--- a/src/HOL/Hoare/Hoare.ML Tue Nov 05 15:59:17 2002 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,217 +0,0 @@
-(* Title: HOL/Hoare/Hoare.ML
- ID: $Id$
- Author: Leonor Prensa Nieto & Tobias Nipkow
- Copyright 1998 TUM
-
-Derivation of the proof rules and, most importantly, the VCG tactic.
-*)
-
-(*** The proof rules ***)
-
-Goalw [thm "Valid_def"] "p <= q ==> Valid p (Basic id) q";
-by (Auto_tac);
-qed "SkipRule";
-
-Goalw [thm "Valid_def"] "p <= {s. (f s):q} ==> Valid p (Basic f) q";
-by (Auto_tac);
-qed "BasicRule";
-
-Goalw [thm "Valid_def"] "Valid P c1 Q ==> Valid Q c2 R ==> Valid P (c1;c2) R";
-by (Asm_simp_tac 1);
-by (Blast_tac 1);
-qed "SeqRule";
-
-Goalw [thm "Valid_def"]
- "p <= {s. (s:b --> s:w) & (s~:b --> s:w')} \
-\ ==> Valid w c1 q ==> Valid w' c2 q \
-\ ==> Valid p (Cond b c1 c2) q";
-by (Asm_simp_tac 1);
-by (Blast_tac 1);
-qed "CondRule";
-
-Goal "! s s'. Sem c s s' --> s : I Int b --> s' : I ==> \
-\ ! s s'. s : I --> iter n b (Sem c) s s' --> s' : I & s' ~: b";
-by (induct_tac "n" 1);
- by (Asm_simp_tac 1);
-by (Simp_tac 1);
-by (Blast_tac 1);
-val lemma = result() RS spec RS spec RS mp RS mp;
-
-Goalw [thm "Valid_def"]
- "p <= i ==> Valid (i Int b) c i ==> i Int (-b) <= q \
-\ ==> Valid p (While b j c) q";
-by (Asm_simp_tac 1);
-by (Clarify_tac 1);
-by (dtac lemma 1);
-by (assume_tac 2);
-by (Blast_tac 1);
-by (Blast_tac 1);
-qed "WhileRule'";
-
-Goal
- "p <= i ==> Valid (i Int b) c i ==> i Int (-b) <= q \
-\ ==> Valid p (While b i c) q";
-by (rtac WhileRule' 1);
-by (ALLGOALS assume_tac);
-qed "WhileRule";
-
-(*** 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 open HOLogic in
-
-(** maps (%x1 ... xn. t) to [x1,...,xn] **)
-fun abs2list (Const ("split",_) $ (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 ("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", unitT, body)
- | mk_abstupleC (v::w) body = let val (n,T) = dest_Free v
- in if w=[] then absfree (n, T, body)
- else let val z = mk_abstupleC w body;
- val T2 = case z of Abs(_,T,_) => T
- | Const (_, Type (_,[_, Type (_,[T,_])])) $ _ => T;
- in Const ("split", (T --> T2 --> boolT) --> mk_prodT (T,T2) --> boolT)
- $ absfree (n, T, z) end 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 ("Pair", Type ("fun", [_, Type
- ("fun", [_, T])])) $ _ $ _ => T;
- in Const ("Pair", [T, T2] ---> mk_prodT (T, T2)) $ x $ z end;
-
-fun dest_Goal (Const ("Goal", _) $ P) = P;
-
-(** maps a goal of the form:
- 1. [| P |] ==> |- VARS x1 ... xn. {._.} _ {._.} or to [x1,...,xn]**)
-fun get_vars thm = let val c = dest_Goal (concl_of (thm));
- val d = Logic.strip_assums_concl c;
- val Const _ $ pre $ _ $ _ = dest_Trueprop d;
- in mk_vars pre end;
-
-
-(** Makes Collect with type **)
-fun mk_CollectC trm = let val T as Type ("fun",[t,_]) = fastype_of trm
- in Collect_const t $ trm end;
-
-fun inclt ty = Const ("op <=", [ty,ty] ---> boolT);
-
-(** Makes "Mset <= t" **)
-fun Mset_incl t = let val MsetT = fastype_of t
- in mk_Trueprop ((inclt MsetT) $ Free ("Mset", MsetT) $ t) end;
-
-
-fun Mset thm = let val vars = get_vars(thm);
- val varsT = fastype_of (mk_bodyC vars);
- val big_Collect = mk_CollectC (mk_abstupleC vars
- (Free ("P",varsT --> boolT) $ mk_bodyC vars));
- val small_Collect = mk_CollectC (Abs("x",varsT,
- Free ("P",varsT --> boolT) $ Bound 0));
- val impl = implies $ (Mset_incl big_Collect) $
- (Mset_incl small_Collect);
- in Tactic.prove (Thm.sign_of_thm thm) ["Mset", "P"] [] impl (K (CLASET' blast_tac 1)) 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 **)
-(*****************************************************************************)
-
-Goal "-(Collect b) = {x. ~(b x)}";
-by (Fast_tac 1);
-qed "Compl_Collect";
-
-
-(**Simp_tacs**)
-
-val before_set2pred_simp_tac =
- (simp_tac (HOL_basic_ss addsimps [Collect_conj_eq RS sym,Compl_Collect]));
-
-val split_simp_tac = (simp_tac (HOL_basic_ss addsimps [split_conv]));
-
-(*****************************************************************************)
-(** set2pred 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 i thm = let fun mk_string [] = ""
- | mk_string (x::xs) = x^" "^mk_string xs;
- val vars=get_vars(thm);
- val var_string = mk_string (map (fst o dest_Free) vars);
- in ((before_set2pred_simp_tac i) THEN_MAYBE
- (EVERY [rtac subsetI i,
- rtac CollectI i,
- dtac CollectD i,
- (TRY(split_all_tac i)) THEN_MAYBE
- ((rename_tac var_string i) THEN
- (full_simp_tac (HOL_basic_ss addsimps [split_conv]) i)) ])) thm
- end;
-
-(*****************************************************************************)
-(** 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 tac = FIRST'[rtac subset_refl, set2pred THEN_MAYBE' tac];
-
-fun BasicSimpTac tac =
- simp_tac
- (HOL_basic_ss addsimps [mem_Collect_eq,split_conv] addsimprocs [record_simproc])
- THEN_MAYBE' MaxSimpTac tac;
-
-(** HoareRuleTac **)
-
-fun WlpTac Mlem tac i = rtac SeqRule i THEN HoareRuleTac Mlem tac false (i+1)
-and HoareRuleTac Mlem tac pre_cond i st = st |>
- (*abstraction over st prevents looping*)
- ( (WlpTac Mlem tac i THEN HoareRuleTac Mlem tac pre_cond i)
- ORELSE
- (FIRST[rtac SkipRule i,
- EVERY[rtac BasicRule i,
- rtac Mlem i,
- split_simp_tac i],
- EVERY[rtac CondRule i,
- HoareRuleTac Mlem tac false (i+2),
- HoareRuleTac Mlem tac false (i+1)],
- EVERY[rtac WhileRule i,
- BasicSimpTac tac (i+2),
- HoareRuleTac Mlem tac true (i+1)] ]
- THEN (if pre_cond then (BasicSimpTac tac i) else (rtac subset_refl i)) ));
-
-
-(** tac:(int -> tactic) is the tactic the user chooses to solve or simplify **)
-(** the final verification conditions **)
-
-fun hoare_tac tac i thm =
- let val Mlem = Mset(thm)
- in SELECT_GOAL(EVERY[HoareRuleTac Mlem tac true 1]) i thm end;