diff --git a/src/HOL/Hoare/hoare_tac.ML b/src/HOL/Hoare/hoare_tac.ML --- a/src/HOL/Hoare/hoare_tac.ML +++ b/src/HOL/Hoare/hoare_tac.ML @@ -1,234 +1,237 @@ (* 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_>\case_prod _ _ _ for \Abs (x, T, t)\\ = Free (x, T) :: abs2list t | abs2list (Abs (x, T, _)) = [Free (x, T)] | abs2list _ = []; (** maps {(x1,...,xn). t} to [x1,...,xn] **) fun mk_vars \<^Const_>\Collect _ for 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", \<^Type>\unit\) 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>\case_prod T T2 \<^Type>\bool\ for \absfree (x, T) z\\ end; (** maps [x1,...,xn] to (x1,...,xn) and types**) fun mk_bodyC [] = \<^Const>\Unity\ | 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_>\Pair A B for _ _\ => \<^Type>\prod A B\); in \<^Const>\Pair T T2 for 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; + val pre = + case HOLogic.dest_Trueprop d of + Const _ \$ pre \$ _ \$ _ \$ _ => pre + | Const _ \$ pre \$ _ \$ _ => pre \ \support for \<^file>\~~/src/HOL/Isar_Examples/Hoare.thy\\ in mk_vars pre end; fun mk_CollectC tm = let val \<^Type>\fun t _\ = fastype_of tm; in \<^Const>\Collect t for tm\ end; fun inclt ty = \<^Const>\less_eq ty\; 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>\bool\) \$ mk_bodyC vars)); val small_Collect = mk_CollectC (Abs ("x", varsT, Free (P, varsT --> \<^Type>\bool\) \$ 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>\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 [ resolve_tac ctxt (get_thms \<^named_theorems>\SkipRule\) i, resolve_tac ctxt (get_thms \<^named_theorems>\AbortRule\) i, EVERY [ resolve_tac ctxt (get_thms \<^named_theorems>\BasicRule\) i, resolve_tac ctxt [Mlem] i, split_simp_tac ctxt i], EVERY [ resolve_tac ctxt (get_thms \<^named_theorems>\CondRule\) i, rule_tac false (i + 2), rule_tac false (i + 1)], EVERY [ resolve_tac ctxt (get_thms \<^named_theorems>\WhileRule\) 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>\SeqRuleTC\) 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>\SkipRuleTC\) i, EVERY [ resolve_tac ctxt (get_thms \<^named_theorems>\BasicRuleTC\) i, resolve_tac ctxt [Mlem] i, split_simp_tac ctxt i], EVERY [ resolve_tac ctxt (get_thms \<^named_theorems>\CondRuleTC\) i, rule_tac false (i + 2), rule_tac false (i + 1)], EVERY [ resolve_tac ctxt (get_thms \<^named_theorems>\WhileRuleTC\) 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;