x. P \<longleftrightarrow> Q(x))"

   290   oops \<comment> \<open>NOT PROVED; needs quantifier duplication\<close>

   291

   292 text\<open>22\<close>

   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>)

   295

   296 text\<open>\<open>\<not>\<not>\<close>23\<close>

   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>)

   299

   300 text\<open>24\<close>

   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>

   306   Not clear why \<open>fast_tac\<close>, \<open>best_tac\<close>, \<open>ASTAR\<close> and

   307   \<open>ITER_DEEPEN\<close> all take forever.

   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

   314

   315 text\<open>25\<close>

   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>)

   323

   324 text\<open>\<open>\<not>\<not>\<close>26\<close>

   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)))"

   329   oops  \<comment>\<open>NOT PROVED\<close>

   330

   331 text\<open>27\<close>

   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>)

   339

   340 text\<open>\<open>\<not>\<not>\<close>28. AMENDED\<close>

   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>)

   347

   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>)

   354

   355 text\<open>\<open>\<not>\<not>\<close>30\<close>

   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>)

   361

   362 text\<open>31\<close>

   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>)

   369

   370 text\<open>32\<close>

   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>)

   377

   378 text\<open>\<open>\<not>\<not>\<close>33\<close>

   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

   384

   385

   386 text\<open>36\<close>

   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>)

   393

   394 text\<open>37\<close>

   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))"

   401   oops  \<comment>\<open>NOT PROVED\<close>

   402

   403 text\<open>39\<close>

   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>)

   406

   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>)

   412

   413 text\<open>44\<close>

   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>)

   420

   421 text\<open>48\<close>

   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>)

   424

   425 text\<open>51\<close>

   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>)

   430

   431 text\<open>52\<close>

   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>)

   437

   438 text\<open>56\<close>

   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>)

   441

   442 text\<open>57\<close>

   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>)

   447

   448 text\<open>60\<close>

   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>)

   451

   452 end