(* 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 ctxt 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;