# Theory Pure-ex.Guess

```(*  Title:      Pure/ex/Guess.thy
Author:     Makarius

Improper proof command 'guess': variant of 'obtain' based on tactical result.

<chain_facts>
guess x <proof body> <proof end> ≡

{
fix thesis
<chain_facts> have "PROP ?guess"
apply magic      ― ‹turn goal into ‹thesis ⟹ #thesis››
<proof body>
apply_end magic  ― ‹turn 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"
*)

section ‹Improper proof command 'guess'›

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

text ‹
The @{command guess} 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
‹⋀\<^vec>x. \<^vec>A \<^vec>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 ‹\<^vec>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 = pos, proof_state = true}
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);
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;