(* Title: Pure/Isar/obtain.ML
Author: Markus Wenzel, TU Muenchen
Generalized existence and cases rules within Isar proof text.
*)
signature OBTAIN =
sig
val obtain_thesis: Proof.context -> ((string * typ) * term) * Proof.context
val obtains_attributes: ('typ, 'term) Element.obtain list -> attribute list
val obtains_attribs: ('typ, 'term) Element.obtain list -> Token.src list
val read_obtains: Proof.context -> term -> Element.obtains -> (binding * term) list
val cert_obtains: Proof.context -> term -> Element.obtains_i -> (binding * term) list
val parse_obtains: Proof.context -> term -> Element.obtains -> (binding * term) list
val consider: Element.obtains_i -> bool -> Proof.state -> Proof.state
val consider_cmd: Element.obtains -> bool -> Proof.state -> Proof.state
val obtain: binding -> (binding * typ option * mixfix) list ->
(binding * typ option * mixfix) list -> (term * term list) list list ->
(Thm.binding * (term * term list) list) list -> bool -> Proof.state -> Proof.state
val obtain_cmd: binding -> (binding * string option * mixfix) list ->
(binding * string option * mixfix) list -> (string * string list) list 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
(** specification elements **)
(* 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 (Thm.cprop_of 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);
(* result declaration *)
fun case_names (obtains: ('typ, 'term) Element.obtain list) =
obtains |> map_index (fn (i, (b, _)) =>
if Binding.is_empty b then string_of_int (i + 1) else Name_Space.base_name b);
fun obtains_attributes obtains =
[Rule_Cases.consumes (~ (length obtains)), Rule_Cases.case_names (case_names obtains)];
fun obtains_attribs obtains =
[Attrib.consumes (~ (length obtains)), Attrib.case_names (case_names obtains)];
(* obtain thesis *)
fun obtain_thesis ctxt =
let
val ([x], ctxt') =
Proof_Context.add_fixes [(Binding.name Auto_Bind.thesisN, NONE, NoSyn)] ctxt;
val t = Object_Logic.fixed_judgment ctxt x;
val v = dest_Free (Object_Logic.drop_judgment ctxt t);
in ((v, t), ctxt') end;
(* obtain clauses *)
local
val mk_all_external = Logic.all_constraint o Variable.default_type;
fun mk_all_internal ctxt (y, z) t =
let
val T =
(case AList.lookup (op =) (Term.add_frees t []) z of
SOME T => T
| NONE => the_default dummyT (Variable.default_type ctxt z));
in Logic.all_const T $ Term.lambda_name (y, Free (z, T)) t end;
fun prepare_clause prep_var parse_prop mk_all ctxt thesis raw_vars raw_props =
let
val ((xs', vars), 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;
in
Logic.list_implies (map (parse_prop ctxt') raw_props, thesis)
|> fold_rev (mk_all ctxt') (xs ~~ xs')
end;
fun prepare_obtains prep_clause check_terms
ctxt thesis (raw_obtains: ('typ, 'term) Element.obtain list) =
let
val clauses = raw_obtains
|> map (fn (_, (raw_vars, raw_props)) => prep_clause ctxt thesis raw_vars raw_props)
|> check_terms ctxt;
in map fst raw_obtains ~~ clauses end;
val parse_clause = prepare_clause Proof_Context.read_var Syntax.parse_prop mk_all_external;
val cert_clause = prepare_clause Proof_Context.cert_var (K I) mk_all_internal;
in
val read_obtains = prepare_obtains parse_clause Syntax.check_terms;
val cert_obtains = prepare_obtains cert_clause (K I);
val parse_obtains = prepare_obtains parse_clause (K I);
end;
(** consider: generalized elimination and cases rule **)
(*
consider (a) x where "A x" | (b) y where "B y" | ... \<equiv>
have thesis
if a [intro?]: "\<And>x. A x \<Longrightarrow> thesis"
and b [intro?]: "\<And>y. B y \<Longrightarrow> thesis"
and ...
for thesis
apply (insert that)
*)
local
fun gen_consider prep_obtains raw_obtains int state =
let
val _ = Proof.assert_forward_or_chain state;
val ctxt = Proof.context_of state;
val ((_, thesis), thesis_ctxt) = obtain_thesis ctxt;
val obtains = prep_obtains thesis_ctxt thesis raw_obtains;
val atts = Rule_Cases.cases_open :: obtains_attributes raw_obtains;
in
state
|> Proof.have true NONE (K I)
[(Binding.name Auto_Bind.thesisN, NONE, NoSyn)]
(map (fn (a, A) => ((a, [Context_Rules.intro_query NONE]), [(A, [])])) obtains)
[((Binding.empty, atts), [(thesis, [])])] int
|-> Proof.refine_insert
end;
in
val consider = gen_consider cert_obtains;
val consider_cmd = gen_consider read_obtains;
end;
(** obtain: augmented context based on generalized existence rule **)
(*
obtain (a) x where "A x" <proof> \<equiv>
have thesis if a [intro?]: "\<And>x. A x \<Longrightarrow> thesis" for thesis
apply (insert that)
<proof>
fix x assm <<obtain_export>> "A x"
*)
local
fun gen_obtain prep_stmt prep_att that_binding raw_decls raw_fixes raw_prems raw_concls int state =
let
val _ = Proof.assert_forward_or_chain state;
val ((_, thesis), thesis_ctxt) = obtain_thesis (Proof.context_of state);
val ({vars, propss, binds, result_binds, ...}, params_ctxt) =
prep_stmt (raw_decls @ raw_fixes) (raw_prems @ map #2 raw_concls) thesis_ctxt;
val (decls, fixes) = chop (length raw_decls) vars ||> map #2;
val (premss, conclss) = chop (length raw_prems) propss;
val propss' = (map o map) (Logic.close_prop fixes (flat premss)) conclss;
val that_prop =
Logic.list_rename_params (map (#1 o #2) decls)
(fold_rev (Logic.all o #2 o #2) decls (Logic.list_implies (flat propss', thesis)));
val cparams = map (Thm.cterm_of params_ctxt o #2 o #2) decls;
val asms =
map (fn ((b, raw_atts), _) => (b, map (prep_att params_ctxt) raw_atts)) raw_concls ~~
map (map (rpair [])) propss';
fun after_qed (result_ctxt, results) state' =
let val [rule] = Proof_Context.export result_ctxt (Proof.context_of state') (flat results) in
state'
|> Proof.fix (map #1 decls)
|> Proof.map_context (fold (Variable.bind_term o apsnd (Logic.close_term fixes)) binds)
|> Proof.assm (obtain_export params_ctxt rule cparams) [] [] asms
end;
in
state
|> Proof.have true NONE after_qed
[(Binding.name Auto_Bind.thesisN, NONE, NoSyn)]
[((that_binding, [Context_Rules.intro_query NONE]), [(that_prop, [])])]
[(Binding.empty_atts, [(thesis, [])])] int
|-> Proof.refine_insert
|> Proof.map_context (fold Variable.bind_term result_binds)
end;
in
val obtain = gen_obtain Proof_Context.cert_stmt (K I);
val obtain_cmd = gen_obtain Proof_Context.read_stmt Attrib.attribute_cmd;
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" ^ Thm.string_of_thm ctxt th)
| [] => error "Goal solved -- nothing guessed"
| _ => error ("Guess split into several cases:\n" ^ Thm.string_of_thm ctxt th));
fun result tac facts ctxt =
let
val ((thesis_var, thesis), thesis_ctxt) = obtain_thesis ctxt;
val st = Goal.init (Thm.cterm_of ctxt thesis);
val rule =
(case SINGLE (Method.insert_tac thesis_ctxt 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 NONE (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: obtain based on tactical result **)
(*
<chain_facts>
guess x <proof body> <proof end> \<equiv>
{
fix thesis
<chain_facts> have "PROP ?guess"
apply magic \<comment> \<open>turn goal into \<open>thesis \<Longrightarrow> #thesis\<close>\<close>
<proof body>
apply_end magic \<comment> \<open>turn final \<open>(\<And>x. P x \<Longrightarrow> thesis) \<Longrightarrow> #thesis\<close> into\<close>
\<comment> \<open>\<open>#((\<And>x. A x \<Longrightarrow> thesis) \<Longrightarrow> thesis)\<close> which is a finished goal state\<close>
<proof end>
}
fix x assm <<obtain_export>> "A x"
*)
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" ^ Thm.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 (fn v => (v, Thm.ctyp_of ctxt (norm_type (TVar v))));
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 (obtain_thesis ctxt);
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))
[] [] [(Binding.empty_atts, 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 (result_ctxt, results) state' =
let val [_, res] = Proof_Context.export result_ctxt (Proof.context_of state') (flat results)
in
state'
|> Proof.end_block
|> guess_context (check_result ctxt thesis res)
end;
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 true "guess"
(SOME before_qed) after_qed
[] [] [(Binding.empty_atts, [(Logic.mk_term goal, []), (goal, [])])]
|> snd
|> Proof.refine_singleton
(Method.primitive_text (fn _ => fn _ => Goal.init (Thm.cterm_of ctxt thesis)))
end;
in
val guess = gen_guess Proof_Context.cert_var;
val guess_cmd = gen_guess Proof_Context.read_var;
end;
end;