src/HOL/Hoare/Hoare_Logic.thy
 author huffman Fri Mar 30 12:32:35 2012 +0200 (2012-03-30) changeset 47220 52426c62b5d0 parent 42174 d0be2722ce9f child 48891 c0eafbd55de3 permissions -rw-r--r--
replace lemmas eval_nat_numeral with a simpler reformulation
```     1 (*  Title:      HOL/Hoare/Hoare_Logic.thy
```
```     2     Author:     Leonor Prensa Nieto & Tobias Nipkow
```
```     3     Copyright   1998 TUM
```
```     4
```
```     5 Sugared semantic embedding of Hoare logic.
```
```     6 Strictly speaking a shallow embedding (as implemented by Norbert Galm
```
```     7 following Mike Gordon) would suffice. Maybe the datatype com comes in useful
```
```     8 later.
```
```     9 *)
```
```    10
```
```    11 theory Hoare_Logic
```
```    12 imports Main
```
```    13 uses ("hoare_syntax.ML") ("hoare_tac.ML")
```
```    14 begin
```
```    15
```
```    16 type_synonym 'a bexp = "'a set"
```
```    17 type_synonym 'a assn = "'a set"
```
```    18
```
```    19 datatype
```
```    20  'a com = Basic "'a \<Rightarrow> 'a"
```
```    21    | Seq "'a com" "'a com"               ("(_;/ _)"      [61,60] 60)
```
```    22    | Cond "'a bexp" "'a com" "'a com"    ("(1IF _/ THEN _ / ELSE _/ FI)"  [0,0,0] 61)
```
```    23    | While "'a bexp" "'a assn" "'a com"  ("(1WHILE _/ INV {_} //DO _ /OD)"  [0,0,0] 61)
```
```    24
```
```    25 abbreviation annskip ("SKIP") where "SKIP == Basic id"
```
```    26
```
```    27 type_synonym 'a sem = "'a => 'a => bool"
```
```    28
```
```    29 inductive Sem :: "'a com \<Rightarrow> 'a sem"
```
```    30 where
```
```    31   "Sem (Basic f) s (f s)"
```
```    32 | "Sem c1 s s'' \<Longrightarrow> Sem c2 s'' s' \<Longrightarrow> Sem (c1;c2) s s'"
```
```    33 | "s \<in> b \<Longrightarrow> Sem c1 s s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) s s'"
```
```    34 | "s \<notin> b \<Longrightarrow> Sem c2 s s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) s s'"
```
```    35 | "s \<notin> b \<Longrightarrow> Sem (While b x c) s s"
```
```    36 | "s \<in> b \<Longrightarrow> Sem c s s'' \<Longrightarrow> Sem (While b x c) s'' s' \<Longrightarrow>
```
```    37    Sem (While b x c) s s'"
```
```    38
```
```    39 inductive_cases [elim!]:
```
```    40   "Sem (Basic f) s s'" "Sem (c1;c2) s s'"
```
```    41   "Sem (IF b THEN c1 ELSE c2 FI) s s'"
```
```    42
```
```    43 definition Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool"
```
```    44   where "Valid p c q \<longleftrightarrow> (!s s'. Sem c s s' --> s : p --> s' : q)"
```
```    45
```
```    46
```
```    47 syntax
```
```    48   "_assign" :: "idt => 'b => 'a com"  ("(2_ :=/ _)" [70, 65] 61)
```
```    49
```
```    50 syntax
```
```    51  "_hoare_vars" :: "[idts, 'a assn,'a com,'a assn] => bool"
```
```    52                  ("VARS _// {_} // _ // {_}" [0,0,55,0] 50)
```
```    53 syntax ("" output)
```
```    54  "_hoare"      :: "['a assn,'a com,'a assn] => bool"
```
```    55                  ("{_} // _ // {_}" [0,55,0] 50)
```
```    56
```
```    57 use "hoare_syntax.ML"
```
```    58 parse_translation {* [(@{syntax_const "_hoare_vars"}, Hoare_Syntax.hoare_vars_tr)] *}
```
```    59 print_translation {* [(@{const_syntax Valid}, Hoare_Syntax.spec_tr' @{syntax_const "_hoare"})] *}
```
```    60
```
```    61
```
```    62 lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"
```
```    63 by (auto simp:Valid_def)
```
```    64
```
```    65 lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q"
```
```    66 by (auto simp:Valid_def)
```
```    67
```
```    68 lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (c1;c2) R"
```
```    69 by (auto simp:Valid_def)
```
```    70
```
```    71 lemma CondRule:
```
```    72  "p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}
```
```    73   \<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q"
```
```    74 by (auto simp:Valid_def)
```
```    75
```
```    76 lemma While_aux:
```
```    77   assumes "Sem (WHILE b INV {i} DO c OD) s s'"
```
```    78   shows "\<forall>s s'. Sem c s s' \<longrightarrow> s \<in> I \<and> s \<in> b \<longrightarrow> s' \<in> I \<Longrightarrow>
```
```    79     s \<in> I \<Longrightarrow> s' \<in> I \<and> s' \<notin> b"
```
```    80   using assms
```
```    81   by (induct "WHILE b INV {i} DO c OD" s s') auto
```
```    82
```
```    83 lemma WhileRule:
```
```    84  "p \<subseteq> i \<Longrightarrow> Valid (i \<inter> b) c i \<Longrightarrow> i \<inter> (-b) \<subseteq> q \<Longrightarrow> Valid p (While b i c) q"
```
```    85 apply (clarsimp simp:Valid_def)
```
```    86 apply(drule While_aux)
```
```    87   apply assumption
```
```    88  apply blast
```
```    89 apply blast
```
```    90 done
```
```    91
```
```    92
```
```    93 lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"
```
```    94   by blast
```
```    95
```
```    96 lemmas AbortRule = SkipRule  -- "dummy version"
```
```    97 use "hoare_tac.ML"
```
```    98
```
```    99 method_setup vcg = {*
```
```   100   Scan.succeed (fn ctxt => SIMPLE_METHOD' (hoare_tac ctxt (K all_tac))) *}
```
```   101   "verification condition generator"
```
```   102
```
```   103 method_setup vcg_simp = {*
```
```   104   Scan.succeed (fn ctxt =>
```
```   105     SIMPLE_METHOD' (hoare_tac ctxt (asm_full_simp_tac (simpset_of ctxt)))) *}
```
```   106   "verification condition generator plus simplification"
```
```   107
```
```   108 end
```