(* Title: Tools/coherent.ML
Author: Stefan Berghofer, TU Muenchen
Author: Marc Bezem, Institutt for Informatikk, Universitetet i Bergen
Prover for coherent logic, see e.g.
Marc Bezem and Thierry Coquand, Automating Coherent Logic, LPAR 2005
for a description of the algorithm.
*)
signature COHERENT_DATA =
sig
val atomize_elimL: thm
val atomize_exL: thm
val atomize_conjL: thm
val atomize_disjL: thm
val operator_names: string list
end;
signature COHERENT =
sig
val verbose: bool ref
val show_facts: bool ref
val coherent_tac: thm list -> Proof.context -> int -> tactic
val coherent_meth: thm list -> Proof.context -> Proof.method
val setup: theory -> theory
end;
functor CoherentFun(Data: COHERENT_DATA) : COHERENT =
struct
val verbose = ref false;
fun message f = if !verbose then tracing (f ()) else ();
datatype cl_prf =
ClPrf of thm * (Type.tyenv * Envir.tenv) * ((indexname * typ) * term) list *
int list * (term list * cl_prf) list;
val is_atomic = not o exists_Const (member (op =) Data.operator_names o #1);
local open Conv in
fun rulify_elim_conv ct =
if is_atomic (Logic.strip_imp_concl (term_of ct)) then all_conv ct
else concl_conv (length (Logic.strip_imp_prems (term_of ct)))
(rewr_conv (symmetric Data.atomize_elimL) then_conv
MetaSimplifier.rewrite true (map symmetric
[Data.atomize_exL, Data.atomize_conjL, Data.atomize_disjL])) ct
end;
fun rulify_elim th = MetaSimplifier.norm_hhf (Conv.fconv_rule rulify_elim_conv th);
(* Decompose elimination rule of the form
A1 ==> ... ==> Am ==> (!!xs1. Bs1 ==> P) ==> ... ==> (!!xsn. Bsn ==> P) ==> P
*)
fun dest_elim prop =
let
val prems = Logic.strip_imp_prems prop;
val concl = Logic.strip_imp_concl prop;
val (prems1, prems2) =
take_suffix (fn t => Logic.strip_assums_concl t = concl) prems;
in
(prems1,
if null prems2 then [([], [concl])]
else map (fn t =>
(map snd (Logic.strip_params t), Logic.strip_assums_hyp t)) prems2)
end;
fun mk_rule th =
let
val th' = rulify_elim th;
val (prems, cases) = dest_elim (prop_of th')
in (th', prems, cases) end;
fun mk_dom ts = fold (fn t =>
Typtab.map_default (fastype_of t, []) (fn us => us @ [t])) ts Typtab.empty;
val empty_env = (Vartab.empty, Vartab.empty);
(* Find matcher that makes conjunction valid in given state *)
fun valid_conj ctxt facts env [] = Seq.single (env, [])
| valid_conj ctxt facts env (t :: ts) =
Seq.maps (fn (u, x) => Seq.map (apsnd (cons x))
(valid_conj ctxt facts
(Pattern.match (ProofContext.theory_of ctxt) (t, u) env) ts
handle Pattern.MATCH => Seq.empty))
(Seq.of_list (sort (int_ord o pairself snd) (Net.unify_term facts t)));
(* Instantiate variables that only occur free in conlusion *)
fun inst_extra_vars ctxt dom cs =
let
val vs = fold Term.add_vars (maps snd cs) [];
fun insts [] inst = Seq.single inst
| insts ((ixn, T) :: vs') inst = Seq.maps
(fn t => insts vs' (((ixn, T), t) :: inst))
(Seq.of_list (case Typtab.lookup dom T of
NONE => error ("Unknown domain: " ^
Syntax.string_of_typ ctxt T ^ "\nfor term(s) " ^
commas (maps (map (Syntax.string_of_term ctxt) o snd) cs))
| SOME ts => ts))
in Seq.map (fn inst =>
(inst, map (apsnd (map (subst_Vars (map (apfst fst) inst)))) cs))
(insts vs [])
end;
(* Check whether disjunction is valid in given state *)
fun is_valid_disj ctxt facts [] = false
| is_valid_disj ctxt facts ((Ts, ts) :: ds) =
let val vs = rev (map_index (fn (i, T) => Var (("x", i), T)) Ts)
in case Seq.pull (valid_conj ctxt facts empty_env
(map (fn t => subst_bounds (vs, t)) ts)) of
SOME _ => true
| NONE => is_valid_disj ctxt facts ds
end;
val show_facts = ref false;
fun string_of_facts ctxt s facts = space_implode "\n"
(s :: map (Syntax.string_of_term ctxt)
(map fst (sort (int_ord o pairself snd) (Net.content facts)))) ^ "\n\n";
fun print_facts ctxt facts =
if !show_facts then message (fn () => string_of_facts ctxt "Facts:" facts)
else ();
fun valid ctxt rules goal dom facts nfacts nparams =
let val seq = Seq.of_list rules |> Seq.maps (fn (th, ps, cs) =>
valid_conj ctxt facts empty_env ps |> Seq.maps (fn (env as (tye, _), is) =>
let val cs' = map (fn (Ts, ts) =>
(map (Envir.typ_subst_TVars tye) Ts, map (Envir.subst_vars env) ts)) cs
in
inst_extra_vars ctxt dom cs' |>
Seq.map_filter (fn (inst, cs'') =>
if is_valid_disj ctxt facts cs'' then NONE
else SOME (th, env, inst, is, cs''))
end))
in
case Seq.pull seq of
NONE => (tracing (string_of_facts ctxt "Countermodel found:" facts); NONE)
| SOME ((th, env, inst, is, cs), _) =>
if cs = [([], [goal])] then SOME (ClPrf (th, env, inst, is, []))
else
(case valid_cases ctxt rules goal dom facts nfacts nparams cs of
NONE => NONE
| SOME prfs => SOME (ClPrf (th, env, inst, is, prfs)))
end
and valid_cases ctxt rules goal dom facts nfacts nparams [] = SOME []
| valid_cases ctxt rules goal dom facts nfacts nparams ((Ts, ts) :: ds) =
let
val _ = message (fn () => "case " ^ commas (map (Syntax.string_of_term ctxt) ts));
val params = rev (map_index (fn (i, T) =>
Free ("par" ^ string_of_int (nparams + i), T)) Ts);
val ts' = map_index (fn (i, t) =>
(subst_bounds (params, t), nfacts + i)) ts;
val dom' = fold (fn (T, p) =>
Typtab.map_default (T, []) (fn ps => ps @ [p]))
(Ts ~~ params) dom;
val facts' = fold (fn (t, i) => Net.insert_term op =
(t, (t, i))) ts' facts
in
case valid ctxt rules goal dom' facts'
(nfacts + length ts) (nparams + length Ts) of
NONE => NONE
| SOME prf => (case valid_cases ctxt rules goal dom facts nfacts nparams ds of
NONE => NONE
| SOME prfs => SOME ((params, prf) :: prfs))
end;
(** proof replaying **)
fun thm_of_cl_prf thy goal asms (ClPrf (th, (tye, env), insts, is, prfs)) =
let
val _ = message (fn () => space_implode "\n"
("asms:" :: map Display.string_of_thm asms) ^ "\n\n");
val th' = Drule.implies_elim_list
(Thm.instantiate
(map (fn (ixn, (S, T)) =>
(Thm.ctyp_of thy (TVar ((ixn, S))), Thm.ctyp_of thy T))
(Vartab.dest tye),
map (fn (ixn, (T, t)) =>
(Thm.cterm_of thy (Var (ixn, Envir.typ_subst_TVars tye T)),
Thm.cterm_of thy t)) (Vartab.dest env) @
map (fn (ixnT, t) =>
(Thm.cterm_of thy (Var ixnT), Thm.cterm_of thy t)) insts) th)
(map (nth asms) is);
val (_, cases) = dest_elim (prop_of th')
in
case (cases, prfs) of
([([], [_])], []) => th'
| ([([], [_])], [([], prf)]) => thm_of_cl_prf thy goal (asms @ [th']) prf
| _ => Drule.implies_elim_list
(Thm.instantiate (Thm.match
(Drule.strip_imp_concl (cprop_of th'), goal)) th')
(map (thm_of_case_prf thy goal asms) (prfs ~~ cases))
end
and thm_of_case_prf thy goal asms ((params, prf), (_, asms')) =
let
val cparams = map (cterm_of thy) params;
val asms'' = map (cterm_of thy o curry subst_bounds (rev params)) asms'
in
Drule.forall_intr_list cparams (Drule.implies_intr_list asms''
(thm_of_cl_prf thy goal (asms @ map Thm.assume asms'') prf))
end;
(** external interface **)
fun coherent_tac rules ctxt = SUBPROOF (fn {prems, concl, params, context, ...} =>
rtac (rulify_elim_conv concl RS equal_elim_rule2) 1 THEN
SUBPROOF (fn {prems = prems', concl, context, ...} =>
let val xs = map term_of params @
map (fn (_, s) => Free (s, the (Variable.default_type context s)))
(Variable.fixes_of context)
in
case valid context (map mk_rule (prems' @ prems @ rules)) (term_of concl)
(mk_dom xs) Net.empty 0 0 of
NONE => no_tac
| SOME prf =>
rtac (thm_of_cl_prf (ProofContext.theory_of context) concl [] prf) 1
end) context 1) ctxt;
fun coherent_meth rules ctxt =
METHOD (fn facts => coherent_tac (facts @ rules) ctxt 1);
val setup = Method.add_method
("coherent", Method.thms_ctxt_args coherent_meth, "prove coherent formula");
end;