theory Lambda
imports "../Nominal2" 
begin


atom_decl name

nominal_datatype lam =
  Var "name"
| App "lam" "lam"
| Lam x::"name" l::"lam"  bind x in l

thm lam.distinct
thm lam.induct
thm lam.exhaust lam.strong_exhaust
thm lam.fv_defs
thm lam.bn_defs
thm lam.perm_simps
thm lam.eq_iff
thm lam.fv_bn_eqvt
thm lam.size_eqvt

ML {*
fun mk_cminus p = Thm.capply @{cterm "uminus::perm \<Rightarrow> perm"} p 

fun minus_permute_intro_tac p = 
  rtac (Drule.instantiate' [] [SOME (mk_cminus p)] @{thm permute_boolE})

fun minus_permute_elim p thm = 
  thm RS (Drule.instantiate' [] [NONE, SOME (mk_cminus p)] @{thm permute_boolI})
*}

ML {*
fun real_head_of (@{term Trueprop} $ t) = real_head_of t
  | real_head_of (Const ("==>", _) $ _ $ t) = real_head_of t
  | real_head_of (Const (@{const_name all}, _) $ Abs (_, _, t)) = real_head_of t
  | real_head_of (Const (@{const_name All}, _) $ Abs (_, _, t)) = real_head_of t
  | real_head_of (Const ("HOL.induct_forall", _) $ Abs (_, _, t)) = real_head_of t
  | real_head_of t = head_of t  
*}

ML {* 
fun mk_vc_compat (avoid, avoid_trm) prems concl_args params = 
  let
    val vc_goal = concl_args
      |> HOLogic.mk_tuple
      |> mk_fresh_star avoid_trm 
      |> HOLogic.mk_Trueprop
      |> (curry Logic.list_implies) prems
      |> (curry list_all_free) params
  in 
    if null avoid then [] else [vc_goal]
  end
*}

ML {*
fun map_term prop f trm =
  if prop trm 
  then f trm
  else case trm of
    (t1 $ t2) => map_term prop f t1 $ map_term prop f t2
  | Abs (x, T, t) => Abs (x, T, map_term prop f t)
  | _ => trm
*}

ML {*
fun add_p_c p (c, c_ty) trm =
  let
    val (P, args) = strip_comb trm
    val (P_name, P_ty) = dest_Free P
    val (ty_args, bool) = strip_type P_ty
    val args' = map (mk_perm p) args
  in
    list_comb (Free (P_name, (c_ty :: ty_args) ---> bool),  c :: args')
    |> (fn t => HOLogic.all_const c_ty $ lambda c t )
    |> (fn t => HOLogic.all_const @{typ perm} $  lambda p t)
  end
*}

ML {*
fun induct_forall_const T = Const ("HOL.induct_forall", (T --> @{typ bool}) --> @{typ bool})
fun mk_induct_forall (a, T) t =  induct_forall_const T $ Abs (a, T, t)
*}

ML {*
fun add_c_prop qnt Ps (c, c_name, c_ty) trm =
  let
    fun add t = 
      let
        val (P, args) = strip_comb t
        val (P_name, P_ty) = dest_Free P
        val (ty_args, bool) = strip_type P_ty
        val args' = args
          |> qnt ? map (incr_boundvars 1)
      in
        list_comb (Free (P_name, (c_ty :: ty_args) ---> bool), c :: args')
        |> qnt ? mk_induct_forall (c_name, c_ty)
      end
  in
    map_term (member (op =) Ps o head_of) add trm
  end
*}

ML {*
fun prep_prem Ps c_name c_ty (avoid, avoid_trm) (params, prems, concl) =
  let
    val prems' = prems
      |> map (incr_boundvars 1) 
      |> map (add_c_prop true Ps (Bound 0, c_name, c_ty))

    val avoid_trm' = avoid_trm
      |> (curry list_abs_free) (params @ [(c_name, c_ty)])
      |> strip_abs_body
      |> (fn t => mk_fresh_star_ty c_ty t (Bound 0))
      |> HOLogic.mk_Trueprop

    val prems'' = 
      if null avoid 
      then prems' 
      else avoid_trm' :: prems'

    val concl' = concl
      |> incr_boundvars 1 
      |> add_c_prop false Ps (Bound 0, c_name, c_ty)  
  in
    mk_full_horn (params @ [(c_name, c_ty)]) prems'' concl'
  end
*}


ML {*
fun same_name (Free (a1, _), Free (a2, _)) = (a1 = a2)
  | same_name (Var (a1, _), Var (a2, _)) = (a1 = a2)
  | same_name (Const (a1, _), Const (a2, _)) = (a1 = a2)
  | same_name _ = false
*}

ML {*
(* local abbreviations *)
fun eqvt_stac ctxt = Nominal_Permeq.eqvt_strict_tac ctxt @{thms permute_minus_cancel} []  
fun eqvt_srule ctxt = Nominal_Permeq.eqvt_strict_rule ctxt @{thms permute_minus_cancel} []  
*}

ML {*
val all_elims = 
  let
     fun spec' ct = Drule.instantiate' [SOME (ctyp_of_term ct)] [NONE, SOME ct] @{thm spec}
  in
    fold (fn ct => fn th => th RS spec' ct)
  end
*}

ML {*
fun helper_tac flag prm p ctxt =
  Subgoal.SUBPROOF (fn {context, prems, ...} =>
    let
      val prems' = prems
        |> map (minus_permute_elim p)
        |> map (eqvt_srule context)
     
      val prm' = (prems' MRS prm)
        |> flag ? (all_elims [p])
        |> flag ? (simplify (HOL_basic_ss addsimps @{thms permute_minus_cancel}))
    in
      simp_tac (HOL_ss addsimps (prm' :: @{thms induct_forall_def })) 1
    end) ctxt
*}

ML {*
fun non_binder_tac prem intr_cvars Ps ctxt = 
  Subgoal.SUBPROOF (fn {context, params, prems, ...} =>
    let
      val thy = ProofContext.theory_of context
      val (prms, p, _) = split_last2 (map snd params)
      val prm_tys = map (fastype_of o term_of) prms
      val cperms = map (cterm_of thy o perm_const) prm_tys
      val p_prms = map2 (fn ct1 => fn ct2 => Thm.mk_binop ct1 p ct2) cperms prms 
      val prem' = cterm_instantiate (intr_cvars ~~ p_prms) prem

      (* for inductive-premises*)
      fun tac1 prm = helper_tac true prm p context

      (* for non-inductive premises *)   
      fun tac2 prm =  
        EVERY' [ minus_permute_intro_tac p, 
                 eqvt_stac context, 
                 helper_tac false prm p context ]

      fun select prm (t, i) =
        (if member same_name Ps (real_head_of t) then tac1 prm else tac2 prm) i
    in
      EVERY1 [rtac prem', RANGE (map (SUBGOAL o select) prems) ]
    end) ctxt
*}


ML {*
fun fresh_thm ctxt fresh_thms p c prms avoid_trm =
  let
    val conj1 = 
      mk_fresh_star (mk_perm (Bound 0) (mk_perm p avoid_trm)) c
    val conj2 =
      mk_fresh_star_ty @{typ perm} (mk_supp (HOLogic.mk_tuple (map (mk_perm p) prms))) (Bound 0)
    val fresh_goal = mk_exists ("q", @{typ perm}) (HOLogic.mk_conj (conj1, conj2))
      |> HOLogic.mk_Trueprop

    val _ = tracing ("fresh goal: " ^ Syntax.string_of_term ctxt fresh_goal)
  in 
    Goal.prove ctxt [] [] fresh_goal
      (K (HEADGOAL (rtac @{thm at_set_avoiding2})))
  end
*}

ML {*
fun binder_tac prem intr_cvars param_trms Ps fresh_thms avoid avoid_trm ctxt = 
  Subgoal.FOCUS (fn {context, params, ...} =>
    let
      val thy = ProofContext.theory_of context
      val (prms, p, c) = split_last2 (map snd params)
      val prm_trms = map term_of prms
      val prm_tys = map fastype_of prm_trms
      val cperms = map (cterm_of thy o perm_const) prm_tys
      val p_prms = map2 (fn ct1 => fn ct2 => Thm.mk_binop ct1 p ct2) cperms prms 
      val prem' = cterm_instantiate (intr_cvars ~~ p_prms) prem
      val avoid_trm' = subst_free (param_trms ~~ prm_trms) avoid_trm      

      val fthm = fresh_thm context fresh_thms (term_of p) (term_of c) (map term_of prms) avoid_trm'
    in
      Skip_Proof.cheat_tac thy
      (* EVERY1 [rtac prem'] *)  
    end) ctxt
*}

ML {*
fun case_tac ctxt fresh_thms Ps (avoid, avoid_trm) intr_cvars param_trms prem =
  let
    val tac1 = non_binder_tac prem intr_cvars Ps ctxt
    val tac2 = binder_tac prem intr_cvars param_trms Ps fresh_thms avoid avoid_trm ctxt
  in 
    EVERY' [ rtac @{thm allI}, rtac @{thm allI}, eqvt_stac ctxt,
             if null avoid then tac1 else tac2 ]
  end
*}

ML {*
fun prove_sinduct_tac raw_induct fresh_thms Ps avoids avoid_trms intr_cvars param_trms {prems, context} =
  let
    val cases_tac = map4 (case_tac context fresh_thms Ps) (avoids ~~avoid_trms) intr_cvars param_trms prems
  in 
    EVERY1 [ DETERM o rtac raw_induct, RANGE cases_tac ]
  end
*}

ML {*
val normalise = @{lemma "(Q --> (!p c. P p c)) ==> (!!c. Q ==> P (0::perm) c)" by simp}
*}

ML {*
fun prove_strong_inductive rule_names avoids raw_induct intrs ctxt =
  let
    val thy = ProofContext.theory_of ctxt
    val ((_, [raw_induct']), ctxt') = Variable.import true [raw_induct] ctxt

    val (ind_prems, ind_concl) = raw_induct'
      |> prop_of
      |> Logic.strip_horn
      |>> map strip_full_horn
    val params = map (fn (x, _, _) => x) ind_prems
    val param_trms = (map o map) Free params  

    val intr_vars_tys = map (fn t => rev (Term.add_vars (prop_of t) [])) intrs
    val intr_vars = (map o map) fst intr_vars_tys
    val intr_vars_substs = map2 (curry (op ~~)) intr_vars param_trms
    val intr_cvars = (map o map) (cterm_of thy o Var) intr_vars_tys      

    val (intr_prems, intr_concls) = intrs
      |> map prop_of
      |> map2 subst_Vars intr_vars_substs
      |> map Logic.strip_horn
      |> split_list

    val intr_concls_args = map (snd o strip_comb o HOLogic.dest_Trueprop) intr_concls 
      
    val avoid_trms = avoids
      |> (map o map) (setify ctxt') 
      |> map fold_union

    val vc_compat_goals = 
      map4 mk_vc_compat (avoids ~~ avoid_trms) intr_prems intr_concls_args params

    val ([c_name, a, p], ctxt'') = Variable.variant_fixes ["c", "'a", "p"] ctxt'
    val c_ty = TFree (a, @{sort fs})
    val c = Free (c_name, c_ty)
    val p = Free (p, @{typ perm})

    val (preconds, ind_concls) = ind_concl
      |> HOLogic.dest_Trueprop
      |> HOLogic.dest_conj 
      |> map HOLogic.dest_imp
      |> split_list

    val Ps = map (fst o strip_comb) ind_concls

    val ind_concl' = ind_concls
      |> map (add_p_c p (c, c_ty))
      |> (curry (op ~~)) preconds  
      |> map HOLogic.mk_imp
      |> fold_conj
      |> HOLogic.mk_Trueprop

    val ind_prems' = ind_prems
      |> map2 (prep_prem Ps c_name c_ty) (avoids ~~ avoid_trms)   

    fun after_qed ctxt_outside fresh_thms ctxt = 
      let
        val thms = Goal.prove ctxt [] ind_prems' ind_concl' 
          (prove_sinduct_tac raw_induct fresh_thms Ps avoids avoid_trms intr_cvars param_trms) 
          |> singleton (ProofContext.export ctxt ctxt_outside)
          |> Datatype_Aux.split_conj_thm
          |> map (fn thm => thm RS normalise)
          |> map (asm_full_simplify (HOL_basic_ss addsimps @{thms permute_zero induct_rulify})) 
          |> map (Drule.rotate_prems (length ind_prems'))
          |> map zero_var_indexes

        val _ = tracing ("RESULTS\n" ^ cat_lines (map (Syntax.string_of_term ctxt o prop_of) thms))
      in
        ctxt
      end 
  in
    Proof.theorem NONE (after_qed ctxt) ((map o map) (rpair []) vc_compat_goals) ctxt''
  end
*}

ML {*
fun prove_strong_inductive_cmd (pred_name, avoids) ctxt =
  let
    val thy = ProofContext.theory_of ctxt;
    val ({names, ...}, {raw_induct, intrs, ...}) =
      Inductive.the_inductive ctxt (Sign.intern_const thy pred_name);

    val rule_names = 
      hd names
      |> the o Induct.lookup_inductP ctxt
      |> fst o Rule_Cases.get
      |> map fst

    val _ = (case duplicates (op = o pairself fst) avoids of
        [] => ()
      | xs => error ("Duplicate case names: " ^ commas_quote (map fst xs)))

    val _ = (case subtract (op =) rule_names (map fst avoids) of
        [] => ()
      | xs => error ("No such case(s) in inductive definition: " ^ commas_quote xs))

    val avoids_ordered = order_default (op =) [] rule_names avoids
      
    fun read_avoids avoid_trms intr =
      let
        (* fixme hack *)
        val (((_, ctrms), _), ctxt') = Variable.import true [intr] ctxt
        val trms = map (term_of o snd) ctrms
        val ctxt'' = fold Variable.declare_term trms ctxt' 
      in
        map (Syntax.read_term ctxt'') avoid_trms 
      end 

    val avoid_trms = map2 read_avoids avoids_ordered intrs
  in
    prove_strong_inductive rule_names avoid_trms raw_induct intrs ctxt
  end
*}

ML {*
(* outer syntax *)
local
  structure P = Parse;
  structure S = Scan
  
  val _ = Keyword.keyword "avoids"

  val single_avoid_parser = 
    P.name -- (P.$$$ ":" |-- P.and_list1 P.term)

  val avoids_parser = 
    S.optional (P.$$$ "avoids" |-- P.enum1 "|" single_avoid_parser) []

  val main_parser = P.xname -- avoids_parser
in
  val _ =
  Outer_Syntax.local_theory_to_proof "nominal_inductive"
    "prove strong induction theorem for inductive predicate involving nominal datatypes"
      Keyword.thy_goal (main_parser >> prove_strong_inductive_cmd)
end
*}

inductive
  Acc :: "('a::pt \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
where
  AccI: "(\<And>y. R y x \<Longrightarrow> Acc R y) \<Longrightarrow> Acc R x"

(*
equivariance Acc
*)

lemma Acc_eqvt [eqvt]:
  fixes p::"perm"
  assumes a: "Acc R x"
  shows "Acc (p \<bullet> R) (p \<bullet> x)"
using a
apply(induct)
apply(rule AccI)
apply(rotate_tac 1)
apply(drule_tac x="-p \<bullet> y" in meta_spec)
apply(simp)
apply(drule meta_mp)
apply(rule_tac p="p" in permute_boolE)
apply(perm_simp add: permute_minus_cancel)
apply(assumption)
apply(assumption)
done
 

nominal_inductive Acc .

section {* Typing *}

nominal_datatype ty =
  TVar string
| TFun ty ty ("_ \<rightarrow> _") 

lemma ty_fresh:
  fixes x::"name"
  and   T::"ty"
  shows "atom x \<sharp> T"
apply (nominal_induct T rule: ty.strong_induct)
apply (simp_all add: ty.fresh pure_fresh)
done



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; \<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"

thm typing.intros
thm typing.induct



equivariance valid
equivariance typing


nominal_inductive typing
  avoids t_Lam: "x"
      (* | t_Var: "x" *)
  apply -
  apply(simp_all add: fresh_star_def ty_fresh lam.fresh)?
  done



lemma
  fixes c::"'a::fs"
  assumes a: "\<Gamma> \<turnstile> t : T" 
  and a1: "\<And>\<Gamma> x T c. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P c \<Gamma> (Var x) T"
  and a2: "\<And>\<Gamma> t1 T1 T2 t2 c. \<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<And>d. P d \<Gamma> t1 T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1; \<And>d. P d \<Gamma> t2 T1\<rbrakk> 
           \<Longrightarrow> P c \<Gamma> (App t1 t2) T2"
  and a3: "\<And>x \<Gamma> T1 t T2 c. \<lbrakk>{atom x} \<sharp>* c; atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2; \<And>d. P d ((x, T1) # \<Gamma>) t T2\<rbrakk> 
           \<Longrightarrow> P c \<Gamma> (Lam x t) T1 \<rightarrow> T2"
  shows "P c \<Gamma> t T"
proof -
  from a have "\<And>p c. P c (p \<bullet> \<Gamma>) (p \<bullet> t) (p \<bullet> T)"
  proof (induct)
    case (t_Var \<Gamma> x T p c)
    then show ?case
      apply -
      apply(perm_strict_simp)
      thm a1
      apply(rule a1)
      apply(drule_tac p="p" in permute_boolI)
      apply(perm_strict_simp add: permute_minus_cancel)
      apply(assumption)
      apply(rotate_tac 1)
      apply(drule_tac p="p" in permute_boolI)
      apply(perm_strict_simp add: permute_minus_cancel)
      apply(assumption)
      done
  next
    case (t_App \<Gamma> t1 T1 T2 t2 p c)
    then show ?case
      apply -
      apply(perm_strict_simp)
      apply(rule a2)
      apply(drule_tac p="p" in permute_boolI)
      apply(perm_strict_simp add: permute_minus_cancel)
      apply(assumption)
      apply(assumption)
      apply(rotate_tac 2)
      apply(drule_tac p="p" in permute_boolI)
      apply(perm_strict_simp add: permute_minus_cancel)
      apply(assumption)
      apply(assumption)
      done
  next
    case (t_Lam x \<Gamma> T1 t T2 p c)
    then show ?case
      apply -
      apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom x}) \<sharp>* c \<and> 
        supp (p \<bullet> \<Gamma>, p \<bullet> Lam x t, p \<bullet> (T1 \<rightarrow> T2)) \<sharp>* q")
      apply(erule exE)
      apply(rule_tac t="p \<bullet> \<Gamma>" and  s="(q + p) \<bullet> \<Gamma>" in subst)
      apply(simp only: permute_plus)
      apply(rule supp_perm_eq)
      apply(simp add: supp_Pair fresh_star_Un)
      apply(rule_tac t="p \<bullet> Lam x t" and  s="(q + p) \<bullet> Lam x t" in subst)
      apply(simp only: permute_plus)
      apply(rule supp_perm_eq)
      apply(simp add: supp_Pair fresh_star_Un)
      apply(rule_tac t="p \<bullet> (T1 \<rightarrow> T2)" and  s="(q + p) \<bullet> (T1 \<rightarrow> T2)" in subst)
      apply(simp only: permute_plus)
      apply(rule supp_perm_eq)
      apply(simp add: supp_Pair fresh_star_Un)
      (* apply(perm_simp) *) 
      apply(simp (no_asm) only: eqvts)
      apply(rule a3)
      apply(simp only: eqvts permute_plus)
      apply(rule_tac p="- (q + p)" in permute_boolE)
      apply(perm_strict_simp add: permute_minus_cancel)
      apply(assumption)
       apply(rule_tac p="- (q + p)" in permute_boolE)
      apply(perm_strict_simp add: permute_minus_cancel)
      apply(assumption)
      apply(perm_strict_simp)
      apply(simp only:)
      thm at_set_avoiding2
      apply(rule at_set_avoiding2)
      apply(simp add: finite_supp)
      apply(simp add: finite_supp)
      apply(simp add: finite_supp)
      apply(rule_tac p="-p" in permute_boolE)
      apply(perm_strict_simp add: permute_minus_cancel)
	--"supplied by the user"
      apply(simp add: fresh_star_Pair)
      sorry
  qed
  then have "P c (0 \<bullet> \<Gamma>) (0 \<bullet> t) (0 \<bullet> T)" .
  then show "P c \<Gamma> t T" by simp
qed

*)


end