(* Title: HOL/Hoare/hoare_tac.ML
Author: Leonor Prensa Nieto & Tobias Nipkow
Author: Walter Guttmann (extension to total-correctness proofs)
Derivation of the proof rules and, most importantly, the VCG tactic.
*)
signature HOARE_TAC =
sig
val hoare_rule_tac: Proof.context -> term list * thm -> (int -> tactic) -> bool ->
int -> tactic
val hoare_tac: Proof.context -> (int -> tactic) -> int -> tactic
val hoare_tc_tac: Proof.context -> (int -> tactic) -> int -> tactic
end;
structure Hoare_Tac: HOARE_TAC =
struct
(*** 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_>\<open>case_prod _ _ _ for \<open>Abs (x, T, t)\<close>\<close> = Free (x, T) :: abs2list t
| abs2list (Abs (x, T, _)) = [Free (x, T)]
| abs2list _ = [];
(** maps {(x1,...,xn). t} to [x1,...,xn] **)
fun mk_vars \<^Const_>\<open>Collect _ for T\<close> = 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", \<^Type>\<open>unit\<close>) 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>\<open>case_prod T T2 \<open>\<^Type>\<open>bool\<close>\<close> for \<open>absfree (x, T) z\<close>\<close>
end;
(** maps [x1,...,xn] to (x1,...,xn) and types**)
fun mk_bodyC [] = \<^Const>\<open>Unity\<close>
| mk_bodyC [x] = x
| mk_bodyC (x :: xs) =
let
val (_, T) = dest_Free x;
val z = mk_bodyC xs;
val T2 =
(case z of
Free (_, T) => T
| \<^Const_>\<open>Pair A B for _ _\<close> => \<^Type>\<open>prod A B\<close>);
in \<^Const>\<open>Pair T T2 for x z\<close> 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 tm =
let val \<^Type>\<open>fun t _\<close> = fastype_of tm;
in \<^Const>\<open>Collect t for tm\<close> end;
fun inclt ty = \<^Const>\<open>less_eq ty\<close>;
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 --> \<^Type>\<open>bool\<close>) $ mk_bodyC vars));
val small_Collect =
mk_CollectC (Abs ("x", varsT, Free (P, varsT --> \<^Type>\<open>bool\<close>) $ 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}]);
(*****************************************************************************)
(** set_to_pred_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 set_to_pred_tac ctxt var_names = SUBGOAL (fn (_, i) =>
before_set2pred_simp_tac ctxt i THEN_MAYBE
EVERY [
resolve_tac ctxt [subsetI] i,
resolve_tac ctxt [CollectI] i,
dresolve_tac ctxt [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)]);
(*******************************************************************************)
(** basic_simp_tac is called to simplify all verification conditions. It does **)
(** a light simplification by applying "mem_Collect_eq", then it calls **)
(** max_simp_tac, 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 max_simp_tac ctxt var_names tac =
FIRST' [resolve_tac ctxt [subset_refl],
set_to_pred_tac ctxt var_names THEN_MAYBE' tac];
fun basic_simp_tac ctxt var_names tac =
simp_tac
(put_simpset HOL_basic_ss ctxt
addsimps [mem_Collect_eq, @{thm split_conv}] addsimprocs [Record.simproc])
THEN_MAYBE' max_simp_tac ctxt var_names tac;
(** hoare_rule_tac **)
fun hoare_rule_tac ctxt (vars, Mlem) tac =
let
val get_thms = Proof_Context.get_thms ctxt;
val var_names = map (fst o dest_Free) vars;
fun wlp_tac i =
resolve_tac ctxt (get_thms \<^named_theorems>\<open>SeqRule\<close>) 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 [
resolve_tac ctxt (get_thms \<^named_theorems>\<open>SkipRule\<close>) i,
resolve_tac ctxt (get_thms \<^named_theorems>\<open>AbortRule\<close>) i,
EVERY [
resolve_tac ctxt (get_thms \<^named_theorems>\<open>BasicRule\<close>) i,
resolve_tac ctxt [Mlem] i,
split_simp_tac ctxt i],
EVERY [
resolve_tac ctxt (get_thms \<^named_theorems>\<open>CondRule\<close>) i,
rule_tac false (i + 2),
rule_tac false (i + 1)],
EVERY [
resolve_tac ctxt (get_thms \<^named_theorems>\<open>WhileRule\<close>) i,
basic_simp_tac ctxt var_names tac (i + 2),
rule_tac true (i + 1)]]
THEN (
if pre_cond then basic_simp_tac ctxt var_names tac i
else resolve_tac ctxt [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 = SUBGOAL (fn (goal, i) =>
SELECT_GOAL (hoare_rule_tac ctxt (Mset ctxt goal) tac true 1) i);
(* total correctness rules *)
fun hoare_tc_rule_tac ctxt (vars, Mlem) tac =
let
val get_thms = Proof_Context.get_thms ctxt;
val var_names = map (fst o dest_Free) vars;
fun wlp_tac i =
resolve_tac ctxt (get_thms \<^named_theorems>\<open>SeqRuleTC\<close>) 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 [
resolve_tac ctxt (get_thms \<^named_theorems>\<open>SkipRuleTC\<close>) i,
EVERY [
resolve_tac ctxt (get_thms \<^named_theorems>\<open>BasicRuleTC\<close>) i,
resolve_tac ctxt [Mlem] i,
split_simp_tac ctxt i],
EVERY [
resolve_tac ctxt (get_thms \<^named_theorems>\<open>CondRuleTC\<close>) i,
rule_tac false (i + 2),
rule_tac false (i + 1)],
EVERY [
resolve_tac ctxt (get_thms \<^named_theorems>\<open>WhileRuleTC\<close>) i,
basic_simp_tac ctxt var_names tac (i + 2),
rule_tac true (i + 1)]]
THEN (
if pre_cond then basic_simp_tac ctxt var_names tac i
else resolve_tac ctxt [subset_refl] i)));
in rule_tac end;
fun hoare_tc_tac ctxt tac = SUBGOAL (fn (goal, i) =>
SELECT_GOAL (hoare_tc_rule_tac ctxt (Mset ctxt goal) tac true 1) i);
end;