File ‹ferrante_rackoff.ML›

```(* 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_global)
(funpow 2 Thm.dest_arg (Thm.cprop_of th)) |> Thm.dest_arg

fun ferrack_conv ctxt
(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 Thm.term_of p of
\<^Const_>‹HOL.conj for _ _› =>
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_>‹HOL.disj for _ _› =>
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_global)
(funpow 4 Thm.dest_arg (Thm.cprop_of (hd minf)))
|> Thm.dest_binop |> apply2 Thm.dest_binop |> apfst (apply2 Thm.dest_fun)
|> apply2 (apply2 (dest_Var o Thm.term_of))

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
Drule.instantiate_normalize (TVars.empty, Vars.make [(p1_v, p1),(p2_v, p2),(mp1_v, mp1), (mp2_v, mp2)]) th
else Drule.instantiate_normalize (TVars.empty, Vars.make [(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 = dest_Var (Thm.term_of (Thm.dest_arg (Thm.dest_arg (Thm.dest_arg1 (Thm.cprop_of qe)))))
fun main vs p =
let
val ((xn,ce),(x,fm)) = (case Thm.term_of p of
\<^Const_>‹Ex _ for ‹Abs (xn, _, _)›› =>
Thm.dest_comb p ||> Thm.dest_abs_global |>> pair xn
| _ => raise CTERM ("main QE only treats existential quantifiers!", [p]))
val cT = Thm.ctyp_of_cterm 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 (Thm.instantiate' [SOME cT] []) @{thms "finite.intros"}
fun ins x th =
Thm.implies_elim (Thm.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 = Thm.instantiate' [SOME cT] [] @{thm "insertI1"}
val insI2 = Thm.instantiate' [SOME cT] [] @{thm "insertI2"}
in
fun provein x S =
case Thm.term_of S of
\<^Const_>‹Orderings.bot _› => raise CTERM ("provein : not a member!", [S])
| \<^Const_>‹insert _ for y _› =>
let val (cy,S') = Thm.dest_binop S
in if Thm.term_of x aconv y then Thm.instantiate' [] [SOME x, SOME S'] insI1
else Thm.implies_elim (Thm.instantiate' [] [SOME x, SOME S', SOME cy] insI2)
(provein x S')
end
end
val tabU = fold (fn t => fn tab => Termtab.update (Thm.term_of t, provein t cU) tab)
u Termtab.empty
val U = the o Termtab.lookup tabU o Thm.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 (Drule.instantiate_normalize (TVars.empty, Vars.make1 (U_v,cU))) nmi
val [npi_conj, npi_disj, npi_eq, npi_neq, npi_lt,
npi_le, npi_gt, npi_ge, npi_P] = map (Drule.instantiate_normalize (TVars.empty, Vars.make1 (U_v,cU))) npi
val [ld_conj, ld_disj, ld_eq, ld_neq, ld_lt,
ld_le, ld_gt, ld_ge, ld_P] = map (Drule.instantiate_normalize (TVars.empty, Vars.make1 (U_v,cU))) ld

fun decomp_mpinf fm =
case Thm.term_of fm of
\<^Const_>‹HOL.conj for _ _› =>
let val (p,q) = Thm.dest_binop fm
in ([p,q], myfwd (minf_conj,pinf_conj, nmi_conj, npi_conj,ld_conj)
(Thm.lambda x p) (Thm.lambda x q))
end
| \<^Const_>‹HOL.disj for _ _› =>
let val (p,q) = Thm.dest_binop fm
in ([p,q],myfwd (minf_disj, pinf_disj, nmi_disj, npi_disj,ld_disj)
(Thm.lambda x p) (Thm.lambda 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 raise Fail "decomp_mpinf: Impossible case!!") fm
val [mi_th, pi_th, nmi_th, npi_th, ld_th] =
if c = Nox then map (Thm.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 (Thm.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))
(Thm.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 (put_simpset HOL_basic_ss ctxt addsimps @{thms simp_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 ctxt tm =
(case Thm.term_of tm of
\<^Const_>‹HOL.eq \<^Type>‹bool› for _ _› => find_args ctxt tm
| \<^Const_>‹Not for _› => h ctxt (Thm.dest_arg tm)
| \<^Const_>‹All _ for _› => find_body ctxt (Thm.dest_arg tm)
| \<^Const_>‹Ex _ for _› => find_body ctxt (Thm.dest_arg tm)
| \<^Const_>‹conj for _ _› => find_args ctxt tm
| \<^Const_>‹disj for _ _› => find_args ctxt tm
| \<^Const_>‹implies for _ _› => find_args ctxt tm
| \<^Const_>‹Pure.imp for _ _› => find_args ctxt tm
| \<^Const_>‹Pure.eq _ for _ _› => find_args ctxt tm
| \<^Const_>‹Pure.all _ for _› => find_body ctxt (Thm.dest_arg tm)
| \<^Const_>‹Trueprop for _› => h ctxt (Thm.dest_arg tm)
| _ => Thm.dest_fun2 tm)
and find_args ctxt tm =
(h ctxt (Thm.dest_arg tm) handle CTERM _ => Thm.dest_arg1 tm)
and find_body ctxt b =
let val ((_, b'), ctxt') = Variable.dest_abs_cterm b ctxt
in h ctxt' b' end;
in h end;

fun raw_ferrack_qe_conv ctxt (thy, {isolate_conv, whatis, simpset = ss}) tm =
let
val ss' =
merge_ss (simpset_of
@{thms simp_thms ex_simps all_simps not_all all_not_ex ex_disj_distrib}), ss);
val pcv = Simplifier.rewrite (put_simpset ss' ctxt);
val postcv = Simplifier.rewrite (put_simpset ss ctxt);
val nnf = K (nnf_conv ctxt then_conv postcv)
val env = Cterms.list_set_rev (Cterms.build (Drule.add_frees_cterm tm))
val qe_conv = Qelim.gen_qelim_conv ctxt pcv postcv pcv cons env
(isolate_conv ctxt) nnf
(fn vs => ferrack_conv ctxt (thy,{isolate_conv = isolate_conv ctxt,
whatis = whatis, simpset = ss}) vs
then_conv postcv)
in (Simplifier.rewrite (put_simpset ss ctxt) then_conv qe_conv) tm end;

fun dlo_instance ctxt tm =
Ferrante_Rackoff_Data.match ctxt (grab_atom_bop ctxt 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 ctxt i THEN
simp_tac (put_simpset (#simpset (snd instance)) ctxt) i THEN  (* FIXME already part of raw_ferrack_qe_conv? *)
CONVERSION (Object_Logic.judgment_conv ctxt (raw_ferrack_qe_conv ctxt instance)) i THEN
simp_tac ctxt i));  (* FIXME really? *)

end;
```