File ‹langford.ML›
signature LANGFORD =
sig
val dlo_tac : Proof.context -> int -> tactic
val dlo_conv : Proof.context -> cterm -> thm
end
structure Langford: LANGFORD =
struct
val dest_set =
let
fun h acc ct =
(case Thm.term_of ct of
\<^Const_>‹bot _› => acc
| \<^Const_>‹insert _ for _ _› => h (Thm.dest_arg1 ct :: acc) (Thm.dest_arg ct));
in h [] end;
fun prove_finite cT u =
let
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
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 Thm.term_of ep of
\<^Const_>‹Ex _ for _› =>
let
val p = Thm.dest_arg ep
val ths =
simplify (put_simpset HOL_basic_ss ctxt addsimps gather)
(Thm.instantiate' [] [SOME p] stupid)
val (L, U) =
let val (_, q) = Thm.dest_abs_global (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 = Thm.ctyp_of_cterm a
val ne = \<^instantiate>‹'a = cT and a and A in lemma ‹insert a A ≠ {}› by simp›
val f = prove_finite cT (dest_set S)
in (ne, f) end
val qe =
(case (Thm.term_of L, Thm.term_of U) of
(\<^Const_>‹bot _›, _) =>
let val (neU, fU) = proveneF U
in simp_rule ctxt (Thm.transitive ths (dlo_qeth_nolb OF [neU, fU])) end
| (_, \<^Const_>‹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 Thm.term_of ct of
\<^Const>‹HOL.conj for _ _› => h (h acc (Thm.dest_arg ct)) (Thm.dest_arg1 ct)
| _ => ct :: acc)
in h [] end;
fun conjuncts ct =
(case Thm.term_of ct of
\<^Const>‹HOL.conj for _ _› => Thm.dest_arg1 ct :: conjuncts (Thm.dest_arg ct)
| _ => [ct]);
val list_conj =
foldr1 (fn (A, B) => \<^instantiate>‹A and B in cterm ‹A ∧ B››);
fun mk_conj_tab th =
let
fun h acc th =
(case Thm.prop_of th of
\<^Const_>‹Trueprop for \<^Const_>‹HOL.conj for p q›› =>
h (h acc (th RS conjunct2)) (th RS conjunct1)
| \<^Const_>‹Trueprop for p› => (p, th) :: acc)
in fold (Termtab.insert Thm.eq_thm) (h [] th) Termtab.empty end;
fun is_conj \<^Const_>‹HOL.conj for _ _› = true
| is_conj _ = false;
fun prove_conj tab cjs =
(case cjs of
[c] =>
if is_conj (Thm.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)) (Thm.term_of c))
fun tabr c = the (Termtab.lookup (mk_conj_tab (Thm.assume r)) (Thm.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>‹P = ll and Q = rr in lemma ‹(P ⟹ Q) ⟹ (Q ⟹ P) ⟹ P ⟷ Q› by (rule 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 (Thm.term_of ct)) (Thm.term_of x);
fun is_eqx x eq =
(case Thm.term_of eq of
\<^Const_>‹HOL.eq _ for l r› =>
l aconv Thm.term_of x orelse r aconv Thm.term_of x
| _ => false);
local
fun reduce_ex_proc ctxt ct =
(case Thm.term_of ct of
\<^Const_>‹Ex _ for ‹Abs _›› =>
let
val e = Thm.dest_fun ct
val (x,p) = Thm.dest_abs_global (Thm.dest_arg ct)
val Free (xn, _) = Thm.term_of x
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
\<^instantiate>‹A = p and B = ‹list_conj (ndx @ dx)› in cterm ‹Trueprop A ≡ Trueprop B››
|> 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
\<^instantiate>‹A = p and B = ‹list_conj (eqs @ neqs)› in cterm ‹Trueprop A ≡ Trueprop B››
|> 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 = \<^simproc_setup>‹passive reduce_ex ("∃x. P x") = ‹K reduce_ex_proc››;
end;
fun raw_dlo_conv ctxt dlo_ss ({qe_bnds, qe_nolb, qe_noub, gst, gs, ...}: Langford_Data.entry) =
let
val ctxt' =
Context_Position.set_visible false (put_simpset dlo_ss ctxt)
addsimps @{thms dnf_simps} |> Simplifier.add_proc 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})
val mk_env = Cterms.list_set_rev o Cterms.build o Drule.add_frees_cterm
in
fn p =>
Qelim.gen_qelim_conv ctxt pcv pcv dnfex_conv cons
(mk_env 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 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_>‹Pure.all _ for _› => find_body ctxt (Thm.dest_arg tm)
| \<^Const_>‹Ex _ for _› => find_body ctxt (Thm.dest_arg tm)
| \<^Const_>‹HOL.conj for _ _› => find_args ctxt tm
| \<^Const_>‹HOL.disj for _ _› => find_args ctxt tm
| \<^Const_>‹HOL.implies for _ _› => find_args ctxt tm
| \<^Const_>‹Pure.imp for _ _› => find_args ctxt tm
| \<^Const_>‹Pure.eq _ for _ _› => find_args ctxt 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 _ => h ctxt (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 dlo_instance ctxt tm =
(fst (Langford_Data.get ctxt), Langford_Data.match ctxt (grab_atom_bop ctxt 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 ctxt f = CSUBGOAL (fn (p, _) => PRIMITIVE (fn st =>
let
fun all x t =
Thm.apply (Thm.cterm_of ctxt (Logic.all_const (Thm.typ_of_cterm x))) (Thm.lambda x t)
val ts = sort Thm.fast_term_ord (f p)
val p' = fold_rev all 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 Thm.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_global 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 ctxt (cfrees (#atoms instance))) i
THEN Object_Logic.full_atomize_tac ctxt i
THEN CONVERSION (Object_Logic.judgment_conv ctxt (raw_dlo_conv ctxt ss instance)) i
THEN (simp_tac (put_simpset ss ctxt) i)));
end;