(* Title:      HOL/Decision_Procs/ferrante_rackoff.ML

   Author:     Amine Chaieb, TU Muenchen

Ferrante and Rackoff's algorithm for quantifier elimination in dense

linear orders.  Proof-synthesis and tactic.

*)

signature FERRANTE_RACKOFF =

sig

  val dlo_conv: Proof.context -> conv

  val dlo_tac: Proof.context -> int -> tactic

end;

structure FerranteRackoff: FERRANTE_RACKOFF =

struct

open Ferrante_Rackoff_Data;

open Conv;

type entry = {minf: thm list, pinf: thm list, nmi: thm list, npi: thm list,

   ld: thm list, qe: thm, atoms : cterm list} *

  {isolate_conv: cterm list -> cterm -> thm,

                 whatis : cterm -> cterm -> ord,

                 simpset : simpset};

fun get_p1 th =

  funpow 2 (Thm.dest_arg o snd o Thm.dest_abs NONE)

    (funpow 2 Thm.dest_arg (cprop_of th)) |> Thm.dest_arg

fun ferrack_conv

   (entr as ({minf = minf, pinf = pinf, nmi = nmi, npi = npi,

              ld = ld, qe = qe, atoms = atoms},

             {isolate_conv = icv, whatis = wi, simpset = simpset}):entry) =

let

 fun uset (vars as (x::vs)) p = case term_of p of

   Const("op &", _)$ _ $ _ =>

     let

       val ((b,l),r) = Thm.dest_comb p |>> Thm.dest_comb

       val (lS,lth) = uset vars l  val (rS, rth) = uset vars r

     in (lS@rS, Drule.binop_cong_rule b lth rth) end

 |  Const("op |", _)$ _ $ _ =>

     let

       val ((b,l),r) = Thm.dest_comb p |>> Thm.dest_comb

       val (lS,lth) = uset vars l  val (rS, rth) = uset vars r

     in (lS@rS, Drule.binop_cong_rule b lth rth) end

 | _ =>

    let

      val th = icv vars p

      val p' = Thm.rhs_of th

      val c = wi x p'

      val S = (if member (op =) [Lt, Le, Eq] c then single o Thm.dest_arg

               else if member (op =) [Gt, Ge] c then single o Thm.dest_arg1

               else if c = NEq then single o Thm.dest_arg o Thm.dest_arg

               else K []) p'

    in (S,th) end

 val ((p1_v,p2_v),(mp1_v,mp2_v)) =

   funpow 2 (Thm.dest_arg o snd o Thm.dest_abs NONE)

     (funpow 4 Thm.dest_arg (cprop_of (hd minf)))

   |> Thm.dest_binop |> pairself Thm.dest_binop |> apfst (pairself Thm.dest_fun)

 fun myfwd (th1, th2, th3, th4, th5) p1 p2

      [(th_1,th_2,th_3,th_4,th_5), (th_1',th_2',th_3',th_4',th_5')] =

  let

   val (mp1, mp2) = (get_p1 th_1, get_p1 th_1')

   val (pp1, pp2) = (get_p1 th_2, get_p1 th_2')

   fun fw mi th th' th'' =

     let

      val th0 = if mi then

           instantiate ([],[(p1_v, p1),(p2_v, p2),(mp1_v, mp1), (mp2_v, mp2)]) th

        else instantiate ([],[(p1_v, p1),(p2_v, p2),(mp1_v, pp1), (mp2_v, pp2)]) th

     in Thm.implies_elim (Thm.implies_elim th0 th') th'' end

  in (fw true th1 th_1 th_1', fw false th2 th_2 th_2',

      fw true th3 th_3 th_3', fw false th4 th_4 th_4', fw true th5 th_5 th_5')

  end

 val U_v = (Thm.dest_arg o Thm.dest_arg o Thm.dest_arg1) (cprop_of qe)

 fun main vs p =

  let

   val ((xn,ce),(x,fm)) = (case term_of p of

                   Const("Ex",_)$Abs(xn,xT,_) =>

                        Thm.dest_comb p ||> Thm.dest_abs (SOME xn) |>> pair xn

                 | _ => raise CTERM ("main QE only treats existential quantifiers!", [p]))

   val cT = ctyp_of_term x

   val (u,nth) = uset (x::vs) fm |>> distinct (op aconvc)

   val nthx = Thm.abstract_rule xn x nth

   val q = Thm.rhs_of nth

   val qx = Thm.rhs_of nthx

   val enth = Drule.arg_cong_rule ce nthx

   val [th0,th1] = map (instantiate' [SOME cT] []) @{thms "finite.intros"}

   fun ins x th =

      Thm.implies_elim (instantiate' [] [(SOME o Thm.dest_arg o Thm.dest_arg)

                                       (Thm.cprop_of th), SOME x] th1) th

   val fU = fold ins u th0

   val cU = funpow 2 Thm.dest_arg (Thm.cprop_of fU)

   local

     val insI1 = instantiate' [SOME cT] [] @{thm "insertI1"}

     val insI2 = instantiate' [SOME cT] [] @{thm "insertI2"}

   in

    fun provein x S =

     case term_of S of

        Const(@{const_name Orderings.bot}, _) => raise CTERM ("provein : not a member!", [S])

      | Const(@{const_name insert}, _) $ y $_ =>

         let val (cy,S') = Thm.dest_binop S

         in if term_of x aconv y then instantiate' [] [SOME x, SOME S'] insI1

         else Thm.implies_elim (instantiate' [] [SOME x, SOME S', SOME cy] insI2)

                           (provein x S')

         end

   end

   val tabU = fold (fn t => fn tab => Termtab.update (term_of t, provein t cU) tab)

                   u Termtab.empty

   val U = the o Termtab.lookup tabU o term_of

   val [minf_conj, minf_disj, minf_eq, minf_neq, minf_lt,

        minf_le, minf_gt, minf_ge, minf_P] = minf

   val [pinf_conj, pinf_disj, pinf_eq, pinf_neq, pinf_lt,

        pinf_le, pinf_gt, pinf_ge, pinf_P] = pinf

   val [nmi_conj, nmi_disj, nmi_eq, nmi_neq, nmi_lt,

        nmi_le, nmi_gt, nmi_ge, nmi_P] = map (instantiate ([],[(U_v,cU)])) nmi

   val [npi_conj, npi_disj, npi_eq, npi_neq, npi_lt,

        npi_le, npi_gt, npi_ge, npi_P] = map (instantiate ([],[(U_v,cU)])) npi

   val [ld_conj, ld_disj, ld_eq, ld_neq, ld_lt,

        ld_le, ld_gt, ld_ge, ld_P] = map (instantiate ([],[(U_v,cU)])) ld

   fun decomp_mpinf fm =

     case term_of fm of

       Const("op &",_)$_$_ =>

        let val (p,q) = Thm.dest_binop fm

        in ([p,q], myfwd (minf_conj,pinf_conj, nmi_conj, npi_conj,ld_conj)

                         (Thm.cabs x p) (Thm.cabs x q))

        end

     | Const("op |",_)$_$_ =>

        let val (p,q) = Thm.dest_binop fm

        in ([p,q],myfwd (minf_disj, pinf_disj, nmi_disj, npi_disj,ld_disj)

                         (Thm.cabs x p) (Thm.cabs x q))

        end

     | _ =>

        (let val c = wi x fm

             val t = (if c=Nox then I

                      else if member (op =) [Lt, Le, Eq] c then Thm.dest_arg

                      else if member (op =) [Gt, Ge] c then Thm.dest_arg1

                      else if c = NEq then (Thm.dest_arg o Thm.dest_arg)

                      else sys_error "decomp_mpinf: Impossible case!!") fm

             val [mi_th, pi_th, nmi_th, npi_th, ld_th] =

               if c = Nox then map (instantiate' [] [SOME fm])

                                    [minf_P, pinf_P, nmi_P, npi_P, ld_P]

               else

                let val [mi_th,pi_th,nmi_th,npi_th,ld_th] =

                 map (instantiate' [] [SOME t])

                 (case c of Lt => [minf_lt, pinf_lt, nmi_lt, npi_lt, ld_lt]

                          | Le => [minf_le, pinf_le, nmi_le, npi_le, ld_le]

                          | Gt => [minf_gt, pinf_gt, nmi_gt, npi_gt, ld_gt]

                          | Ge => [minf_ge, pinf_ge, nmi_ge, npi_ge, ld_ge]

                          | Eq => [minf_eq, pinf_eq, nmi_eq, npi_eq, ld_eq]

                          | NEq => [minf_neq, pinf_neq, nmi_neq, npi_neq, ld_neq])

                    val tU = U t

                    fun Ufw th = Thm.implies_elim th tU

                 in [mi_th, pi_th, Ufw nmi_th, Ufw npi_th, Ufw ld_th]

                 end

         in ([], K (mi_th, pi_th, nmi_th, npi_th, ld_th)) end)

   val (minf_th, pinf_th, nmi_th, npi_th, ld_th) = divide_and_conquer decomp_mpinf q

   val qe_th = Drule.implies_elim_list

                  ((fconv_rule (Thm.beta_conversion true))

                   (instantiate' [] (map SOME [cU, qx, get_p1 minf_th, get_p1 pinf_th])

                        qe))

                  [fU, ld_th, nmi_th, npi_th, minf_th, pinf_th]

    val bex_conv =

      Simplifier.rewrite (HOL_basic_ss addsimps simp_thms@(@{thms "bex_simps" (1-5)}))

    val result_th = fconv_rule (arg_conv bex_conv) (Thm.transitive enth qe_th)

   in result_th

   end

in main

end;

val grab_atom_bop =

 let

  fun h bounds tm =

   (case term_of tm of

     Const ("op =", T) $ _ $ _ =>

       if domain_type T = HOLogic.boolT then find_args bounds tm

       else Thm.dest_fun2 tm

   | Const ("Not", _) $ _ => h bounds (Thm.dest_arg tm)

   | Const ("All", _) $ _ => find_body bounds (Thm.dest_arg tm)

   | Const ("Ex", _) $ _ => find_body bounds (Thm.dest_arg tm)

   | Const ("op &", _) $ _ $ _ => find_args bounds tm

   | Const ("op |", _) $ _ $ _ => find_args bounds tm

   | Const ("op -->", _) $ _ $ _ => find_args bounds tm

   | Const ("==>", _) $ _ $ _ => find_args bounds tm

   | Const ("==", _) $ _ $ _ => find_args bounds tm

   | Const ("all", _) $ _ => find_body bounds (Thm.dest_arg tm)

   | Const ("Trueprop", _) $ _ => h bounds (Thm.dest_arg tm)

   | _ => Thm.dest_fun2 tm)

  and find_args bounds tm =

           (h bounds (Thm.dest_arg tm) handle CTERM _ => Thm.dest_arg1 tm)

 and find_body bounds b =

   let val (_, b') = Thm.dest_abs (SOME (Name.bound bounds)) b

   in h (bounds + 1) b' end;

in h end;

fun raw_ferrack_qe_conv ctxt (thy, {isolate_conv, whatis, simpset}) tm =

 let

   val ss = simpset

   val ss' =

     merge_ss (HOL_basic_ss addsimps (simp_thms @ ex_simps @ all_simps)

              @ [not_all,@{thm "all_not_ex"}, ex_disj_distrib], ss)

     |> Simplifier.inherit_context ss

   val pcv = Simplifier.rewrite ss'     

   val postcv = Simplifier.rewrite ss

   val nnf = K (nnf_conv then_conv postcv)

   val qe_conv = Qelim.gen_qelim_conv pcv postcv pcv cons (Thm.add_cterm_frees tm [])

                  (isolate_conv ctxt) nnf

                  (fn vs => ferrack_conv (thy,{isolate_conv = isolate_conv ctxt,

                                               whatis = whatis, simpset = simpset}) vs

                   then_conv postcv)

 in (Simplifier.rewrite ss then_conv qe_conv) tm end;

fun dlo_instance ctxt tm =

  Ferrante_Rackoff_Data.match ctxt (grab_atom_bop 0 tm);

fun dlo_conv ctxt tm =

  (case dlo_instance ctxt tm of

    NONE => raise CTERM ("ferrackqe_conv: no corresponding instance in context!", [tm])

  | SOME instance => raw_ferrack_qe_conv ctxt instance tm);

fun dlo_tac ctxt = CSUBGOAL (fn (p, i) =>

  (case dlo_instance ctxt p of

    NONE => no_tac

  | SOME instance =>

      Object_Logic.full_atomize_tac i THEN

      simp_tac (#simpset (snd instance)) i THEN  (* FIXME already part of raw_ferrack_qe_conv? *)

      CONVERSION (Object_Logic.judgment_conv (raw_ferrack_qe_conv ctxt instance)) i THEN

      simp_tac (simpset_of ctxt) i));  (* FIXME really? *)

end;