(* Title: Pure/Isar/obtain.ML
Author: Markus Wenzel, TU Muenchen
The 'obtain' and 'guess' language elements -- generalized existence at
the level of proof texts: 'obtain' involves a proof that certain
fixes/assumes may be introduced into the present context; 'guess' is
similar, but derives these elements from the course of reasoning!
<chain_facts>
obtain x where "A x" <proof> ==
have "!!thesis. (!!x. A x ==> thesis) ==> thesis"
proof succeed
fix thesis
assume that [intro?]: "!!x. A x ==> thesis"
<chain_facts>
show thesis
apply (insert that)
<proof>
qed
fix x assm <<obtain_export>> "A x"
<chain_facts>
guess x <proof body> <proof end> ==
{
fix thesis
<chain_facts> have "PROP ?guess"
apply magic -- {* turns goal into "thesis ==> #thesis" *}
<proof body>
apply_end magic -- {* turns final "(!!x. P x ==> thesis) ==> #thesis" into
"#((!!x. A x ==> thesis) ==> thesis)" which is a finished goal state *}
<proof end>
}
fix x assm <<obtain_export>> "A x"
*)
signature OBTAIN =
sig
val thatN: string
val obtain: string -> (binding * string option * mixfix) list ->
(Attrib.binding * (string * string list) list) list -> bool -> Proof.state -> Proof.state
val obtain_i: string -> (binding * typ option * mixfix) list ->
(Thm.binding * (term * term list) list) list -> bool -> Proof.state -> Proof.state
val result: (Proof.context -> tactic) -> thm list -> Proof.context ->
(cterm list * thm list) * Proof.context
val guess: (binding * string option * mixfix) list -> bool -> Proof.state -> Proof.state
val guess_i: (binding * typ option * mixfix) list -> bool -> Proof.state -> Proof.state
end;
structure Obtain: OBTAIN =
struct
(** obtain_export **)
(*
[x, A x]
:
B
--------
B
*)
fun eliminate_term ctxt xs tm =
let
val vs = map (dest_Free o Thm.term_of) xs;
val bads = Term.fold_aterms (fn t as Free v =>
if member (op =) vs v then insert (op aconv) t else I | _ => I) tm [];
val _ = null bads orelse
error ("Result contains obtained parameters: " ^
space_implode " " (map (Syntax.string_of_term ctxt) bads));
in tm end;
fun eliminate fix_ctxt rule xs As thm =
let
val thy = ProofContext.theory_of fix_ctxt;
val _ = eliminate_term fix_ctxt xs (Thm.full_prop_of thm);
val _ = ObjectLogic.is_judgment thy (Thm.concl_of thm) orelse
error "Conclusion in obtained context must be object-logic judgment";
val ((_, [thm']), ctxt') = Variable.import_thms true [thm] fix_ctxt;
val prems = Drule.strip_imp_prems (#prop (Thm.crep_thm thm'));
in
((Drule.implies_elim_list thm' (map Thm.assume prems)
|> Drule.implies_intr_list (map Drule.norm_hhf_cterm As)
|> Drule.forall_intr_list xs)
COMP rule)
|> Drule.implies_intr_list prems
|> singleton (Variable.export ctxt' fix_ctxt)
end;
fun obtain_export ctxt rule xs _ As =
(eliminate ctxt rule xs As, eliminate_term ctxt xs);
(** obtain **)
fun bind_judgment ctxt name =
let
val (bind, ctxt') = ProofContext.bind_fixes [name] ctxt;
val (t as _ $ Free v) = bind (ObjectLogic.fixed_judgment (ProofContext.theory_of ctxt) name);
in ((v, t), ctxt') end;
val thatN = "that";
local
fun gen_obtain prep_att prep_vars prep_propp
name raw_vars raw_asms int state =
let
val _ = Proof.assert_forward_or_chain state;
val thy = Proof.theory_of state;
val cert = Thm.cterm_of thy;
val ctxt = Proof.context_of state;
val chain_facts = if can Proof.assert_chain state then Proof.the_facts state else [];
(*obtain vars*)
val (vars, vars_ctxt) = prep_vars raw_vars ctxt;
val (_, fix_ctxt) = vars_ctxt |> ProofContext.add_fixes_i vars;
val xs = map (Binding.base_name o #1) vars;
(*obtain asms*)
val (asms_ctxt, proppss) = prep_propp (fix_ctxt, map snd raw_asms);
val asm_props = maps (map fst) proppss;
val asms = map fst (Attrib.map_specs (prep_att thy) raw_asms) ~~ proppss;
val _ = Variable.warn_extra_tfrees fix_ctxt asms_ctxt;
(*obtain statements*)
val thesisN = Name.variant xs AutoBind.thesisN;
val (thesis_var, thesis) = #1 (bind_judgment fix_ctxt thesisN);
val asm_frees = fold Term.add_frees asm_props [];
val parms = xs |> map (fn x =>
let val x' = ProofContext.get_skolem fix_ctxt x
in (x', the_default propT (AList.lookup (op =) asm_frees x')) end);
val that_name = if name = "" then thatN else name;
val that_prop =
Term.list_all_free (parms, Logic.list_implies (asm_props, thesis))
|> Library.curry Logic.list_rename_params xs;
val obtain_prop =
Logic.list_rename_params ([AutoBind.thesisN],
Term.list_all_free ([thesis_var], Logic.mk_implies (that_prop, thesis)));
fun after_qed _ =
Proof.local_qed (NONE, false)
#> `Proof.the_fact #-> (fn rule =>
Proof.fix_i vars
#> Proof.assm_i (obtain_export fix_ctxt rule (map (cert o Free) parms)) asms);
in
state
|> Proof.enter_forward
|> Proof.have_i NONE (K I) [(Thm.empty_binding, [(obtain_prop, [])])] int
|> Proof.proof (SOME Method.succeed_text) |> Seq.hd
|> Proof.fix_i [(Binding.name thesisN, NONE, NoSyn)]
|> Proof.assume_i
[((Binding.name that_name, [ContextRules.intro_query NONE]), [(that_prop, [])])]
|> `Proof.the_facts
||> Proof.chain_facts chain_facts
||> Proof.show_i NONE after_qed [(Thm.empty_binding, [(thesis, [])])] false
|-> Proof.refine_insert
end;
in
val obtain = gen_obtain Attrib.attribute ProofContext.read_vars ProofContext.read_propp;
val obtain_i = gen_obtain (K I) ProofContext.cert_vars ProofContext.cert_propp;
end;
(** tactical result **)
fun check_result ctxt thesis th =
(case Thm.prems_of th of
[prem] =>
if Thm.concl_of th aconv thesis andalso
Logic.strip_assums_concl prem aconv thesis then th
else error ("Guessed a different clause:\n" ^ ProofContext.string_of_thm ctxt th)
| [] => error "Goal solved -- nothing guessed."
| _ => error ("Guess split into several cases:\n" ^ ProofContext.string_of_thm ctxt th));
fun result tac facts ctxt =
let
val thy = ProofContext.theory_of ctxt;
val cert = Thm.cterm_of thy;
val ((thesis_var, thesis), thesis_ctxt) = bind_judgment ctxt AutoBind.thesisN;
val rule =
(case SINGLE (Method.insert_tac facts 1 THEN tac thesis_ctxt) (Goal.init (cert thesis)) of
NONE => raise THM ("Obtain.result: tactic failed", 0, facts)
| SOME th => check_result ctxt thesis (MetaSimplifier.norm_hhf (Goal.conclude th)));
val closed_rule = Thm.forall_intr (cert (Free thesis_var)) rule;
val ((_, [rule']), ctxt') = Variable.import_thms false [closed_rule] ctxt;
val obtain_rule = Thm.forall_elim (cert (Logic.varify (Free thesis_var))) rule';
val ((params, stmt), fix_ctxt) = Variable.focus (Thm.cprem_of obtain_rule 1) ctxt';
val (prems, ctxt'') =
Assumption.add_assms (obtain_export fix_ctxt obtain_rule params)
(Drule.strip_imp_prems stmt) fix_ctxt;
in ((params, prems), ctxt'') end;
(** guess **)
local
fun unify_params vars thesis_var raw_rule ctxt =
let
val thy = ProofContext.theory_of ctxt;
val certT = Thm.ctyp_of thy;
val cert = Thm.cterm_of thy;
val string_of_typ = Syntax.string_of_typ ctxt;
val string_of_term = setmp show_types true (Syntax.string_of_term ctxt);
fun err msg th = error (msg ^ ":\n" ^ ProofContext.string_of_thm ctxt th);
val maxidx = fold (Term.maxidx_typ o snd o fst) vars ~1;
val rule = Thm.incr_indexes (maxidx + 1) raw_rule;
val params = RuleCases.strip_params (Logic.nth_prem (1, Thm.prop_of rule));
val m = length vars;
val n = length params;
val _ = m <= n orelse err "More variables than parameters in obtained rule" rule;
fun unify ((x, T), (y, U)) (tyenv, max) = Sign.typ_unify thy (T, U) (tyenv, max)
handle Type.TUNIFY =>
err ("Failed to unify variable " ^
string_of_term (Free (x, Envir.norm_type tyenv T)) ^ " against parameter " ^
string_of_term (Syntax.mark_boundT (y, Envir.norm_type tyenv U)) ^ " in") rule;
val (tyenv, _) = fold unify (map #1 vars ~~ Library.take (m, params))
(Vartab.empty, Int.max (maxidx, Thm.maxidx_of rule));
val norm_type = Envir.norm_type tyenv;
val xs = map (apsnd norm_type o fst) vars;
val ys = map (apsnd norm_type) (Library.drop (m, params));
val ys' = map Name.internal (Name.variant_list (map fst xs) (map fst ys)) ~~ map #2 ys;
val terms = map (Drule.mk_term o cert o Free) (xs @ ys');
val instT =
fold (Term.add_tvarsT o #2) params []
|> map (TVar #> (fn T => (certT T, certT (norm_type T))));
val closed_rule = rule
|> Thm.forall_intr (cert (Free thesis_var))
|> Thm.instantiate (instT, []);
val ((_, rule' :: terms'), ctxt') = Variable.import_thms false (closed_rule :: terms) ctxt;
val vars' =
map (dest_Free o Thm.term_of o Drule.dest_term) terms' ~~
(map snd vars @ replicate (length ys) NoSyn);
val rule'' = Thm.forall_elim (cert (Logic.varify (Free thesis_var))) rule';
in ((vars', rule''), ctxt') end;
fun inferred_type (binding, _, mx) ctxt =
let
val x = Binding.base_name binding;
val (T, ctxt') = ProofContext.inferred_param x ctxt
in ((x, T, mx), ctxt') end;
fun polymorphic ctxt vars =
let val Ts = map Logic.dest_type (Variable.polymorphic ctxt (map (Logic.mk_type o #2) vars))
in map2 (fn (x, _, mx) => fn T => ((x, T), mx)) vars Ts end;
fun gen_guess prep_vars raw_vars int state =
let
val _ = Proof.assert_forward_or_chain state;
val thy = Proof.theory_of state;
val cert = Thm.cterm_of thy;
val ctxt = Proof.context_of state;
val chain_facts = if can Proof.assert_chain state then Proof.the_facts state else [];
val (thesis_var, thesis) = #1 (bind_judgment ctxt AutoBind.thesisN);
val vars = ctxt |> prep_vars raw_vars |-> fold_map inferred_type |> fst |> polymorphic ctxt;
fun guess_context raw_rule state' =
let
val ((parms, rule), ctxt') =
unify_params vars thesis_var raw_rule (Proof.context_of state');
val (bind, _) = ProofContext.bind_fixes (map (#1 o #1) parms) ctxt';
val ts = map (bind o Free o #1) parms;
val ps = map dest_Free ts;
val asms =
Logic.strip_assums_hyp (Logic.nth_prem (1, Thm.prop_of rule))
|> map (fn asm => (Term.betapplys (Term.list_abs (ps, asm), ts), []));
val _ = not (null asms) orelse error "Trivial result -- nothing guessed";
in
state'
|> Proof.map_context (K ctxt')
|> Proof.fix_i (map (fn ((x, T), mx) => (Binding.name x, SOME T, mx)) parms)
|> `Proof.context_of |-> (fn fix_ctxt => Proof.assm_i
(obtain_export fix_ctxt rule (map cert ts)) [(Thm.empty_binding, asms)])
|> Proof.add_binds_i AutoBind.no_facts
end;
val goal = Var (("guess", 0), propT);
fun print_result ctxt' (k, [(s, [_, th])]) =
ProofDisplay.print_results int ctxt' (k, [(s, [th])]);
val before_qed = SOME (Method.primitive_text (Goal.conclude #> MetaSimplifier.norm_hhf #>
(fn th => Goal.protect (Conjunction.intr (Drule.mk_term (Thm.cprop_of th)) th))));
fun after_qed [[_, res]] =
Proof.end_block #> guess_context (check_result ctxt thesis res);
in
state
|> Proof.enter_forward
|> Proof.begin_block
|> Proof.fix_i [(Binding.name AutoBind.thesisN, NONE, NoSyn)]
|> Proof.chain_facts chain_facts
|> Proof.local_goal print_result (K I) (apsnd (rpair I))
"guess" before_qed after_qed [(Thm.empty_binding, [Logic.mk_term goal, goal])]
|> Proof.refine (Method.primitive_text (K (Goal.init (cert thesis)))) |> Seq.hd
end;
in
val guess = gen_guess ProofContext.read_vars;
val guess_i = gen_guess ProofContext.cert_vars;
end;
end;