theory Lambda

imports "../Parser"

begin

atom_decl name

nominal_datatype lam =

  Var "name"

| App "lam" "lam"

| Lam x::"name" l::"lam"  bind x in l

lemmas supp_fn' = lam.fv[simplified lam.supp]

declare lam.perm[eqvt]

section {* Strong Induction Principles*}

(* 

  Old way of establishing strong induction

  principles by chosing a fresh name.

*)

lemma

  fixes c::"'a::fs"

  assumes a1: "\<And>name c. P c (Var name)"

  and     a2: "\<And>lam1 lam2 c. \<lbrakk>\<And>d. P d lam1; \<And>d. P d lam2\<rbrakk> \<Longrightarrow> P c (App lam1 lam2)"

  and     a3: "\<And>name lam c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lam\<rbrakk> \<Longrightarrow> P c (Lam name lam)"

  shows "P c lam"

proof -

  have "\<And>p. P c (p \<bullet> lam)"

    apply(induct lam arbitrary: c rule: lam.induct)

    apply(perm_simp)

    apply(rule a1)

    apply(perm_simp)

    apply(rule a2)

    apply(assumption)

    apply(assumption)

    apply(subgoal_tac "\<exists>new::name. (atom new) \<sharp> (c, Lam (p \<bullet> name) (p \<bullet> lam))")

    defer

    apply(simp add: fresh_def)

    apply(rule_tac X="supp (c, Lam (p \<bullet> name) (p \<bullet> lam))" in obtain_at_base)

    apply(simp add: supp_Pair finite_supp)

    apply(blast)

    apply(erule exE)

    apply(rule_tac t="p \<bullet> Lam name lam" and 

                   s="(((p \<bullet> name) \<leftrightarrow> new) + p) \<bullet> Lam name lam" in subst)

    apply(simp del: lam.perm)

    apply(subst lam.perm)

    apply(subst (2) lam.perm)

    apply(rule flip_fresh_fresh)

    apply(simp add: fresh_def)

    apply(simp only: supp_fn')

    apply(simp)

    apply(simp add: fresh_Pair)

    apply(simp)

    apply(rule a3)

    apply(simp add: fresh_Pair)

    apply(drule_tac x="((p \<bullet> name) \<leftrightarrow> new) + p" in meta_spec)

    apply(simp)

    done

  then have "P c (0 \<bullet> lam)" by blast

  then show "P c lam" by simp

qed

(* 

  New way of establishing strong induction

  principles by using a appropriate permutation.

*)

lemma

  fixes c::"'a::fs"

  assumes a1: "\<And>name c. P c (Var name)"

  and     a2: "\<And>lam1 lam2 c. \<lbrakk>\<And>d. P d lam1; \<And>d. P d lam2\<rbrakk> \<Longrightarrow> P c (App lam1 lam2)"

  and     a3: "\<And>name lam c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lam\<rbrakk> \<Longrightarrow> P c (Lam name lam)"

  shows "P c lam"

proof -

  have "\<And>p. P c (p \<bullet> lam)"

    apply(induct lam arbitrary: c rule: lam.induct)

    apply(perm_simp)

    apply(rule a1)

    apply(perm_simp)

    apply(rule a2)

    apply(assumption)

    apply(assumption)

    apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom name}) \<sharp>* c \<and> supp (p \<bullet> Lam name lam) \<sharp>* q")

    apply(erule exE)

    apply(rule_tac t="p \<bullet> Lam name lam" and 

                   s="q \<bullet> p \<bullet> Lam name lam" in subst)

    defer

    apply(simp)

    apply(rule a3)

    apply(simp add: eqvts fresh_star_def)

    apply(drule_tac x="q + p" in meta_spec)

    apply(simp)

    apply(rule at_set_avoiding2)

    apply(simp add: finite_supp)

    apply(simp add: finite_supp)

    apply(simp add: finite_supp)

    apply(perm_simp)

    apply(simp add: fresh_star_def fresh_def supp_fn')

    apply(rule supp_perm_eq)

    apply(simp)

    done

  then have "P c (0 \<bullet> lam)" by blast

  then show "P c lam" by simp

qed

section {* Typing *}

nominal_datatype ty =

  TVar string

| TFun ty ty

notation

 TFun ("_ \<rightarrow> _") 

declare ty.perm[eqvt]

inductive

  valid :: "(name \<times> ty) list \<Rightarrow> bool"

where

  "valid []"

| "\<lbrakk>atom x \<sharp> Gamma; valid Gamma\<rbrakk> \<Longrightarrow> valid ((x, T)#Gamma)"

inductive

  typing :: "(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool" ("_ \<turnstile> _ : _" [60,60,60] 60) 

where

    t_Var[intro]: "\<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Var x : T"

  | t_App[intro]: "\<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2 \<or> \<Gamma> \<turnstile> t2 : T1\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : T2"

  | t_Lam[intro]: "\<lbrakk>atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam x t : T1 \<rightarrow> T2"

ML {*

fun map_term f t = 

  (case f t of

     NONE => map_term' f t 

   | x => x)

and map_term' f (t $ u) = 

    (case (map_term f t, map_term f u) of

        (NONE, NONE) => NONE

      | (SOME t'', NONE) => SOME (t'' $ u)

      | (NONE, SOME u'') => SOME (t $ u'')

      | (SOME t'', SOME u'') => SOME (t'' $ u''))

  | map_term' f (Abs (s, T, t)) = 

      (case map_term f t of

        NONE => NONE

      | SOME t'' => SOME (Abs (s, T, t'')))

  | map_term' _ _  = NONE;

fun map_thm_tac ctxt tac thm =

let

  val monos = Inductive.get_monos ctxt

in

  EVERY [cut_facts_tac [thm] 1, etac rev_mp 1,

    REPEAT_DETERM (FIRSTGOAL (resolve_tac monos)),

    REPEAT_DETERM (rtac impI 1 THEN (atac 1 ORELSE tac))]

end

(* 

 proves F[f t] from F[t] where F[t] is the given theorem  

  - F needs to be monotone

  - f returns either SOME for a term it fires 

    and NONE elsewhere 

*)

fun map_thm ctxt f tac thm =

let

  val opt_goal_trm = map_term f (prop_of thm)

  fun prove goal = 

    Goal.prove ctxt [] [] goal (fn _ => map_thm_tac ctxt tac thm)

in

  case opt_goal_trm of

    NONE => thm

  | SOME goal => prove goal

end

fun transform_prem ctxt names thm =

let

  fun split_conj names (Const ("op &", _) $ p $ q) = 

      (case head_of p of

         Const (name, _) => if name mem names then SOME q else NONE

       | _ => NONE)

  | split_conj _ _ = NONE;

in

  map_thm ctxt (split_conj names) (etac conjunct2 1) thm

end

*}

ML {*

open Nominal_Permeq

*}

ML {* 

fun single_case_tac ctxt pred_names pi intro  = 

let

in

  eqvt_strict_tac ctxt [] [] THEN' 

  SUBPROOF (fn {prems, context as ctxt, ...} =>

    let

      val prems' = map (transform_prem ctxt pred_names) prems

      val side_cond_tac = EVERY' 

        [ rtac rule, 

          eqvt_strict_tac ctxt @{thms permute_minus_cancel(2)} [],

          resolve_tac prems' ]

    in

      HEADGOAL (rtac intro THEN_ALL_NEW (resolve_tac prems' ORELSE' side_cond_tac)) 

    end) ctxt

end

*}

ML {*

let

  val thy = ProofContext.theory_of ctxt

  val ({names, ...}, {raw_induct, intrs, ...}) =

    Inductive.the_inductive ctxt (Sign.intern_const thy pred_name)

    ctxt |> Variable.import_terms false [concl_of raw_induct] 

  val preds = map (fst o HOLogic.dest_imp)

    (HOLogic.dest_conj (HOLogic.dest_Trueprop raw_concl));

in

end

*}

*}

thm eqvts

thm eqvts_raw

inductive

 alpha_lam_raw'

where

  "name = namea \<Longrightarrow> alpha_lam_raw' (Var_raw name) (Var_raw namea)"

| "\<lbrakk>alpha_lam_raw' lam_raw1 lam_raw1a; alpha_lam_raw' lam_raw2 lam_raw2a\<rbrakk> \<Longrightarrow>

   alpha_lam_raw' (App_raw lam_raw1 lam_raw2) (App_raw lam_raw1a lam_raw2a)"

| "\<exists>pi. ({atom name}, lam_raw) \<approx>gen alpha_lam_raw fv_lam_raw pi ({atom namea}, lam_rawa) \<Longrightarrow>

   alpha_lam_raw' (Lam_raw name lam_raw) (Lam_raw namea lam_rawa)"

lemma

  assumes a: "alpha_lam_raw' t1 t2"

  shows "alpha_lam_raw' (p \<bullet> t1) (p \<bullet> t2)"

using a

apply(induct)

oops

section {* size function *}

lemma size_eqvt_raw:

  fixes t::"lam_raw"

  shows "size (pi \<bullet> t)  = size t"

  apply (induct rule: lam_raw.inducts)

  apply simp_all

  done

instantiation lam :: size 

begin

quotient_definition

  "size_lam :: lam \<Rightarrow> nat"

is

  "size :: lam_raw \<Rightarrow> nat"

lemma size_rsp:

  "alpha_lam_raw x y \<Longrightarrow> size x = size y"

  apply (induct rule: alpha_lam_raw.inducts)

  apply (simp_all only: lam_raw.size)

  apply (simp_all only: alphas)

  apply clarify

  apply (simp_all only: size_eqvt_raw)

  done

lemma [quot_respect]:

  "(alpha_lam_raw ===> op =) size size"

  by (simp_all add: size_rsp)

lemma [quot_preserve]:

  "(rep_lam ---> id) size = size"

  by (simp_all add: size_lam_def)

instance

  by default

end

lemmas size_lam[simp] = 

  lam_raw.size(4)[quot_lifted]

  lam_raw.size(5)[quot_lifted]

  lam_raw.size(6)[quot_lifted]

(* is this needed? *)

lemma [measure_function]: 

  "is_measure (size::lam\<Rightarrow>nat)" 

  by (rule is_measure_trivial)

section {* Matching *}

definition

  MATCH :: "('c::pt \<Rightarrow> (bool * 'a::pt * 'b::pt)) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"

where

  "MATCH M d x \<equiv> if (\<exists>!r. \<exists>q. M q = (True, x, r)) then (THE r. \<exists>q. M q = (True, x, r)) else d"

(*

lemma MATCH_eqvt: