1 theory LamEx

     2 imports Nominal QuotMain

     3 begin

     4 

     5 atom_decl name

     6 

     7 thm abs_fresh(1)

     8 

     9 nominal_datatype rlam =

    10   rVar "name"

    11 | rApp "rlam" "rlam"

    12 | rLam "name" "rlam"

    13 

    14 print_theorems

    15 

    16 function

    17   rfv :: "rlam \<Rightarrow> name set"

    18 where

    19   rfv_var: "rfv (rVar a) = {a}"

    20 | rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)"

    21 | rfv_lam: "rfv (rLam a t) = (rfv t) - {a}"

    22 sorry

    23 

    24 termination rfv sorry

    25 

    26 inductive

    27   alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100)

    28 where

    29   a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)"

    30 | a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2"

    31 | a3: "\<lbrakk>t \<approx> ([(a,b)]\<bullet>s); a \<notin> rfv (rLam b t)\<rbrakk> \<Longrightarrow> rLam a t \<approx> rLam b s"

    32 

    33 print_theorems

    34 

    35 lemma alpha_refl:

    36   fixes t::"rlam"

    37   shows "t \<approx> t"

    38   apply(induct t rule: rlam.induct)

    39   apply(simp add: a1)

    40   apply(simp add: a2)

    41   apply(rule a3)

    42   apply(subst pt_swap_bij'')

    43   apply(rule pt_name_inst)

    44   apply(rule at_name_inst)

    45   apply(simp)

    46   apply(simp)

    47   done

    48 

    49 lemma alpha_equivp:

    50   shows "equivp alpha"

    51 sorry

    52 

    53 quotient lam = rlam / alpha

    54   apply(rule alpha_equivp)

    55   done

    56 

    57 print_quotients

    58 

    59 quotient_def 

    60   Var :: "name \<Rightarrow> lam"

    61 where

    62   "Var \<equiv> rVar"

    63 

    64 quotient_def 

    65   App :: "lam \<Rightarrow> lam \<Rightarrow> lam"

    66 where

    67   "App \<equiv> rApp"

    68 

    69 quotient_def 

    70   Lam :: "name \<Rightarrow> lam \<Rightarrow> lam"

    71 where

    72   "Lam \<equiv> rLam"

    73 

    74 thm Var_def

    75 thm App_def

    76 thm Lam_def

    77 

    78 quotient_def 

    79   fv :: "lam \<Rightarrow> name set"

    80 where

    81   "fv \<equiv> rfv"

    82 

    83 thm fv_def

    84 

    85 (* definition of overloaded permutation function *)

    86 (* for the lifted type lam                       *)

    87 overloading

    88   perm_lam \<equiv> "perm :: 'x prm \<Rightarrow> lam \<Rightarrow> lam"   (unchecked)

    89 begin

    90 

    91 quotient_def 

    92   perm_lam :: "'x prm \<Rightarrow> lam \<Rightarrow> lam"

    93 where

    94   "perm_lam \<equiv> (perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam)"

    95 

    96 end

    97 

    98 (*quotient_def (for lam)

    99   abs_fun_lam :: "'x prm \<Rightarrow> lam \<Rightarrow> lam"

   100 where

   101   "perm_lam \<equiv> (perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam)"*)

   102 

   103 

   104 thm perm_lam_def

   105 

   106 (* lemmas that need to lift *)

   107 lemma pi_var_com:

   108   fixes pi::"'x prm"

   109   shows "(pi\<bullet>rVar a) \<approx> rVar (pi\<bullet>a)"

   110   sorry

   111 

   112 lemma pi_app_com:

   113   fixes pi::"'x prm"

   114   shows "(pi\<bullet>rApp t1 t2) \<approx> rApp (pi\<bullet>t1) (pi\<bullet>t2)"

   115   sorry

   116 

   117 lemma pi_lam_com:

   118   fixes pi::"'x prm"

   119   shows "(pi\<bullet>rLam a t) \<approx> rLam (pi\<bullet>a) (pi\<bullet>t)"

   120   sorry

   121 

   122 

   123 

   124 lemma real_alpha:

   125   assumes a: "t = [(a,b)]\<bullet>s" "a\<sharp>[b].s"

   126   shows "Lam a t = Lam b s"

   127 using a

   128 unfolding fresh_def supp_def

   129 sorry

   130 

   131 lemma perm_rsp[quotient_rsp]:

   132   "(op = ===> alpha ===> alpha) op \<bullet> op \<bullet>"

   133   apply(auto)

   134   (* this is propably true if some type conditions are imposed ;o) *)

   135   sorry

   136 

   137 lemma fresh_rsp:

   138   "(op = ===> alpha ===> op =) fresh fresh"

   139   apply(auto)

   140   (* this is probably only true if some type conditions are imposed *)

   141   sorry

   142 

   143 lemma rVar_rsp[quotient_rsp]:

   144   "(op = ===> alpha) rVar rVar"

   145 by (auto intro:a1)

   146 

   147 lemma rApp_rsp[quotient_rsp]: "(alpha ===> alpha ===> alpha) rApp rApp"

   148 by (auto intro:a2)

   149 

   150 lemma rLam_rsp[quotient_rsp]: "(op = ===> alpha ===> alpha) rLam rLam"

   151   apply(auto)

   152   apply(rule a3)

   153   apply(rule_tac t="[(x,x)]\<bullet>y" and s="y" in subst)

   154   apply(rule sym)

   155   apply(rule trans)

   156   apply(rule pt_name3)

   157   apply(rule at_ds1[OF at_name_inst])

   158   apply(simp add: pt_name1)

   159   apply(assumption)

   160   apply(simp add: abs_fresh)

   161   done

   162 

   163 lemma rfv_rsp[quotient_rsp]: "(alpha ===> op =) rfv rfv"

   164   sorry

   165 

   166 lemma rvar_inject: "rVar a \<approx> rVar b = (a = b)"

   167 apply (auto)

   168 apply (erule alpha.cases)

   169 apply (simp_all add: rlam.inject alpha_refl)

   170 done

   171 

   172 ML {* val qty = @{typ "lam"} *}

   173 ML {* val rsp_thms = @{thms perm_rsp fresh_rsp rVar_rsp rApp_rsp rLam_rsp rfv_rsp} *}

   174 

   175 ML {* val (rty, rel, rel_refl, rel_eqv) = lookup_quot_data @{context} qty *}

   176 ML {* val (trans2, reps_same, absrep, quot) = lookup_quot_thms @{context} "lam" *}

   177 ML {* fun lift_tac_lam lthy t = lift_tac lthy t *}

   178 

   179 lemma pi_var: "(pi\<Colon>('x \<times> 'x) list) \<bullet> Var a = Var (pi \<bullet> a)"

   180 apply (tactic {* lift_tac_lam @{context} @{thm pi_var_com} 1 *})

   181 done

   182 

   183 lemma pi_app: "(pi\<Colon>('x \<times> 'x) list) \<bullet> App (x\<Colon>lam) (xa\<Colon>lam) = App (pi \<bullet> x) (pi \<bullet> xa)"

   184 apply (tactic {* lift_tac_lam @{context} @{thm pi_app_com} 1 *})

   185 done

   186 

   187 lemma pi_lam: "(pi\<Colon>('x \<times> 'x) list) \<bullet> Lam (a\<Colon>name) (x\<Colon>lam) = Lam (pi \<bullet> a) (pi \<bullet> x)"

   188 apply (tactic {* lift_tac_lam @{context} @{thm pi_lam_com} 1 *})

   189 done

   190 

   191 lemma fv_var: "fv (Var (a\<Colon>name)) = {a}"

   192 apply (tactic {* lift_tac_lam @{context} @{thm rfv_var} 1 *})

   193 done

   194 

   195 lemma fv_app: "fv (App (x\<Colon>lam) (xa\<Colon>lam)) = fv x \<union> fv xa"

   196 apply (tactic {* lift_tac_lam @{context} @{thm rfv_app} 1 *})

   197 done

   198 

   199 lemma fv_lam: "fv (Lam (a\<Colon>name) (x\<Colon>lam)) = fv x - {a}"

   200 apply (tactic {* lift_tac_lam @{context} @{thm rfv_lam} 1 *})

   201 done

   202 

   203 lemma a1: "(a\<Colon>name) = (b\<Colon>name) \<Longrightarrow> Var a = Var b"

   204 apply (tactic {* lift_tac_lam @{context} @{thm a1} 1 *})

   205 done

   206 

   207 lemma a2: "\<lbrakk>(x\<Colon>lam) = (xa\<Colon>lam); (xb\<Colon>lam) = (xc\<Colon>lam)\<rbrakk> \<Longrightarrow> App x xb = App xa xc"

   208 apply (tactic {* lift_tac_lam @{context} @{thm a2} 1 *})

   209 done

   210 

   211 lemma a3: "\<lbrakk>(x\<Colon>lam) = [(a\<Colon>name, b\<Colon>name)] \<bullet> (xa\<Colon>lam); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> Lam a x = Lam b xa"

   212 apply (tactic {* lift_tac_lam @{context} @{thm a3} 1 *})

   213 done

   214 

   215 lemma alpha_cases: "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P;

   216      \<And>x xa xb xc. \<lbrakk>a1 = App x xb; a2 = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P;

   217      \<And>x a b xa. \<lbrakk>a1 = Lam a x; a2 = Lam b xa; x = [(a, b)] \<bullet> xa; a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> P\<rbrakk>

   218     \<Longrightarrow> P"

   219 apply (tactic {* lift_tac_lam @{context} @{thm alpha.cases} 1 *})

   220 done

   221 

   222 lemma alpha_induct: "\<lbrakk>(qx\<Colon>lam) = (qxa\<Colon>lam); \<And>(a\<Colon>name) b\<Colon>name. a = b \<Longrightarrow> (qxb\<Colon>lam \<Rightarrow> lam \<Rightarrow> bool) (Var a) (Var b);

   223      \<And>(x\<Colon>lam) (xa\<Colon>lam) (xb\<Colon>lam) xc\<Colon>lam. \<lbrakk>x = xa; qxb x xa; xb = xc; qxb xb xc\<rbrakk> \<Longrightarrow> qxb (App x xb) (App xa xc);

   224      \<And>(x\<Colon>lam) (a\<Colon>name) (b\<Colon>name) xa\<Colon>lam.

   225         \<lbrakk>x = [(a, b)] \<bullet> xa; qxb x ([(a, b)] \<bullet> xa); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> qxb (Lam a x) (Lam b xa)\<rbrakk>

   226     \<Longrightarrow> qxb qx qxa"

   227 apply (tactic {* lift_tac_lam @{context} @{thm alpha.induct} 1 *})

   228 done

   229 

   230 lemma var_inject: "(Var a = Var b) = (a = b)"

   231 apply (tactic {* lift_tac_lam @{context} @{thm rvar_inject} 1 *})

   232 done

   233 

   234 lemma lam_induct:" \<lbrakk>\<And>name. P (Var name); \<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (App lam1 lam2);

   235               \<And>name lam. P lam \<Longrightarrow> P (Lam name lam)\<rbrakk> \<Longrightarrow> P lam"

   236 apply (tactic {* lift_tac_lam @{context} @{thm rlam.induct} 1 *})

   237 done

   238 

   239 lemma var_supp:

   240   shows "supp (Var a) = ((supp a)::name set)"

   241   apply(simp add: supp_def)

   242   apply(simp add: pi_var)

   243   apply(simp add: var_inject)

   244   done

   245 

   246 lemma var_fresh:

   247   fixes a::"name"

   248   shows "(a\<sharp>(Var b)) = (a\<sharp>b)"

   249   apply(simp add: fresh_def)

   250   apply(simp add: var_supp)

   251   done

   252