single uniqueness theorems for map, (un)fold, (co)rec for mutual (co)datatypes
(* Title: HOL/Hoare/hoare_tac.ML Author: Leonor Prensa Nieto & Tobias NipkowDerivation of the proof rules and, most importantly, the VCG tactic.*)signature HOARE =sig val hoare_rule_tac: Proof.context -> term list * thm -> (int -> tactic) -> bool -> int -> tactic val hoare_tac: Proof.context -> (int -> tactic) -> int -> tacticend;structure Hoare: HOARE =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 (@{const_name case_prod}, _) $ 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 (@{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 case_prod}, (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] = 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 (@{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 tm = let val T as Type ("fun",[t,_]) = fastype_of tm; in HOLogic.Collect_const t $ tm end;fun inclt ty = Const (@{const_name Orderings.less_eq}, [ty,ty] ---> HOLogic.boolT);infun 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}]);(*****************************************************************************)(** 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 var_names = map (fst o dest_Free) vars; fun wlp_tac i = resolve_tac ctxt @{thms 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 @{thms SkipRule} i, resolve_tac ctxt @{thms AbortRule} i, EVERY [ resolve_tac ctxt @{thms BasicRule} i, resolve_tac ctxt [Mlem] i, split_simp_tac ctxt i], EVERY [ resolve_tac ctxt @{thms CondRule} i, rule_tac false (i + 2), rule_tac false (i + 1)], EVERY [ resolve_tac ctxt @{thms 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);end;