author | wenzelm |
Tue, 22 Mar 2011 16:39:34 +0100 | |
changeset 42054 | 8cd4783904d8 |
parent 41959 | b460124855b8 |
child 42152 | 6c17259724b2 |
permissions | -rw-r--r-- |
41959 | 1 |
(* Title: HOL/Hoare/Hoare_Logic_Abort.thy |
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Author: Leonor Prensa Nieto & Tobias Nipkow |
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Copyright 2003 TUM |
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Like Hoare.thy, but with an Abort statement for modelling run time errors. |
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*) |
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theory Hoare_Logic_Abort |
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imports Main |
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major cleanup of hoare_tac.ML: just one copy for Hoare.thy and HoareAbort.thy (only 1 line different), refrain from inspecting the main goal, proper context;
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uses ("hoare_tac.ML") |
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begin |
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types |
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'a bexp = "'a set" |
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'a assn = "'a set" |
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datatype |
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'a com = Basic "'a \<Rightarrow> 'a" |
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| Abort |
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| Seq "'a com" "'a com" ("(_;/ _)" [61,60] 60) |
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| Cond "'a bexp" "'a com" "'a com" ("(1IF _/ THEN _ / ELSE _/ FI)" [0,0,0] 61) |
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| While "'a bexp" "'a assn" "'a com" ("(1WHILE _/ INV {_} //DO _ /OD)" [0,0,0] 61) |
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abbreviation annskip ("SKIP") where "SKIP == Basic id" |
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types 'a sem = "'a option => 'a option => bool" |
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inductive Sem :: "'a com \<Rightarrow> 'a sem" |
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where |
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"Sem (Basic f) None None" |
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| "Sem (Basic f) (Some s) (Some (f s))" |
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| "Sem Abort s None" |
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| "Sem c1 s s'' \<Longrightarrow> Sem c2 s'' s' \<Longrightarrow> Sem (c1;c2) s s'" |
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| "Sem (IF b THEN c1 ELSE c2 FI) None None" |
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| "s \<in> b \<Longrightarrow> Sem c1 (Some s) s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) (Some s) s'" |
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| "s \<notin> b \<Longrightarrow> Sem c2 (Some s) s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) (Some s) s'" |
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| "Sem (While b x c) None None" |
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| "s \<notin> b \<Longrightarrow> Sem (While b x c) (Some s) (Some s)" |
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| "s \<in> b \<Longrightarrow> Sem c (Some s) s'' \<Longrightarrow> Sem (While b x c) s'' s' \<Longrightarrow> |
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Sem (While b x c) (Some s) s'" |
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inductive_cases [elim!]: |
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"Sem (Basic f) s s'" "Sem (c1;c2) s s'" |
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"Sem (IF b THEN c1 ELSE c2 FI) s s'" |
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definition Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool" where |
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"Valid p c q == \<forall>s s'. Sem c s s' \<longrightarrow> s : Some ` p \<longrightarrow> s' : Some ` q" |
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(** parse translations **) |
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syntax |
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"_assign" :: "id => 'b => 'a com" ("(2_ :=/ _)" [70,65] 61) |
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syntax |
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"_hoare_abort_vars" :: "[idts, 'a assn,'a com,'a assn] => bool" |
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("VARS _// {_} // _ // {_}" [0,0,55,0] 50) |
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syntax ("" output) |
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"_hoare_abort" :: "['a assn,'a com,'a assn] => bool" |
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("{_} // _ // {_}" [0,55,0] 50) |
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ML {* |
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local |
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fun free a = Free(a,dummyT) |
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fun abs((a,T),body) = |
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let val a = absfree(a, dummyT, body) |
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in if T = Bound 0 then a else Const(Syntax.constrainAbsC,dummyT) $ a $ T end |
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in |
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fun mk_abstuple [x] body = abs (x, body) |
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| mk_abstuple (x::xs) body = |
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Syntax.const @{const_syntax prod_case} $ abs (x, mk_abstuple xs body); |
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fun mk_fbody a e [x as (b,_)] = if a=b then e else free b |
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| mk_fbody a e ((b,_)::xs) = |
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Syntax.const @{const_syntax Pair} $ (if a=b then e else free b) $ mk_fbody a e xs; |
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fun mk_fexp a e xs = mk_abstuple xs (mk_fbody a e xs) |
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end |
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*} |
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(* bexp_tr & assn_tr *) |
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(*all meta-variables for bexp except for TRUE are translated as if they |
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were boolean expressions*) |
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ML{* |
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fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE" (* FIXME !? *) |
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| bexp_tr b xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs b; |
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fun assn_tr r xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs r; |
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*} |
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(* com_tr *) |
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ML{* |
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fun com_tr (t as (Const (@{syntax_const "_assign"},_) $ x $ e)) xs = |
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(case Syntax.strip_positions x of |
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Free (a, _) => Syntax.const @{const_syntax Basic} $ mk_fexp a e xs |
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| _ => t) |
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| com_tr (Const (@{const_syntax Basic},_) $ f) xs = Syntax.const @{const_syntax Basic} $ f |
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| com_tr (Const (@{const_syntax Seq},_) $ c1 $ c2) xs = |
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Syntax.const @{const_syntax Seq} $ com_tr c1 xs $ com_tr c2 xs |
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| com_tr (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) xs = |
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Syntax.const @{const_syntax Cond} $ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs |
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| com_tr (Const (@{const_syntax While},_) $ b $ I $ c) xs = |
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Syntax.const @{const_syntax While} $ bexp_tr b xs $ assn_tr I xs $ com_tr c xs |
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| com_tr t _ = t (* if t is just a Free/Var *) |
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*} |
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parents:
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(* triple_tr *) (* FIXME does not handle "_idtdummy" *) |
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ML{* |
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local |
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fun var_tr (Free (a, _)) = (a, Bound 0) (* Bound 0 = dummy term *) |
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| var_tr (Const (@{syntax_const "_constrain"}, _) $ Free (a, _) $ T) = (a, T); |
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parents:
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fun vars_tr (Const (@{syntax_const "_idts"}, _) $ idt $ vars) = |
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var_tr (Syntax.strip_positions idt) :: vars_tr vars |
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| vars_tr t = [var_tr (Syntax.strip_positions t)] |
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in |
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fun hoare_vars_tr [vars, pre, prg, post] = |
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let val xs = vars_tr vars |
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in Syntax.const @{const_syntax Valid} $ |
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assn_tr pre xs $ com_tr prg xs $ assn_tr post xs |
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end |
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| hoare_vars_tr ts = raise TERM ("hoare_vars_tr", ts); |
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end |
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*} |
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parse_translation {* [(@{syntax_const "_hoare_abort_vars"}, hoare_vars_tr)] *} |
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(*****************************************************************************) |
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(*** print translations ***) |
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ML{* |
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fun dest_abstuple (Const (@{const_syntax prod_case},_) $ (Abs(v,_, body))) = |
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subst_bound (Syntax.free v, dest_abstuple body) |
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| dest_abstuple (Abs(v,_, body)) = subst_bound (Syntax.free v, body) |
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| dest_abstuple trm = trm; |
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fun abs2list (Const (@{const_syntax prod_case},_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t |
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| abs2list (Abs(x,T,t)) = [Free (x, T)] |
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| abs2list _ = []; |
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fun mk_ts (Const (@{const_syntax prod_case},_) $ (Abs(x,_,t))) = mk_ts t |
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| mk_ts (Abs(x,_,t)) = mk_ts t |
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| mk_ts (Const (@{const_syntax Pair},_) $ a $ b) = a::(mk_ts b) |
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| mk_ts t = [t]; |
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fun mk_vts (Const (@{const_syntax prod_case},_) $ (Abs(x,_,t))) = |
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((Syntax.free x)::(abs2list t), mk_ts t) |
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| mk_vts (Abs(x,_,t)) = ([Syntax.free x], [t]) |
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| mk_vts t = raise Match; |
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fun find_ch [] i xs = (false, (Syntax.free "not_ch", Syntax.free "not_ch")) |
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| find_ch ((v,t)::vts) i xs = |
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if t = Bound i then find_ch vts (i-1) xs |
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else (true, (v, subst_bounds (xs,t))); |
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fun is_f (Const (@{const_syntax prod_case},_) $ (Abs(x,_,t))) = true |
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| is_f (Abs(x,_,t)) = true |
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| is_f t = false; |
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*} |
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(* assn_tr' & bexp_tr'*) |
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ML{* |
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fun assn_tr' (Const (@{const_syntax Collect},_) $ T) = dest_abstuple T |
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| assn_tr' (Const (@{const_syntax inter},_) $ (Const (@{const_syntax Collect},_) $ T1) $ |
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(Const (@{const_syntax Collect},_) $ T2)) = |
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Syntax.const @{const_syntax inter} $ dest_abstuple T1 $ dest_abstuple T2 |
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| assn_tr' t = t; |
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fun bexp_tr' (Const (@{const_syntax Collect},_) $ T) = dest_abstuple T |
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| bexp_tr' t = t; |
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*} |
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(*com_tr' *) |
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ML{* |
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fun mk_assign f = |
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let val (vs, ts) = mk_vts f; |
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val (ch, which) = find_ch (vs~~ts) ((length vs)-1) (rev vs) |
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in |
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if ch then Syntax.const @{syntax_const "_assign"} $ fst which $ snd which |
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else Syntax.const @{const_syntax annskip} |
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end; |
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fun com_tr' (Const (@{const_syntax Basic},_) $ f) = |
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if is_f f then mk_assign f else Syntax.const @{const_syntax Basic} $ f |
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| com_tr' (Const (@{const_syntax Seq},_) $ c1 $ c2) = |
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Syntax.const @{const_syntax Seq} $ com_tr' c1 $ com_tr' c2 |
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| com_tr' (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) = |
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Syntax.const @{const_syntax Cond} $ bexp_tr' b $ com_tr' c1 $ com_tr' c2 |
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| com_tr' (Const (@{const_syntax While},_) $ b $ I $ c) = |
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Syntax.const @{const_syntax While} $ bexp_tr' b $ assn_tr' I $ com_tr' c |
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| com_tr' t = t; |
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fun spec_tr' [p, c, q] = |
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Syntax.const @{syntax_const "_hoare_abort"} $ assn_tr' p $ com_tr' c $ assn_tr' q |
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*} |
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print_translation {* [(@{const_syntax Valid}, spec_tr')] *} |
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(*** The proof rules ***) |
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lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q" |
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by (auto simp:Valid_def) |
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lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q" |
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by (auto simp:Valid_def) |
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lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (c1;c2) R" |
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by (auto simp:Valid_def) |
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lemma CondRule: |
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"p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')} |
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\<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q" |
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by (fastsimp simp:Valid_def image_def) |
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lemma While_aux: |
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assumes "Sem (WHILE b INV {i} DO c OD) s s'" |
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shows "\<forall>s s'. Sem c s s' \<longrightarrow> s \<in> Some ` (I \<inter> b) \<longrightarrow> s' \<in> Some ` I \<Longrightarrow> |
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s \<in> Some ` I \<Longrightarrow> s' \<in> Some ` (I \<inter> -b)" |
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using assms |
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by (induct "WHILE b INV {i} DO c OD" s s') auto |
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lemma WhileRule: |
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"p \<subseteq> i \<Longrightarrow> Valid (i \<inter> b) c i \<Longrightarrow> i \<inter> (-b) \<subseteq> q \<Longrightarrow> Valid p (While b i c) q" |
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apply(simp add:Valid_def) |
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apply(simp (no_asm) add:image_def) |
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apply clarify |
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apply(drule While_aux) |
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apply assumption |
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apply blast |
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apply blast |
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done |
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lemma AbortRule: "p \<subseteq> {s. False} \<Longrightarrow> Valid p Abort q" |
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by(auto simp:Valid_def) |
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||
24470
41c81e23c08d
removed Hoare/hoare.ML, Hoare/hoareAbort.ML, ex/svc_oracle.ML (which can be mistaken as attached ML script on case-insensitive file-system);
wenzelm
parents:
21588
diff
changeset
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41c81e23c08d
removed Hoare/hoare.ML, Hoare/hoareAbort.ML, ex/svc_oracle.ML (which can be mistaken as attached ML script on case-insensitive file-system);
wenzelm
parents:
21588
diff
changeset
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subsection {* Derivation of the proof rules and, most importantly, the VCG tactic *} |
41c81e23c08d
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changeset
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41c81e23c08d
removed Hoare/hoare.ML, Hoare/hoareAbort.ML, ex/svc_oracle.ML (which can be mistaken as attached ML script on case-insensitive file-system);
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parents:
21588
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changeset
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lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}" |
41c81e23c08d
removed Hoare/hoare.ML, Hoare/hoareAbort.ML, ex/svc_oracle.ML (which can be mistaken as attached ML script on case-insensitive file-system);
wenzelm
parents:
21588
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changeset
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by blast |
41c81e23c08d
removed Hoare/hoare.ML, Hoare/hoareAbort.ML, ex/svc_oracle.ML (which can be mistaken as attached ML script on case-insensitive file-system);
wenzelm
parents:
21588
diff
changeset
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245 |
|
28457
25669513fd4c
major cleanup of hoare_tac.ML: just one copy for Hoare.thy and HoareAbort.thy (only 1 line different), refrain from inspecting the main goal, proper context;
wenzelm
parents:
27330
diff
changeset
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use "hoare_tac.ML" |
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method_setup vcg = {* |
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Scan.succeed (fn ctxt => SIMPLE_METHOD' (hoare_tac ctxt (K all_tac))) *} |
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"verification condition generator" |
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method_setup vcg_simp = {* |
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Scan.succeed (fn ctxt => |
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SIMPLE_METHOD' (hoare_tac ctxt (asm_full_simp_tac (simpset_of ctxt)))) *} |
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"verification condition generator plus simplification" |
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(* Special syntax for guarded statements and guarded array updates: *) |
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syntax |
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"_guarded_com" :: "bool \<Rightarrow> 'a com \<Rightarrow> 'a com" ("(2_ \<rightarrow>/ _)" 71) |
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"_array_update" :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a com" ("(2_[_] :=/ _)" [70, 65] 61) |
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translations |
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"P \<rightarrow> c" == "IF P THEN c ELSE CONST Abort FI" |
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"a[i] := v" => "(i < CONST length a) \<rightarrow> (a := CONST list_update a i v)" |
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(* reverse translation not possible because of duplicate "a" *) |
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text{* Note: there is no special syntax for guarded array access. Thus |
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you must write @{text"j < length a \<rightarrow> a[i] := a!j"}. *} |
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end |