src/HOL/Tools/Function/induction_schema.ML

comment;

(*  Title:      HOL/Tools/Function/induction_schema.ML
    Author:     Alexander Krauss, TU Muenchen

A method to prove induction schemas.
*)

signature INDUCTION_SCHEMA =
sig
  val mk_ind_tac : (int -> tactic) -> (int -> tactic) -> (int -> tactic)
                   -> Proof.context -> thm list -> tactic
  val induction_schema_tac : Proof.context -> thm list -> tactic
  val setup : theory -> theory
end


structure Induction_Schema : INDUCTION_SCHEMA =
struct

open Function_Lib

type rec_call_info = int * (string * typ) list * term list * term list

datatype scheme_case = SchemeCase of
 {bidx : int,
  qs: (string * typ) list,
  oqnames: string list,
  gs: term list,
  lhs: term list,
  rs: rec_call_info list}

datatype scheme_branch = SchemeBranch of
 {P : term,
  xs: (string * typ) list,
  ws: (string * typ) list,
  Cs: term list}

datatype ind_scheme = IndScheme of
 {T: typ, (* sum of products *)
  branches: scheme_branch list,
  cases: scheme_case list}

val ind_atomize = Raw_Simplifier.rewrite true @{thms induct_atomize}
val ind_rulify = Raw_Simplifier.rewrite true @{thms induct_rulify}

fun meta thm = thm RS eq_reflection

val sum_prod_conv = Raw_Simplifier.rewrite true
  (map meta (@{thm split_conv} :: @{thms sum.cases}))

fun term_conv thy cv t =
  cv (cterm_of thy t)
  |> prop_of |> Logic.dest_equals |> snd

fun mk_relT T = HOLogic.mk_setT (HOLogic.mk_prodT (T, T))

fun dest_hhf ctxt t =
  let
    val ((params, imp), ctxt') = Variable.focus t ctxt
  in
    (ctxt', map #2 params, Logic.strip_imp_prems imp, Logic.strip_imp_concl imp)
  end

fun mk_scheme' ctxt cases concl =
  let
    fun mk_branch concl =
      let
        val (_, ws, Cs, _ $ Pxs) = dest_hhf ctxt concl
        val (P, xs) = strip_comb Pxs
      in
        SchemeBranch { P=P, xs=map dest_Free xs, ws=ws, Cs=Cs }
      end

    val (branches, cases') = (* correction *)
      case Logic.dest_conjunctions concl of
        [conc] =>
        let
          val _ $ Pxs = Logic.strip_assums_concl conc
          val (P, _) = strip_comb Pxs
          val (cases', conds) =
            take_prefix (Term.exists_subterm (curry op aconv P)) cases
          val concl' = fold_rev (curry Logic.mk_implies) conds conc
        in
          ([mk_branch concl'], cases')
        end
      | concls => (map mk_branch concls, cases)

    fun mk_case premise =
      let
        val (ctxt', qs, prems, _ $ Plhs) = dest_hhf ctxt premise
        val (P, lhs) = strip_comb Plhs

        fun bidx Q =
          find_index (fn SchemeBranch {P=P',...} => Q aconv P') branches

        fun mk_rcinfo pr =
          let
            val (_, Gvs, Gas, _ $ Phyp) = dest_hhf ctxt' pr
            val (P', rcs) = strip_comb Phyp
          in
            (bidx P', Gvs, Gas, rcs)
          end

        fun is_pred v = exists (fn SchemeBranch {P,...} => v aconv P) branches

        val (gs, rcprs) =
          take_prefix (not o Term.exists_subterm is_pred) prems
      in
        SchemeCase {bidx=bidx P, qs=qs, oqnames=map fst qs(*FIXME*),
          gs=gs, lhs=lhs, rs=map mk_rcinfo rcprs}
      end

    fun PT_of (SchemeBranch { xs, ...}) =
      foldr1 HOLogic.mk_prodT (map snd xs)

    val ST = Balanced_Tree.make (uncurry SumTree.mk_sumT) (map PT_of branches)
  in
    IndScheme {T=ST, cases=map mk_case cases', branches=branches }
  end

fun mk_completeness ctxt (IndScheme {cases, branches, ...}) bidx =
  let
    val SchemeBranch { xs, ws, Cs, ... } = nth branches bidx
    val relevant_cases = filter (fn SchemeCase {bidx=bidx', ...} => bidx' = bidx) cases

    val allqnames = fold (fn SchemeCase {qs, ...} => fold (insert (op =) o Free) qs) relevant_cases []
    val (Pbool :: xs') = map Free (Variable.variant_frees ctxt allqnames (("P", HOLogic.boolT) :: xs))
    val Cs' = map (Pattern.rewrite_term (Proof_Context.theory_of ctxt) (filter_out (op aconv) (map Free xs ~~ xs')) []) Cs

    fun mk_case (SchemeCase {qs, oqnames, gs, lhs, ...}) =
      HOLogic.mk_Trueprop Pbool
      |> fold_rev (fn x_l => curry Logic.mk_implies (HOLogic.mk_Trueprop(HOLogic.mk_eq x_l)))
           (xs' ~~ lhs)
      |> fold_rev (curry Logic.mk_implies) gs
      |> fold_rev mk_forall_rename (oqnames ~~ map Free qs)
  in
    HOLogic.mk_Trueprop Pbool
    |> fold_rev (curry Logic.mk_implies o mk_case) relevant_cases
    |> fold_rev (curry Logic.mk_implies) Cs'
    |> fold_rev (Logic.all o Free) ws
    |> fold_rev mk_forall_rename (map fst xs ~~ xs')
    |> mk_forall_rename ("P", Pbool)
  end

fun mk_wf R (IndScheme {T, ...}) =
  HOLogic.Trueprop $ (Const (@{const_name wf}, mk_relT T --> HOLogic.boolT) $ R)

fun mk_ineqs R (IndScheme {T, cases, branches}) =
  let
    fun inject i ts =
       SumTree.mk_inj T (length branches) (i + 1) (foldr1 HOLogic.mk_prod ts)

    val thesis = Free ("thesis", HOLogic.boolT) (* FIXME *)

    fun mk_pres bdx args =
      let
        val SchemeBranch { xs, ws, Cs, ... } = nth branches bdx
        fun replace (x, v) t = betapply (lambda (Free x) t, v)
        val Cs' = map (fold replace (xs ~~ args)) Cs
        val cse =
          HOLogic.mk_Trueprop thesis
          |> fold_rev (curry Logic.mk_implies) Cs'
          |> fold_rev (Logic.all o Free) ws
      in
        Logic.mk_implies (cse, HOLogic.mk_Trueprop thesis)
      end

    fun f (SchemeCase {bidx, qs, oqnames, gs, lhs, rs, ...}) =
      let
        fun g (bidx', Gvs, Gas, rcarg) =
          let val export =
            fold_rev (curry Logic.mk_implies) Gas
            #> fold_rev (curry Logic.mk_implies) gs
            #> fold_rev (Logic.all o Free) Gvs
            #> fold_rev mk_forall_rename (oqnames ~~ map Free qs)
          in
            (HOLogic.mk_mem (HOLogic.mk_prod (inject bidx' rcarg, inject bidx lhs), R)
             |> HOLogic.mk_Trueprop
             |> export,
             mk_pres bidx' rcarg
             |> export
             |> Logic.all thesis)
          end
      in
        map g rs
      end
  in
    map f cases
  end


fun mk_ind_goal thy branches =
  let
    fun brnch (SchemeBranch { P, xs, ws, Cs, ... }) =
      HOLogic.mk_Trueprop (list_comb (P, map Free xs))
      |> fold_rev (curry Logic.mk_implies) Cs
      |> fold_rev (Logic.all o Free) ws
      |> term_conv thy ind_atomize
      |> Object_Logic.drop_judgment thy
      |> HOLogic.tupled_lambda (foldr1 HOLogic.mk_prod (map Free xs))
  in
    SumTree.mk_sumcases HOLogic.boolT (map brnch branches)
  end

fun mk_induct_rule ctxt R x complete_thms wf_thm ineqss
  (IndScheme {T, cases=scases, branches}) =
  let
    val n = length branches
    val scases_idx = map_index I scases

    fun inject i ts =
      SumTree.mk_inj T n (i + 1) (foldr1 HOLogic.mk_prod ts)
    val P_of = nth (map (fn (SchemeBranch { P, ... }) => P) branches)

    val thy = Proof_Context.theory_of ctxt
    val cert = cterm_of thy

    val P_comp = mk_ind_goal thy branches

    (* Inductive Hypothesis: !!z. (z,x):R ==> P z *)
    val ihyp = Logic.all_const T $ Abs ("z", T,
      Logic.mk_implies
        (HOLogic.mk_Trueprop (
          Const (@{const_name Set.member}, HOLogic.mk_prodT (T, T) --> mk_relT T --> HOLogic.boolT) 
          $ (HOLogic.pair_const T T $ Bound 0 $ x)
          $ R),
         HOLogic.mk_Trueprop (P_comp $ Bound 0)))
      |> cert

    val aihyp = Thm.assume ihyp

    (* Rule for case splitting along the sum types *)
    val xss = map (fn (SchemeBranch { xs, ... }) => map Free xs) branches
    val pats = map_index (uncurry inject) xss
    val sum_split_rule =
      Pat_Completeness.prove_completeness thy [x] (P_comp $ x) xss (map single pats)

    fun prove_branch (bidx, (SchemeBranch { P, xs, ws, Cs, ... }, (complete_thm, pat))) =
      let
        val fxs = map Free xs
        val branch_hyp = Thm.assume (cert (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, pat))))

        val C_hyps = map (cert #> Thm.assume) Cs

        val (relevant_cases, ineqss') =
          (scases_idx ~~ ineqss)
          |> filter (fn ((_, SchemeCase {bidx=bidx', ...}), _) => bidx' = bidx)
          |> split_list

        fun prove_case (cidx, SchemeCase {qs, gs, lhs, rs, ...}) ineq_press =
          let
            val case_hyps =
              map (Thm.assume o cert o HOLogic.mk_Trueprop o HOLogic.mk_eq) (fxs ~~ lhs)

            val cqs = map (cert o Free) qs
            val ags = map (Thm.assume o cert) gs

            val replace_x_ss = HOL_basic_ss addsimps (branch_hyp :: case_hyps)
            val sih = full_simplify replace_x_ss aihyp

            fun mk_Prec (idx, Gvs, Gas, rcargs) (ineq, pres) =
              let
                val cGas = map (Thm.assume o cert) Gas
                val cGvs = map (cert o Free) Gvs
                val import = fold Thm.forall_elim (cqs @ cGvs)
                  #> fold Thm.elim_implies (ags @ cGas)
                val ipres = pres
                  |> Thm.forall_elim (cert (list_comb (P_of idx, rcargs)))
                  |> import
              in
                sih
                |> Thm.forall_elim (cert (inject idx rcargs))
                |> Thm.elim_implies (import ineq) (* Psum rcargs *)
                |> Conv.fconv_rule sum_prod_conv
                |> Conv.fconv_rule ind_rulify
                |> (fn th => th COMP ipres) (* P rs *)
                |> fold_rev (Thm.implies_intr o cprop_of) cGas
                |> fold_rev Thm.forall_intr cGvs
              end

            val P_recs = map2 mk_Prec rs ineq_press   (*  [P rec1, P rec2, ... ]  *)

            val step = HOLogic.mk_Trueprop (list_comb (P, lhs))
              |> fold_rev (curry Logic.mk_implies o prop_of) P_recs
              |> fold_rev (curry Logic.mk_implies) gs
              |> fold_rev (Logic.all o Free) qs
              |> cert

            val Plhs_to_Pxs_conv =
              foldl1 (uncurry Conv.combination_conv)
                (Conv.all_conv :: map (fn ch => K (Thm.symmetric (ch RS eq_reflection))) case_hyps)

            val res = Thm.assume step
              |> fold Thm.forall_elim cqs
              |> fold Thm.elim_implies ags
              |> fold Thm.elim_implies P_recs (* P lhs *)
              |> Conv.fconv_rule (Conv.arg_conv Plhs_to_Pxs_conv) (* P xs *)
              |> fold_rev (Thm.implies_intr o cprop_of) (ags @ case_hyps)
              |> fold_rev Thm.forall_intr cqs (* !!qs. Gas ==> xs = lhss ==> P xs *)
          in
            (res, (cidx, step))
          end

        val (cases, steps) = split_list (map2 prove_case relevant_cases ineqss')

        val bstep = complete_thm
          |> Thm.forall_elim (cert (list_comb (P, fxs)))
          |> fold (Thm.forall_elim o cert) (fxs @ map Free ws)
          |> fold Thm.elim_implies C_hyps
          |> fold Thm.elim_implies cases (* P xs *)
          |> fold_rev (Thm.implies_intr o cprop_of) C_hyps
          |> fold_rev (Thm.forall_intr o cert o Free) ws

        val Pxs = cert (HOLogic.mk_Trueprop (P_comp $ x))
          |> Goal.init
          |> (Simplifier.rewrite_goals_tac (map meta (branch_hyp :: @{thm split_conv} :: @{thms sum.cases}))
              THEN CONVERSION ind_rulify 1)
          |> Seq.hd
          |> Thm.elim_implies (Conv.fconv_rule Drule.beta_eta_conversion bstep)
          |> Goal.finish ctxt
          |> Thm.implies_intr (cprop_of branch_hyp)
          |> fold_rev (Thm.forall_intr o cert) fxs
      in
        (Pxs, steps)
      end

    val (branches, steps) =
      map_index prove_branch (branches ~~ (complete_thms ~~ pats))
      |> split_list |> apsnd flat

    val istep = sum_split_rule
      |> fold (fn b => fn th => Drule.compose_single (b, 1, th)) branches
      |> Thm.implies_intr ihyp
      |> Thm.forall_intr (cert x) (* "!!x. (!!y<x. P y) ==> P x" *)

    val induct_rule =
      @{thm "wf_induct_rule"}
      |> (curry op COMP) wf_thm
      |> (curry op COMP) istep

    val steps_sorted = map snd (sort (int_ord o pairself fst) steps)
  in
    (steps_sorted, induct_rule)
  end


fun mk_ind_tac comp_tac pres_tac term_tac ctxt facts =
  (* FIXME proper use of facts!? *)
  (ALLGOALS (Method.insert_tac facts)) THEN HEADGOAL (SUBGOAL (fn (t, i) =>
  let
    val (ctxt', _, cases, concl) = dest_hhf ctxt t
    val scheme as IndScheme {T=ST, branches, ...} = mk_scheme' ctxt' cases concl
    val ([Rn,xn], ctxt'') = Variable.variant_fixes ["R","x"] ctxt'
    val R = Free (Rn, mk_relT ST)
    val x = Free (xn, ST)
    val cert = cterm_of (Proof_Context.theory_of ctxt)

    val ineqss = mk_ineqs R scheme
      |> map (map (pairself (Thm.assume o cert)))
    val complete =
      map_range (mk_completeness ctxt scheme #> cert #> Thm.assume) (length branches)
    val wf_thm = mk_wf R scheme |> cert |> Thm.assume

    val (descent, pres) = split_list (flat ineqss)
    val newgoals = complete @ pres @ wf_thm :: descent

    val (steps, indthm) =
      mk_induct_rule ctxt'' R x complete wf_thm ineqss scheme

    fun project (i, SchemeBranch {xs, ...}) =
      let
        val inst = (foldr1 HOLogic.mk_prod (map Free xs))
          |> SumTree.mk_inj ST (length branches) (i + 1)
          |> cert
      in
        indthm
        |> Drule.instantiate' [] [SOME inst]
        |> simplify SumTree.sumcase_split_ss
        |> Conv.fconv_rule ind_rulify
      end

    val res = Conjunction.intr_balanced (map_index project branches)
      |> fold_rev Thm.implies_intr (map cprop_of newgoals @ steps)
      |> Drule.generalize ([], [Rn])

    val nbranches = length branches
    val npres = length pres
  in
    Thm.compose_no_flatten false (res, length newgoals) i
    THEN term_tac (i + nbranches + npres)
    THEN (EVERY (map (TRY o pres_tac) ((i + nbranches + npres - 1) downto (i + nbranches))))
    THEN (EVERY (map (TRY o comp_tac) ((i + nbranches - 1) downto i)))
  end))


fun induction_schema_tac ctxt =
  mk_ind_tac (K all_tac) (assume_tac APPEND' Goal.assume_rule_tac ctxt) (K all_tac) ctxt;

val setup =
  Method.setup @{binding induction_schema} (Scan.succeed (RAW_METHOD o induction_schema_tac))
    "proves an induction principle"

end