(*  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: bool -> 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 = 

  Raw_Simplifier.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));

(** 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_prem2 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 *)

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 note_flag 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 = Goal.prove ctxt' [] [] goal 

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

      |> Datatype_Aux.split_conj_thm 

      |> ProofContext.export ctxt' ctxt

      |> map (fn th => th RS mp)

      |> map zero_var_indexes

  in

    if note_flag

    then fold_map note_named_thm (pred_names ~~ thms) ctxt 

    else (thms, ctxt) 

  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 true preds raw_induct intrs ctxt |> snd

  end

local structure P = Parse and K = Keyword in

val _ =

  Outer_Syntax.local_theory "equivariance"

    "Proves equivariance for inductive predicate involving nominal datatypes." 

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

end;

end (* structure *)