(* 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 * typ option * mixfix) list ->
(Thm.binding * (term * term list) list) list -> bool -> Proof.state -> Proof.state
val obtain_cmd: string -> (binding * string option * mixfix) list ->
(Attrib.binding * (string * string list) list) list -> bool -> Proof.state -> Proof.state
val result: (Proof.context -> tactic) -> thm list -> Proof.context ->
((string * cterm) list * thm list) * Proof.context
val guess: (binding * typ option * mixfix) list -> bool -> Proof.state -> Proof.state
val guess_cmd: (binding * string 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 ctxt rule xs As thm =
let
val _ = eliminate_term ctxt xs (Thm.full_prop_of thm);
val _ = Object_Logic.is_judgment ctxt (Thm.concl_of thm) orelse
error "Conclusion in obtained context must be object-logic judgment";
val ((_, [thm']), ctxt') = Variable.import true [thm] 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 ctxt') As)
|> Drule.forall_intr_list xs)
COMP rule)
|> Drule.implies_intr_list prems
|> singleton (Variable.export ctxt' 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 ([x], ctxt') = Proof_Context.add_fixes [(Binding.name name, NONE, NoSyn)] ctxt;
val (t as _ $ Free v) = Object_Logic.fixed_judgment ctxt x;
in ((v, t), ctxt') end;
val thatN = "that";
local
fun gen_obtain prep_att prep_var prep_propp
name raw_vars raw_asms int state =
let
val _ = Proof.assert_forward_or_chain state;
val ctxt = Proof.context_of state;
val chain_facts = if can Proof.assert_chain state then Proof.the_facts state else [];
(*obtain vars*)
val ((xs', vars), fix_ctxt) = ctxt
|> fold_map prep_var raw_vars
|-> (fn vars => Proof_Context.add_fixes vars ##>> pair vars);
val xs = map (Variable.check_name o #1) vars;
(*obtain asms*)
val (asm_propss, binds) = prep_propp fix_ctxt (map snd raw_asms);
val asm_props = flat asm_propss;
val declare_asms = fold Variable.declare_term asm_props #> Proof_Context.bind_terms binds;
val asms =
map (fn ((b, raw_atts), _) => (b, map (prep_att fix_ctxt) raw_atts)) raw_asms ~~
map (map (rpair [])) asm_propss;
(*obtain params*)
val (Ts, params_ctxt) = fold_map Proof_Context.inferred_param xs' (declare_asms fix_ctxt);
val params = map Free (xs' ~~ Ts);
val cparams = map (Thm.cterm_of params_ctxt) params;
val binds' = map (apsnd (fold_rev Term.dependent_lambda_name (xs ~~ params))) binds;
val _ = Variable.warn_extra_tfrees fix_ctxt params_ctxt;
(*obtain statements*)
val thesisN = singleton (Name.variant_list xs) Auto_Bind.thesisN;
val (thesis_var, thesis) = #1 (bind_judgment params_ctxt thesisN);
val that_name = if name = "" then thatN else name;
val that_prop =
Logic.list_rename_params xs
(fold_rev Logic.all params (Logic.list_implies (asm_props, thesis)));
val obtain_prop =
Logic.list_rename_params [Auto_Bind.thesisN]
(Logic.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 vars
#> Proof.map_context declare_asms
#> Proof.assm (obtain_export params_ctxt rule cparams) asms);
in
state
|> Proof.enter_forward
|> Proof.have NONE (K I) [] [(Thm.empty_binding, [(obtain_prop, [])])] int
|> Proof.map_context (fold Variable.bind_term binds')
|> Proof.proof (SOME Method.succeed_text) |> Seq.hd
|> Proof.fix [(Binding.name thesisN, NONE, NoSyn)]
|> Proof.assume
[((Binding.name that_name, [Context_Rules.intro_query NONE]), [(that_prop, [])])]
|> `Proof.the_facts
||> Proof.chain_facts chain_facts
||> Proof.show NONE after_qed [] [(Thm.empty_binding, [(thesis, [])])] false
|-> Proof.refine_insert
end;
in
val obtain = gen_obtain (K I) Proof_Context.cert_var Proof_Context.cert_propp;
val obtain_cmd = gen_obtain Attrib.attribute_cmd Proof_Context.read_var Proof_Context.read_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" ^ Display.string_of_thm ctxt th)
| [] => error "Goal solved -- nothing guessed"
| _ => error ("Guess split into several cases:\n" ^ Display.string_of_thm ctxt th));
fun result tac facts ctxt =
let
val ((thesis_var, thesis), thesis_ctxt) = bind_judgment ctxt Auto_Bind.thesisN;
val st = Goal.init (Thm.cterm_of ctxt thesis);
val rule =
(case SINGLE (Method.insert_tac facts 1 THEN tac thesis_ctxt) st of
NONE => raise THM ("Obtain.result: tactic failed", 0, facts)
| SOME th =>
check_result thesis_ctxt thesis (Raw_Simplifier.norm_hhf thesis_ctxt (Goal.conclude th)));
val closed_rule = Thm.forall_intr (Thm.cterm_of ctxt (Free thesis_var)) rule;
val ((_, [rule']), ctxt') = Variable.import false [closed_rule] ctxt;
val obtain_rule =
Thm.forall_elim (Thm.cterm_of ctxt (Logic.varify_global (Free thesis_var))) rule';
val ((params, stmt), fix_ctxt) = Variable.focus_cterm (Thm.cprem_of obtain_rule 1) ctxt';
val (prems, ctxt'') =
Assumption.add_assms (obtain_export fix_ctxt obtain_rule (map #2 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 = Proof_Context.theory_of ctxt;
val string_of_term = Syntax.string_of_term (Config.put show_types true ctxt);
fun err msg th = error (msg ^ ":\n" ^ Display.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 = Rule_Cases.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_Trans.mark_bound_abs (y, Envir.norm_type tyenv U)) ^ " in") rule;
val (tyenv, _) = fold unify (map #1 vars ~~ 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) (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 Thm.cterm_of ctxt o Free) (xs @ ys');
val instT =
fold (Term.add_tvarsT o #2) params []
|> map (TVar #> (fn T => apply2 (Thm.ctyp_of ctxt) (T, norm_type T)));
val closed_rule = rule
|> Thm.forall_intr (Thm.cterm_of ctxt (Free thesis_var))
|> Thm.instantiate (instT, []);
val ((_, rule' :: terms'), ctxt') = Variable.import 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 (Thm.cterm_of ctxt' (Logic.varify_global (Free thesis_var))) rule';
in ((vars', rule''), ctxt') end;
fun inferred_type (binding, _, mx) ctxt =
let
val x = Variable.check_name binding;
val (T, ctxt') = Proof_Context.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_var raw_vars int state =
let
val _ = Proof.assert_forward_or_chain state;
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 Auto_Bind.thesisN);
val vars = ctxt
|> fold_map prep_var 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 (xs, _) = Variable.add_fixes (map (#1 o #1) parms) ctxt';
val ps = xs ~~ map (#2 o #1) parms;
val ts = map Free ps;
val asms =
Logic.strip_assums_hyp (Logic.nth_prem (1, Thm.prop_of rule))
|> map (fn asm => (Term.betapplys (fold_rev Term.abs ps asm, ts), []));
val _ = not (null asms) orelse error "Trivial result -- nothing guessed";
in
state'
|> Proof.map_context (K ctxt')
|> Proof.fix (map (fn ((x, T), mx) => (Binding.name x, SOME T, mx)) parms)
|> `Proof.context_of |-> (fn fix_ctxt => Proof.assm
(obtain_export fix_ctxt rule (map (Thm.cterm_of ctxt) ts))
[(Thm.empty_binding, asms)])
|> Proof.map_context (fold Variable.unbind_term Auto_Bind.no_facts)
end;
val goal = Var (("guess", 0), propT);
val pos = Position.thread_data ();
fun print_result ctxt' (k, [(s, [_, th])]) =
Proof_Display.print_results int pos ctxt' (k, [(s, [th])]);
val before_qed =
Method.primitive_text (fn ctxt =>
Goal.conclude #> Raw_Simplifier.norm_hhf ctxt #>
(fn th => Goal.protect 0 (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 [(Binding.name Auto_Bind.thesisN, NONE, NoSyn)]
|> Proof.chain_facts chain_facts
|> Proof.internal_goal print_result Proof_Context.mode_schematic "guess"
(SOME before_qed) after_qed [] [(Thm.empty_binding, [(Logic.mk_term goal, []), (goal, [])])]
|> Proof.refine (Method.primitive_text (fn _ => fn _ =>
Goal.init (Thm.cterm_of ctxt thesis))) |> Seq.hd
end;
in
val guess = gen_guess Proof_Context.cert_var;
val guess_cmd = gen_guess Proof_Context.read_var;
end;
end;