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src/Cube/Example.thy
changeset 61337 4645502c3c64
parent 59499 14095f771781
child 61388 92e97b800d1e
equal deleted inserted replaced
61336:fa4ebbd350ae 61337:4645502c3c64 
    23     DEPTH_SOLVE_1 (assume_tac ctxt 1 ORELSE resolve_tac ctxt (facts @ thms) 1))))\<close>
 
    23     DEPTH_SOLVE_1 (assume_tac ctxt 1 ORELSE resolve_tac ctxt (facts @ thms) 1))))\<close>
 
    24 
    24 
    25 
    25 
    26 subsection \<open>Simple types\<close>
 
    26 subsection \<open>Simple types\<close>
 
    27 
    27 
    28 schematic_lemma "A:* \<turnstile> A\<rightarrow>A : ?T"
 
    28 schematic_goal "A:* \<turnstile> A\<rightarrow>A : ?T"
 
    29   by (depth_solve rules)
 
    29   by (depth_solve rules)
 
    30 
    30 
    31 schematic_lemma "A:* \<turnstile> \<Lambda> a:A. a : ?T"
 
    31 schematic_goal "A:* \<turnstile> \<Lambda> a:A. a : ?T"
 
    32   by (depth_solve rules)
 
    32   by (depth_solve rules)
 
    33 
    33 
    34 schematic_lemma "A:* B:* b:B \<turnstile> \<Lambda> x:A. b : ?T"
 
    34 schematic_goal "A:* B:* b:B \<turnstile> \<Lambda> x:A. b : ?T"
 
    35   by (depth_solve rules)
 
    35   by (depth_solve rules)
 
    36 
    36 
    37 schematic_lemma "A:* b:A \<turnstile> (\<Lambda> a:A. a)^b: ?T"
 
    37 schematic_goal "A:* b:A \<turnstile> (\<Lambda> a:A. a)^b: ?T"
 
    38   by (depth_solve rules)
 
    38   by (depth_solve rules)
 
    39 
    39 
    40 schematic_lemma "A:* B:* c:A b:B \<turnstile> (\<Lambda> x:A. b)^ c: ?T"
 
    40 schematic_goal "A:* B:* c:A b:B \<turnstile> (\<Lambda> x:A. b)^ c: ?T"
 
    41   by (depth_solve rules)
 
    41   by (depth_solve rules)
 
    42 
    42 
    43 schematic_lemma "A:* B:* \<turnstile> \<Lambda> a:A. \<Lambda> b:B. a : ?T"
 
    43 schematic_goal "A:* B:* \<turnstile> \<Lambda> a:A. \<Lambda> b:B. a : ?T"
 
    44   by (depth_solve rules)
 
    44   by (depth_solve rules)
 
    45 
    45 
    46 
    46 
    47 subsection \<open>Second-order types\<close>
 
    47 subsection \<open>Second-order types\<close>
 
    48 
    48 
    49 schematic_lemma (in L2) "\<turnstile> \<Lambda> A:*. \<Lambda> a:A. a : ?T"
 
    49 schematic_goal (in L2) "\<turnstile> \<Lambda> A:*. \<Lambda> a:A. a : ?T"
 
    50   by (depth_solve rules)
 
    50   by (depth_solve rules)
 
    51 
    51 
    52 schematic_lemma (in L2) "A:* \<turnstile> (\<Lambda> B:*.\<Lambda> b:B. b)^A : ?T"
 
    52 schematic_goal (in L2) "A:* \<turnstile> (\<Lambda> B:*.\<Lambda> b:B. b)^A : ?T"
 
    53   by (depth_solve rules)
 
    53   by (depth_solve rules)
 
    54 
    54 
    55 schematic_lemma (in L2) "A:* b:A \<turnstile> (\<Lambda> B:*.\<Lambda> b:B. b) ^ A ^ b: ?T"
 
    55 schematic_goal (in L2) "A:* b:A \<turnstile> (\<Lambda> B:*.\<Lambda> b:B. b) ^ A ^ b: ?T"
 
    56   by (depth_solve rules)
 
    56   by (depth_solve rules)
 
    57 
    57 
    58 schematic_lemma (in L2) "\<turnstile> \<Lambda> B:*.\<Lambda> a:(\<Pi> A:*.A).a ^ ((\<Pi> A:*.A)\<rightarrow>B) ^ a: ?T"
 
    58 schematic_goal (in L2) "\<turnstile> \<Lambda> B:*.\<Lambda> a:(\<Pi> A:*.A).a ^ ((\<Pi> A:*.A)\<rightarrow>B) ^ a: ?T"
 
    59   by (depth_solve rules)
 
    59   by (depth_solve rules)
 
    60 
    60 
    61 
    61 
    62 subsection \<open>Weakly higher-order propositional logic\<close>
 
    62 subsection \<open>Weakly higher-order propositional logic\<close>
 
    63 
    63 
    64 schematic_lemma (in Lomega) "\<turnstile> \<Lambda> A:*.A\<rightarrow>A : ?T"
 
    64 schematic_goal (in Lomega) "\<turnstile> \<Lambda> A:*.A\<rightarrow>A : ?T"
 
    65   by (depth_solve rules)
 
    65   by (depth_solve rules)
 
    66 
    66 
    67 schematic_lemma (in Lomega) "B:* \<turnstile> (\<Lambda> A:*.A\<rightarrow>A) ^ B : ?T"
 
    67 schematic_goal (in Lomega) "B:* \<turnstile> (\<Lambda> A:*.A\<rightarrow>A) ^ B : ?T"
 
    68   by (depth_solve rules)
 
    68   by (depth_solve rules)
 
    69 
    69 
    70 schematic_lemma (in Lomega) "B:* b:B \<turnstile> (\<Lambda> y:B. b): ?T"
 
    70 schematic_goal (in Lomega) "B:* b:B \<turnstile> (\<Lambda> y:B. b): ?T"
 
    71   by (depth_solve rules)
 
    71   by (depth_solve rules)
 
    72 
    72 
    73 schematic_lemma (in Lomega) "A:* F:*\<rightarrow>* \<turnstile> F^(F^A): ?T"
 
    73 schematic_goal (in Lomega) "A:* F:*\<rightarrow>* \<turnstile> F^(F^A): ?T"
 
    74   by (depth_solve rules)
 
    74   by (depth_solve rules)
 
    75 
    75 
    76 schematic_lemma (in Lomega) "A:* \<turnstile> \<Lambda> F:*\<rightarrow>*.F^(F^A): ?T"
 
    76 schematic_goal (in Lomega) "A:* \<turnstile> \<Lambda> F:*\<rightarrow>*.F^(F^A): ?T"
 
    77   by (depth_solve rules)
 
    77   by (depth_solve rules)
 
    78 
    78 
    79 
    79 
    80 subsection \<open>LP\<close>
 
    80 subsection \<open>LP\<close>
 
    81 
    81 
    82 schematic_lemma (in LP) "A:* \<turnstile> A \<rightarrow> * : ?T"
 
    82 schematic_goal (in LP) "A:* \<turnstile> A \<rightarrow> * : ?T"
 
    83   by (depth_solve rules)
 
    83   by (depth_solve rules)
 
    84 
    84 
    85 schematic_lemma (in LP) "A:* P:A\<rightarrow>* a:A \<turnstile> P^a: ?T"
 
    85 schematic_goal (in LP) "A:* P:A\<rightarrow>* a:A \<turnstile> P^a: ?T"
 
    86   by (depth_solve rules)
 
    86   by (depth_solve rules)
 
    87 
    87 
    88 schematic_lemma (in LP) "A:* P:A\<rightarrow>A\<rightarrow>* a:A \<turnstile> \<Pi> a:A. P^a^a: ?T"
 
    88 schematic_goal (in LP) "A:* P:A\<rightarrow>A\<rightarrow>* a:A \<turnstile> \<Pi> a:A. P^a^a: ?T"
 
    89   by (depth_solve rules)
 
    89   by (depth_solve rules)
 
    90 
    90 
    91 schematic_lemma (in LP) "A:* P:A\<rightarrow>* Q:A\<rightarrow>* \<turnstile> \<Pi> a:A. P^a \<rightarrow> Q^a: ?T"
 
    91 schematic_goal (in LP) "A:* P:A\<rightarrow>* Q:A\<rightarrow>* \<turnstile> \<Pi> a:A. P^a \<rightarrow> Q^a: ?T"
 
    92   by (depth_solve rules)
 
    92   by (depth_solve rules)
 
    93 
    93 
    94 schematic_lemma (in LP) "A:* P:A\<rightarrow>* \<turnstile> \<Pi> a:A. P^a \<rightarrow> P^a: ?T"
 
    94 schematic_goal (in LP) "A:* P:A\<rightarrow>* \<turnstile> \<Pi> a:A. P^a \<rightarrow> P^a: ?T"
 
    95   by (depth_solve rules)
 
    95   by (depth_solve rules)
 
    96 
    96 
    97 schematic_lemma (in LP) "A:* P:A\<rightarrow>* \<turnstile> \<Lambda> a:A. \<Lambda> x:P^a. x: ?T"
 
    97 schematic_goal (in LP) "A:* P:A\<rightarrow>* \<turnstile> \<Lambda> a:A. \<Lambda> x:P^a. x: ?T"
 
    98   by (depth_solve rules)
 
    98   by (depth_solve rules)
 
    99 
    99 
   100 schematic_lemma (in LP) "A:* P:A\<rightarrow>* Q:* \<turnstile> (\<Pi> a:A. P^a\<rightarrow>Q) \<rightarrow> (\<Pi> a:A. P^a) \<rightarrow> Q : ?T"
 
   100 schematic_goal (in LP) "A:* P:A\<rightarrow>* Q:* \<turnstile> (\<Pi> a:A. P^a\<rightarrow>Q) \<rightarrow> (\<Pi> a:A. P^a) \<rightarrow> Q : ?T"
 
   101   by (depth_solve rules)
 
   101   by (depth_solve rules)
 
   102 
   102 
   103 schematic_lemma (in LP) "A:* P:A\<rightarrow>* Q:* a0:A \<turnstile>
 
   103 schematic_goal (in LP) "A:* P:A\<rightarrow>* Q:* a0:A \<turnstile>
 
   104         \<Lambda> x:\<Pi> a:A. P^a\<rightarrow>Q. \<Lambda> y:\<Pi> a:A. P^a. x^a0^(y^a0): ?T"
 
   104         \<Lambda> x:\<Pi> a:A. P^a\<rightarrow>Q. \<Lambda> y:\<Pi> a:A. P^a. x^a0^(y^a0): ?T"
 
   105   by (depth_solve rules)
 
   105   by (depth_solve rules)
 
   106 
   106 
   107 
   107 
   108 subsection \<open>Omega-order types\<close>
 
   108 subsection \<open>Omega-order types\<close>
 
   109 
   109 
   110 schematic_lemma (in L2) "A:* B:* \<turnstile> \<Pi> C:*.(A\<rightarrow>B\<rightarrow>C)\<rightarrow>C : ?T"
 
   110 schematic_goal (in L2) "A:* B:* \<turnstile> \<Pi> C:*.(A\<rightarrow>B\<rightarrow>C)\<rightarrow>C : ?T"
 
   111   by (depth_solve rules)
 
   111   by (depth_solve rules)
 
   112 
   112 
   113 schematic_lemma (in Lomega2) "\<turnstile> \<Lambda> A:*.\<Lambda> B:*.\<Pi> C:*.(A\<rightarrow>B\<rightarrow>C)\<rightarrow>C : ?T"
 
   113 schematic_goal (in Lomega2) "\<turnstile> \<Lambda> A:*.\<Lambda> B:*.\<Pi> C:*.(A\<rightarrow>B\<rightarrow>C)\<rightarrow>C : ?T"
 
   114   by (depth_solve rules)
 
   114   by (depth_solve rules)
 
   115 
   115 
   116 schematic_lemma (in Lomega2) "\<turnstile> \<Lambda> A:*.\<Lambda> B:*.\<Lambda> x:A. \<Lambda> y:B. x : ?T"
 
   116 schematic_goal (in Lomega2) "\<turnstile> \<Lambda> A:*.\<Lambda> B:*.\<Lambda> x:A. \<Lambda> y:B. x : ?T"
 
   117   by (depth_solve rules)
 
   117   by (depth_solve rules)
 
   118 
   118 
   119 schematic_lemma (in Lomega2) "A:* B:* \<turnstile> ?p : (A\<rightarrow>B) \<rightarrow> ((B\<rightarrow>\<Pi> P:*.P)\<rightarrow>(A\<rightarrow>\<Pi> P:*.P))"
 
   119 schematic_goal (in Lomega2) "A:* B:* \<turnstile> ?p : (A\<rightarrow>B) \<rightarrow> ((B\<rightarrow>\<Pi> P:*.P)\<rightarrow>(A\<rightarrow>\<Pi> P:*.P))"
 
   120   apply (strip_asms rules)
 
   120   apply (strip_asms rules)
 
   121   apply (rule lam_ss)
 
   121   apply (rule lam_ss)
 
   122     apply (depth_solve1 rules)
 
   122     apply (depth_solve1 rules)
 
   123    prefer 2
 
   123    prefer 2
 
   124    apply (depth_solve1 rules)
 
   124    apply (depth_solve1 rules)
 
   138   done
 
   138   done
 
   139 
   139 
   140 
   140 
   141 subsection \<open>Second-order Predicate Logic\<close>
 
   141 subsection \<open>Second-order Predicate Logic\<close>
 
   142 
   142 
   143 schematic_lemma (in LP2) "A:* P:A\<rightarrow>* \<turnstile> \<Lambda> a:A. P^a\<rightarrow>(\<Pi> A:*.A) : ?T"
 
   143 schematic_goal (in LP2) "A:* P:A\<rightarrow>* \<turnstile> \<Lambda> a:A. P^a\<rightarrow>(\<Pi> A:*.A) : ?T"
 
   144   by (depth_solve rules)
 
   144   by (depth_solve rules)
 
   145 
   145 
   146 schematic_lemma (in LP2) "A:* P:A\<rightarrow>A\<rightarrow>* \<turnstile>
 
   146 schematic_goal (in LP2) "A:* P:A\<rightarrow>A\<rightarrow>* \<turnstile>
 
   147     (\<Pi> a:A. \<Pi> b:A. P^a^b\<rightarrow>P^b^a\<rightarrow>\<Pi> P:*.P) \<rightarrow> \<Pi> a:A. P^a^a\<rightarrow>\<Pi> P:*.P : ?T"
 
   147     (\<Pi> a:A. \<Pi> b:A. P^a^b\<rightarrow>P^b^a\<rightarrow>\<Pi> P:*.P) \<rightarrow> \<Pi> a:A. P^a^a\<rightarrow>\<Pi> P:*.P : ?T"
 
   148   by (depth_solve rules)
 
   148   by (depth_solve rules)
 
   149 
   149 
   150 schematic_lemma (in LP2) "A:* P:A\<rightarrow>A\<rightarrow>* \<turnstile>
 
   150 schematic_goal (in LP2) "A:* P:A\<rightarrow>A\<rightarrow>* \<turnstile>
 
   151     ?p: (\<Pi> a:A. \<Pi> b:A. P^a^b\<rightarrow>P^b^a\<rightarrow>\<Pi> P:*.P) \<rightarrow> \<Pi> a:A. P^a^a\<rightarrow>\<Pi> P:*.P"
 
   151     ?p: (\<Pi> a:A. \<Pi> b:A. P^a^b\<rightarrow>P^b^a\<rightarrow>\<Pi> P:*.P) \<rightarrow> \<Pi> a:A. P^a^a\<rightarrow>\<Pi> P:*.P"
 
   152   -- \<open>Antisymmetry implies irreflexivity:\<close>
 
   152   -- \<open>Antisymmetry implies irreflexivity:\<close>
 
   153   apply (strip_asms rules)
 
   153   apply (strip_asms rules)
 
   154   apply (rule lam_ss)
 
   154   apply (rule lam_ss)
 
   155     apply (depth_solve1 rules)
 
   155     apply (depth_solve1 rules)
 
   167   done
 
   167   done
 
   168 
   168 
   169 
   169 
   170 subsection \<open>LPomega\<close>
 
   170 subsection \<open>LPomega\<close>
 
   171 
   171 
   172 schematic_lemma (in LPomega) "A:* \<turnstile> \<Lambda> P:A\<rightarrow>A\<rightarrow>*.\<Lambda> a:A. P^a^a : ?T"
 
   172 schematic_goal (in LPomega) "A:* \<turnstile> \<Lambda> P:A\<rightarrow>A\<rightarrow>*.\<Lambda> a:A. P^a^a : ?T"
 
   173   by (depth_solve rules)
 
   173   by (depth_solve rules)
 
   174 
   174 
   175 schematic_lemma (in LPomega) "\<turnstile> \<Lambda> A:*.\<Lambda> P:A\<rightarrow>A\<rightarrow>*.\<Lambda> a:A. P^a^a : ?T"
 
   175 schematic_goal (in LPomega) "\<turnstile> \<Lambda> A:*.\<Lambda> P:A\<rightarrow>A\<rightarrow>*.\<Lambda> a:A. P^a^a : ?T"
 
   176   by (depth_solve rules)
 
   176   by (depth_solve rules)
 
   177 
   177 
   178 
   178 
   179 subsection \<open>Constructions\<close>
 
   179 subsection \<open>Constructions\<close>
 
   180 
   180 
   181 schematic_lemma (in CC) "\<turnstile> \<Lambda> A:*.\<Lambda> P:A\<rightarrow>*.\<Lambda> a:A. P^a\<rightarrow>\<Pi> P:*.P: ?T"
 
   181 schematic_goal (in CC) "\<turnstile> \<Lambda> A:*.\<Lambda> P:A\<rightarrow>*.\<Lambda> a:A. P^a\<rightarrow>\<Pi> P:*.P: ?T"
 
   182   by (depth_solve rules)
 
   182   by (depth_solve rules)
 
   183 
   183 
   184 schematic_lemma (in CC) "\<turnstile> \<Lambda> A:*.\<Lambda> P:A\<rightarrow>*.\<Pi> a:A. P^a: ?T"
 
   184 schematic_goal (in CC) "\<turnstile> \<Lambda> A:*.\<Lambda> P:A\<rightarrow>*.\<Pi> a:A. P^a: ?T"
 
   185   by (depth_solve rules)
 
   185   by (depth_solve rules)
 
   186 
   186 
   187 schematic_lemma (in CC) "A:* P:A\<rightarrow>* a:A \<turnstile> ?p : (\<Pi> a:A. P^a)\<rightarrow>P^a"
 
   187 schematic_goal (in CC) "A:* P:A\<rightarrow>* a:A \<turnstile> ?p : (\<Pi> a:A. P^a)\<rightarrow>P^a"
 
   188   apply (strip_asms rules)
 
   188   apply (strip_asms rules)
 
   189   apply (rule lam_ss)
 
   189   apply (rule lam_ss)
 
   190     apply (depth_solve1 rules)
 
   190     apply (depth_solve1 rules)
 
   191    prefer 2
 
   191    prefer 2
 
   192    apply (depth_solve1 rules)
 
   192    apply (depth_solve1 rules)
 
   194   done
 
   194   done
 
   195 
   195 
   196 
   196 
   197 subsection \<open>Some random examples\<close>
 
   197 subsection \<open>Some random examples\<close>
 
   198 
   198 
   199 schematic_lemma (in LP2) "A:* c:A f:A\<rightarrow>A \<turnstile>
 
   199 schematic_goal (in LP2) "A:* c:A f:A\<rightarrow>A \<turnstile>
 
   200     \<Lambda> a:A. \<Pi> P:A\<rightarrow>*.P^c \<rightarrow> (\<Pi> x:A. P^x\<rightarrow>P^(f^x)) \<rightarrow> P^a : ?T"
 
   200     \<Lambda> a:A. \<Pi> P:A\<rightarrow>*.P^c \<rightarrow> (\<Pi> x:A. P^x\<rightarrow>P^(f^x)) \<rightarrow> P^a : ?T"
 
   201   by (depth_solve rules)
 
   201   by (depth_solve rules)
 
   202 
   202 
   203 schematic_lemma (in CC) "\<Lambda> A:*.\<Lambda> c:A. \<Lambda> f:A\<rightarrow>A.
 
   203 schematic_goal (in CC) "\<Lambda> A:*.\<Lambda> c:A. \<Lambda> f:A\<rightarrow>A.
 
   204     \<Lambda> a:A. \<Pi> P:A\<rightarrow>*.P^c \<rightarrow> (\<Pi> x:A. P^x\<rightarrow>P^(f^x)) \<rightarrow> P^a : ?T"
 
   204     \<Lambda> a:A. \<Pi> P:A\<rightarrow>*.P^c \<rightarrow> (\<Pi> x:A. P^x\<rightarrow>P^(f^x)) \<rightarrow> P^a : ?T"
 
   205   by (depth_solve rules)
 
   205   by (depth_solve rules)
 
   206 
   206 
   207 schematic_lemma (in LP2)
 
   207 schematic_goal (in LP2)
 
   208   "A:* a:A b:A \<turnstile> ?p: (\<Pi> P:A\<rightarrow>*.P^a\<rightarrow>P^b) \<rightarrow> (\<Pi> P:A\<rightarrow>*.P^b\<rightarrow>P^a)"
 
   208   "A:* a:A b:A \<turnstile> ?p: (\<Pi> P:A\<rightarrow>*.P^a\<rightarrow>P^b) \<rightarrow> (\<Pi> P:A\<rightarrow>*.P^b\<rightarrow>P^a)"
 
   209   -- \<open>Symmetry of Leibnitz equality\<close>
 
   209   -- \<open>Symmetry of Leibnitz equality\<close>
 
   210   apply (strip_asms rules)
 
   210   apply (strip_asms rules)
 
   211   apply (rule lam_ss)
 
   211   apply (rule lam_ss)
 
   212     apply (depth_solve1 rules)
 
   212     apply (depth_solve1 rules)
isabelle