(*  Title:      HOL/Hoare/hoare_tac.ML

    Author:     Leonor Prensa Nieto & Tobias Nipkow

Derivation of the proof rules and, most importantly, the VCG tactic.

*)

(* FIXME structure Hoare: HOARE *)

(*** The tactics ***)

(*****************************************************************************)

(** The function Mset makes the theorem                                     **)

(** "?Mset <= {(x1,...,xn). ?P (x1,...,xn)} ==> ?Mset <= {s. ?P s}",        **)

(** where (x1,...,xn) are the variables of the particular program we are    **)

(** working on at the moment of the call                                    **)

(*****************************************************************************)

local

(** maps (%x1 ... xn. t) to [x1,...,xn] **)

fun abs2list (Const (@{const_name prod_case}, _) $ Abs (x, T, t)) = Free (x, T) :: abs2list t

  | abs2list (Abs (x, T, t)) = [Free (x, T)]

  | abs2list _ = [];

(** maps {(x1,...,xn). t} to [x1,...,xn] **)

fun mk_vars (Const (@{const_name Collect},_) $ T) = abs2list T

  | mk_vars _ = [];

(** abstraction of body over a tuple formed from a list of free variables.

Types are also built **)

fun mk_abstupleC []     body = absfree ("x", HOLogic.unitT, body)

  | mk_abstupleC (v::w) body = let val (n,T) = dest_Free v

                               in if w=[] then absfree (n, T, body)

        else let val z  = mk_abstupleC w body;

                 val T2 = case z of Abs(_,T,_) => T

                        | Const (_, Type (_,[_, Type (_,[T,_])])) $ _ => T;

       in

        Const (@{const_name prod_case},

          (T --> T2 --> HOLogic.boolT) --> HOLogic.mk_prodT (T,T2) --> HOLogic.boolT)

            $ absfree (n, T, z)

       end end;

(** maps [x1,...,xn] to (x1,...,xn) and types**)

fun mk_bodyC []      = HOLogic.unit

  | mk_bodyC (x::xs) = if xs=[] then x

               else let val (n, T) = dest_Free x ;

                        val z = mk_bodyC xs;

                        val T2 = case z of Free(_, T) => T

                                         | Const (@{const_name Pair}, Type ("fun", [_, Type

                                            ("fun", [_, T])])) $ _ $ _ => T;

                 in Const (@{const_name Pair}, [T, T2] ---> HOLogic.mk_prodT (T, T2)) $ x $ z end;

(** maps a subgoal of the form:

        VARS x1 ... xn {._.} _ {._.} or to [x1,...,xn]**)

fun get_vars c =

  let

    val d = Logic.strip_assums_concl c;

    val Const _ $ pre $ _ $ _ = HOLogic.dest_Trueprop d;

  in mk_vars pre end;

fun mk_CollectC trm =

  let val T as Type ("fun",[t,_]) = fastype_of trm

  in HOLogic.Collect_const t $ trm end;

fun inclt ty = Const (@{const_name Orderings.less_eq}, [ty,ty] ---> HOLogic.boolT);

in

fun Mset ctxt prop =

  let

    val [(Mset, _), (P, _)] = Variable.variant_frees ctxt [] [("Mset", ()), ("P", ())];

    val vars = get_vars prop;

    val varsT = fastype_of (mk_bodyC vars);

    val big_Collect = mk_CollectC (mk_abstupleC vars (Free (P, varsT --> HOLogic.boolT) $ mk_bodyC vars));

    val small_Collect = mk_CollectC (Abs ("x", varsT, Free (P, varsT --> HOLogic.boolT) $ Bound 0));

    val MsetT = fastype_of big_Collect;

    fun Mset_incl t = HOLogic.mk_Trueprop (inclt MsetT $ Free (Mset, MsetT) $ t);

    val impl = Logic.mk_implies (Mset_incl big_Collect, Mset_incl small_Collect);

    val th = Goal.prove ctxt [Mset, P] [] impl (fn _ => blast_tac ctxt 1);

 in (vars, th) end;

end;

(*****************************************************************************)

(** Simplifying:                                                            **)

(** Some useful lemmata, lists and simplification tactics to control which  **)

(** theorems are used to simplify at each moment, so that the original      **)

(** input does not suffer any unexpected transformation                     **)

(*****************************************************************************)

(**Simp_tacs**)

val before_set2pred_simp_tac =

  (simp_tac (HOL_basic_ss addsimps [Collect_conj_eq RS sym, @{thm Compl_Collect}]));

val split_simp_tac = (simp_tac (HOL_basic_ss addsimps [@{thm split_conv}]));

(*****************************************************************************)

(** set2pred_tac transforms sets inclusion into predicates implication,     **)

(** maintaining the original variable names.                                **)

(** Ex. "{x. x=0} <= {x. x <= 1}" -set2pred-> "x=0 --> x <= 1"              **)

(** Subgoals containing intersections (A Int B) or complement sets (-A)     **)

(** are first simplified by "before_set2pred_simp_tac", that returns only   **)

(** subgoals of the form "{x. P x} <= {x. Q x}", which are easily           **)

(** transformed.                                                            **)

(** This transformation may solve very easy subgoals due to a ligth         **)

(** simplification done by (split_all_tac)                                  **)

(*****************************************************************************)

fun set2pred_tac var_names = SUBGOAL (fn (goal, i) =>

  before_set2pred_simp_tac i THEN_MAYBE

  EVERY [

    rtac subsetI i,

    rtac CollectI i,

    dtac CollectD i,

    TRY (split_all_tac i) THEN_MAYBE

     (rename_tac var_names i THEN full_simp_tac (HOL_basic_ss addsimps [@{thm split_conv}]) i)]);

(*****************************************************************************)

(** BasicSimpTac is called to simplify all verification conditions. It does **)

(** a light simplification by applying "mem_Collect_eq", then it calls      **)

(** MaxSimpTac, which solves subgoals of the form "A <= A",                 **)

(** and transforms any other into predicates, applying then                 **)

(** the tactic chosen by the user, which may solve the subgoal completely.  **)

(*****************************************************************************)

fun MaxSimpTac var_names tac = FIRST'[rtac subset_refl, set2pred_tac var_names THEN_MAYBE' tac];

fun BasicSimpTac var_names tac =

  simp_tac

    (HOL_basic_ss addsimps [mem_Collect_eq, @{thm split_conv}] addsimprocs [Record.simproc])

  THEN_MAYBE' MaxSimpTac var_names tac;

(** hoare_rule_tac **)

fun hoare_rule_tac (vars, Mlem) tac =

  let

    val var_names = map (fst o dest_Free) vars;

    fun wlp_tac i =

      rtac @{thm SeqRule} i THEN rule_tac false (i + 1)

    and rule_tac pre_cond i st = st |> (*abstraction over st prevents looping*)

      ((wlp_tac i THEN rule_tac pre_cond i)

        ORELSE

        (FIRST [

          rtac @{thm SkipRule} i,

          rtac @{thm AbortRule} i,

          EVERY [

            rtac @{thm BasicRule} i,

            rtac Mlem i,

            split_simp_tac i],

          EVERY [

            rtac @{thm CondRule} i,

            rule_tac false (i + 2),

            rule_tac false (i + 1)],

          EVERY [

            rtac @{thm WhileRule} i,

            BasicSimpTac var_names tac (i + 2),

            rule_tac true (i + 1)]]

         THEN (if pre_cond then BasicSimpTac var_names tac i else rtac subset_refl i)));

  in rule_tac end;

(** tac is the tactic the user chooses to solve or simplify **)

(** the final verification conditions                       **)

fun hoare_tac ctxt (tac: int -> tactic) = SUBGOAL (fn (goal, i) =>

  SELECT_GOAL (hoare_rule_tac (Mset ctxt goal) tac true 1) i);