src/Cube/Example.thy
author wenzelm
Sun Nov 02 18:21:45 2014 +0100 (2014-11-02)
changeset 58889 5b7a9633cfa8
parent 58617 4f169d2cf6f3
child 59498 50b60f501b05
permissions -rw-r--r--
modernized header uniformly as section;
     1 section \<open>Lambda Cube Examples\<close>
     2 
     3 theory Example
     4 imports Cube
     5 begin
     6 
     7 text \<open>Examples taken from:
     8 
     9   H. Barendregt. Introduction to Generalised Type Systems.
    10   J. Functional Programming.\<close>
    11 
    12 method_setup depth_solve =
    13   \<open>Attrib.thms >> (fn thms => K (METHOD (fn facts =>
    14     (DEPTH_SOLVE (HEADGOAL (ares_tac (facts @ thms)))))))\<close>
    15 
    16 method_setup depth_solve1 =
    17   \<open>Attrib.thms >> (fn thms => K (METHOD (fn facts =>
    18     (DEPTH_SOLVE_1 (HEADGOAL (ares_tac (facts @ thms)))))))\<close>
    19 
    20 method_setup strip_asms =
    21   \<open>Attrib.thms >> (fn thms => K (METHOD (fn facts =>
    22     REPEAT (resolve_tac [@{thm strip_b}, @{thm strip_s}] 1 THEN
    23     DEPTH_SOLVE_1 (ares_tac (facts @ thms) 1)))))\<close>
    24 
    25 
    26 subsection \<open>Simple types\<close>
    27 
    28 schematic_lemma "A:* \<turnstile> A\<rightarrow>A : ?T"
    29   by (depth_solve rules)
    30 
    31 schematic_lemma "A:* \<turnstile> \<Lambda> a:A. a : ?T"
    32   by (depth_solve rules)
    33 
    34 schematic_lemma "A:* B:* b:B \<turnstile> \<Lambda> x:A. b : ?T"
    35   by (depth_solve rules)
    36 
    37 schematic_lemma "A:* b:A \<turnstile> (\<Lambda> a:A. a)^b: ?T"
    38   by (depth_solve rules)
    39 
    40 schematic_lemma "A:* B:* c:A b:B \<turnstile> (\<Lambda> x:A. b)^ c: ?T"
    41   by (depth_solve rules)
    42 
    43 schematic_lemma "A:* B:* \<turnstile> \<Lambda> a:A. \<Lambda> b:B. a : ?T"
    44   by (depth_solve rules)
    45 
    46 
    47 subsection \<open>Second-order types\<close>
    48 
    49 schematic_lemma (in L2) "\<turnstile> \<Lambda> A:*. \<Lambda> a:A. a : ?T"
    50   by (depth_solve rules)
    51 
    52 schematic_lemma (in L2) "A:* \<turnstile> (\<Lambda> B:*.\<Lambda> b:B. b)^A : ?T"
    53   by (depth_solve rules)
    54 
    55 schematic_lemma (in L2) "A:* b:A \<turnstile> (\<Lambda> B:*.\<Lambda> b:B. b) ^ A ^ b: ?T"
    56   by (depth_solve rules)
    57 
    58 schematic_lemma (in L2) "\<turnstile> \<Lambda> B:*.\<Lambda> a:(\<Pi> A:*.A).a ^ ((\<Pi> A:*.A)\<rightarrow>B) ^ a: ?T"
    59   by (depth_solve rules)
    60 
    61 
    62 subsection \<open>Weakly higher-order propositional logic\<close>
    63 
    64 schematic_lemma (in Lomega) "\<turnstile> \<Lambda> A:*.A\<rightarrow>A : ?T"
    65   by (depth_solve rules)
    66 
    67 schematic_lemma (in Lomega) "B:* \<turnstile> (\<Lambda> A:*.A\<rightarrow>A) ^ B : ?T"
    68   by (depth_solve rules)
    69 
    70 schematic_lemma (in Lomega) "B:* b:B \<turnstile> (\<Lambda> y:B. b): ?T"
    71   by (depth_solve rules)
    72 
    73 schematic_lemma (in Lomega) "A:* F:*\<rightarrow>* \<turnstile> F^(F^A): ?T"
    74   by (depth_solve rules)
    75 
    76 schematic_lemma (in Lomega) "A:* \<turnstile> \<Lambda> F:*\<rightarrow>*.F^(F^A): ?T"
    77   by (depth_solve rules)
    78 
    79 
    80 subsection \<open>LP\<close>
    81 
    82 schematic_lemma (in LP) "A:* \<turnstile> A \<rightarrow> * : ?T"
    83   by (depth_solve rules)
    84 
    85 schematic_lemma (in LP) "A:* P:A\<rightarrow>* a:A \<turnstile> P^a: ?T"
    86   by (depth_solve rules)
    87 
    88 schematic_lemma (in LP) "A:* P:A\<rightarrow>A\<rightarrow>* a:A \<turnstile> \<Pi> a:A. P^a^a: ?T"
    89   by (depth_solve rules)
    90 
    91 schematic_lemma (in LP) "A:* P:A\<rightarrow>* Q:A\<rightarrow>* \<turnstile> \<Pi> a:A. P^a \<rightarrow> Q^a: ?T"
    92   by (depth_solve rules)
    93 
    94 schematic_lemma (in LP) "A:* P:A\<rightarrow>* \<turnstile> \<Pi> a:A. P^a \<rightarrow> P^a: ?T"
    95   by (depth_solve rules)
    96 
    97 schematic_lemma (in LP) "A:* P:A\<rightarrow>* \<turnstile> \<Lambda> a:A. \<Lambda> x:P^a. x: ?T"
    98   by (depth_solve rules)
    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"
   101   by (depth_solve rules)
   102 
   103 schematic_lemma (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"
   105   by (depth_solve rules)
   106 
   107 
   108 subsection \<open>Omega-order types\<close>
   109 
   110 schematic_lemma (in L2) "A:* B:* \<turnstile> \<Pi> C:*.(A\<rightarrow>B\<rightarrow>C)\<rightarrow>C : ?T"
   111   by (depth_solve rules)
   112 
   113 schematic_lemma (in Lomega2) "\<turnstile> \<Lambda> A:*.\<Lambda> B:*.\<Pi> C:*.(A\<rightarrow>B\<rightarrow>C)\<rightarrow>C : ?T"
   114   by (depth_solve rules)
   115 
   116 schematic_lemma (in Lomega2) "\<turnstile> \<Lambda> A:*.\<Lambda> B:*.\<Lambda> x:A. \<Lambda> y:B. x : ?T"
   117   by (depth_solve rules)
   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))"
   120   apply (strip_asms rules)
   121   apply (rule lam_ss)
   122     apply (depth_solve1 rules)
   123    prefer 2
   124    apply (depth_solve1 rules)
   125   apply (rule lam_ss)
   126     apply (depth_solve1 rules)
   127    prefer 2
   128    apply (depth_solve1 rules)
   129   apply (rule lam_ss)
   130     apply assumption
   131    prefer 2
   132    apply (depth_solve1 rules)
   133   apply (erule pi_elim)
   134    apply assumption
   135   apply (erule pi_elim)
   136    apply assumption
   137   apply assumption
   138   done
   139 
   140 
   141 subsection \<open>Second-order Predicate Logic\<close>
   142 
   143 schematic_lemma (in LP2) "A:* P:A\<rightarrow>* \<turnstile> \<Lambda> a:A. P^a\<rightarrow>(\<Pi> A:*.A) : ?T"
   144   by (depth_solve rules)
   145 
   146 schematic_lemma (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"
   148   by (depth_solve rules)
   149 
   150 schematic_lemma (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"
   152   -- \<open>Antisymmetry implies irreflexivity:\<close>
   153   apply (strip_asms rules)
   154   apply (rule lam_ss)
   155     apply (depth_solve1 rules)
   156    prefer 2
   157    apply (depth_solve1 rules)
   158   apply (rule lam_ss)
   159     apply assumption
   160    prefer 2
   161    apply (depth_solve1 rules)
   162   apply (rule lam_ss)
   163     apply (depth_solve1 rules)
   164    prefer 2
   165    apply (depth_solve1 rules)
   166   apply (erule pi_elim, assumption, assumption?)+
   167   done
   168 
   169 
   170 subsection \<open>LPomega\<close>
   171 
   172 schematic_lemma (in LPomega) "A:* \<turnstile> \<Lambda> P:A\<rightarrow>A\<rightarrow>*.\<Lambda> a:A. P^a^a : ?T"
   173   by (depth_solve rules)
   174 
   175 schematic_lemma (in LPomega) "\<turnstile> \<Lambda> A:*.\<Lambda> P:A\<rightarrow>A\<rightarrow>*.\<Lambda> a:A. P^a^a : ?T"
   176   by (depth_solve rules)
   177 
   178 
   179 subsection \<open>Constructions\<close>
   180 
   181 schematic_lemma (in CC) "\<turnstile> \<Lambda> A:*.\<Lambda> P:A\<rightarrow>*.\<Lambda> a:A. P^a\<rightarrow>\<Pi> P:*.P: ?T"
   182   by (depth_solve rules)
   183 
   184 schematic_lemma (in CC) "\<turnstile> \<Lambda> A:*.\<Lambda> P:A\<rightarrow>*.\<Pi> a:A. P^a: ?T"
   185   by (depth_solve rules)
   186 
   187 schematic_lemma (in CC) "A:* P:A\<rightarrow>* a:A \<turnstile> ?p : (\<Pi> a:A. P^a)\<rightarrow>P^a"
   188   apply (strip_asms rules)
   189   apply (rule lam_ss)
   190     apply (depth_solve1 rules)
   191    prefer 2
   192    apply (depth_solve1 rules)
   193   apply (erule pi_elim, assumption, assumption)
   194   done
   195 
   196 
   197 subsection \<open>Some random examples\<close>
   198 
   199 schematic_lemma (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"
   201   by (depth_solve rules)
   202 
   203 schematic_lemma (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"
   205   by (depth_solve rules)
   206 
   207 schematic_lemma (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)"
   209   -- \<open>Symmetry of Leibnitz equality\<close>
   210   apply (strip_asms rules)
   211   apply (rule lam_ss)
   212     apply (depth_solve1 rules)
   213    prefer 2
   214    apply (depth_solve1 rules)
   215   apply (erule_tac a = "\<Lambda> x:A. \<Pi> Q:A\<rightarrow>*.Q^x\<rightarrow>Q^a" in pi_elim)
   216    apply (depth_solve1 rules)
   217   apply (unfold beta)
   218   apply (erule imp_elim)
   219    apply (rule lam_bs)
   220      apply (depth_solve1 rules)
   221     prefer 2
   222     apply (depth_solve1 rules)
   223    apply (rule lam_ss)
   224      apply (depth_solve1 rules)
   225     prefer 2
   226     apply (depth_solve1 rules)
   227    apply assumption
   228   apply assumption
   229   done
   230 
   231 end