1 theory Lambda 

     2   imports "Nominal" 

     3 begin

     4 

     5 atom_decl name

     6 

     7 section {* Alpha-Equated Lambda-Terms *}

     8 

     9 nominal_datatype lam =

    10   Var "name"

    11 | App "lam" "lam" 

    12 | Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._")

    13 

    14 section {* Capture-Avoiding Substitution *}

    15 

    16 consts subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam"  ("_[_::=_]")

    17 

    18 nominal_primrec

    19   "(Var x)[y::=s] = (if x=y then s else (Var x))"

    20   "(App t1 t2)[y::=s] = App (t1[y::=s]) (t2[y::=s])"

    21   "x\<sharp>(y,s) \<Longrightarrow> (Lam [x].t)[y::=s] = Lam [x].(t[y::=s])"

    22 apply(finite_guess)+

    23 apply(rule TrueI)+

    24 apply(simp add: abs_fresh)+

    25 apply(fresh_guess)+

    26 done

    27 

    28 lemma  subst_eqvt[eqvt]:

    29   fixes pi::"name prm"

    30   shows "pi\<bullet>(t1[x::=t2]) = (pi\<bullet>t1)[(pi\<bullet>x)::=(pi\<bullet>t2)]"

    31 by (nominal_induct t1 avoiding: x t2 rule: lam.strong_induct)

    32    (auto simp add: perm_bij fresh_atm fresh_bij)

    33 

    34 lemma forget: 

    35   assumes a: "x\<sharp>L"

    36   shows "L[x::=P] = L"

    37   using a 

    38 by (nominal_induct L avoiding: x P rule: lam.strong_induct)

    39    (auto simp add: abs_fresh fresh_atm)

    40 

    41 lemma fresh_fact:

    42   fixes z::"name"

    43   shows "\<lbrakk>z\<sharp>s; (z=y \<or> z\<sharp>t)\<rbrakk> \<Longrightarrow> z\<sharp>t[y::=s]"

    44 by (nominal_induct t avoiding: z y s rule: lam.strong_induct)

    45    (auto simp add: abs_fresh fresh_prod fresh_atm)

    46 

    47 lemma subst_rename: 

    48   assumes a: "y\<sharp>t"

    49   shows "t[x::=s] = ([(y,x)]\<bullet>t)[y::=s]"

    50 using a 

    51 by (nominal_induct t avoiding: x y s rule: lam.strong_induct)

    52      (auto simp add: calc_atm fresh_atm abs_fresh)

    53 

    54 text {* 

    55   The purpose of the two lemmas below is to work

    56   around some quirks in Isabelle's handling of 

    57   meta_quantifiers and meta_implications. 

    58   *} 

    59 

    60 lemma meta_impCE:

    61   assumes major: "P ==> PROP Q"

    62     and 1: "~ P ==> R"

    63     and 2: "PROP Q ==> R"

    64   shows R

    65 proof (cases P)

    66   assume P

    67   then have "PROP Q" by (rule major)

    68   then show R by (rule 2)

    69 next

    70   assume "~ P"

    71   then show R by (rule 1)

    72 qed

    73 

    74 declare meta_allE [elim]

    75     and meta_impCE [elim!]

    76 

    77 end    

    78    

    79 

    80 

    81 

    82   

    83 

    84   

    85   

    86 

    87