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 {* 

  let

      |> mk_fresh_star avoid_trm 

      |> HOLogic.mk_Trueprop

      |> (curry Logic.list_implies) prems

      |> (curry list_all_free) params

  in 

  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 {*

  Subgoal.FOCUS (fn {context, params, ...} =>

    let

      val thy = ProofContext.theory_of context

      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

    in

      Skip_Proof.cheat_tac thy

      (* EVERY1 [rtac prem'] *)  

    end) ctxt

*}

ML {*

  let

    val tac1 = non_binder_tac prem intr_cvars Ps ctxt

  in 

    EVERY' [ rtac @{thm allI}, rtac @{thm allI}, eqvt_stac ctxt,

             if null avoid then tac1 else tac2 ]

  end

*}

ML {*

  let

  in 

    EVERY1 [ DETERM o rtac raw_induct, RANGE cases_tac ]

  end

*}

ML {*

*}

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_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' 

          |> 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"

  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)