(* 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 **) (*****************************************************************************)
(** 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 \<^Type>\<open>bool\<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 pre = case HOLogic.dest_Trueprop d of Const _ $ pre $ _ $ _ $ _ => pre
| Const _ $ pre $ _ $ _ => pre \<comment> \<open>support for \<^file>\<open>~~/src/HOL/Isar_Examples/Hoare.thy\<close>\<close> in mk_vars pre end;
fun mk_CollectC tm = letval \<^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] = Name.variants (Variable.names_of 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 **) (*****************************************************************************)
(*****************************************************************************) (** 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 @{thms subsetI} i,
resolve_tac ctxt @{thms CollectI} i,
dresolve_tac ctxt @{thms 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 |> Simplifier.add_simp @{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 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 @{thms subset_refl} i))); in rule_tac end;
(** tac is the tactic the user chooses to solve or simplify **) (** the final verification conditions **)
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 @{thms subset_refl} i))); in rule_tac end;
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Bemerkung:
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