src/HOL/Tools/Qelim/ferrante_rackoff.ML

Generic QE need no Context anymore

(* Title:      HOL/Tools/ferrante_rackoff.ML
   ID:         $Id$
   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_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 binop_cong b th1 th2 = Thm.combination (Drule.arg_cong_rule b th1) th2;
val is_refl = op aconv o Logic.dest_equals o Thm.prop_of;
fun C f x y = f y x

fun get_p1 th = 
 let 
    fun appair f (x,y) = (f x, f y)
  in funpow 2 (Thm.dest_arg o snd o Thm.dest_abs NONE) 
     (funpow 2 Thm.dest_arg (cprop_of th)) |> Thm.dest_arg 
end;

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, binop_cong 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, binop_cong b lth rth) end
 | _ => 
    let 
      val th = icv vars p 
      val p' = Thm.rhs_of th
      val c = wi x p'
      val S = (if c mem [Lt, Le, Eq] then single o Thm.dest_arg
               else if c mem [Gt, Ge] 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)) = 
  let
   fun appair f (x,y) = (f x, f y)
  in funpow 2 (Thm.dest_arg o snd o Thm.dest_abs NONE) 
       (funpow 4 Thm.dest_arg (cprop_of (hd minf))) 
     |> Thm.dest_binop |> appair Thm.dest_binop |> apfst (appair Thm.dest_fun)  
  end

 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 implies_elim (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",_)$Abs(xn,xT,_) =>  
                        Thm.dest_comb p ||> Thm.dest_abs (SOME xn) |>> pair xn
                 | _ => error "main QE only trats existential quantifiers!")
   val cT = ctyp_of_term x
   val (u,nth) = uset (x::vs) fm |>> distinct (op aconvc)
   val nthx = Thm.abstract_rule xn x nth
   val q = Thm.rhs_of nth
   val qx = Thm.rhs_of nthx
   val enth = Drule.arg_cong_rule ce nthx
   val [th0,th1] = map (instantiate' [SOME cT] []) @{thms "finite.intros"}
   fun ins x th = 
      implies_elim (instantiate' [] [(SOME o Thm.dest_arg o Thm.dest_arg) 
                                       (Thm.cprop_of th), SOME x] th1) th
   val fU = fold ins u th0
   val cU = funpow 2 Thm.dest_arg (Thm.cprop_of fU)
   local 
     val insI1 = instantiate' [SOME cT] [] @{thm "insertI1"}
     val insI2 = instantiate' [SOME cT] [] @{thm "insertI2"}
   in
    fun provein x S = 
     case term_of S of
        Const("{}",_) => error "provein : not a member!"
      | Const("insert",_)$y$_ => 
         let val (cy,S') = Thm.dest_binop S
         in if term_of x aconv y then instantiate' [] [SOME x, SOME S'] insI1
         else 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 = valOf 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 c mem [Lt, Le, Eq] then Thm.dest_arg
                      else if c mem [Gt,Ge] then Thm.dest_arg1
                      else if c = NEq then (Thm.dest_arg o Thm.dest_arg) 
                      else 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 = 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 = fold (C implies_elim)  [fU, ld_th, nmi_th, npi_th, minf_th, pinf_th] 
                  ((fconv_rule (Thm.beta_conversion true)) 
                   (instantiate' [] (map SOME [cU, qx, get_p1 minf_th, get_p1 pinf_th]) 
                        qe))
    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) (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 ("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 pcv = Simplifier.rewrite 
     (merge_ss (HOL_basic_ss addsimps (simp_thms @ ex_simps @ all_simps)
              @ [not_all,@{thm "all_not_ex"}, ex_disj_distrib], 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 ferrackqe_conv ctxt tm = 
 case Ferrante_Rackoff_Data.match ctxt (grab_atom_bop 0 tm) of
  NONE => error "ferrackqe_conv : no corresponding instance in context!"
| SOME res => raw_ferrack_qe_conv ctxt res tm


fun core_ferrack_tac ctxt res i st =
 let val p = nth (cprems_of st) (i - 1)
     val th = symmetric (arg_conv (raw_ferrack_qe_conv ctxt res) p)
     val p' = Thm.lhs_of th
     val th' = implies_intr p' (equal_elim th (assume p')) 
     val _ = print_thm th
  in (rtac th' i) st 
  end

fun dlo_tac ctxt i st = 
 let 
   val instance = (case Ferrante_Rackoff_Data.match ctxt 
                           (grab_atom_bop 0 (nth (cprems_of st) (i - 1))) of 
                    NONE => error "ferrackqe_conv : no corresponding instance in context!"
                  | SOME r => r)
   val ss = #simpset (snd instance)
   in
   (ObjectLogic.full_atomize_tac i THEN 
    simp_tac ss i THEN
    core_ferrack_tac ctxt instance i THEN 
    (TRY (simp_tac (Simplifier.local_simpset_of ctxt) i))) st
  end;

end;