(*  Title:      nominal_eqvt.ML

    Author:     Stefan Berghofer (original code)

    Author:     Christian Urban

    Automatic proofs for equivariance of inductive predicates.

*)

signature NOMINAL_EQVT =

sig

  val eqvt_rel_tac: Proof.context -> string list -> term -> thm -> thm list -> int -> tactic

  val eqvt_rel_single_case_tac: Proof.context -> string list -> term -> thm -> int -> tactic

  val equivariance: term list -> thm -> thm list -> Proof.context -> thm list * local_theory

  val equivariance_cmd: string -> Proof.context -> local_theory

end

structure Nominal_Eqvt : NOMINAL_EQVT =

struct

open Nominal_Permeq;

open Nominal_ThmDecls;

val atomize_conv = 

  MetaSimplifier.rewrite_cterm (true, false, false) (K (K NONE))

    (HOL_basic_ss addsimps @{thms induct_atomize});

val atomize_intr = Conv.fconv_rule (Conv.prems_conv ~1 atomize_conv);

fun atomize_induct ctxt = Conv.fconv_rule (Conv.prems_conv ~1

  (Conv.params_conv ~1 (K (Conv.prems_conv ~1 atomize_conv)) ctxt));

(** 

 given the theorem F[t]; proves the theorem F[f t] 

  - F needs to be monotone

  - f returns either SOME for a term it fires on 

    and NONE elsewhere 

**)

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

  val simps = HOL_basic_ss addsimps @{thms split_def}

in

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

    REPEAT_DETERM (FIRSTGOAL (simp_tac simps THEN' resolve_tac monos)),

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

end

fun map_thm ctxt f tac thm =

let

  val opt_goal_trm = map_term f (prop_of thm)

in

  case opt_goal_trm of

    NONE => thm

  | SOME goal =>

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

end

(*

 inductive premises can be of the form

 R ... /\ P ...; split_conj picks out

 the part P ...

*)

fun transform_prem ctxt names thm =

let

  fun split_conj names (Const ("op &", _) $ f1 $ f2) = 

      (case head_of f1 of

         Const (name, _) => if member (op =) names name then SOME f2 else NONE

       | _ => NONE)

  | split_conj _ _ = NONE;

in

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

end

(** equivariance tactics **)

val perm_boolE = @{thm permute_boolE}

val perm_cancel = @{thms permute_minus_cancel(2)}

fun eqvt_rel_single_case_tac ctxt pred_names pi intro  = 

let

  val thy = ProofContext.theory_of ctxt

  val cpi = Thm.cterm_of thy (mk_minus pi)

  val pi_intro_rule = Drule.instantiate' [] [SOME cpi] perm_boolE

  val simps1 = HOL_basic_ss addsimps @{thms permute_fun_def minus_minus split_paired_all}

  val simps2 = HOL_basic_ss addsimps @{thms permute_bool_def}

in

  eqvt_strict_tac ctxt [] pred_names THEN'

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

    let

      val prems' = map (transform_prem ctxt pred_names) prems

      val tac1 = resolve_tac prems'

      val tac2 = EVERY' [ rtac pi_intro_rule, 

            eqvt_strict_tac ctxt perm_cancel pred_names, resolve_tac prems' ]

      val tac3 = EVERY' [ rtac pi_intro_rule, 

            eqvt_strict_tac ctxt perm_cancel pred_names, simp_tac simps1, 

            simp_tac simps2, resolve_tac prems']

    in

      (rtac intro THEN_ALL_NEW FIRST' [tac1, tac2, tac3]) 1 

    end) ctxt

end

fun eqvt_rel_tac ctxt pred_names pi induct intros =

let

  val cases = map (eqvt_rel_single_case_tac ctxt pred_names pi) intros

in

  EVERY' (rtac induct :: cases)

end

(** equivariance procedure *)

(* sets up goal and makes sure parameters

   are untouched PROBLEM: this violates the 

   form of eqvt lemmas *)

fun prepare_goal pi pred =

let

  val (c, xs) = strip_comb pred;

in

  HOLogic.mk_imp (pred, list_comb (c, map (mk_perm pi) xs))

end

(* stores thm under name.eqvt and adds [eqvt]-attribute *)

fun note_named_thm (name, thm) ctxt = 

let

  val thm_name = Binding.qualified_name 

    (Long_Name.qualify (Long_Name.base_name name) "eqvt")

  val attr = Attrib.internal (K eqvt_add)

  val ((_, [thm']), ctxt') =  Local_Theory.note ((thm_name, [attr]), [thm]) ctxt

in

  (thm', ctxt')

end

fun equivariance pred_trms raw_induct intrs ctxt = 

let

  val is_already_eqvt = 

    filter (is_eqvt ctxt) pred_trms

    |> map (Syntax.string_of_term ctxt)

  val _ = if null is_already_eqvt then ()

    else error ("Already equivariant: " ^ commas is_already_eqvt)

  val pred_names = map (fst o dest_Const) pred_trms

  val raw_induct' = atomize_induct ctxt raw_induct

  val intrs' = map atomize_intr intrs

  val (([raw_concl], [raw_pi]), ctxt') = 

    ctxt 

    |> Variable.import_terms false [concl_of raw_induct'] 

    ||>> Variable.variant_fixes ["p"]

  val pi = Free (raw_pi, @{typ perm})

  val preds = map (fst o HOLogic.dest_imp)

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

  val goal = HOLogic.mk_Trueprop 

    (foldr1 HOLogic.mk_conj (map (prepare_goal pi) preds))

  val thms = Datatype_Aux.split_conj_thm (Goal.prove ctxt' [] [] goal 

    (fn {context,...} => eqvt_rel_tac context pred_names pi raw_induct' intrs' 1)

    |> singleton (ProofContext.export ctxt' ctxt))

  val thms' = map (fn th => zero_var_indexes (th RS mp)) thms

in

  ctxt |> fold_map note_named_thm (pred_names ~~ thms')   

end

fun equivariance_cmd pred_name ctxt =

let

  val thy = ProofContext.theory_of ctxt

  val (_, {preds, raw_induct, intrs, ...}) =

    Inductive.the_inductive ctxt (Sign.intern_const thy pred_name)

in

  equivariance preds raw_induct intrs ctxt |> snd

end

local structure P = OuterParse and K = OuterKeyword in

val _ =

  OuterSyntax.local_theory "equivariance"

    "Proves equivariance for inductive predicate involving nominal datatypes." 

      K.thy_decl (P.xname >> equivariance_cmd);

end;

end (* structure *)