Quelle  Guess.thy

(*  Title:      Pure/ex/Guess.thy
  Author: Makarius
 
 Improper proof command 'guess': variant of 'obtain' based on tactical result.
 
  🪙
  guess x 🚫body> 🚫end> ≡
 
  {
  fix thesis
  🪙 have "PROP ?guess"
  apply magic 🍋 ‹turn goal into ‹thesis ==> #thesis›\›
  🚫body>
  apply_end magic 🍋 ‹turn final ‹(∧x. P x ==> thesis) ==> #thesis› into›
  🍋 ‹‹#((∧x. A x ==> thesis) ==> thesis)› which is a finished goal state›
  🚫end>
  }
  fix x assm 🚫>> "A x"
*)

section ‹Improper proof command 'guess'›

theory Guess
  imports Pure
  keywords "guess" :: prf_script_asm_goal % "proof"
begin

text ‹
  The is similar to @{command obtain}, but it derives the obtained context
  elements from the course of tactical reasoning in the proof. Thus it can
  considerably obscure the proof: it is provided here as 🪙‹improper› and
  experimental feature.
 
  A proof with @{command guess} starts with a fixed goal ‹thesis›. The
  subsequent refinement steps may turn this to anything of the form
  ‹∧🪙x. 🪙A 🪙x ==> thesis›, but without splitting into new
  subgoals. The final goal state is then used as reduction rule for the obtain
  pattern described above. Obtained parameters ‹🪙x› are marked as
  internal by default, and thus inaccessible in the proof text. The variable
  names and type constraints given as arguments for @{command "guess"} specify
  a prefix of accessible parameters.
 
  Some examples are in 🍋‹Guess_Examples.thy›.
 ›

ML ‹
 signature GUESS =
 sig
  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 Guess: GUESS =
 struct
 
 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 =
  TVars.build
  (params |> fold (#2 #> Term.fold_atyps (fn T => fn instT =>
  (case T of
  TVar v =>
  if TVars.defined instT v then instT
  else TVars.add (v, Thm.ctyp_of ctxt (norm_type T)) instT
  | _ => instT))));
  val closed_rule = rule
  |> Thm.forall_intr (Thm.cterm_of ctxt (Free thesis_var))
  |> Thm.instantiate (instT, Vars.empty);
 
  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.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.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 guess = (("guess", 0), propT);
  val goal = Var guess;
  fun print_result ctxt' (k, [(s, [_, th])]) =
  Proof_Display.print_results {interactive = int, pos = Position.thread_data ()}
  ctxt' (k, [(s, [th])]);
  val before_qed =
  Method.primitive_text (fn ctxt =>
  Goal.conclude #> 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);
  val res' = res RS Conjunction.conjunctionD2;
  in
  state'
  |> Proof.end_block
  |> guess_context (Obtain.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.Basic (fn _ => fn _ => CONTEXT_TACTIC
  (PRIMITIVE (Thm.instantiate (TVars.empty, Vars.make1 (guess, Thm.cterm_of ctxt thesis)))
  THEN Goal.conjunction_tac 1
  THEN resolve_tac ctxt [Drule.termI] 1)))
  end;
 
 in
 
 val guess = gen_guess Proof_Context.cert_var;
 val guess_cmd = gen_guess Proof_Context.read_var;
 
 val _ =
  Outer_Syntax.command 🍋‹guess› "wild guessing (unstructured)"
  (Scan.optional Parse.vars [] >> (Toplevel.proof' o guess_cmd));
 
 end;
 
 end;
 ›

end