(* Title: HOL/Decision_Procs/langford.ML
Author: Amine Chaieb, TU Muenchen
*)
signature LANGFORD_QE =
sig
val dlo_tac : Proof.context -> int -> tactic
val dlo_conv : Proof.context -> cterm -> thm
end
structure LangfordQE: LANGFORD_QE =
struct
val dest_set =
let
fun h acc ct =
(case term_of ct of
Const (@{const_name Orderings.bot}, _) => acc
| Const (@{const_name insert}, _) $ _ $ t => h (Thm.dest_arg1 ct :: acc) (Thm.dest_arg ct));
in h [] end;
fun prove_finite cT u =
let
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
in fold ins u th0 end;
fun simp_rule ctxt =
Conv.fconv_rule
(Conv.arg_conv
(Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps @{thms ball_simps simp_thms})));
fun basic_dloqe ctxt stupid dlo_qeth dlo_qeth_nolb dlo_qeth_noub gather ep =
case term_of ep of
Const(@{const_name Ex},_)$_ =>
let
val p = Thm.dest_arg ep
val ths =
simplify (put_simpset HOL_basic_ss ctxt addsimps gather) (instantiate' [] [SOME p] stupid)
val (L,U) =
let
val (_, q) = Thm.dest_abs NONE (Thm.dest_arg (Thm.rhs_of ths))
in (Thm.dest_arg1 q |> Thm.dest_arg1, Thm.dest_arg q |> Thm.dest_arg1)
end
fun proveneF S =
let val (a,A) = Thm.dest_comb S |>> Thm.dest_arg
val cT = ctyp_of_term a
val ne = instantiate' [SOME cT] [SOME a, SOME A]
@{thm insert_not_empty}
val f = prove_finite cT (dest_set S)
in (ne, f) end
val qe = case (term_of L, term_of U) of
(Const (@{const_name Orderings.bot}, _),_) =>
let
val (neU,fU) = proveneF U
in simp_rule ctxt (Thm.transitive ths (dlo_qeth_nolb OF [neU, fU])) end
| (_,Const (@{const_name Orderings.bot}, _)) =>
let
val (neL,fL) = proveneF L
in simp_rule ctxt (Thm.transitive ths (dlo_qeth_noub OF [neL, fL])) end
| (_,_) =>
let
val (neL,fL) = proveneF L
val (neU,fU) = proveneF U
in simp_rule ctxt (Thm.transitive ths (dlo_qeth OF [neL, neU, fL, fU]))
end
in qe end
| _ => error "dlo_qe : Not an existential formula";
val all_conjuncts =
let fun h acc ct =
case term_of ct of
@{term HOL.conj}$_$_ => h (h acc (Thm.dest_arg ct)) (Thm.dest_arg1 ct)
| _ => ct::acc
in h [] end;
fun conjuncts ct =
case term_of ct of
@{term HOL.conj}$_$_ => (Thm.dest_arg1 ct)::(conjuncts (Thm.dest_arg ct))
| _ => [ct];
fun fold1 f = foldr1 (uncurry f);
val list_conj = fold1 (fn c => fn c' => Thm.apply (Thm.apply @{cterm HOL.conj} c) c') ;
fun mk_conj_tab th =
let fun h acc th =
case prop_of th of
@{term "Trueprop"}$(@{term HOL.conj}$p$q) =>
h (h acc (th RS conjunct2)) (th RS conjunct1)
| @{term "Trueprop"}$p => (p,th)::acc
in fold (Termtab.insert Thm.eq_thm) (h [] th) Termtab.empty end;
fun is_conj (@{term HOL.conj}$_$_) = true
| is_conj _ = false;
fun prove_conj tab cjs =
case cjs of
[c] => if is_conj (term_of c) then prove_conj tab (conjuncts c) else tab c
| c::cs => conjI OF [prove_conj tab [c], prove_conj tab cs];
fun conj_aci_rule eq =
let
val (l,r) = Thm.dest_equals eq
fun tabl c = the (Termtab.lookup (mk_conj_tab (Thm.assume l)) (term_of c))
fun tabr c = the (Termtab.lookup (mk_conj_tab (Thm.assume r)) (term_of c))
val ll = Thm.dest_arg l
val rr = Thm.dest_arg r
val thl = prove_conj tabl (conjuncts rr)
|> Drule.implies_intr_hyps
val thr = prove_conj tabr (conjuncts ll)
|> Drule.implies_intr_hyps
val eqI = instantiate' [] [SOME ll, SOME rr] @{thm iffI}
in Thm.implies_elim (Thm.implies_elim eqI thl) thr |> mk_meta_eq end;
fun contains x ct = member (op aconv) (Misc_Legacy.term_frees (term_of ct)) (term_of x);
fun is_eqx x eq = case term_of eq of
Const(@{const_name HOL.eq},_)$l$r => l aconv term_of x orelse r aconv term_of x
| _ => false ;
local
fun proc ctxt ct =
case term_of ct of
Const(@{const_name Ex},_)$Abs (xn,_,_) =>
let
val e = Thm.dest_fun ct
val (x,p) = Thm.dest_abs (SOME xn) (Thm.dest_arg ct)
val Pp = Thm.apply @{cterm "Trueprop"} p
val (eqs,neqs) = List.partition (is_eqx x) (all_conjuncts p)
in case eqs of
[] =>
let
val (dx,ndx) = List.partition (contains x) neqs
in case ndx of [] => NONE
| _ =>
conj_aci_rule (Thm.mk_binop @{cterm "op == :: prop => _"} Pp
(Thm.apply @{cterm Trueprop} (list_conj (ndx @dx))))
|> Thm.abstract_rule xn x |> Drule.arg_cong_rule e
|> Conv.fconv_rule (Conv.arg_conv
(Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps @{thms simp_thms ex_simps})))
|> SOME
end
| _ => conj_aci_rule (Thm.mk_binop @{cterm "op == :: prop => _"} Pp
(Thm.apply @{cterm Trueprop} (list_conj (eqs@neqs))))
|> Thm.abstract_rule xn x |> Drule.arg_cong_rule e
|> Conv.fconv_rule (Conv.arg_conv
(Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps @{thms simp_thms ex_simps})))
|> SOME
end
| _ => NONE;
in val reduce_ex_simproc =
Simplifier.make_simproc
{lhss = [@{cpat "EX x. ?P x"}] , name = "reduce_ex_simproc",
proc = K proc, identifier = []}
end;
fun raw_dlo_conv ctxt dlo_ss ({qe_bnds, qe_nolb, qe_noub, gst, gs, ...}: Langford_Data.entry) =
let
val ctxt' = put_simpset dlo_ss ctxt addsimps @{thms "dnf_simps"} addsimprocs [reduce_ex_simproc]
val dnfex_conv = Simplifier.rewrite ctxt'
val pcv =
Simplifier.rewrite
(put_simpset dlo_ss ctxt
addsimps @{thms simp_thms ex_simps all_simps all_not_ex not_all ex_disj_distrib})
in fn p =>
Qelim.gen_qelim_conv pcv pcv dnfex_conv cons
(Thm.add_cterm_frees p []) (K Thm.reflexive) (K Thm.reflexive)
(K (basic_dloqe ctxt gst qe_bnds qe_nolb qe_noub gs)) p
end;
val grab_atom_bop =
let
fun h bounds tm =
(case term_of tm of
Const (@{const_name HOL.eq}, T) $ _ $ _ =>
if domain_type T = HOLogic.boolT then find_args bounds tm
else Thm.dest_fun2 tm
| Const (@{const_name Not}, _) $ _ => h bounds (Thm.dest_arg tm)
| Const (@{const_name All}, _) $ _ => find_body bounds (Thm.dest_arg tm)
| Const ("all", _) $ _ => find_body bounds (Thm.dest_arg tm)
| Const (@{const_name Ex}, _) $ _ => find_body bounds (Thm.dest_arg tm)
| Const (@{const_name HOL.conj}, _) $ _ $ _ => find_args bounds tm
| Const (@{const_name HOL.disj}, _) $ _ $ _ => find_args bounds tm
| Const (@{const_name HOL.implies}, _) $ _ $ _ => find_args bounds tm
| Const ("==>", _) $ _ $ _ => find_args bounds tm
| Const ("==", _) $ _ $ _ => find_args bounds tm
| Const (@{const_name Trueprop}, _) $ _ => h bounds (Thm.dest_arg tm)
| _ => Thm.dest_fun2 tm)
and find_args bounds tm =
(h bounds (Thm.dest_arg tm) handle CTERM _ => h bounds (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 dlo_instance ctxt tm =
(fst (Langford_Data.get ctxt),
Langford_Data.match ctxt (grab_atom_bop 0 tm));
fun dlo_conv ctxt tm =
(case dlo_instance ctxt tm of
(_, NONE) => raise CTERM ("dlo_conv (langford): no corresponding instance in context!", [tm])
| (ss, SOME instance) => raw_dlo_conv ctxt ss instance tm);
fun generalize_tac f = CSUBGOAL (fn (p, _) => PRIMITIVE (fn st =>
let
fun all T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "all"}
fun gen x t = Thm.apply (all (ctyp_of_term x)) (Thm.lambda x t)
val ts = sort (fn (a,b) => Term_Ord.fast_term_ord (term_of a, term_of b)) (f p)
val p' = fold_rev gen ts p
in Thm.implies_intr p' (Thm.implies_elim st (fold Thm.forall_elim ts (Thm.assume p'))) end));
fun cfrees ats ct =
let
val ins = insert (op aconvc)
fun h acc t =
case (term_of t) of
_$_$_ => if member (op aconvc) ats (Thm.dest_fun2 t)
then ins (Thm.dest_arg t) (ins (Thm.dest_arg1 t) acc)
else h (h acc (Thm.dest_arg t)) (Thm.dest_fun t)
| _$_ => h (h acc (Thm.dest_arg t)) (Thm.dest_fun t)
| Abs(_,_,_) => Thm.dest_abs NONE t ||> h acc |> uncurry (remove (op aconvc))
| Free _ => if member (op aconvc) ats t then acc else ins t acc
| Var _ => if member (op aconvc) ats t then acc else ins t acc
| _ => acc
in h [] ct end
fun dlo_tac ctxt = CSUBGOAL (fn (p, i) =>
(case dlo_instance ctxt p of
(ss, NONE) => simp_tac (put_simpset ss ctxt) i
| (ss, SOME instance) =>
Object_Logic.full_atomize_tac ctxt i THEN
simp_tac (put_simpset ss ctxt) i
THEN (CONVERSION Thm.eta_long_conversion) i
THEN (TRY o generalize_tac (cfrees (#atoms instance))) i
THEN Object_Logic.full_atomize_tac ctxt i
THEN CONVERSION (Object_Logic.judgment_conv (raw_dlo_conv ctxt ss instance)) i
THEN (simp_tac (put_simpset ss ctxt) i)));
end;