--- 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;
--- a/src/HOL/Hoare/Hoare.thy Tue Nov 05 15:59:17 2002 +0100
+++ b/src/HOL/Hoare/Hoare.thy Wed Nov 06 14:01:38 2002 +0100
@@ -10,7 +10,7 @@
*)
theory Hoare = Main
-files ("Hoare.ML"):
+files ("hoare.ML"):
types
'a bexp = "'a set"
@@ -193,7 +193,7 @@
print_translation {* [("Valid", spec_tr')] *}
-use "Hoare.ML"
+use "hoare.ML"
method_setup vcg = {*
Method.no_args
--- a/src/HOL/Hoare/Pointers.thy Tue Nov 05 15:59:17 2002 +0100
+++ b/src/HOL/Hoare/Pointers.thy Wed Nov 06 14:01:38 2002 +0100
@@ -16,6 +16,30 @@
theory Pointers = Hoare:
+(* field access and update *)
+syntax
+ "@faccess" :: "'a option => ('a \<Rightarrow> 'v option) => 'v"
+ ("_^:_" [65,1000] 65)
+ "@fassign" :: "'p option => id => 'v => 's com"
+ ("(2_^._ :=/ _)" [70,1000,65] 61)
+translations
+ "p^:f" == "f(the p)"
+ "p^.f := e" => "f := fun_upd f (the p) e"
+
+
+text{* An example due to Suzuki: *}
+
+lemma "|- VARS v n.
+ {w = Some w0 & x = Some x0 & y = Some y0 & z = Some z0 &
+ distinct[w0,x0,y0,z0]}
+ w^.v := (1::int); w^.n := x;
+ x^.v := 2; x^.n := y;
+ y^.v := 3; y^.n := z;
+ z^.v := 4; x^.n := z
+ {w^:n^:n^:v = 4}"
+by vcg_simp
+
+
section"The heap"
subsection"Paths in the heap"
@@ -33,12 +57,18 @@
done
lemma [simp]: "path h (Some a) as z =
- (as = [] \<and> z = Some a \<or> (\<exists>bs. as = a#bs \<and> path h (h a) bs z))"
+ (as = [] \<and> z = Some a \<or> (\<exists>bs. as = a#bs \<and> path h (h a) bs z))"
apply(case_tac as)
apply fastsimp
apply fastsimp
done
+lemma [simp]: "\<And>x. path f x (as@bs) z = (\<exists>y. path f x as y \<and> path f y bs z)"
+by(induct as, simp+)
+
+lemma [simp]: "\<And>x. u \<notin> set as \<Longrightarrow> path (f(u\<mapsto>v)) x as y = path f x as y"
+by(induct as, simp, simp add:eq_sym_conv)
+
subsection "Lists on the heap"
constdefs
@@ -66,13 +96,13 @@
lemma list_app: "\<And>x. list h x (as@bs) = (\<exists>y. path h x as y \<and> list h y bs)"
by(induct as, simp, clarsimp)
-lemma list_hd_not_in_tl: "list h (h a) as \<Longrightarrow> a \<notin> set as"
+lemma list_hd_not_in_tl[simp]: "list h (h a) as \<Longrightarrow> a \<notin> set as"
apply (clarsimp simp add:in_set_conv_decomp)
apply(frule list_app[THEN iffD1])
apply(fastsimp dest:list_app[THEN iffD1] list_unique)
done
-lemma list_distinct: "\<And>x. list h x as \<Longrightarrow> distinct as"
+lemma list_distinct[simp]: "\<And>x. list h x as \<Longrightarrow> distinct as"
apply(induct as, simp)
apply(fastsimp dest:list_hd_not_in_tl)
done
@@ -101,7 +131,7 @@
WHILE p ~= None
INV {\<exists>As' Bs'. list tl p As' \<and> list tl q Bs' \<and> set As' \<inter> set Bs' = {} \<and>
rev As' @ Bs' = rev As @ Bs}
- DO r := p; p := tl(the p); tl := tl(the r := q); q := r OD
+ DO r := p; p := p^:tl; r^.tl := q; q := r OD
{list tl q (rev As @ Bs)}"
apply vcg_simp
@@ -113,7 +143,6 @@
apply clarify
apply(rename_tac As' b Bs')
-apply(frule list_distinct)
apply clarsimp
apply(rename_tac As'')
apply(rule_tac x = As'' in exI)
@@ -132,9 +161,9 @@
lemma "|- VARS tl p.
{list tl p As \<and> X \<in> set As}
- WHILE p ~= None & p ~= Some X
- INV {p ~= None & (\<exists>As'. list tl p As' \<and> X \<in> set As')}
- DO p := tl(the p) OD
+ WHILE p \<noteq> None \<and> p \<noteq> Some X
+ INV {p \<noteq> None \<and> (\<exists>As'. list tl p As' \<and> X \<in> set As')}
+ DO p := p^:tl OD
{p = Some X}"
apply vcg_simp
apply(case_tac p)
@@ -150,9 +179,9 @@
lemma "|- VARS tl p.
{path tl p As (Some X)}
- WHILE p ~= None & p ~= Some X
- INV {p ~= None & (\<exists>As'. path tl p As' (Some X))}
- DO p := tl(the p) OD
+ WHILE p \<noteq> None \<and> p \<noteq> Some X
+ INV {p \<noteq> None \<and> (\<exists>As'. path tl p As' (Some X))}
+ DO p := p^:tl OD
{p = Some X}"
apply vcg_simp
apply(case_tac p)
@@ -183,9 +212,9 @@
lemma "|- VARS tl p.
{Some X \<in> ({(Some x,tl x) |x. True}^* `` {p})}
- WHILE p ~= None & p ~= Some X
- INV {p ~= None & Some X \<in> ({(Some x,tl x) |x. True}^* `` {p})}
- DO p := tl(the p) OD
+ WHILE p \<noteq> None \<and> p \<noteq> Some X
+ INV {p \<noteq> None \<and> Some X \<in> ({(Some x,tl x) |x. True}^* `` {p})}
+ DO p := p^:tl OD
{p = Some X}"
apply vcg_simp
apply(case_tac p)
@@ -201,10 +230,10 @@
text{*Finally, the simplest version, based on a relation on type @{typ 'a}:*}
lemma "|- VARS tl p.
- {p ~= None & X \<in> ({(x,y). tl x = Some y}^* `` {the p})}
- WHILE p ~= None & p ~= Some X
- INV {p ~= None & X \<in> ({(x,y). tl x = Some y}^* `` {the p})}
- DO p := tl(the p) OD
+ {p \<noteq> None \<and> X \<in> ({(x,y). tl x = Some y}^* `` {the p})}
+ WHILE p \<noteq> None \<and> p \<noteq> Some X
+ INV {p \<noteq> None \<and> X \<in> ({(x,y). tl x = Some y}^* `` {the p})}
+ DO p := p^:tl OD
{p = Some X}"
apply vcg_simp
apply clarsimp
@@ -214,4 +243,116 @@
apply clarsimp
done
+subsection{*Merging two lists*}
+
+consts merge :: "'a list * 'a list * ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list"
+
+recdef merge "measure(%(xs,ys,f). size xs + size ys)"
+"merge(x#xs,y#ys,f) = (if f x y then x # merge(xs,y#ys,f)
+ else y # merge(x#xs,ys,f))"
+"merge(x#xs,[],f) = x # merge(xs,[],f)"
+"merge([],y#ys,f) = y # merge([],ys,f)"
+"merge([],[],f) = []"
+
+lemma imp_disjCL: "(P|Q \<longrightarrow> R) = ((P \<longrightarrow> R) \<and> (~P \<longrightarrow> Q \<longrightarrow> R))"
+by blast
+
+declare imp_disjL[simp del] imp_disjCL[simp]
+
+lemma "|- VARS hd tl p q r s.
+ {list tl p Ps \<and> list tl q Qs \<and> set Ps \<inter> set Qs = {} \<and>
+ (p \<noteq> None \<or> q \<noteq> None)}
+ IF q = None \<or> p \<noteq> None \<and> p^:hd \<le> q^:hd
+ THEN r := p; p := p^:tl ELSE r := q; q := q^:tl FI;
+ s := r;
+ WHILE p \<noteq> None \<or> q \<noteq> None
+ INV {EX rs ps qs a. path tl r rs s \<and> list tl p ps \<and> list tl q qs \<and>
+ distinct(a # ps @ qs @ rs) \<and> s = Some a \<and>
+ merge(Ps,Qs,\<lambda>x y. hd x \<le> hd y) =
+ rs @ a # merge(ps,qs,\<lambda>x y. hd x \<le> hd y) \<and>
+ (tl a = p \<or> tl a = q)}
+ DO IF q = None \<or> p \<noteq> None \<and> p^:hd \<le> q^:hd
+ THEN s^.tl := p; p := p^:tl ELSE s^.tl := q; q := q^:tl FI;
+ s := s^:tl
+ OD
+ {list tl r (merge(Ps,Qs,\<lambda>x y. hd x \<le> hd y))}"
+apply vcg_simp
+
+apply clarsimp
+apply(rule conjI)
+apply clarsimp
+apply(rule exI, rule conjI, rule disjI1, rule refl)
+apply (fastsimp)
+apply(rule conjI)
+apply clarsimp
+apply(rule exI, rule conjI, rule disjI1, rule refl)
+apply clarsimp
+apply(rule exI)
+apply(rule conjI)
+apply assumption
+apply(rule exI)
+apply(rule conjI)
+apply(rule exI)
+apply(rule conjI)
+apply(rule refl)
+apply assumption
+apply (fastsimp)
+apply(case_tac p)
+apply clarsimp
+apply(rule exI, rule conjI, rule disjI1, rule refl)
+apply (fastsimp)
+apply clarsimp
+apply(rule exI, rule conjI, rule disjI1, rule refl)
+apply(rule exI)
+apply(rule conjI)
+apply(rule exI)
+apply(rule conjI)
+apply(rule refl)
+apply assumption
+apply (fastsimp)
+
+apply clarsimp
+apply(rule conjI)
+apply clarsimp
+apply(rule_tac x = "rs @ [a]" in exI)
+apply simp
+apply(rule_tac x = "bs" in exI)
+apply (fastsimp simp:eq_sym_conv)
+
+apply(rule conjI)
+apply clarsimp
+apply(rule_tac x = "rs @ [a]" in exI)
+apply simp
+apply(rule_tac x = "bs" in exI)
+apply(rule conjI)
+apply (simp add:eq_sym_conv)
+apply(rule exI)
+apply(rule conjI)
+apply(rule_tac x = bsa in exI)
+apply(rule conjI)
+apply(rule refl)
+apply (simp add:eq_sym_conv)
+apply (fastsimp simp:eq_sym_conv)
+apply(case_tac p)
+apply clarsimp
+apply(rule_tac x = "rs @ [a]" in exI)
+apply simp
+apply(rule_tac x = "bs" in exI)
+apply (fastsimp simp:eq_sym_conv)
+
+apply clarsimp
+apply(rule_tac x = "rs @ [a]" in exI)
+apply simp
+apply(rule exI)
+apply(rule conjI)
+apply(rule_tac x = bs in exI)
+apply(rule conjI)
+apply(rule refl)
+apply (simp add:eq_sym_conv)
+apply(rule_tac x = bsa in exI)
+apply (fastsimp simp:eq_sym_conv)
+
+apply(clarsimp simp add:list_app)
+done
+
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Hoare/hoare.ML Wed Nov 06 14:01:38 2002 +0100
@@ -0,0 +1,217 @@
+(* 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;