tuned proofs -- clarified flow of facts wrt. calculation;
(* 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] body = absfree (dest_Free v) body
| mk_abstupleC (v :: w) body =
let
val (x, T) = dest_Free v;
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 (x, T) z
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**)
fun before_set2pred_simp_tac ctxt =
simp_tac (put_simpset HOL_basic_ss ctxt addsimps [Collect_conj_eq RS sym, @{thm Compl_Collect}]);
fun split_simp_tac ctxt =
simp_tac (put_simpset HOL_basic_ss ctxt 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 ctxt var_names = SUBGOAL (fn (goal, i) =>
before_set2pred_simp_tac ctxt i THEN_MAYBE
EVERY [
rtac subsetI i,
rtac CollectI i,
dtac CollectD i,
TRY (split_all_tac ctxt i) THEN_MAYBE
(rename_tac var_names i THEN
full_simp_tac (put_simpset HOL_basic_ss ctxt 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 ctxt var_names tac =
FIRST'[rtac subset_refl, set2pred_tac ctxt var_names THEN_MAYBE' tac];
fun BasicSimpTac ctxt var_names tac =
simp_tac
(put_simpset HOL_basic_ss ctxt
addsimps [mem_Collect_eq, @{thm split_conv}] addsimprocs [Record.simproc])
THEN_MAYBE' MaxSimpTac ctxt var_names tac;
(** hoare_rule_tac **)
fun hoare_rule_tac ctxt (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 ctxt i],
EVERY [
rtac @{thm CondRule} i,
rule_tac false (i + 2),
rule_tac false (i + 1)],
EVERY [
rtac @{thm WhileRule} i,
BasicSimpTac ctxt var_names tac (i + 2),
rule_tac true (i + 1)]]
THEN (if pre_cond then BasicSimpTac ctxt 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 ctxt (Mset ctxt goal) tac true 1) i);