New many-sorted version.
--- a/src/HOL/Hoare/Examples.ML Wed Oct 14 11:51:11 1998 +0200
+++ b/src/HOL/Hoare/Examples.ML Wed Oct 14 15:26:31 1998 +0200
@@ -1,94 +1,178 @@
(* Title: HOL/Hoare/Examples.thy
ID: $Id$
- Author: Norbert Galm
- Copyright 1995 TUM
-
-Various arithmetic examples.
+ Author: Norbert Galm & Tobias Nipkow
+ Copyright 1998 TUM
*)
-open Examples;
+(*** ARITHMETIC ***)
(*** multiplication by successive addition ***)
-Goal
- "{m=0 & s=0} \
-\ WHILE m ~= a DO {s = m*b} s := s+b; m := Suc(m) END\
-\ {s = a*b}";
-by (hoare_tac 1);
-by (ALLGOALS (asm_full_simp_tac (simpset() addsimps add_ac)));
+Goal "|- VARS m s. \
+\ {m=0 & s=0} \
+\ WHILE m~=a \
+\ INV {s=m*b} \
+\ DO s := s+b; m := m+1 OD \
+\ {s = a*b}";
+by(hoare_tac (Asm_full_simp_tac) 1);
qed "multiply_by_add";
-
(*** Euclid's algorithm for GCD ***)
-Goal
-" {0<A & 0<B & a=A & b=B} \
-\ WHILE a ~= b \
-\ DO {0<a & 0<b & gcd A B = gcd a b} \
-\ IF a<b \
-\ THEN \
-\ b:=b-a \
-\ ELSE \
-\ a:=a-b \
-\ END \
-\ END \
+Goal "|- VARS a b. \
+\ {0<A & 0<B & a=A & b=B} \
+\ WHILE a~=b \
+\ INV {0<a & 0<b & gcd A B = gcd a b} \
+\ DO IF a<b THEN b := b-a ELSE a := a-b FI OD \
\ {a = gcd A B}";
+by (hoare_tac (K all_tac) 1);
-by (hoare_tac 1);
(*Now prove the verification conditions*)
by Auto_tac;
by (etac gcd_nnn 4);
by (asm_full_simp_tac (simpset() addsimps [not_less_iff_le, gcd_diff_l]) 3);
by (force_tac (claset(),
- simpset() addsimps [not_less_iff_le, order_le_less]) 2);
-by (asm_simp_tac (simpset() addsimps [less_imp_le, gcd_diff_r]) 1);
+ simpset() addsimps [not_less_iff_le, le_eq_less_or_eq]) 2);
+by (asm_simp_tac (simpset() addsimps [gcd_diff_r,less_imp_le]) 1);
qed "Euclid_GCD";
-
-(*** Power by interated squaring and multiplication ***)
+(*** Power by iterated squaring and multiplication ***)
-Goal
-" {a=A & b=B} \
-\ c:=1; \
-\ WHILE b~=0 \
-\ DO {A^B = c * a^b} \
-\ WHILE b mod 2=0 \
-\ DO {A^B = c * a^b} \
-\ a:=a*a; \
-\ b:=b div 2 \
-\ END; \
-\ c:=c*a; \
-\ b:= b - 1 \
-\ END \
+Goal "|- VARS a b c. \
+\ {a=A & b=B} \
+\ c := 1; \
+\ WHILE b ~= 0 \
+\ INV {A^B = c * a^b} \
+\ DO WHILE b mod 2 = 0 \
+\ INV {A^B = c * a^b} \
+\ DO a := a*a; b := b div 2 OD; \
+\ c := c*a; b := b-1 \
+\ OD \
\ {c = A^B}";
-
-by (hoare_tac 1);
-
+by(hoare_tac (Asm_full_simp_tac) 1);
by (exhaust_tac "b" 1);
by (hyp_subst_tac 1);
by (asm_full_simp_tac (simpset() addsimps [mod_less]) 1);
by (asm_simp_tac (simpset() addsimps [mult_assoc]) 1);
-
qed "power_by_mult";
-(*** factorial ***)
-
-Goal
-" {a=A} \
-\ b:=1; \
-\ WHILE a~=0 \
-\ DO {fac A = b*fac a} \
-\ b:=b*a; \
-\ a:=a-1 \
-\ END \
+Goal "|- VARS a b. \
+\ {a=A} \
+\ b := 1; \
+\ WHILE a ~= 0 \
+\ INV {fac A = b * fac a} \
+\ DO b := b*a; a := a-1 OD \
\ {b = fac A}";
-
-by (hoare_tac 1);
+by (hoare_tac Asm_full_simp_tac 1);
by Safe_tac;
by (exhaust_tac "a" 1);
by (ALLGOALS
(asm_simp_tac
(simpset() addsimps [add_mult_distrib,add_mult_distrib2,mult_assoc])));
by (Fast_tac 1);
+qed"factorial";
-qed"factorial";
+(*** LISTS ***)
+
+Goal "|- VARS y x. \
+\ {x=X} \
+\ y:=[]; \
+\ WHILE x ~= [] \
+\ INV {rev(x)@y = rev(X)} \
+\ DO y := (hd x # y); x := tl x OD \
+\ {y=rev(X)}";
+by (hoare_tac Asm_full_simp_tac 1);
+by (asm_full_simp_tac (simpset() addsimps [neq_Nil_conv]) 1);
+by Safe_tac;
+by (ALLGOALS(Asm_full_simp_tac ));
+qed "imperative_reverse";
+
+Goal
+"|- VARS x y. \
+\ {x=X & y=Y} \
+\ x := rev(x); \
+\ WHILE x~=[] \
+\ INV {rev(x)@y = X@Y} \
+\ DO y := (hd x # y); \
+\ x := tl x \
+\ OD \
+\ {y = X@Y}";
+by (hoare_tac Asm_full_simp_tac 1);
+by (asm_full_simp_tac (simpset() addsimps [neq_Nil_conv]) 1);
+by Safe_tac;
+by (ALLGOALS(Asm_full_simp_tac));
+qed "imperative_append";
+
+
+(*** ARRAYS ***)
+
+(* Search for 0 *)
+Goal
+"|- VARS A i. \
+\ {True} \
+\ i := 0; \
+\ WHILE i < length A & A!i ~= 0 \
+\ INV {!j. j<i --> A!j ~= 0} \
+\ DO i := i+1 OD \
+\ {(i < length A --> A!i = 0) & \
+\ (i = length A --> (!j. j < length A --> A!j ~= 0))}";
+by (hoare_tac Asm_full_simp_tac 1);
+by(blast_tac (claset() addSEs [less_SucE]) 1);
+qed "zero_search";
+
+(*
+The `partition' procedure for quicksort.
+`A' is the array to be sorted (modelled as a list).
+Elements of A must be of class order to infer at the end
+that the elements between u and l are equal to pivot.
+
+Ambiguity warnings of parser are due to := being used
+both for assignment and list update.
+*)
+Goal
+"[| leq == %A i. !k. k<i --> A!k <= pivot; \
+\ geq == %A i. !k. i<k & k<length A --> pivot <= A!k |] ==> \
+\ |- VARS A u l.\
+\ {0 < length(A::('a::order)list)} \
+\ l := 0; u := length A - 1; \
+\ WHILE l <= u \
+\ INV {leq A l & geq A u & u<length A & l<=length A} \
+\ DO WHILE l < length A & A!l <= pivot \
+\ INV {leq A l & geq A u & u<length A & l<=length A} \
+\ DO l := l+1 OD; \
+\ WHILE 0 < u & pivot <= A!u \
+\ INV {leq A l & geq A u & u<length A & l<=length A} \
+\ DO u := u-1 OD; \
+\ IF l <= u THEN A := A[l := A!u, u := A!l] ELSE SKIP FI \
+\ OD \
+\ {leq A u & (!k. u<k & k<l --> A!k = pivot) & geq A l}";
+(* expand and delete abbreviations first *)
+by(asm_simp_tac HOL_basic_ss 1);
+by(REPEAT(etac thin_rl 1));
+by (hoare_tac Asm_full_simp_tac 1);
+ by(asm_full_simp_tac (simpset() addsimps [neq_Nil_conv]) 1);
+ by(Clarify_tac 1);
+ by(asm_full_simp_tac (simpset() addsimps [nth_list_update]
+ addSolver cut_trans_tac) 1);
+ by(blast_tac (claset() addSEs [less_SucE] addIs [Suc_leI]) 1);
+ br conjI 1;
+ by(Clarify_tac 1);
+ bd (pred_less_imp_le RS le_imp_less_Suc) 1;
+ by(blast_tac (claset() addSEs [less_SucE]) 1);
+ br less_imp_diff_less 1;
+ by(Blast_tac 1);
+ by(Clarify_tac 1);
+ by(asm_simp_tac (simpset() addsimps [nth_list_update]
+ addSolver cut_trans_tac) 1);
+ by(Clarify_tac 1);
+ by(trans_tac 1);
+by(Clarify_tac 1);
+by(asm_simp_tac (simpset() addSolver cut_trans_tac) 1);
+br conjI 1;
+ by(Clarify_tac 1);
+ br order_antisym 1;
+ by(asm_simp_tac (simpset() addSolver cut_trans_tac) 1);
+ by(asm_simp_tac (simpset() addSolver cut_trans_tac) 1);
+by(Clarify_tac 1);
+by(asm_simp_tac (simpset() addSolver cut_trans_tac) 1);
+qed "Partition";
--- a/src/HOL/Hoare/Examples.thy Wed Oct 14 11:51:11 1998 +0200
+++ b/src/HOL/Hoare/Examples.thy Wed Oct 14 15:26:31 1998 +0200
@@ -1,9 +1,9 @@
(* Title: HOL/Hoare/Examples.thy
ID: $Id$
Author: Norbert Galm
- Copyright 1995 TUM
+ Copyright 1998 TUM
-Various arithmetic examples.
+Various examples.
*)
Examples = Hoare + Arith2
--- a/src/HOL/Hoare/Hoare.ML Wed Oct 14 11:51:11 1998 +0200
+++ b/src/HOL/Hoare/Hoare.ML Wed Oct 14 15:26:31 1998 +0200
@@ -1,226 +1,210 @@
(* Title: HOL/Hoare/Hoare.ML
ID: $Id$
- Author: Norbert Galm & Tobias Nipkow
- Copyright 1995 TUM
+ Author: Leonor Prensa Nieto & Tobias Nipkow
+ Copyright 1998 TUM
-The verification condition generation tactics.
+Derivation of the proof rules and, most importantly, the VCG tactic.
*)
-open Hoare;
-
-(*** Hoare rules ***)
+(*** The proof rules ***)
-val SkipRule = prove_goalw thy [Spec_def,Skip_def]
- "(!!s. p(s) ==> q(s)) ==> Spec p Skip q"
- (fn prems => [fast_tac (claset() addIs prems) 1]);
+Goalw [Valid_def] "p <= q ==> Valid p SKIP q";
+by(Auto_tac);
+qed "SkipRule";
-val AssignRule = prove_goalw thy [Spec_def,Assign_def]
- "(!!s. p s ==> q(%x. if x=v then e s else s x)) ==> Spec p (Assign v e) q"
- (fn prems => [fast_tac (claset() addIs prems) 1]);
+Goalw [Valid_def] "p <= {s. (f s):q} ==> Valid p (Basic f) q";
+by(Auto_tac);
+qed "BasicRule";
-val SeqRule = prove_goalw thy [Spec_def,Seq_def]
- "[| Spec p c (%s. q s); Spec (%s. q s) c' r |] ==> Spec p (Seq c c') r"
- (fn prems => [cut_facts_tac prems 1, Fast_tac 1]);
+Goalw [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";
-val IfRule = prove_goalw thy [Spec_def,Cond_def]
- "[| !!s. p s ==> (b s --> q s) & (~b s --> q' s); \
-\ Spec (%s. q s) c r; Spec (%s. q' s) c' r |] \
-\ ==> Spec p (Cond b c c') r"
- (fn [prem1,prem2,prem3] =>
- [REPEAT (rtac allI 1),
- REPEAT (rtac impI 1),
- dtac prem1 1,
- cut_facts_tac [prem2,prem3] 1,
- fast_tac (claset() addIs [prem1]) 1]);
-
-val strenthen_pre = prove_goalw thy [Spec_def]
- "[| !!s. p s ==> p' s; Spec p' c q |] ==> Spec p c q"
- (fn [prem1,prem2] =>[cut_facts_tac [prem2] 1,
- fast_tac (claset() addIs [prem1]) 1]);
+Goalw [Valid_def]
+ "[| p <= {s. (s:b --> s:w) & (s~:b --> s:w')}; \
+\ Valid w c1 q; Valid w' c2 q |] \
+\ ==> Valid p (IF b THEN c1 ELSE c2 FI) q";
+by(Asm_simp_tac 1);
+by(Blast_tac 1);
+qed "CondRule";
-val lemma = prove_goalw thy [Spec_def,While_def]
- "[| Spec (%s. I s & b s) c I; !!s. [| I s; ~b s |] ==> q s |] \
-\ ==> Spec I (While b I c) q"
- (fn [prem1,prem2] =>
- [REPEAT(rtac allI 1), rtac impI 1, etac exE 1, rtac mp 1, atac 2,
- etac thin_rl 1, res_inst_tac[("x","s")]spec 1,
- res_inst_tac[("x","s'")]spec 1, induct_tac "n" 1,
- Simp_tac 1,
- fast_tac (claset() addIs [prem2]) 1,
- simp_tac (simpset() addsimps [Seq_def]) 1,
- cut_facts_tac [prem1] 1, fast_tac (claset() addIs [prem2]) 1]);
+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;
-val WhileRule = lemma RSN (2,strenthen_pre);
+Goalw [Valid_def]
+ "[| p <= i; Valid (i Int b) c i; (i Int -b) <= q |] \
+\ ==> Valid p (WHILE b INV {i} DO c OD) q";
+by(Asm_simp_tac 1);
+by(Clarify_tac 1);
+bd lemma 1;
+ba 2;
+by(Blast_tac 1);
+by(Blast_tac 1);
+qed "WhileRule";
-
-(*** meta_spec used in StateElimTac ***)
+(*** The tactics ***)
-val meta_spec = prove_goal HOL.thy
- "(!!s x. PROP P s x) ==> (!!s. PROP P s (x s))"
- (fn prems => [resolve_tac prems 1]);
-
-
-(**************************************************************************************************)
-(*** Funktion zum Generieren eines Theorems durch Umbennenen von Namen von Lambda-Abstraktionen ***)
-(*** in einem bestehenden Theorem. Bsp.: "!a.?P(a) ==> ?P(?x)" aus "!x.?P(x) ==> ?P(?x)" ***)
-(**************************************************************************************************)
+(*****************************************************************************)
+(** 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. For instance, (found,x,y) are **)
+(** the variables of the Zero Search program. **)
+(*****************************************************************************)
-(* rename_abs:term (von:string,nach:string,trm:term) benennt in trm alle Lambda-Abstraktionen
- mit Namen von in nach um *)
+local open HOLogic in
-fun rename_abs (von,nach,Abs (s,t,trm)) =
- if von=s
- then Abs (nach,t,rename_abs (von,nach,trm))
- else Abs (s,t,rename_abs (von,nach,trm))
- | rename_abs (von,nach,trm1 $ trm2) =rename_abs (von,nach,trm1) $ rename_abs (von,nach,trm2)
- | rename_abs (_,_,trm) =trm;
+(** 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 _ = [];
-(* ren_abs_thm:thm (von:string,nach:string,theorem:thm) benennt in theorem alle Lambda-Abstraktionen
- mit Namen von in nach um. Vorgehen:
- - Term t zu thoerem bestimmen
- - Term t' zu t durch Umbenennen der Namen generieren
- - Certified Term ct' zu t' erstellen
- - Thoerem ct'==ct' anlegen
- - Nach der Regel "[|P==Q; P|] ==> Q" wird aus "ct'==ct'" und theorem das Theorem zu ct'
- abgeleitet (ist moeglich, da t' mit t unifiziert werden kann, da nur Umnbenennungen) *)
-
-fun ren_abs_thm (von,nach,theorem) =
- equal_elim
- (reflexive (cterm_of (#sign (rep_thm theorem))
- (rename_abs (von,nach,#prop (rep_thm theorem)))))
- theorem;
-
+(** 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;
-(****************************************************************************)
-(*** Taktik zum Anwenden eines Theorems theorem auf ein Subgoal i durch ***)
-(*** - Umbenennen von Lambda-Abstraktionen im Theorem ***)
-(*** - Instanziieren von freien Variablen im Theorem ***)
-(*** - Composing des Subgoals mit dem Theorem ***)
-(****************************************************************************)
+(** maps [x1,...,xn] to (x1,...,xn) and types**)
+fun mk_bodyC [] = Const ("()", unitT)
+ | 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;
-(* - rens:(string*string) list, d.h. es koennen verschiedene Lambda-Abstraktionen umbenannt werden
- - insts:(cterm*cterm) list, d.h. es koennen verschiedene Variablen instanziiert werden *)
+fun dest_Goal (Const ("Goal", _) $ P) = P;
-fun comp_inst_ren_tac rens insts theorem i =
- let fun compose_inst_ren_tac [] insts theorem i =
- compose_tac (false,
- cterm_instantiate insts theorem,nprems_of theorem) i
- | compose_inst_ren_tac ((von,nach)::rl) insts theorem i =
- compose_inst_ren_tac rl insts
- (ren_abs_thm (von,nach,theorem)) i
- in compose_inst_ren_tac rens insts theorem i end;
+(** 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;
-(*************************************************************** *********)
-(*** Taktik zum Eliminieren des Zustandes waehrend Hoare-Beweisen ***)
-(*** Bsp.: "!!s. s(Suc(0))=0 --> s(Suc(0))+1=1" wird zu "!!s b. b=0 --> b+1=1" ***)
-(****************************************************************************)
+(** 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;
+
-(* pvars_of_term:term list (name:string,trm:term) gibt die Liste aller Programm-Variablen
- aus trm zurueck. name gibt dabei den Namen der Zustandsvariablen an.
- Bsp.: bei name="s" und dem Term "s(Suc(Suc(0)))=s(0)" (entspricht "c=a")
- wird [0,Suc(Suc(0))] geliefert (Liste ist i.A. unsortiert) *)
+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);
+ val cimpl = cterm_of (#sign (rep_thm thm)) impl
+ in prove_goalw_cterm [] cimpl (fn prems =>
+ [cut_facts_tac prems 1,Blast_tac 1]) end;
-fun pvars_of_term (name,trm) =
- let fun add_vars (name,Free (s,t) $ trm,vl) =
- if name=s then if trm mem vl then vl else trm::vl
- else add_vars (name,trm,vl)
- | add_vars (name,Abs (s,t,trm),vl) =add_vars (name,trm,vl)
- | add_vars (name,trm1 $ trm2,vl) =add_vars (name,trm2,add_vars (name,trm1,vl))
- | add_vars (_,_,vl) =vl
- in add_vars (name,trm,[]) end;
+end;
-(* VarsElimTac: Taktik zum Eliminieren von bestimmten Programmvariablen aus dem Subgoal i
- - v::vl:(term) list Liste der zu eliminierenden Programmvariablen
- - meta_spec:thm Theorem, welches zur Entfernung der Variablen benutzt wird
- z.B.: "(!!s x. PROP P(s,x)) ==> (!!s. PROP P(s,x(s)))"
- - namexAll:string Name von ^ (hier "x")
- - varx:term Term zu ^ (hier Var(("x",0),...))
- - varP:term Term zu ^ (hier Var(("P",0),...))
- - type_pvar:typ Typ der Programmvariablen (d.h. 'a bei 'a prog, z.B.: nat, bool, ...)
+(*****************************************************************************)
+(** 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 **)
+(*****************************************************************************)
+
+val Compl_Collect = prove_goal thy "-(Collect b) = {x. ~(b x)}"
+ (fn _ => [Fast_tac 1]);
+
+(**Simp_tacs**)
- Vorgehen:
- - eliminiere jede pvar durch Anwendung von comp_inst_ren_tac. Dazu:
- - Unbenennung in meta_spec: namexAll wird in den Namen der Prog.-Var. umbenannt
- z.B.: fuer die Prog.-Var. mit Namen "a" ergibt sich
- meta_spec zu "(!! s a. PROP P(s,a)) ==> (!! s. PROP P(s,x(s)))"
- - Instanziierungen in meta_spec:
- varx wird mit "%s:(type_pvar) state. s(pvar)" instanziiert
- varP wird entsprechend instanziiert. Beispiel fuer Prog.-Var. "a":
- - zu Subgoal "!!s. s(Suc(0)) = s(0) ==> s(0) = 1" bestimme Term ohne "!!s.":
- trm0 = "s(Suc(0)) = s(0) ==> s(0) = 1" (s ist hier freie Variable)
- - substituiere alle Vorkommen von s(pvar) durch eine freie Var. xs:
- trm1 = "s(Suc(0)) = xs ==> xs = 1"
- - abstrahiere ueber xs:
- trm2 = "%xs. s(Suc(0)) = xs ==> xs = 1"
- - abstrahiere ueber restliche Vorkommen von s:
- trm3 = "%s xs. s(Suc(0)) = xs ==> xs = 1"
- - instanziiere varP mit trm3
-*)
+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]));
+
+(*****************************************************************************)
+(** 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) **)
+(*****************************************************************************)
-(* StateElimTac: tactic to eliminate all program variable from subgoal i
- - applies to subgoals of the form "!!s:('a) state. P(s)",
- i.e. the term Const("all",_) $ Abs ("s",pvar --> 'a,_)
- - meta_spec has the form "(!!s x. PROP P(s,x)) ==> (!!s. PROP P(s,x(s)))"
-*)
+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]) 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];
-val StateElimTac = SUBGOAL (fn (Bi,i) =>
- let val Const _ $ Abs (_,Type ("fun",[_,type_pvar]),trm) = Bi
- val _ $ (_ $ Abs (_,_,_ $ Abs (namexAll,_,_))) $
- (_ $ Abs (_,_,varP $ _ $ (varx $ _))) =
- #prop (rep_thm meta_spec)
- fun vtac v i st = st |>
- let val cterm = cterm_of (#sign (rep_thm st))
- val (_,_,_ $ Abs (_,_,trm),_) = dest_state (st,i);
- val (sname,trm0) = variant_abs ("s",dummyT,trm);
- val xsname = variant_name ("xs",trm0);
- val trm1 = subst_term (Free (sname,dummyT) $ v,
- Syntax.free xsname,trm0)
- val trm2 = Abs ("xs", type_pvar,
- abstract_over (Syntax.free xsname,trm1))
- in
- comp_inst_ren_tac
- [(namexAll,pvar2pvarID v)]
- [(cterm varx,
- cterm (Abs ("s",Type ("nat",[]) --> type_pvar,
- Bound 0 $ v))),
- (cterm varP,
- cterm (Abs ("s", Type ("nat",[]) --> type_pvar,
- abstract_over (Free (sname,dummyT),trm2))))]
- meta_spec i
- end
- fun vars_tac [] i = all_tac
- | vars_tac (v::vl) i = vtac v i THEN vars_tac vl i
- in
- vars_tac (pvars_of_term (variant_abs ("s",dummyT,trm))) i
- end);
+fun BasicSimpTac tac =
+ simp_tac (HOL_basic_ss addsimps [mem_Collect_eq,split])
+ 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)) ));
-(*** tactics for verification condition generation ***)
-
-(* pre_cond:bool gibt an, ob das Subgoal von der Form Spec(?Q,c,p) ist oder nicht. Im Fall
- von pre_cond=false besteht die Vorbedingung nur nur aus einer scheme-Variable. Die dann
- generierte Verifikationsbedingung hat die Form "!!s.?Q --> ...". "?Q" kann deshalb zu gegebenen
- Zeitpunkt mittels "rtac impI" und "atac" gebunden werden, die Bedingung loest sich dadurch auf. *)
-
-fun WlpTac i = (rtac SeqRule i) THEN (HoareRuleTac (i+1) false)
-and HoareRuleTac i pre_cond st = st |>
- (*abstraction over st prevents looping*)
- ( (WlpTac i THEN HoareRuleTac i pre_cond)
- ORELSE
- (FIRST[rtac SkipRule i,
- rtac AssignRule i,
- EVERY[rtac IfRule i,
- HoareRuleTac (i+2) false,
- HoareRuleTac (i+1) false],
- EVERY[rtac WhileRule i,
- Asm_full_simp_tac (i+2),
- HoareRuleTac (i+1) true]]
- THEN
- (if pre_cond then (Asm_full_simp_tac i) else (atac i))) );
-
-val hoare_tac =
- SELECT_GOAL
- (EVERY[HoareRuleTac 1 true, ALLGOALS StateElimTac, prune_params_tac]);
-
+(** 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 Wed Oct 14 11:51:11 1998 +0200
+++ b/src/HOL/Hoare/Hoare.thy Wed Oct 14 15:26:31 1998 +0200
@@ -1,196 +1,193 @@
(* Title: HOL/Hoare/Hoare.thy
ID: $Id$
- Author: Norbert Galm & Tobias Nipkow
- Copyright 1995 TUM
+ Author: Leonor Prensa Nieto & Tobias Nipkow
+ Copyright 1998 TUM
Sugared semantic embedding of Hoare logic.
+Strictly speaking a shallow embedding (as implemented by Norbert Galm
+following Mike Gordon) would suffice. Maybe the datatype com comes in useful
+later.
*)
-Hoare = Arith +
+Hoare = Main +
types
- pvar = nat (* encoding of program variables ( < 26! ) *)
- 'a state = pvar => 'a (* program state *)
- 'a exp = 'a state => 'a (* denotation of expressions *)
- 'a bexp = 'a state => bool (* denotation of boolean expressions *)
- 'a com = 'a state => 'a state => bool (* denotation of programs *)
+ 'a bexp = 'a set
+ 'a assn = 'a set
+ 'a fexp = 'a =>'a
+
+datatype
+ 'a com = Basic ('a fexp)
+ | Seq ('a com) ('a com) ("(_;/_)" [61,60] 60)
+ | Cond ('a bexp) ('a com) ('a com) ("(1IF _/ THEN _ / ELSE _/ FI)" [0,0,0] 61)
+ | While ('a bexp) ('a assn) ('a com) ("(1WHILE _/ INV {_} //DO _ /OD)" [0,0,0] 61)
+
+syntax
+ "@assign" :: id => 'b => 'a com ("(2_ :=/ _ )" [70,65] 61)
+ "@annskip" :: 'a com ("SKIP")
+
+translations
+ "SKIP" == "Basic id"
+
+types 'a sem = 'a => 'a => bool
+
+consts iter :: nat => 'a bexp => 'a sem => 'a sem
+primrec
+"iter 0 b S = (%s s'. s ~: b & (s=s'))"
+"iter (Suc n) b S = (%s s'. s : b & (? s''. S s s'' & iter n b S s'' s'))"
+
+consts Sem :: 'a com => 'a sem
+primrec
+"Sem(Basic f) s s' = (s' = f s)"
+"Sem(c1;c2) s s' = (? s''. Sem c1 s s'' & Sem c2 s'' s')"
+"Sem(IF b THEN c1 ELSE c2 FI) s s' = ((s : b --> Sem c1 s s') &
+ (s ~: b --> Sem c2 s s'))"
+"Sem(While b x c) s s' = (? n. iter n b (Sem c) s s')"
+
+constdefs Valid :: ['a bexp, 'a com, 'a bexp] => bool
+ "Valid p c q == !s s'. Sem c s s' --> s : p --> s' : q"
-(* program syntax *)
-
nonterminals
- prog
+ vars
syntax
- "@Skip" :: prog ("SKIP")
- "@Assign" :: [id, 'a] => prog ("_ := _")
- "@Seq" :: [prog, prog] => prog ("_;//_" [0,1000] 999)
- "@If" :: [bool, prog, prog] => prog ("IF _//THEN// (_)//ELSE// (_)//END")
- "@While" :: [bool, bool, prog] => prog ("WHILE _//DO {_}// (_)//END")
- "@Spec" :: [bool, prog, bool] => bool ("{_}//_//{_}")
-
-
-(* denotational semantics *)
-
-constdefs
- Skip :: 'a com
- "Skip s s' == (s=s')"
-
- Assign :: [pvar, 'a exp] => 'a com
- "Assign v e s s' == (s' = (%x. if x=v then e(s) else s(x)))"
+ "" :: "id => vars" ("_")
+ "_vars" :: "[id, vars] => vars" ("_ _")
- Seq :: ['a com, 'a com] => 'a com
- "Seq c c' s s' == ? s''. c s s'' & c' s'' s'"
-
- Cond :: ['a bexp, 'a com, 'a com] => 'a com
- "Cond b c c' s s' == (b(s) --> c s s') & (~b s --> c' s s')"
-
-consts
- Iter :: [nat, 'a bexp, 'a com] => 'a com
-
-primrec
- "Iter 0 b c = (%s s'.~b(s) & (s=s'))"
- "Iter (Suc n) b c = (%s s'. b(s) & Seq c (Iter n b c) s s')"
-
-constdefs
- While :: ['a bexp, 'a bexp, 'a com] => 'a com
- "While b I c s s' == ? n. Iter n b c s s'"
-
- Spec :: ['a bexp, 'a com, 'a bexp] => bool
- "Spec p c q == !s s'. c s s' --> p s --> q s'"
+syntax
+ "@hoare_vars" :: [vars, 'a assn,'a com,'a assn] => bool
+ ("|- VARS _.// {_} // _ // {_}" [0,0,55,0] 50)
+syntax ("" output)
+ "@hoare" :: ['a assn,'a com,'a assn] => bool
+ ("|- {_} // _ // {_}" [0,55,0] 50)
end
ML
-
-(*** term manipulation ***)
-
-(* name_in_term:bool (name:string,trm:term) bestimmt, ob in trm der Name name als Konstante,
- freie Var., scheme-Variable oder als Name fuer eine Lambda-Abstraktion vorkommt *)
+(** parse translations **)
-fun name_in_term (name,Const (s,t)) = (name=s)
- | name_in_term (name,Free (s,t)) = (name=s)
- | name_in_term (name,Var (ix,t)) = (name= string_of_indexname ix)
- | name_in_term (name,Abs (s,t,trm)) = (name=s) orelse
- (name_in_term (name,trm))
- | name_in_term (name,trm1 $ trm2) = (name_in_term (name,trm1)) orelse
- (name_in_term (name,trm2))
- | name_in_term (_,_) = false;
-
-(* variant_name:string (name:string,trm:term) liefert einen von name
- abgewandelten Namen (durch Anhaengen von genuegend vielen "_"), der nicht
- in trm vorkommt. Im Gegensatz zu variant_abs beruecktsichtigt es auch Namen
- von scheme-Variablen und von Lambda-Abstraktionen in trm *)
+fun mk_abstuple [] body = absfree ("x", dummyT, body)
+ | mk_abstuple [v] body = absfree ((fst o dest_Free) v, dummyT, body)
+ | mk_abstuple (v::w) body = Syntax.const "split" $
+ absfree ((fst o dest_Free) v, dummyT, mk_abstuple w body);
-(*This could be done more simply by calling Term.variant, supplying a list of
- all free, bound and scheme variables in the term.*)
-fun variant_name (name,trm) = if name_in_term (name,trm)
- then variant_name (name ^ "_",trm)
- else name;
-
-(* subst_term:term (von:term,nach:term,trm:term) liefert den Term, der aus
-trm entsteht, wenn alle Teilterme, die gleich von sind, durch nach ersetzt
-wurden *)
+
+fun mk_fbody v e [] = Syntax.const "()"
+ | mk_fbody v e [x] = if v=x then e else x
+ | mk_fbody v e (x::xs) = Syntax.const "Pair" $ (if v=x then e else x) $
+ mk_fbody v e xs;
-fun subst_term (von,nach,Abs (s,t,trm)) =if von=Abs (s,t,trm)
- then nach
- else Abs (s,t,subst_term (von,nach,trm))
- | subst_term (von,nach,trm1 $ trm2) =if von=trm1 $ trm2
- then nach
- else subst_term (von,nach,trm1) $ subst_term (von,nach,trm2)
- | subst_term (von,nach,trm) =if von=trm
- then nach
- else trm;
+fun mk_fexp v e xs = mk_abstuple xs (mk_fbody v e xs);
-(* Translation between program vars ("a" - "z") and their encoding as
- natural numbers: "a" <==> 0, "b" <==> Suc(0), ..., "z" <==> 25 *)
+(* bexp_tr & assn_tr *)
+(*all meta-variables for bexp except for TRUE and FALSE are translated as if they
+ were boolean expressions*)
+
+fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE"
+ | bexp_tr b xs = Syntax.const "Collect" $ mk_abstuple xs b;
+
+fun assn_tr r xs = Syntax.const "Collect" $ mk_abstuple xs r;
-fun is_pvarID s = size s=1 andalso "a"<=s andalso s<="z";
+(* com_tr *)
+
+fun assign_tr [Free (V,_),E] xs = Syntax.const "Basic" $
+ mk_fexp (Free(V,dummyT)) E xs
+ | assign_tr ts _ = raise TERM ("assign_tr", ts);
-fun pvarID2pvar s =
- let fun rest2pvar (i,arg) =
- if i=0 then arg else rest2pvar(i-1, Syntax.const "Suc" $ arg)
- in rest2pvar(ord s - ord "a", Syntax.const "0") end;
+fun com_tr (Const("@assign",_) $ Free (V,_) $ E) xs =
+ assign_tr [Free (V,dummyT),E] xs
+ | com_tr (Const ("Basic",_) $ f) xs = Syntax.const "Basic" $ f
+ | com_tr (Const ("Seq",_) $ c1 $ c2) xs = Syntax.const "Seq" $
+ com_tr c1 xs $ com_tr c2 xs
+ | com_tr (Const ("Cond",_) $ b $ c1 $ c2) xs = Syntax.const "Cond" $
+ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs
+ | com_tr (Const ("While",_) $ b $ I $ c) xs = Syntax.const "While" $
+ bexp_tr b xs $ assn_tr I xs $ com_tr c xs
+ | com_tr trm _ = trm;
+
+(* triple_tr *)
-fun pvar2pvarID trm =
- let
- fun rest2pvarID (Const("0",_),i) =chr (i + ord "a")
- | rest2pvarID (Const("Suc",_) $ trm,i) =rest2pvarID (trm,i+1)
- in
- rest2pvarID (trm,0)
- end;
+fun vars_tr (x as Free _) = [x]
+ | vars_tr (Const ("_vars", _) $ (x as Free _) $ vars) = x :: vars_tr vars
+ | vars_tr t = raise TERM ("vars_tr", [t]);
+
+fun hoare_vars_tr [vars, pre, prg, post] =
+ let val xs = vars_tr vars
+ in Syntax.const "Valid" $
+ assn_tr pre xs $ com_tr prg xs $ assn_tr post xs
+ end
+ | hoare_vars_tr ts = raise TERM ("hoare_vars_tr", ts);
+
+
+
+val parse_translation = [("@hoare_vars", hoare_vars_tr)];
-(*** parse translations: syntax -> semantics ***)
+(*****************************************************************************)
+
+(*** print translations ***)
-(* term_tr:term (name:string,trm:term) ersetzt in trm alle freien Variablen, die eine pvarID
- darstellen, durch name $ pvarID2pvar(pvarID). Beispiel:
- bei name="s" und dem Term "a=b & a=X" wird der Term "s(0)=s(Suc(0)) & s(0)=X" geliefert *)
+fun dest_abstuple (Const ("split",_) $ (Abs(v,_, body))) =
+ subst_bound (Syntax.free v, dest_abstuple body)
+ | dest_abstuple (Abs(v,_, body)) = subst_bound (Syntax.free v, body)
+ | dest_abstuple trm = trm;
-fun term_tr (name,Free (s,t)) = if is_pvarID s
- then Syntax.free name $ pvarID2pvar s
- else Free (s,t)
- | term_tr (name,Abs (s,t,trm)) = Abs (s,t,term_tr (name,trm))
- | term_tr (name,trm1 $ trm2) = term_tr (name,trm1) $ term_tr (name,trm2)
- | term_tr (name,trm) = trm;
+fun abs2list (Const ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
+ | abs2list (Abs(x,T,t)) = [Free (x, T)]
+ | abs2list _ = [];
-fun bool_tr B =
- let val name = variant_name("s",B)
- in Abs (name,dummyT,abstract_over (Syntax.free name,term_tr (name,B))) end;
+fun mk_ts (Const ("split",_) $ (Abs(x,_,t))) = mk_ts t
+ | mk_ts (Abs(x,_,t)) = mk_ts t
+ | mk_ts (Const ("Pair",_) $ a $ b) = a::(mk_ts b)
+ | mk_ts t = [t];
-fun exp_tr E =
- let val name = variant_name("s",E)
- in Abs (name,dummyT,abstract_over (Syntax.free name,term_tr (name,E))) end;
+fun mk_vts (Const ("split",_) $ (Abs(x,_,t))) =
+ ((Syntax.free x)::(abs2list t), mk_ts t)
+ | mk_vts (Abs(x,_,t)) = ([Syntax.free x], [t])
+ | mk_vts t = raise Match;
+
+fun find_ch [] i xs = (false, (Syntax.free "not_ch",Syntax.free "not_ch" ))
+ | find_ch ((v,t)::vts) i xs = if t=(Bound i) then find_ch vts (i-1) xs
+ else (true, (v, subst_bounds (xs,t)));
+
+fun is_f (Const ("split",_) $ (Abs(x,_,t))) = true
+ | is_f (Abs(x,_,t)) = true
+ | is_f t = false;
+
+(* assn_tr' & bexp_tr'*)
+
+fun assn_tr' (Const ("Collect",_) $ T) = dest_abstuple T
+ | assn_tr' (Const ("op Int",_) $ (Const ("Collect",_) $ T1) $
+ (Const ("Collect",_) $ T2)) =
+ Syntax.const "op Int" $ dest_abstuple T1 $ dest_abstuple T2
+ | assn_tr' t = t;
-fun prog_tr (Const ("@Skip",_)) = Syntax.const "Skip"
- | prog_tr (Const ("@Assign",_) $ Free (V,_) $ E) =
- if is_pvarID V
- then Syntax.const "Assign" $ pvarID2pvar V $ exp_tr E
- else error("Not a valid program variable: " ^ quote V)
- | prog_tr (Const ("@Seq",_) $ C $ C') =
- Syntax.const "Seq" $ prog_tr C $ prog_tr C'
- | prog_tr (Const ("@If",_) $ B $ C $ C') =
- Syntax.const "Cond" $ bool_tr B $ prog_tr C $ prog_tr C'
- | prog_tr (Const ("@While",_) $ B $ INV $ C) =
- Syntax.const "While" $ bool_tr B $ bool_tr INV $ prog_tr C;
+fun bexp_tr' (Const ("Collect",_) $ T) = dest_abstuple T
+ | bexp_tr' t = t;
+
+(*com_tr' *)
-fun spec_tr [P,C,Q] = Syntax.const "Spec" $ bool_tr P $ prog_tr C $ bool_tr Q;
+fun mk_assign f =
+ let val (vs, ts) = mk_vts f;
+ val (ch, which) = find_ch (vs~~ts) ((length vs)-1) (rev vs)
+ in if ch then Syntax.const "@assign" $ fst(which) $ snd(which)
+ else Syntax.const "@skip" end;
-val parse_translation = [("@Spec",spec_tr)];
+fun com_tr' (Const ("Basic",_) $ f) = if is_f f then mk_assign f
+ else Syntax.const "Basic" $ f
+ | com_tr' (Const ("Seq",_) $ c1 $ c2) = Syntax.const "Seq" $
+ com_tr' c1 $ com_tr' c2
+ | com_tr' (Const ("Cond",_) $ b $ c1 $ c2) = Syntax.const "Cond" $
+ bexp_tr' b $ com_tr' c1 $ com_tr' c2
+ | com_tr' (Const ("While",_) $ b $ I $ c) = Syntax.const "While" $
+ bexp_tr' b $ assn_tr' I $ com_tr' c
+ | com_tr' t = t;
-(*** print translations: semantics -> syntax ***)
-
-(* Note: does not mark tokens *)
-
-(* term_tr':term (name:string,trm:term) ersetzt in trm alle Vorkommen von name $ pvar durch
- entsprechende freie Variablen, welche die pvarID zu pvar darstellen. Beispiel:
- bei name="s" und dem Term "s(0)=s(Suc(0)) & s(0)=X" wird der Term "a=b & a=X" geliefert *)
-
-fun term_tr' (name,Free (s,t) $ trm) =
- if name=s then Syntax.free (pvar2pvarID trm)
- else Free (s,t) $ term_tr' (name,trm)
- | term_tr' (name,Abs (s,t,trm)) = Abs (s,t,term_tr' (name,trm))
- | term_tr' (name,trm1 $ trm2) = term_tr' (name,trm1) $ term_tr' (name,trm2)
- | term_tr' (name,trm) = trm;
-
-fun bexp_tr' (Abs(_,_,b)) = term_tr' (variant_abs ("s",dummyT,b));
-
-fun exp_tr' (Abs(_,_,e)) = term_tr' (variant_abs ("s",dummyT,e));
-
-fun com_tr' (Const ("Skip",_)) = Syntax.const "@Skip"
- | com_tr' (Const ("Assign",_) $ v $ e) =
- Syntax.const "@Assign" $ Syntax.free (pvar2pvarID v) $ exp_tr' e
- | com_tr' (Const ("Seq",_) $ c $ c') =
- Syntax.const "@Seq" $ com_tr' c $ com_tr' c'
- | com_tr' (Const ("Cond",_) $ b $ c $ c') =
- Syntax.const "@If" $ bexp_tr' b $ com_tr' c $ com_tr' c'
- | com_tr' (Const ("While",_) $ b $ inv $ c) =
- Syntax.const "@While" $ bexp_tr' b $ bexp_tr' inv $ com_tr' c;
-
-fun spec_tr' [p,c,q] =
- Syntax.const "@Spec" $ bexp_tr' p $ com_tr' c $ bexp_tr' q;
-
-val print_translation = [("Spec",spec_tr')];
+fun spec_tr' [p, c, q] =
+ Syntax.const "@hoare" $ assn_tr' p $ com_tr' c $ assn_tr' q
+
+val print_translation = [("Valid", spec_tr')];
--- a/src/HOL/Hoare/List_Examples.ML Wed Oct 14 11:51:11 1998 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,27 +0,0 @@
-Goal
-"{x=X} \
-\ y := []; \
-\ WHILE x ~= [] \
-\ DO { rev(x)@y = rev(X)} \
-\ y := hd x # y; x := tl x \
-\ END \
-\{y=rev(X)}";
-by (hoare_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [neq_Nil_conv]) 1);
-by Safe_tac;
-by (Asm_full_simp_tac 1);
-qed "imperative_reverse";
-
-Goal
-"{x=X & y = Y} \
-\ x := rev(x); \
-\ WHILE x ~= [] \
-\ DO { rev(x)@y = X@Y} \
-\ y := hd x # y; x := tl x \
-\ END \
-\{y = X@Y}";
-by (hoare_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [neq_Nil_conv]) 1);
-by Safe_tac;
-by (Asm_full_simp_tac 1);
-qed "imperative_append";
--- a/src/HOL/Hoare/List_Examples.thy Wed Oct 14 11:51:11 1998 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1 +0,0 @@
-List_Examples = Hoare + List
--- a/src/HOL/Hoare/README.html Wed Oct 14 11:51:11 1998 +0200
+++ b/src/HOL/Hoare/README.html Wed Oct 14 15:26:31 1998 +0200
@@ -1,6 +1,59 @@
<HTML><HEAD><TITLE>HOL/Hoare/ReadMe</TITLE></HEAD><BODY>
-<H2>Semantic Embedding of Hoare Logic</H2>
+<H2>Hoare Logic for a Simple WHILE Language</H2>
+
+<H1>The language and logic<H1>
+
+This directory contains an implementation of Hoare logic for a simple WHILE
+language. The are
+<UL>
+<LI> SKIP
+<LI> _ := _
+<LI> _ ; _
+<LI> <kbd>IF _ THEN _ ELSE _ FI<kbd>
+<LI> WHILE _ INV {_} DO _ OD
+</UL>
+Note that each WHILE-loop must be annotated with an invariant.
+<P>
+
+After loading theory Hoare, you can state goals of the form
+<PRE>
+|- VARS x y ... . {P} prog {Q}
+</PRE>
+where <kbd>prog</kbd> is a program in the above language, <kbd>P</kbd> is the
+precondition, <kbd>Q</kbd> the postcondition, and <kbd>x y ...<kbd> is the
+list of all <i>program variables</i> in <kbd>prog</kbd>. The latter list must
+be nonempty and it must include all variables that occur on the left-hand
+side of an assignment in <kbd>prof</kbd>. Example:
+<PRE>
+|- VARS x. {x = a} x := x+1 {x = a+1}
+</PRE>
+The (normal) variable <kbd>a</kbd> is merely used to record the initial
+value of <kbd>x</kbd> and is not a program variable. Pre and postconditions
+can be arbitrary HOL formulae mentioning both program variables and normal
+variables.
+<P>
+
+The implementation hides reasoning in Hoare logic completely and provides a
+tactic hoare_tac for generating the verification conditions in ordinary
+logic:
+<PRE>
+by(hoare_tac tac i);
+</PRE>
+applies the tactic to subgoal <kbd>i</kbd> and applies the parameter
+<kbd>tac</kbd> to all generated verification conditions. A typical call is
+<PRE>
+by(hoare_tac Asm_full_simp_tac 1);
+</PRE>
+which, given the example goal above, solves it completely.
+<P>
+
+IMPORTANT:
+This is a logic of partial correctness. You can only prove that your program
+does the right thing <i>if</i> it terminates, but not <i>that</i> it
+terminates.
+
+<H1>Notes on the implementation</H1>
This directory contains a sugared shallow semantic embedding of Hoare logic
for a while language. The implementation closely follows<P>
--- a/src/HOL/Hoare/ROOT.ML Wed Oct 14 11:51:11 1998 +0200
+++ b/src/HOL/Hoare/ROOT.ML Wed Oct 14 15:26:31 1998 +0200
@@ -1,10 +1,9 @@
(* Title: HOL/IMP/ROOT.ML
ID: $Id$
Author: Tobias Nipkow
- Copyright 1995 TUM
+ Copyright 1998 TUM
*)
HOL_build_completed; (*Make examples fail if HOL did*)
use_thy "Examples";
-use_thy "List_Examples";