New many-sorted version.
authornipkow
Wed Oct 14 15:26:31 1998 +0200 (1998-10-14)
changeset 56467c2ddbaf8b8c
parent 5645 b872b209db69
child 5647 4e8837255b87
New many-sorted version.
src/HOL/Hoare/Examples.ML
src/HOL/Hoare/Examples.thy
src/HOL/Hoare/Hoare.ML
src/HOL/Hoare/Hoare.thy
src/HOL/Hoare/List_Examples.ML
src/HOL/Hoare/List_Examples.thy
src/HOL/Hoare/README.html
src/HOL/Hoare/ROOT.ML
     1.1 --- a/src/HOL/Hoare/Examples.ML	Wed Oct 14 11:51:11 1998 +0200
     1.2 +++ b/src/HOL/Hoare/Examples.ML	Wed Oct 14 15:26:31 1998 +0200
     1.3 @@ -1,94 +1,178 @@
     1.4  (*  Title:      HOL/Hoare/Examples.thy
     1.5      ID:         $Id$
     1.6 -    Author:     Norbert Galm
     1.7 -    Copyright   1995 TUM
     1.8 -
     1.9 -Various arithmetic examples.
    1.10 +    Author:     Norbert Galm & Tobias Nipkow
    1.11 +    Copyright   1998 TUM
    1.12  *)
    1.13  
    1.14 -open Examples;
    1.15 +(*** ARITHMETIC ***)
    1.16  
    1.17  (*** multiplication by successive addition ***)
    1.18  
    1.19 -Goal
    1.20 - "{m=0 & s=0} \
    1.21 -\ WHILE m ~= a DO {s = m*b} s := s+b; m := Suc(m) END\
    1.22 -\ {s = a*b}";
    1.23 -by (hoare_tac 1);
    1.24 -by (ALLGOALS (asm_full_simp_tac (simpset() addsimps add_ac)));
    1.25 +Goal "|- VARS m s. \
    1.26 +\  {m=0 & s=0} \
    1.27 +\  WHILE m~=a \
    1.28 +\  INV {s=m*b} \  
    1.29 +\  DO s := s+b; m := m+1 OD \
    1.30 +\  {s = a*b}";
    1.31 +by(hoare_tac (Asm_full_simp_tac) 1);
    1.32  qed "multiply_by_add";
    1.33  
    1.34 -
    1.35  (*** Euclid's algorithm for GCD ***)
    1.36  
    1.37 -Goal
    1.38 -" {0<A & 0<B & a=A & b=B}   \
    1.39 -\ WHILE a ~= b  \
    1.40 -\ DO  {0<a & 0<b & gcd A B = gcd a b} \
    1.41 -\      IF a<b   \
    1.42 -\      THEN   \
    1.43 -\           b:=b-a   \
    1.44 -\      ELSE   \
    1.45 -\           a:=a-b   \
    1.46 -\      END   \
    1.47 -\ END   \
    1.48 +Goal "|- VARS a b. \
    1.49 +\ {0<A & 0<B & a=A & b=B} \
    1.50 +\ WHILE  a~=b  \
    1.51 +\ INV {0<a & 0<b & gcd A B = gcd a b} \
    1.52 +\ DO IF a<b THEN b := b-a ELSE a := a-b FI OD \
    1.53  \ {a = gcd A B}";
    1.54 +by (hoare_tac (K all_tac) 1);
    1.55  
    1.56 -by (hoare_tac 1);
    1.57  (*Now prove the verification conditions*)
    1.58  by Auto_tac;
    1.59  by (etac gcd_nnn 4);
    1.60  by (asm_full_simp_tac (simpset() addsimps [not_less_iff_le, gcd_diff_l]) 3);
    1.61  by (force_tac (claset(),
    1.62 -	       simpset() addsimps [not_less_iff_le, order_le_less]) 2);
    1.63 -by (asm_simp_tac (simpset() addsimps [less_imp_le, gcd_diff_r]) 1);
    1.64 +               simpset() addsimps [not_less_iff_le, le_eq_less_or_eq]) 2);
    1.65 +by (asm_simp_tac (simpset() addsimps [gcd_diff_r,less_imp_le]) 1);
    1.66  qed "Euclid_GCD";
    1.67  
    1.68 -
    1.69 -(*** Power by interated squaring and multiplication ***)
    1.70 +(*** Power by iterated squaring and multiplication ***)
    1.71  
    1.72 -Goal
    1.73 -" {a=A & b=B}   \
    1.74 -\ c:=1;   \
    1.75 -\ WHILE b~=0   \
    1.76 -\ DO {A^B = c * a^b}   \
    1.77 -\      WHILE b mod 2=0   \
    1.78 -\      DO  {A^B = c * a^b}  \
    1.79 -\           a:=a*a;   \
    1.80 -\           b:=b div 2   \
    1.81 -\      END;   \
    1.82 -\      c:=c*a;   \
    1.83 -\      b:= b - 1 \
    1.84 -\ END   \
    1.85 +Goal "|- VARS a b c. \
    1.86 +\ {a=A & b=B} \
    1.87 +\ c := 1; \
    1.88 +\ WHILE b ~= 0 \
    1.89 +\ INV {A^B = c * a^b} \
    1.90 +\ DO  WHILE b mod 2 = 0 \
    1.91 +\     INV {A^B = c * a^b} \
    1.92 +\     DO  a := a*a; b := b div 2 OD; \
    1.93 +\     c := c*a; b := b-1 \
    1.94 +\ OD \
    1.95  \ {c = A^B}";
    1.96 -
    1.97 -by (hoare_tac 1);
    1.98 -
    1.99 +by(hoare_tac (Asm_full_simp_tac) 1);
   1.100  by (exhaust_tac "b" 1);
   1.101  by (hyp_subst_tac 1);
   1.102  by (asm_full_simp_tac (simpset() addsimps [mod_less]) 1);
   1.103  by (asm_simp_tac (simpset() addsimps [mult_assoc]) 1);
   1.104 -
   1.105  qed "power_by_mult";
   1.106  
   1.107 -(*** factorial ***)
   1.108 -
   1.109 -Goal
   1.110 -" {a=A}   \
   1.111 -\ b:=1;   \
   1.112 -\ WHILE a~=0    \
   1.113 -\ DO  {fac A = b*fac a} \
   1.114 -\      b:=b*a;   \
   1.115 -\      a:=a-1   \
   1.116 -\ END   \
   1.117 +Goal "|- VARS a b. \
   1.118 +\ {a=A} \
   1.119 +\ b := 1; \
   1.120 +\ WHILE a ~= 0 \
   1.121 +\ INV {fac A = b * fac a} \
   1.122 +\ DO b := b*a; a := a-1 OD \
   1.123  \ {b = fac A}";
   1.124 -
   1.125 -by (hoare_tac 1);
   1.126 +by (hoare_tac Asm_full_simp_tac 1);
   1.127  by Safe_tac;
   1.128  by (exhaust_tac "a" 1);
   1.129  by (ALLGOALS
   1.130      (asm_simp_tac
   1.131       (simpset() addsimps [add_mult_distrib,add_mult_distrib2,mult_assoc])));
   1.132  by (Fast_tac 1);
   1.133 +qed"factorial";
   1.134  
   1.135 -qed"factorial";
   1.136 +(*** LISTS ***)
   1.137 +
   1.138 +Goal "|- VARS y x. \
   1.139 +\ {x=X} \
   1.140 +\ y:=[]; \
   1.141 +\ WHILE x ~= [] \
   1.142 +\ INV {rev(x)@y = rev(X)} \
   1.143 +\ DO y := (hd x # y); x := tl x OD \
   1.144 +\ {y=rev(X)}";
   1.145 +by (hoare_tac Asm_full_simp_tac 1);
   1.146 +by (asm_full_simp_tac (simpset() addsimps [neq_Nil_conv]) 1);
   1.147 +by Safe_tac;
   1.148 +by (ALLGOALS(Asm_full_simp_tac ));
   1.149 +qed "imperative_reverse";
   1.150 +
   1.151 +Goal
   1.152 +"|- VARS x y. \
   1.153 +\ {x=X & y=Y} \
   1.154 +\ x := rev(x); \
   1.155 +\ WHILE x~=[] \
   1.156 +\ INV {rev(x)@y = X@Y} \
   1.157 +\ DO y := (hd x # y); \
   1.158 +\    x := tl x \
   1.159 +\ OD \
   1.160 +\ {y = X@Y}";
   1.161 +by (hoare_tac Asm_full_simp_tac 1);
   1.162 +by (asm_full_simp_tac (simpset() addsimps [neq_Nil_conv]) 1);
   1.163 +by Safe_tac;
   1.164 +by (ALLGOALS(Asm_full_simp_tac));
   1.165 +qed "imperative_append";
   1.166 +
   1.167 +
   1.168 +(*** ARRAYS ***)
   1.169 +
   1.170 +(* Search for 0 *)
   1.171 +Goal
   1.172 +"|- VARS A i. \
   1.173 +\ {True} \
   1.174 +\ i := 0; \
   1.175 +\ WHILE i < length A & A!i ~= 0 \
   1.176 +\ INV {!j. j<i --> A!j ~= 0} \
   1.177 +\ DO i := i+1 OD \
   1.178 +\ {(i < length A --> A!i = 0) & \
   1.179 +\  (i = length A --> (!j. j < length A --> A!j ~= 0))}";
   1.180 +by (hoare_tac Asm_full_simp_tac 1);
   1.181 +by(blast_tac (claset() addSEs [less_SucE]) 1);
   1.182 +qed "zero_search";
   1.183 +
   1.184 +(* 
   1.185 +The `partition' procedure for quicksort.
   1.186 +`A' is the array to be sorted (modelled as a list).
   1.187 +Elements of A must be of class order to infer at the end
   1.188 +that the elements between u and l are equal to pivot.
   1.189 +
   1.190 +Ambiguity warnings of parser are due to := being used
   1.191 +both for assignment and list update.
   1.192 +*)
   1.193 +Goal
   1.194 +"[| leq == %A i. !k. k<i --> A!k <= pivot; \
   1.195 +\   geq == %A i. !k. i<k & k<length A --> pivot <= A!k |] ==> \
   1.196 +\ |- VARS A u l.\
   1.197 +\ {0 < length(A::('a::order)list)} \
   1.198 +\ l := 0; u := length A - 1; \
   1.199 +\ WHILE l <= u \
   1.200 +\  INV {leq A l & geq A u & u<length A & l<=length A} \
   1.201 +\  DO WHILE l < length A & A!l <= pivot \
   1.202 +\      INV {leq A l & geq A u & u<length A & l<=length A} \
   1.203 +\      DO l := l+1 OD; \
   1.204 +\     WHILE 0 < u & pivot <= A!u \
   1.205 +\      INV {leq A l & geq A u  & u<length A & l<=length A} \
   1.206 +\      DO u := u-1 OD; \
   1.207 +\     IF l <= u THEN A := A[l := A!u, u := A!l] ELSE SKIP FI \
   1.208 +\  OD \
   1.209 +\ {leq A u & (!k. u<k & k<l --> A!k = pivot) & geq A l}";
   1.210 +(* expand and delete abbreviations first *)
   1.211 +by(asm_simp_tac HOL_basic_ss 1);
   1.212 +by(REPEAT(etac thin_rl 1));
   1.213 +by (hoare_tac Asm_full_simp_tac 1);
   1.214 +    by(asm_full_simp_tac (simpset() addsimps [neq_Nil_conv]) 1);
   1.215 +    by(Clarify_tac 1);
   1.216 +    by(asm_full_simp_tac (simpset() addsimps [nth_list_update]
   1.217 +                                    addSolver cut_trans_tac) 1);
   1.218 +   by(blast_tac (claset() addSEs [less_SucE] addIs [Suc_leI]) 1);
   1.219 +  br conjI 1;
   1.220 +   by(Clarify_tac 1);
   1.221 +   bd (pred_less_imp_le RS le_imp_less_Suc) 1;
   1.222 +   by(blast_tac (claset() addSEs [less_SucE]) 1);
   1.223 +  br less_imp_diff_less 1;
   1.224 +  by(Blast_tac 1);
   1.225 + by(Clarify_tac 1);
   1.226 + by(asm_simp_tac (simpset() addsimps [nth_list_update]
   1.227 +                            addSolver cut_trans_tac) 1);
   1.228 + by(Clarify_tac 1);
   1.229 + by(trans_tac 1);
   1.230 +by(Clarify_tac 1);
   1.231 +by(asm_simp_tac (simpset() addSolver cut_trans_tac) 1);
   1.232 +br conjI 1;
   1.233 + by(Clarify_tac 1);
   1.234 + br order_antisym 1;
   1.235 +  by(asm_simp_tac (simpset() addSolver cut_trans_tac) 1);
   1.236 + by(asm_simp_tac (simpset() addSolver cut_trans_tac) 1);
   1.237 +by(Clarify_tac 1);
   1.238 +by(asm_simp_tac (simpset() addSolver cut_trans_tac) 1);
   1.239 +qed "Partition";
     2.1 --- a/src/HOL/Hoare/Examples.thy	Wed Oct 14 11:51:11 1998 +0200
     2.2 +++ b/src/HOL/Hoare/Examples.thy	Wed Oct 14 15:26:31 1998 +0200
     2.3 @@ -1,9 +1,9 @@
     2.4  (*  Title:      HOL/Hoare/Examples.thy
     2.5      ID:         $Id$
     2.6      Author:     Norbert Galm
     2.7 -    Copyright   1995 TUM
     2.8 +    Copyright   1998 TUM
     2.9  
    2.10 -Various arithmetic examples.
    2.11 +Various examples.
    2.12  *)
    2.13  
    2.14  Examples = Hoare + Arith2
     3.1 --- a/src/HOL/Hoare/Hoare.ML	Wed Oct 14 11:51:11 1998 +0200
     3.2 +++ b/src/HOL/Hoare/Hoare.ML	Wed Oct 14 15:26:31 1998 +0200
     3.3 @@ -1,226 +1,210 @@
     3.4  (*  Title:      HOL/Hoare/Hoare.ML
     3.5      ID:         $Id$
     3.6 -    Author:     Norbert Galm & Tobias Nipkow
     3.7 -    Copyright   1995 TUM
     3.8 +    Author:     Leonor Prensa Nieto & Tobias Nipkow
     3.9 +    Copyright   1998 TUM
    3.10  
    3.11 -The verification condition generation tactics.
    3.12 +Derivation of the proof rules and, most importantly, the VCG tactic.
    3.13  *)
    3.14  
    3.15 -open Hoare;
    3.16 -
    3.17 -(*** Hoare rules ***)
    3.18 +(*** The proof rules ***)
    3.19  
    3.20 -val SkipRule = prove_goalw thy [Spec_def,Skip_def]
    3.21 -  "(!!s. p(s) ==> q(s)) ==> Spec p Skip q"
    3.22 -  (fn prems => [fast_tac (claset() addIs prems) 1]);
    3.23 +Goalw [Valid_def] "p <= q ==> Valid p SKIP q";
    3.24 +by(Auto_tac);
    3.25 +qed "SkipRule";
    3.26  
    3.27 -val AssignRule = prove_goalw thy [Spec_def,Assign_def]
    3.28 -  "(!!s. p s ==> q(%x. if x=v then e s else s x)) ==> Spec p (Assign v e) q"
    3.29 -  (fn prems => [fast_tac (claset() addIs prems) 1]);
    3.30 +Goalw [Valid_def] "p <= {s. (f s):q} ==> Valid p (Basic f) q";
    3.31 +by(Auto_tac);
    3.32 +qed "BasicRule";
    3.33  
    3.34 -val SeqRule = prove_goalw thy [Spec_def,Seq_def]
    3.35 -  "[| Spec p c (%s. q s); Spec (%s. q s) c' r |] ==> Spec p (Seq c c') r"
    3.36 -  (fn prems => [cut_facts_tac prems 1, Fast_tac 1]);
    3.37 +Goalw [Valid_def] "[| Valid P c1 Q; Valid Q c2 R |] ==> Valid P (c1;c2) R";
    3.38 +by(Asm_simp_tac 1);
    3.39 +by(Blast_tac 1);
    3.40 +qed "SeqRule";
    3.41  
    3.42 -val IfRule = prove_goalw thy [Spec_def,Cond_def]
    3.43 -  "[| !!s. p s ==> (b s --> q s) & (~b s --> q' s); \
    3.44 -\     Spec (%s. q s) c r; Spec (%s. q' s) c' r |] \
    3.45 -\  ==> Spec p (Cond b c c') r"
    3.46 -  (fn [prem1,prem2,prem3] =>
    3.47 -     [REPEAT (rtac allI 1),
    3.48 -      REPEAT (rtac impI 1),
    3.49 -      dtac prem1 1,
    3.50 -      cut_facts_tac [prem2,prem3] 1,
    3.51 -      fast_tac (claset() addIs [prem1]) 1]);
    3.52 -
    3.53 -val strenthen_pre = prove_goalw thy [Spec_def]
    3.54 -  "[| !!s. p s ==> p' s; Spec p' c q |] ==> Spec p c q"
    3.55 -  (fn [prem1,prem2] =>[cut_facts_tac [prem2] 1,
    3.56 -                       fast_tac (claset() addIs [prem1]) 1]);
    3.57 +Goalw [Valid_def]
    3.58 + "[| p <= {s. (s:b --> s:w) & (s~:b --> s:w')}; \
    3.59 +\    Valid w c1 q; Valid w' c2 q |] \
    3.60 +\ ==> Valid p (IF b THEN c1 ELSE c2 FI) q";
    3.61 +by(Asm_simp_tac 1);
    3.62 +by(Blast_tac 1);
    3.63 +qed "CondRule";
    3.64  
    3.65 -val lemma = prove_goalw thy [Spec_def,While_def]
    3.66 -  "[| Spec (%s. I s & b s) c I; !!s. [| I s; ~b s |] ==> q s |] \
    3.67 -\  ==> Spec I (While b I c) q"
    3.68 -  (fn [prem1,prem2] =>
    3.69 -     [REPEAT(rtac allI 1), rtac impI 1, etac exE 1, rtac mp 1, atac 2,
    3.70 -      etac thin_rl 1, res_inst_tac[("x","s")]spec 1,
    3.71 -      res_inst_tac[("x","s'")]spec 1, induct_tac "n" 1,
    3.72 -      Simp_tac 1,
    3.73 -      fast_tac (claset() addIs [prem2]) 1,
    3.74 -      simp_tac (simpset() addsimps [Seq_def]) 1,
    3.75 -      cut_facts_tac [prem1] 1, fast_tac (claset() addIs [prem2]) 1]);
    3.76 +Goal "! s s'. Sem c s s' --> s : I Int b --> s' : I ==> \
    3.77 +\     ! s s'. s : I --> iter n b (Sem c) s s' --> s' : I & s' ~: b";
    3.78 +by(induct_tac "n" 1);
    3.79 + by(Asm_simp_tac 1);
    3.80 +by(Simp_tac 1);
    3.81 +by(Blast_tac 1);
    3.82 +val lemma = result() RS spec RS spec RS mp RS mp;
    3.83  
    3.84 -val WhileRule = lemma RSN (2,strenthen_pre);
    3.85 +Goalw [Valid_def]
    3.86 + "[| p <= i; Valid (i Int b) c i; (i Int -b) <= q |] \
    3.87 +\ ==> Valid p (WHILE b INV {i} DO c OD) q";
    3.88 +by(Asm_simp_tac 1);
    3.89 +by(Clarify_tac 1);
    3.90 +bd lemma 1;
    3.91 +ba 2;
    3.92 +by(Blast_tac 1);
    3.93 +by(Blast_tac 1);
    3.94 +qed "WhileRule";
    3.95  
    3.96 -
    3.97 -(*** meta_spec used in StateElimTac ***)
    3.98 +(*** The tactics ***)
    3.99  
   3.100 -val meta_spec = prove_goal HOL.thy
   3.101 -  "(!!s x. PROP P s x) ==> (!!s. PROP P s (x s))"
   3.102 -  (fn prems => [resolve_tac prems 1]);
   3.103 -
   3.104 -
   3.105 -(**************************************************************************************************)
   3.106 -(*** Funktion zum Generieren eines Theorems durch Umbennenen von Namen von Lambda-Abstraktionen ***)
   3.107 -(*** in einem bestehenden Theorem. Bsp.: "!a.?P(a) ==> ?P(?x)" aus "!x.?P(x) ==> ?P(?x)"        ***)
   3.108 -(**************************************************************************************************)
   3.109 +(*****************************************************************************)
   3.110 +(** The function Mset makes the theorem                                     **)
   3.111 +(** "?Mset <= {(x1,...,xn). ?P (x1,...,xn)} ==> ?Mset <= {s. ?P s}",        **)
   3.112 +(** where (x1,...,xn) are the variables of the particular program we are    **)
   3.113 +(** working on at the moment of the call. For instance, (found,x,y) are     **)
   3.114 +(** the variables of the Zero Search program.                               **)
   3.115 +(*****************************************************************************)
   3.116  
   3.117 -(* rename_abs:term (von:string,nach:string,trm:term) benennt in trm alle Lambda-Abstraktionen
   3.118 -   mit Namen von in nach um *)
   3.119 +local open HOLogic in
   3.120  
   3.121 -fun rename_abs (von,nach,Abs (s,t,trm)) =
   3.122 -    if von=s
   3.123 -	then Abs (nach,t,rename_abs (von,nach,trm))
   3.124 -        else Abs (s,t,rename_abs (von,nach,trm))
   3.125 -  | rename_abs (von,nach,trm1 $ trm2)   =rename_abs (von,nach,trm1) $ rename_abs (von,nach,trm2)
   3.126 -  | rename_abs (_,_,trm)                =trm;
   3.127 +(** maps (%x1 ... xn. t) to [x1,...,xn] **)
   3.128 +fun abs2list (Const ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
   3.129 +  | abs2list (Abs(x,T,t)) = [Free (x, T)]
   3.130 +  | abs2list _ = [];
   3.131 +
   3.132 +(** maps {(x1,...,xn). t} to [x1,...,xn] **)
   3.133 +fun mk_vars (Const ("Collect",_) $ T) = abs2list T
   3.134 +  | mk_vars _ = [];
   3.135  
   3.136 -(* ren_abs_thm:thm (von:string,nach:string,theorem:thm) benennt in theorem alle Lambda-Abstraktionen
   3.137 -   mit Namen von in nach um. Vorgehen:
   3.138 -        - Term t zu thoerem bestimmen
   3.139 -        - Term t' zu t durch Umbenennen der Namen generieren
   3.140 -        - Certified Term ct' zu t' erstellen
   3.141 -        - Thoerem ct'==ct' anlegen
   3.142 -        - Nach der Regel "[|P==Q; P|] ==> Q" wird aus "ct'==ct'" und theorem das Theorem zu ct'
   3.143 -          abgeleitet (ist moeglich, da t' mit t unifiziert werden kann, da nur Umnbenennungen) *)
   3.144 -
   3.145 -fun ren_abs_thm (von,nach,theorem)      =
   3.146 -        equal_elim
   3.147 -                (reflexive (cterm_of (#sign (rep_thm theorem))
   3.148 -			    (rename_abs (von,nach,#prop (rep_thm theorem)))))
   3.149 -                theorem;
   3.150 -
   3.151 +(** abstraction of body over a tuple formed from a list of free variables. 
   3.152 +Types are also built **)
   3.153 +fun mk_abstupleC []     body = absfree ("x", unitT, body)
   3.154 +  | mk_abstupleC (v::w) body = let val (n,T) = dest_Free v
   3.155 +                               in if w=[] then absfree (n, T, body)
   3.156 +        else let val z  = mk_abstupleC w body;
   3.157 +                 val T2 = case z of Abs(_,T,_) => T
   3.158 +                        | Const (_, Type (_,[_, Type (_,[T,_])])) $ _ => T;
   3.159 +       in Const ("split", (T --> T2 --> boolT) --> mk_prodT (T,T2) --> boolT) 
   3.160 +          $ absfree (n, T, z) end end;
   3.161  
   3.162 -(****************************************************************************)
   3.163 -(*** Taktik zum Anwenden eines Theorems theorem auf ein Subgoal i durch   ***)
   3.164 -(***  - Umbenennen von Lambda-Abstraktionen im Theorem                    ***)
   3.165 -(***  - Instanziieren von freien Variablen im Theorem                     ***)
   3.166 -(***  - Composing des Subgoals mit dem Theorem                            ***)
   3.167 -(****************************************************************************)
   3.168 +(** maps [x1,...,xn] to (x1,...,xn) and types**)
   3.169 +fun mk_bodyC []      = Const ("()", unitT) 
   3.170 +  | mk_bodyC (x::xs) = if xs=[] then x 
   3.171 +               else let val (n, T) = dest_Free x ;
   3.172 +                        val z = mk_bodyC xs;
   3.173 +                        val T2 = case z of Free(_, T) => T
   3.174 +                                         | Const ("Pair", Type ("fun", [_, Type
   3.175 +                                            ("fun", [_, T])])) $ _ $ _ => T;
   3.176 +                 in Const ("Pair", [T, T2] ---> mk_prodT (T, T2)) $ x $ z end;
   3.177  
   3.178 -(* - rens:(string*string) list, d.h. es koennen verschiedene Lambda-Abstraktionen umbenannt werden
   3.179 -   - insts:(cterm*cterm) list, d.h. es koennen verschiedene Variablen instanziiert werden *)
   3.180 +fun dest_Goal (Const ("Goal", _) $ P) = P;
   3.181  
   3.182 -fun comp_inst_ren_tac rens insts theorem i      =
   3.183 -        let fun compose_inst_ren_tac [] insts theorem i                     =
   3.184 -	      compose_tac (false,
   3.185 -			   cterm_instantiate insts theorem,nprems_of theorem) i
   3.186 -	      | compose_inst_ren_tac ((von,nach)::rl) insts theorem i       =
   3.187 -                        compose_inst_ren_tac rl insts 
   3.188 -			  (ren_abs_thm (von,nach,theorem)) i
   3.189 -        in  compose_inst_ren_tac rens insts theorem i  end;
   3.190 +(** maps a goal of the form:
   3.191 +        1. [| P |] ==> |- VARS x1 ... xn. {._.} _ {._.} or to [x1,...,xn]**) 
   3.192 +fun get_vars thm = let  val c = dest_Goal (concl_of (thm));
   3.193 +                        val d = Logic.strip_assums_concl c;
   3.194 +                        val Const _ $ pre $ _ $ _ = dest_Trueprop d;
   3.195 +      in mk_vars pre end;
   3.196  
   3.197  
   3.198 -(***************************************************************    *********)
   3.199 -(*** Taktik zum Eliminieren des Zustandes waehrend Hoare-Beweisen                               ***)
   3.200 -(***    Bsp.: "!!s. s(Suc(0))=0 --> s(Suc(0))+1=1" wird zu "!!s b. b=0 --> b+1=1"               ***)
   3.201 -(****************************************************************************)
   3.202 +(** Makes Collect with type **)
   3.203 +fun mk_CollectC trm = let val T as Type ("fun",[t,_]) = fastype_of trm 
   3.204 +                      in Collect_const t $ trm end;
   3.205 +
   3.206 +fun inclt ty = Const ("op <=", [ty,ty] ---> boolT);
   3.207 +
   3.208 +(** Makes "Mset <= t" **)
   3.209 +fun Mset_incl t = let val MsetT = fastype_of t 
   3.210 +                 in mk_Trueprop ((inclt MsetT) $ Free ("Mset", MsetT) $ t) end;
   3.211 +
   3.212  
   3.213 -(* pvars_of_term:term list (name:string,trm:term) gibt die Liste aller Programm-Variablen
   3.214 -   aus trm zurueck. name gibt dabei den Namen der Zustandsvariablen an.
   3.215 -        Bsp.: bei name="s" und dem Term "s(Suc(Suc(0)))=s(0)" (entspricht "c=a")
   3.216 -              wird [0,Suc(Suc(0))] geliefert (Liste ist i.A. unsortiert) *)
   3.217 +fun Mset thm = let val vars = get_vars(thm);
   3.218 +                   val varsT = fastype_of (mk_bodyC vars);
   3.219 +                   val big_Collect = mk_CollectC (mk_abstupleC vars 
   3.220 +                         (Free ("P",varsT --> boolT) $ mk_bodyC vars));
   3.221 +                   val small_Collect = mk_CollectC (Abs("x",varsT,
   3.222 +                           Free ("P",varsT --> boolT) $ Bound 0));
   3.223 +                   val impl = implies $ (Mset_incl big_Collect) $ 
   3.224 +                                          (Mset_incl small_Collect);
   3.225 +                   val cimpl = cterm_of (#sign (rep_thm thm)) impl
   3.226 +   in  prove_goalw_cterm [] cimpl (fn prems => 
   3.227 +                              [cut_facts_tac prems 1,Blast_tac 1]) end;
   3.228  
   3.229 -fun pvars_of_term (name,trm)    =
   3.230 -  let fun add_vars (name,Free (s,t) $ trm,vl) =
   3.231 -            if name=s then if trm mem vl then vl else trm::vl
   3.232 -                      else add_vars (name,trm,vl)
   3.233 -	| add_vars (name,Abs (s,t,trm),vl)    =add_vars (name,trm,vl)
   3.234 -	| add_vars (name,trm1 $ trm2,vl)      =add_vars (name,trm2,add_vars (name,trm1,vl))
   3.235 -	| add_vars (_,_,vl)                   =vl
   3.236 -  in add_vars (name,trm,[]) end;
   3.237 +end;
   3.238  
   3.239  
   3.240 -(* VarsElimTac: Taktik zum Eliminieren von bestimmten Programmvariablen aus dem Subgoal i
   3.241 - - v::vl:(term) list  Liste der zu eliminierenden Programmvariablen
   3.242 - - meta_spec:thm      Theorem, welches zur Entfernung der Variablen benutzt wird
   3.243 -		      z.B.: "(!!s x. PROP P(s,x)) ==> (!!s. PROP P(s,x(s)))"
   3.244 - - namexAll:string    Name von    ^                                  (hier "x")
   3.245 - - varx:term          Term zu                                      ^ (hier Var(("x",0),...))
   3.246 - - varP:term          Term zu                                  ^     (hier Var(("P",0),...))
   3.247 - - type_pvar:typ      Typ der Programmvariablen (d.h. 'a bei 'a prog, z.B.: nat, bool, ...)
   3.248 +(*****************************************************************************)
   3.249 +(** Simplifying:                                                            **)
   3.250 +(** Some useful lemmata, lists and simplification tactics to control which  **)
   3.251 +(** theorems are used to simplify at each moment, so that the original      **)
   3.252 +(** input does not suffer any unexpected transformation                     **)
   3.253 +(*****************************************************************************)
   3.254 +
   3.255 +val Compl_Collect = prove_goal thy "-(Collect b) = {x. ~(b x)}"
   3.256 +    (fn _ => [Fast_tac 1]);
   3.257 +
   3.258 +(**Simp_tacs**)
   3.259  
   3.260 - Vorgehen:
   3.261 -      - eliminiere jede pvar durch Anwendung von comp_inst_ren_tac. Dazu:
   3.262 -      - Unbenennung in meta_spec: namexAll wird in den Namen der Prog.-Var. umbenannt
   3.263 -	z.B.: fuer die Prog.-Var. mit Namen "a" ergibt sich
   3.264 -	      meta_spec zu "(!! s a. PROP P(s,a)) ==> (!! s. PROP P(s,x(s)))"
   3.265 -      - Instanziierungen in meta_spec:
   3.266 -	      varx wird mit "%s:(type_pvar) state. s(pvar)" instanziiert
   3.267 -	      varP wird entsprechend instanziiert. Beispiel fuer Prog.-Var. "a":
   3.268 -	 - zu Subgoal "!!s. s(Suc(0)) = s(0) ==> s(0) = 1" bestimme Term ohne "!!s.":
   3.269 -		trm0 = "s(Suc(0)) = s(0) ==> s(0) = 1" (s ist hier freie Variable)
   3.270 -	 - substituiere alle Vorkommen von s(pvar) durch eine freie Var. xs:
   3.271 -		trm1 = "s(Suc(0)) = xs ==> xs = 1"
   3.272 -	 - abstrahiere ueber xs:
   3.273 -		trm2 = "%xs. s(Suc(0)) = xs ==> xs = 1"
   3.274 -	 - abstrahiere ueber restliche Vorkommen von s:
   3.275 -		trm3 = "%s xs. s(Suc(0)) = xs ==> xs = 1"
   3.276 -	 - instanziiere varP mit trm3
   3.277 -*)
   3.278 +val before_set2pred_simp_tac =
   3.279 +  (simp_tac (HOL_basic_ss addsimps [Collect_conj_eq RS sym,Compl_Collect]));
   3.280 +
   3.281 +val split_simp_tac = (simp_tac (HOL_basic_ss addsimps [split]));
   3.282 +
   3.283 +(*****************************************************************************)
   3.284 +(** set2pred transforms sets inclusion into predicates implication,         **)
   3.285 +(** maintaining the original variable names.                                **)
   3.286 +(** Ex. "{x. x=0} <= {x. x <= 1}" -set2pred-> "x=0 --> x <= 1"              **)
   3.287 +(** Subgoals containing intersections (A Int B) or complement sets (-A)     **)
   3.288 +(** are first simplified by "before_set2pred_simp_tac", that returns only   **)
   3.289 +(** subgoals of the form "{x. P x} <= {x. Q x}", which are easily           **)
   3.290 +(** transformed.                                                            **)
   3.291 +(** This transformation may solve very easy subgoals due to a ligth         **)
   3.292 +(** simplification done by (split_all_tac)                                  **)
   3.293 +(*****************************************************************************)
   3.294  
   3.295 -(* StateElimTac: tactic to eliminate all program variable from subgoal i
   3.296 -    - applies to subgoals of the form "!!s:('a) state. P(s)",
   3.297 -        i.e. the term  Const("all",_) $ Abs ("s",pvar --> 'a,_)
   3.298 -    -   meta_spec has the form "(!!s x. PROP P(s,x)) ==> (!!s. PROP P(s,x(s)))"
   3.299 -*)
   3.300 +fun set2pred i thm = let fun mk_string [] = ""
   3.301 +                           | mk_string (x::xs) = x^" "^mk_string xs;
   3.302 +                         val vars=get_vars(thm);
   3.303 +                         val var_string = mk_string (map (fst o dest_Free) vars);
   3.304 +      in ((before_set2pred_simp_tac i) THEN_MAYBE
   3.305 +          (EVERY [rtac subsetI i, 
   3.306 +                  rtac CollectI i,
   3.307 +                  dtac CollectD i,
   3.308 +                  (TRY(split_all_tac i)) THEN_MAYBE 
   3.309 +                  ((rename_tac var_string i) THEN
   3.310 +                   (full_simp_tac (HOL_basic_ss addsimps [split]) i)) ])) thm
   3.311 +      end;
   3.312 +
   3.313 +(*****************************************************************************)
   3.314 +(** BasicSimpTac is called to simplify all verification conditions. It does **)
   3.315 +(** a light simplification by applying "mem_Collect_eq", then it calls      **)
   3.316 +(** MaxSimpTac, which solves subgoals of the form "A <= A",                 **)
   3.317 +(** and transforms any other into predicates, applying then                 **)
   3.318 +(** the tactic chosen by the user, which may solve the subgoal completely.  **)
   3.319 +(*****************************************************************************)
   3.320 +
   3.321 +fun MaxSimpTac tac = FIRST'[rtac subset_refl, set2pred THEN_MAYBE' tac];
   3.322  
   3.323 -val StateElimTac = SUBGOAL (fn (Bi,i) =>
   3.324 -  let val Const _ $ Abs (_,Type ("fun",[_,type_pvar]),trm) = Bi
   3.325 -      val _ $ (_ $ Abs (_,_,_ $ Abs (namexAll,_,_))) $
   3.326 -			    (_ $ Abs (_,_,varP $ _ $ (varx $ _))) = 
   3.327 -			    #prop (rep_thm meta_spec)
   3.328 -      fun vtac v i st = st |>
   3.329 -	  let val cterm = cterm_of (#sign (rep_thm st))
   3.330 -	      val (_,_,_ $ Abs (_,_,trm),_) = dest_state (st,i);
   3.331 -	      val (sname,trm0) = variant_abs ("s",dummyT,trm);
   3.332 -	      val xsname = variant_name ("xs",trm0);
   3.333 -	      val trm1 = subst_term (Free (sname,dummyT) $ v,
   3.334 -				     Syntax.free xsname,trm0)
   3.335 -	      val trm2 = Abs ("xs", type_pvar,
   3.336 -			      abstract_over (Syntax.free xsname,trm1))
   3.337 -	  in
   3.338 -	      comp_inst_ren_tac
   3.339 -		[(namexAll,pvar2pvarID v)]
   3.340 -		[(cterm varx,
   3.341 -		  cterm (Abs  ("s",Type ("nat",[]) --> type_pvar,
   3.342 -			       Bound 0 $ v))),
   3.343 -		 (cterm varP,
   3.344 -		  cterm (Abs ("s", Type ("nat",[]) --> type_pvar,
   3.345 -			      abstract_over (Free (sname,dummyT),trm2))))]
   3.346 -		meta_spec i
   3.347 -	  end
   3.348 -      fun vars_tac [] i      = all_tac
   3.349 -	| vars_tac (v::vl) i = vtac v i THEN vars_tac vl i
   3.350 -  in
   3.351 -      vars_tac (pvars_of_term (variant_abs ("s",dummyT,trm))) i
   3.352 -  end);
   3.353 +fun BasicSimpTac tac =
   3.354 +  simp_tac (HOL_basic_ss addsimps [mem_Collect_eq,split])
   3.355 +  THEN_MAYBE' MaxSimpTac tac;
   3.356 +
   3.357 +(** HoareRuleTac **)
   3.358 +
   3.359 +fun WlpTac Mlem tac i = rtac SeqRule i THEN  HoareRuleTac Mlem tac false (i+1)
   3.360 +and HoareRuleTac Mlem tac pre_cond i st = st |>
   3.361 +        (*abstraction over st prevents looping*)
   3.362 +    ( (WlpTac Mlem tac i THEN HoareRuleTac Mlem tac pre_cond i)
   3.363 +      ORELSE
   3.364 +      (FIRST[rtac SkipRule i,
   3.365 +             EVERY[rtac BasicRule i,
   3.366 +                   rtac Mlem i,
   3.367 +                   split_simp_tac i],
   3.368 +             EVERY[rtac CondRule i,
   3.369 +                   HoareRuleTac Mlem tac false (i+2),
   3.370 +                   HoareRuleTac Mlem tac false (i+1)],
   3.371 +             EVERY[rtac WhileRule i,
   3.372 +                   BasicSimpTac tac (i+2),
   3.373 +                   HoareRuleTac Mlem tac true (i+1)] ] 
   3.374 +       THEN (if pre_cond then (BasicSimpTac tac i) else (rtac subset_refl i)) ));
   3.375  
   3.376  
   3.377 -(*** tactics for verification condition generation ***)
   3.378 -
   3.379 -(* pre_cond:bool gibt an, ob das Subgoal von der Form Spec(?Q,c,p) ist oder nicht. Im Fall
   3.380 -   von pre_cond=false besteht die Vorbedingung nur nur aus einer scheme-Variable. Die dann
   3.381 -   generierte Verifikationsbedingung hat die Form "!!s.?Q --> ...". "?Q" kann deshalb zu gegebenen
   3.382 -   Zeitpunkt mittels "rtac impI" und "atac" gebunden werden, die Bedingung loest sich dadurch auf. *)
   3.383 -
   3.384 -fun WlpTac i = (rtac SeqRule i) THEN (HoareRuleTac (i+1) false)
   3.385 -and HoareRuleTac i pre_cond st = st |>  
   3.386 -	(*abstraction over st prevents looping*)
   3.387 -    ( (WlpTac i THEN HoareRuleTac i pre_cond)
   3.388 -      ORELSE
   3.389 -      (FIRST[rtac SkipRule i,
   3.390 -	     rtac AssignRule i,
   3.391 -	     EVERY[rtac IfRule i,
   3.392 -		   HoareRuleTac (i+2) false,
   3.393 -		   HoareRuleTac (i+1) false],
   3.394 -	     EVERY[rtac WhileRule i,
   3.395 -		   Asm_full_simp_tac (i+2),
   3.396 -		   HoareRuleTac (i+1) true]]
   3.397 -       THEN
   3.398 -       (if pre_cond then (Asm_full_simp_tac i) else (atac i))) );
   3.399 -
   3.400 -val hoare_tac = 
   3.401 -  SELECT_GOAL
   3.402 -    (EVERY[HoareRuleTac 1 true, ALLGOALS StateElimTac, prune_params_tac]);
   3.403 -
   3.404 +(** tac:(int -> tactic) is the tactic the user chooses to solve or simplify **)
   3.405 +(** the final verification conditions                                       **)
   3.406 + 
   3.407 +fun hoare_tac tac i thm =
   3.408 +  let val Mlem = Mset(thm)
   3.409 +  in SELECT_GOAL(EVERY[HoareRuleTac Mlem tac true 1]) i thm end;
     4.1 --- a/src/HOL/Hoare/Hoare.thy	Wed Oct 14 11:51:11 1998 +0200
     4.2 +++ b/src/HOL/Hoare/Hoare.thy	Wed Oct 14 15:26:31 1998 +0200
     4.3 @@ -1,196 +1,193 @@
     4.4  (*  Title:      HOL/Hoare/Hoare.thy
     4.5      ID:         $Id$
     4.6 -    Author:     Norbert Galm & Tobias Nipkow
     4.7 -    Copyright   1995 TUM
     4.8 +    Author:     Leonor Prensa Nieto & Tobias Nipkow
     4.9 +    Copyright   1998 TUM
    4.10  
    4.11  Sugared semantic embedding of Hoare logic.
    4.12 +Strictly speaking a shallow embedding (as implemented by Norbert Galm
    4.13 +following Mike Gordon) would suffice. Maybe the datatype com comes in useful
    4.14 +later.
    4.15  *)
    4.16  
    4.17 -Hoare = Arith +
    4.18 +Hoare  = Main +
    4.19  
    4.20  types
    4.21 -  pvar = nat                                    (* encoding of program variables ( < 26! ) *)
    4.22 -  'a state = pvar => 'a                         (* program state *)
    4.23 -  'a exp = 'a state => 'a                       (* denotation of expressions *)
    4.24 -  'a bexp = 'a state => bool                    (* denotation of boolean expressions *)
    4.25 -  'a com = 'a state => 'a state => bool         (* denotation of programs *)
    4.26 +    'a bexp = 'a set
    4.27 +    'a assn = 'a set
    4.28 +    'a fexp = 'a =>'a
    4.29 +
    4.30 +datatype
    4.31 + 'a com = Basic ('a fexp)         
    4.32 +   | Seq ('a com) ('a com)               ("(_;/_)"      [61,60] 60)
    4.33 +   | Cond ('a bexp) ('a com) ('a com)    ("(1IF _/ THEN _ / ELSE _/ FI)"  [0,0,0] 61)
    4.34 +   | While ('a bexp) ('a assn) ('a com)  ("(1WHILE _/ INV {_} //DO _ /OD)"  [0,0,0] 61)
    4.35 +  
    4.36 +syntax
    4.37 +  "@assign"  :: id => 'b => 'a com        ("(2_ :=/ _ )" [70,65] 61)
    4.38 +  "@annskip" :: 'a com                    ("SKIP")
    4.39 +
    4.40 +translations
    4.41 +            "SKIP" == "Basic id"
    4.42 +
    4.43 +types 'a sem = 'a => 'a => bool
    4.44 +
    4.45 +consts iter :: nat => 'a bexp => 'a sem => 'a sem
    4.46 +primrec
    4.47 +"iter 0 b S = (%s s'. s ~: b & (s=s'))"
    4.48 +"iter (Suc n) b S = (%s s'. s : b & (? s''. S s s'' & iter n b S s'' s'))"
    4.49 +
    4.50 +consts Sem :: 'a com => 'a sem
    4.51 +primrec
    4.52 +"Sem(Basic f) s s' = (s' = f s)"
    4.53 +"Sem(c1;c2) s s' = (? s''. Sem c1 s s'' & Sem c2 s'' s')"
    4.54 +"Sem(IF b THEN c1 ELSE c2 FI) s s' = ((s  : b --> Sem c1 s s') &
    4.55 +                                      (s ~: b --> Sem c2 s s'))"
    4.56 +"Sem(While b x c) s s' = (? n. iter n b (Sem c) s s')"
    4.57 +
    4.58 +constdefs Valid :: ['a bexp, 'a com, 'a bexp] => bool
    4.59 +  "Valid p c q == !s s'. Sem c s s' --> s : p --> s' : q"
    4.60  
    4.61  
    4.62 -(* program syntax *)
    4.63 -
    4.64  nonterminals
    4.65 -  prog
    4.66 +  vars
    4.67  
    4.68  syntax
    4.69 -  "@Skip"       :: prog                         ("SKIP")
    4.70 -  "@Assign"     :: [id, 'a] => prog             ("_ := _")
    4.71 -  "@Seq"        :: [prog, prog] => prog         ("_;//_" [0,1000] 999)
    4.72 -  "@If"         :: [bool, prog, prog] => prog   ("IF _//THEN//  (_)//ELSE//  (_)//END")
    4.73 -  "@While"      :: [bool, bool, prog] => prog   ("WHILE _//DO {_}//  (_)//END")
    4.74 -  "@Spec"       :: [bool, prog, bool] => bool   ("{_}//_//{_}")
    4.75 -
    4.76 -
    4.77 -(* denotational semantics *)
    4.78 -
    4.79 -constdefs
    4.80 -  Skip          :: 'a com
    4.81 -  "Skip s s' == (s=s')"
    4.82 -
    4.83 -  Assign        :: [pvar, 'a exp] => 'a com
    4.84 -  "Assign v e s s' == (s' = (%x. if x=v then e(s) else s(x)))"
    4.85 +  ""		     :: "id => vars"		       ("_")
    4.86 +  "_vars" 	     :: "[id, vars] => vars"	       ("_ _")
    4.87  
    4.88 -  Seq           :: ['a com, 'a com] => 'a com
    4.89 -  "Seq c c' s s' == ? s''. c s s'' & c' s'' s'"
    4.90 -
    4.91 -  Cond          :: ['a bexp, 'a com, 'a com] => 'a com
    4.92 -  "Cond b c c' s s' == (b(s) --> c s s') & (~b s --> c' s s')"
    4.93 -
    4.94 -consts
    4.95 -  Iter          :: [nat, 'a bexp, 'a com] => 'a com
    4.96 -
    4.97 -primrec
    4.98 -  "Iter 0 b c = (%s s'.~b(s) & (s=s'))"
    4.99 -  "Iter (Suc n) b c = (%s s'. b(s) & Seq c (Iter n b c) s s')"
   4.100 -
   4.101 -constdefs
   4.102 -  While         :: ['a bexp, 'a bexp, 'a com] => 'a com
   4.103 -  "While b I c s s' == ? n. Iter n b c s s'"
   4.104 -
   4.105 -  Spec          :: ['a bexp, 'a com, 'a bexp] => bool
   4.106 -  "Spec p c q == !s s'. c s s' --> p s --> q s'"
   4.107 +syntax
   4.108 + "@hoare_vars" :: [vars, 'a assn,'a com,'a assn] => bool
   4.109 +                 ("|- VARS _.// {_} // _ // {_}" [0,0,55,0] 50)
   4.110 +syntax ("" output)
   4.111 + "@hoare"      :: ['a assn,'a com,'a assn] => bool
   4.112 +                 ("|- {_} // _ // {_}" [0,55,0] 50)
   4.113  
   4.114  end
   4.115  
   4.116  ML
   4.117  
   4.118 -
   4.119 -(*** term manipulation ***)
   4.120 -
   4.121 -(* name_in_term:bool (name:string,trm:term) bestimmt, ob in trm der Name name als Konstante,
   4.122 -   freie Var., scheme-Variable oder als Name fuer eine Lambda-Abstraktion vorkommt *)
   4.123 +(** parse translations **)
   4.124  
   4.125 -fun name_in_term (name,Const (s,t))    = (name=s)
   4.126 -  | name_in_term (name,Free (s,t))     = (name=s)
   4.127 -  | name_in_term (name,Var (ix,t))  = (name= string_of_indexname ix)
   4.128 -  | name_in_term (name,Abs (s,t,trm))  = (name=s) orelse
   4.129 -                                         (name_in_term (name,trm))
   4.130 -  | name_in_term (name,trm1 $ trm2)    = (name_in_term (name,trm1)) orelse
   4.131 -                                         (name_in_term (name,trm2))
   4.132 -  | name_in_term (_,_)                 = false;
   4.133 -
   4.134 -(* variant_name:string (name:string,trm:term) liefert einen von name
   4.135 -   abgewandelten Namen (durch Anhaengen von genuegend vielen "_"), der nicht
   4.136 -   in trm vorkommt. Im Gegensatz zu variant_abs beruecktsichtigt es auch Namen
   4.137 -   von scheme-Variablen und von Lambda-Abstraktionen in trm *)
   4.138 +fun mk_abstuple []     body = absfree ("x", dummyT, body)
   4.139 +  | mk_abstuple [v]    body = absfree ((fst o dest_Free) v, dummyT, body)
   4.140 +  | mk_abstuple (v::w) body = Syntax.const "split" $
   4.141 +                              absfree ((fst o dest_Free) v, dummyT, mk_abstuple w body);
   4.142  
   4.143 -(*This could be done more simply by calling Term.variant, supplying a list of
   4.144 -  all free, bound and scheme variables in the term.*)
   4.145 -fun variant_name (name,trm) = if name_in_term (name,trm)
   4.146 -			      then variant_name (name ^ "_",trm)
   4.147 -                              else name;
   4.148 -
   4.149 -(* subst_term:term (von:term,nach:term,trm:term) liefert den Term, der aus
   4.150 -trm entsteht, wenn alle Teilterme, die gleich von sind, durch nach ersetzt
   4.151 -wurden *)
   4.152 +  
   4.153 +fun mk_fbody v e []      = Syntax.const "()"
   4.154 +  | mk_fbody v e [x]     = if v=x then e else x
   4.155 +  | mk_fbody v e (x::xs) = Syntax.const "Pair" $ (if v=x then e else x) $
   4.156 +                           mk_fbody v e xs;
   4.157  
   4.158 -fun subst_term (von,nach,Abs (s,t,trm)) =if von=Abs (s,t,trm)
   4.159 -                                                then nach
   4.160 -                                                else Abs (s,t,subst_term (von,nach,trm))
   4.161 -  | subst_term (von,nach,trm1 $ trm2)   =if von=trm1 $ trm2
   4.162 -                                                then nach
   4.163 -                                                else subst_term (von,nach,trm1) $ subst_term (von,nach,trm2)
   4.164 -  | subst_term (von,nach,trm)           =if von=trm
   4.165 -                                                then nach
   4.166 -                                                else trm;
   4.167 +fun mk_fexp v e xs = mk_abstuple xs (mk_fbody v e xs);
   4.168  
   4.169  
   4.170 -(* Translation between program vars ("a" - "z") and their encoding as
   4.171 -   natural numbers: "a" <==> 0, "b" <==> Suc(0), ..., "z" <==> 25 *)
   4.172 +(* bexp_tr & assn_tr *)
   4.173 +(*all meta-variables for bexp except for TRUE and FALSE are translated as if they
   4.174 +  were boolean expressions*)
   4.175 +  
   4.176 +fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE"
   4.177 +  | bexp_tr b xs = Syntax.const "Collect" $ mk_abstuple xs b;
   4.178 +  
   4.179 +fun assn_tr r xs = Syntax.const "Collect" $ mk_abstuple xs r;
   4.180  
   4.181 -fun is_pvarID s = size s=1 andalso "a"<=s andalso s<="z";
   4.182 +(* com_tr *)
   4.183 +  
   4.184 +fun assign_tr [Free (V,_),E] xs = Syntax.const "Basic" $
   4.185 +                                      mk_fexp (Free(V,dummyT)) E xs
   4.186 +  | assign_tr ts _ = raise TERM ("assign_tr", ts);
   4.187  
   4.188 -fun pvarID2pvar s =
   4.189 -  let fun rest2pvar (i,arg) =
   4.190 -            if i=0 then arg else rest2pvar(i-1, Syntax.const "Suc" $ arg)
   4.191 -  in rest2pvar(ord s - ord "a", Syntax.const "0") end;
   4.192 +fun com_tr (Const("@assign",_) $ Free (V,_) $ E) xs =
   4.193 +               assign_tr [Free (V,dummyT),E] xs
   4.194 +  | com_tr (Const ("Basic",_) $ f) xs = Syntax.const "Basic" $ f
   4.195 +  | com_tr (Const ("Seq",_) $ c1 $ c2) xs = Syntax.const "Seq" $
   4.196 +                                                 com_tr c1 xs $ com_tr c2 xs
   4.197 +  | com_tr (Const ("Cond",_) $ b $ c1 $ c2) xs = Syntax.const "Cond" $
   4.198 +                                  bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs
   4.199 +  | com_tr (Const ("While",_) $ b $ I $ c) xs = Syntax.const "While" $
   4.200 +                                         bexp_tr b xs $ assn_tr I xs $ com_tr c xs
   4.201 +  | com_tr trm _ = trm;
   4.202 +
   4.203 +(* triple_tr *)
   4.204  
   4.205 -fun pvar2pvarID trm     =
   4.206 -        let
   4.207 -                fun rest2pvarID (Const("0",_),i)                =chr (i + ord "a")
   4.208 -                  | rest2pvarID (Const("Suc",_) $ trm,i)        =rest2pvarID (trm,i+1)
   4.209 -        in
   4.210 -                rest2pvarID (trm,0)
   4.211 -        end;
   4.212 +fun vars_tr (x as Free _) = [x]
   4.213 +  | vars_tr (Const ("_vars", _) $ (x as Free _) $ vars) = x :: vars_tr vars
   4.214 +  | vars_tr t = raise TERM ("vars_tr", [t]);
   4.215 +
   4.216 +fun hoare_vars_tr [vars, pre, prg, post] =
   4.217 +      let val xs = vars_tr vars
   4.218 +      in Syntax.const "Valid" $
   4.219 +           assn_tr pre xs $ com_tr prg xs $ assn_tr post xs
   4.220 +      end
   4.221 +  | hoare_vars_tr ts = raise TERM ("hoare_vars_tr", ts);
   4.222 +  
   4.223 +
   4.224 +
   4.225 +val parse_translation = [("@hoare_vars", hoare_vars_tr)];
   4.226  
   4.227  
   4.228 -(*** parse translations: syntax -> semantics ***)
   4.229 +(*****************************************************************************)
   4.230 +
   4.231 +(*** print translations ***)
   4.232  
   4.233 -(* term_tr:term (name:string,trm:term) ersetzt in trm alle freien Variablen, die eine pvarID
   4.234 -   darstellen, durch name $ pvarID2pvar(pvarID). Beispiel:
   4.235 -   bei name="s" und dem Term "a=b & a=X" wird der Term "s(0)=s(Suc(0)) & s(0)=X" geliefert *)
   4.236 +fun dest_abstuple (Const ("split",_) $ (Abs(v,_, body))) =
   4.237 +                            subst_bound (Syntax.free v, dest_abstuple body)
   4.238 +  | dest_abstuple (Abs(v,_, body)) = subst_bound (Syntax.free v, body)
   4.239 +  | dest_abstuple trm = trm;
   4.240  
   4.241 -fun term_tr (name,Free (s,t)) = if is_pvarID s
   4.242 -                                then Syntax.free name $ pvarID2pvar s
   4.243 -                                else Free (s,t)
   4.244 -  | term_tr (name,Abs (s,t,trm)) = Abs (s,t,term_tr (name,trm))
   4.245 -  | term_tr (name,trm1 $ trm2)  = term_tr (name,trm1) $ term_tr (name,trm2)
   4.246 -  | term_tr (name,trm) = trm;
   4.247 +fun abs2list (Const ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
   4.248 +  | abs2list (Abs(x,T,t)) = [Free (x, T)]
   4.249 +  | abs2list _ = [];
   4.250  
   4.251 -fun bool_tr B =
   4.252 -  let val name = variant_name("s",B)
   4.253 -  in Abs (name,dummyT,abstract_over (Syntax.free name,term_tr (name,B))) end;
   4.254 +fun mk_ts (Const ("split",_) $ (Abs(x,_,t))) = mk_ts t
   4.255 +  | mk_ts (Abs(x,_,t)) = mk_ts t
   4.256 +  | mk_ts (Const ("Pair",_) $ a $ b) = a::(mk_ts b)
   4.257 +  | mk_ts t = [t];
   4.258  
   4.259 -fun exp_tr E =
   4.260 -  let val name = variant_name("s",E)
   4.261 -  in Abs (name,dummyT,abstract_over (Syntax.free name,term_tr (name,E))) end;
   4.262 +fun mk_vts (Const ("split",_) $ (Abs(x,_,t))) = 
   4.263 +           ((Syntax.free x)::(abs2list t), mk_ts t)
   4.264 +  | mk_vts (Abs(x,_,t)) = ([Syntax.free x], [t])
   4.265 +  | mk_vts t = raise Match;
   4.266 +  
   4.267 +fun find_ch [] i xs = (false, (Syntax.free "not_ch",Syntax.free "not_ch" ))
   4.268 +  | find_ch ((v,t)::vts) i xs = if t=(Bound i) then find_ch vts (i-1) xs
   4.269 +              else (true, (v, subst_bounds (xs,t)));
   4.270 +  
   4.271 +fun is_f (Const ("split",_) $ (Abs(x,_,t))) = true
   4.272 +  | is_f (Abs(x,_,t)) = true
   4.273 +  | is_f t = false;
   4.274 +  
   4.275 +(* assn_tr' & bexp_tr'*)
   4.276 +  
   4.277 +fun assn_tr' (Const ("Collect",_) $ T) = dest_abstuple T
   4.278 +  | assn_tr' (Const ("op Int",_) $ (Const ("Collect",_) $ T1) $ 
   4.279 +                                   (Const ("Collect",_) $ T2)) =  
   4.280 +            Syntax.const "op Int" $ dest_abstuple T1 $ dest_abstuple T2
   4.281 +  | assn_tr' t = t;
   4.282  
   4.283 -fun prog_tr (Const ("@Skip",_)) = Syntax.const "Skip"
   4.284 -  | prog_tr (Const ("@Assign",_) $ Free (V,_) $ E) =
   4.285 -      if is_pvarID V
   4.286 -      then Syntax.const "Assign" $ pvarID2pvar V $ exp_tr E
   4.287 -      else error("Not a valid program variable: " ^ quote V)
   4.288 -  | prog_tr (Const ("@Seq",_) $ C $ C') =
   4.289 -      Syntax.const "Seq" $ prog_tr C $ prog_tr C'
   4.290 -  | prog_tr (Const ("@If",_) $ B $ C $ C') =
   4.291 -      Syntax.const "Cond" $ bool_tr B $ prog_tr C $ prog_tr C'
   4.292 -  | prog_tr (Const ("@While",_) $ B $ INV $ C) =
   4.293 -      Syntax.const "While" $ bool_tr B $ bool_tr INV $ prog_tr C;
   4.294 +fun bexp_tr' (Const ("Collect",_) $ T) = dest_abstuple T 
   4.295 +  | bexp_tr' t = t;
   4.296 +
   4.297 +(*com_tr' *)
   4.298  
   4.299 -fun spec_tr [P,C,Q] = Syntax.const "Spec" $ bool_tr P $ prog_tr C $ bool_tr Q;
   4.300 +fun mk_assign f =
   4.301 +  let val (vs, ts) = mk_vts f;
   4.302 +      val (ch, which) = find_ch (vs~~ts) ((length vs)-1) (rev vs)
   4.303 +  in if ch then Syntax.const "@assign" $ fst(which) $ snd(which)
   4.304 +     else Syntax.const "@skip" end;
   4.305  
   4.306 -val parse_translation = [("@Spec",spec_tr)];
   4.307 +fun com_tr' (Const ("Basic",_) $ f) = if is_f f then mk_assign f
   4.308 +                                           else Syntax.const "Basic" $ f
   4.309 +  | com_tr' (Const ("Seq",_) $ c1 $ c2) = Syntax.const "Seq" $
   4.310 +                                                 com_tr' c1 $ com_tr' c2
   4.311 +  | com_tr' (Const ("Cond",_) $ b $ c1 $ c2) = Syntax.const "Cond" $
   4.312 +                                           bexp_tr' b $ com_tr' c1 $ com_tr' c2
   4.313 +  | com_tr' (Const ("While",_) $ b $ I $ c) = Syntax.const "While" $
   4.314 +                                               bexp_tr' b $ assn_tr' I $ com_tr' c
   4.315 +  | com_tr' t = t;
   4.316  
   4.317  
   4.318 -(*** print translations: semantics -> syntax ***)
   4.319 -
   4.320 -(* Note: does not mark tokens *)
   4.321 -
   4.322 -(* term_tr':term (name:string,trm:term) ersetzt in trm alle Vorkommen von name $ pvar durch
   4.323 -   entsprechende freie Variablen, welche die pvarID zu pvar darstellen. Beispiel:
   4.324 -        bei name="s" und dem Term "s(0)=s(Suc(0)) & s(0)=X" wird der Term "a=b & a=X" geliefert *)
   4.325 -
   4.326 -fun term_tr' (name,Free (s,t) $ trm) =
   4.327 -      if name=s then Syntax.free (pvar2pvarID trm)
   4.328 -      else Free (s,t) $ term_tr' (name,trm)
   4.329 -  | term_tr' (name,Abs (s,t,trm)) = Abs (s,t,term_tr' (name,trm))
   4.330 -  | term_tr' (name,trm1 $ trm2) = term_tr' (name,trm1) $ term_tr' (name,trm2)
   4.331 -  | term_tr' (name,trm) = trm;
   4.332 -
   4.333 -fun bexp_tr' (Abs(_,_,b)) = term_tr' (variant_abs ("s",dummyT,b));
   4.334 -
   4.335 -fun exp_tr' (Abs(_,_,e)) = term_tr' (variant_abs ("s",dummyT,e));
   4.336 -
   4.337 -fun com_tr' (Const ("Skip",_)) = Syntax.const "@Skip"
   4.338 -  | com_tr' (Const ("Assign",_) $ v $ e) =
   4.339 -      Syntax.const "@Assign" $ Syntax.free (pvar2pvarID v) $ exp_tr' e
   4.340 -  | com_tr' (Const ("Seq",_) $ c $ c') =
   4.341 -      Syntax.const "@Seq" $ com_tr' c $ com_tr' c'
   4.342 -  | com_tr' (Const ("Cond",_) $ b $ c $ c') =
   4.343 -       Syntax.const "@If" $ bexp_tr' b $ com_tr' c $ com_tr' c'
   4.344 -  | com_tr' (Const ("While",_) $ b $ inv $ c) =
   4.345 -       Syntax.const "@While" $ bexp_tr' b $ bexp_tr' inv $ com_tr' c;
   4.346 -
   4.347 -fun spec_tr' [p,c,q] =
   4.348 -       Syntax.const "@Spec" $ bexp_tr' p $ com_tr' c $ bexp_tr' q;
   4.349 -
   4.350 -val print_translation = [("Spec",spec_tr')];
   4.351 +fun spec_tr' [p, c, q] =
   4.352 +  Syntax.const "@hoare" $ assn_tr' p $ com_tr' c $ assn_tr' q
   4.353 + 
   4.354 +val print_translation = [("Valid", spec_tr')];
     5.1 --- a/src/HOL/Hoare/List_Examples.ML	Wed Oct 14 11:51:11 1998 +0200
     5.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     5.3 @@ -1,27 +0,0 @@
     5.4 -Goal
     5.5 -"{x=X} \
     5.6 -\ y := []; \
     5.7 -\ WHILE x ~= [] \
     5.8 -\ DO { rev(x)@y = rev(X)} \
     5.9 -\    y := hd x # y; x := tl x \
    5.10 -\ END \
    5.11 -\{y=rev(X)}";
    5.12 -by (hoare_tac 1);
    5.13 -by (asm_full_simp_tac (simpset() addsimps [neq_Nil_conv]) 1);
    5.14 -by Safe_tac;
    5.15 -by (Asm_full_simp_tac 1);
    5.16 -qed "imperative_reverse";
    5.17 -
    5.18 -Goal
    5.19 -"{x=X & y = Y} \
    5.20 -\ x := rev(x); \
    5.21 -\ WHILE x ~= [] \
    5.22 -\ DO { rev(x)@y = X@Y} \
    5.23 -\    y := hd x # y; x := tl x \
    5.24 -\ END \
    5.25 -\{y = X@Y}";
    5.26 -by (hoare_tac 1);
    5.27 -by (asm_full_simp_tac (simpset() addsimps [neq_Nil_conv]) 1);
    5.28 -by Safe_tac;
    5.29 -by (Asm_full_simp_tac 1);
    5.30 -qed "imperative_append";
     6.1 --- a/src/HOL/Hoare/List_Examples.thy	Wed Oct 14 11:51:11 1998 +0200
     6.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     6.3 @@ -1,1 +0,0 @@
     6.4 -List_Examples = Hoare + List
     7.1 --- a/src/HOL/Hoare/README.html	Wed Oct 14 11:51:11 1998 +0200
     7.2 +++ b/src/HOL/Hoare/README.html	Wed Oct 14 15:26:31 1998 +0200
     7.3 @@ -1,6 +1,59 @@
     7.4  <HTML><HEAD><TITLE>HOL/Hoare/ReadMe</TITLE></HEAD><BODY>
     7.5  
     7.6 -<H2>Semantic Embedding of Hoare Logic</H2>
     7.7 +<H2>Hoare Logic for a Simple WHILE Language</H2>
     7.8 +
     7.9 +<H1>The language and logic<H1>
    7.10 +
    7.11 +This directory contains an implementation of Hoare logic for a simple WHILE
    7.12 +language. The  are
    7.13 +<UL>
    7.14 +<LI> SKIP
    7.15 +<LI> _ := _
    7.16 +<LI> _ ; _
    7.17 +<LI> <kbd>IF _ THEN _ ELSE _ FI<kbd>
    7.18 +<LI> WHILE _ INV {_} DO _ OD
    7.19 +</UL>
    7.20 +Note that each WHILE-loop must be annotated with an invariant.
    7.21 +<P>
    7.22 +
    7.23 +After loading theory Hoare, you can state goals of the form
    7.24 +<PRE>
    7.25 +|- VARS x y ... . {P} prog {Q}
    7.26 +</PRE>
    7.27 +where <kbd>prog</kbd> is a program in the above language, <kbd>P</kbd> is the
    7.28 +precondition, <kbd>Q</kbd> the postcondition, and <kbd>x y ...<kbd> is the
    7.29 +list of all <i>program variables</i> in <kbd>prog</kbd>. The latter list must
    7.30 +be nonempty and it must include all variables that occur on the left-hand
    7.31 +side of an assignment in <kbd>prof</kbd>. Example:
    7.32 +<PRE>
    7.33 +|- VARS x. {x = a} x := x+1 {x = a+1}
    7.34 +</PRE>
    7.35 +The (normal) variable <kbd>a</kbd> is merely used to record the initial
    7.36 +value of <kbd>x</kbd> and is not a program variable. Pre and postconditions
    7.37 +can be arbitrary HOL formulae mentioning both program variables and normal
    7.38 +variables.
    7.39 +<P>
    7.40 +
    7.41 +The implementation hides reasoning in Hoare logic completely and provides a
    7.42 +tactic hoare_tac for generating the verification conditions in ordinary
    7.43 +logic:
    7.44 +<PRE>
    7.45 +by(hoare_tac tac i);
    7.46 +</PRE>
    7.47 +applies the tactic to subgoal <kbd>i</kbd> and applies the parameter
    7.48 +<kbd>tac</kbd> to all generated verification conditions. A typical call is
    7.49 +<PRE>
    7.50 +by(hoare_tac Asm_full_simp_tac 1);
    7.51 +</PRE>
    7.52 +which, given the example goal above, solves it completely.
    7.53 +<P>
    7.54 +
    7.55 +IMPORTANT:
    7.56 +This is a logic of partial correctness. You can only prove that your program
    7.57 +does the right thing <i>if</i> it terminates, but not <i>that</i> it
    7.58 +terminates.
    7.59 +
    7.60 +<H1>Notes on the implementation</H1>
    7.61  
    7.62  This directory contains a sugared shallow semantic embedding of Hoare logic
    7.63  for a while language. The implementation closely follows<P>
     8.1 --- a/src/HOL/Hoare/ROOT.ML	Wed Oct 14 11:51:11 1998 +0200
     8.2 +++ b/src/HOL/Hoare/ROOT.ML	Wed Oct 14 15:26:31 1998 +0200
     8.3 @@ -1,10 +1,9 @@
     8.4  (*  Title:      HOL/IMP/ROOT.ML
     8.5      ID:         $Id$
     8.6      Author:     Tobias Nipkow
     8.7 -    Copyright   1995 TUM
     8.8 +    Copyright   1998 TUM
     8.9  *)
    8.10  
    8.11  HOL_build_completed;    (*Make examples fail if HOL did*)
    8.12  
    8.13  use_thy "Examples";
    8.14 -use_thy "List_Examples";