
diff -r c2860c37e574 -r aa2581752afb src/HOL/IMPP/Hoare.thy
--- a/src/HOL/IMPP/Hoare.thy	Wed Jun 07 01:06:53 2006 +0200
+++ b/src/HOL/IMPP/Hoare.thy	Wed Jun 07 01:51:22 2006 +0200
@@ -102,6 +102,429 @@
          ==> G|-{%Z s. s'=s & P Z (setlocs s newlocs[Loc Arg::=a s])}.
              X:=CALL pn(a) .{Q}"
 
-ML {* use_legacy_bindings (the_context ()) *}
+
+section {* Soundness and relative completeness of Hoare rules wrt operational semantics *}
+
+lemma single_stateE: 
+  "state_not_singleton ==> !t. (!s::state. s = t) --> False"
+apply (unfold state_not_singleton_def)
+apply clarify
+apply (case_tac "ta = t")
+apply blast
+apply (blast dest: not_sym)
+done
+
+declare peek_and_def [simp]
+
+
+subsection "validity"
+
+lemma triple_valid_def2: 
+  "|=n:{P}.c.{Q} = (!Z s. P Z s --> (!s'. <c,s> -n-> s' --> Q Z s'))"
+apply (unfold triple_valid_def)
+apply auto
+done
+
+lemma Body_triple_valid_0: "|=0:{P}. BODY pn .{Q}"
+apply (simp (no_asm) add: triple_valid_def2)
+apply clarsimp
+done
+
+(* only ==> direction required *)
+lemma Body_triple_valid_Suc: "|=n:{P}. the (body pn) .{Q} = |=Suc n:{P}. BODY pn .{Q}"
+apply (simp (no_asm) add: triple_valid_def2)
+apply force
+done
+
+lemma triple_valid_Suc [rule_format (no_asm)]: "|=Suc n:t --> |=n:t"
+apply (unfold triple_valid_def)
+apply (induct_tac t)
+apply simp
+apply (fast intro: evaln_Suc)
+done
+
+lemma triples_valid_Suc: "||=Suc n:ts ==> ||=n:ts"
+apply (fast intro: triple_valid_Suc)
+done
+
+
+subsection "derived rules"
+
+lemma conseq12: "[| G|-{P'}.c.{Q'}; !Z s. P Z s -->  
+                         (!s'. (!Z'. P' Z' s --> Q' Z' s') --> Q Z s') |]  
+       ==> G|-{P}.c.{Q}"
+apply (rule hoare_derivs.conseq)
+apply blast
+done
+
+lemma conseq1: "[| G|-{P'}.c.{Q}; !Z s. P Z s --> P' Z s |] ==> G|-{P}.c.{Q}"
+apply (erule conseq12)
+apply fast
+done
+
+lemma conseq2: "[| G|-{P}.c.{Q'}; !Z s. Q' Z s --> Q Z s |] ==> G|-{P}.c.{Q}"
+apply (erule conseq12)
+apply fast
+done
+
+lemma Body1: "[| G Un (%p. {P p}.      BODY p  .{Q p})`Procs   
+          ||- (%p. {P p}. the (body p) .{Q p})`Procs;  
+    pn:Procs |] ==> G|-{P pn}. BODY pn .{Q pn}"
+apply (drule hoare_derivs.Body)
+apply (erule hoare_derivs.weaken)
+apply fast
+done
+
+lemma BodyN: "(insert ({P}. BODY pn .{Q}) G) |-{P}. the (body pn) .{Q} ==>  
+  G|-{P}. BODY pn .{Q}"
+apply (rule Body1)
+apply (rule_tac [2] singletonI)
+apply clarsimp
+done
+
+lemma escape: "[| !Z s. P Z s --> G|-{%Z s'. s'=s}.c.{%Z'. Q Z} |] ==> G|-{P}.c.{Q}"
+apply (rule hoare_derivs.conseq)
+apply fast
+done
+
+lemma constant: "[| C ==> G|-{P}.c.{Q} |] ==> G|-{%Z s. P Z s & C}.c.{Q}"
+apply (rule hoare_derivs.conseq)
+apply fast
+done
+
+lemma LoopF: "G|-{%Z s. P Z s & ~b s}.WHILE b DO c.{P}"
+apply (rule hoare_derivs.Loop [THEN conseq2])
+apply  (simp_all (no_asm))
+apply (rule hoare_derivs.conseq)
+apply fast
+done
+
+(*
+Goal "[| G'||-ts; G' <= G |] ==> G||-ts"
+by (etac hoare_derivs.cut 1);
+by (etac hoare_derivs.asm 1);
+qed "thin";
+*)
+lemma thin [rule_format]: "G'||-ts ==> !G. G' <= G --> G||-ts"
+apply (erule hoare_derivs.induct)
+apply                (tactic {* ALLGOALS (EVERY'[Clarify_tac, REPEAT o smp_tac 1]) *})
+apply (rule hoare_derivs.empty)
+apply               (erule (1) hoare_derivs.insert)
+apply              (fast intro: hoare_derivs.asm)
+apply             (fast intro: hoare_derivs.cut)
+apply            (fast intro: hoare_derivs.weaken)
+apply           (rule hoare_derivs.conseq, intro strip, tactic "smp_tac 2 1", clarify, tactic "smp_tac 1 1",rule exI, rule exI, erule (1) conjI)
+prefer 7
+apply          (rule_tac hoare_derivs.Body, drule_tac spec, erule_tac mp, fast)
+apply         (tactic {* ALLGOALS (resolve_tac ((funpow 5 tl) (thms "hoare_derivs.intros")) THEN_ALL_NEW CLASET' fast_tac) *})
+done
+
+lemma weak_Body: "G|-{P}. the (body pn) .{Q} ==> G|-{P}. BODY pn .{Q}"
+apply (rule BodyN)
+apply (erule thin)
+apply auto
+done
+
+lemma derivs_insertD: "G||-insert t ts ==> G|-t & G||-ts"
+apply (fast intro: hoare_derivs.weaken)
+done
+
+lemma finite_pointwise [rule_format (no_asm)]: "[| finite U;  
+  !p. G |-     {P' p}.c0 p.{Q' p}       --> G |-     {P p}.c0 p.{Q p} |] ==>  
+      G||-(%p. {P' p}.c0 p.{Q' p}) ` U --> G||-(%p. {P p}.c0 p.{Q p}) ` U"
+apply (erule finite_induct)
+apply simp
+apply clarsimp
+apply (drule derivs_insertD)
+apply (rule hoare_derivs.insert)
+apply  auto
+done
+
+
+subsection "soundness"
+
+lemma Loop_sound_lemma: 
+ "G|={P &> b}. c .{P} ==>  
+  G|={P}. WHILE b DO c .{P &> (Not o b)}"
+apply (unfold hoare_valids_def)
+apply (simp (no_asm_use) add: triple_valid_def2)
+apply (rule allI)
+apply (subgoal_tac "!d s s'. <d,s> -n-> s' --> d = WHILE b DO c --> ||=n:G --> (!Z. P Z s --> P Z s' & ~b s') ")
+apply  (erule thin_rl, fast)
+apply ((rule allI)+, rule impI)
+apply (erule evaln.induct)
+apply (simp_all (no_asm))
+apply fast
+apply fast
+done
+
+lemma Body_sound_lemma: 
+   "[| G Un (%pn. {P pn}.      BODY pn  .{Q pn})`Procs  
+         ||=(%pn. {P pn}. the (body pn) .{Q pn})`Procs |] ==>  
+        G||=(%pn. {P pn}.      BODY pn  .{Q pn})`Procs"
+apply (unfold hoare_valids_def)
+apply (rule allI)
+apply (induct_tac n)
+apply  (fast intro: Body_triple_valid_0)
+apply clarsimp
+apply (drule triples_valid_Suc)
+apply (erule (1) notE impE)
+apply (simp add: ball_Un)
+apply (drule spec, erule impE, erule conjI, assumption)
+apply (fast intro!: Body_triple_valid_Suc [THEN iffD1])
+done
+
+lemma hoare_sound: "G||-ts ==> G||=ts"
+apply (erule hoare_derivs.induct)
+apply              (tactic {* TRYALL (eresolve_tac [thm "Loop_sound_lemma", thm "Body_sound_lemma"] THEN_ALL_NEW atac) *})
+apply            (unfold hoare_valids_def)
+apply            blast
+apply           blast
+apply          (blast) (* asm *)
+apply         (blast) (* cut *)
+apply        (blast) (* weaken *)
+apply       (tactic {* ALLGOALS (EVERY'[REPEAT o thin_tac "?x : hoare_derivs", SIMPSET' simp_tac, CLASET' clarify_tac, REPEAT o smp_tac 1]) *})
+apply       (simp_all (no_asm_use) add: triple_valid_def2)
+apply       (intro strip, tactic "smp_tac 2 1", blast) (* conseq *)
+apply      (tactic {* ALLGOALS (CLASIMPSET' clarsimp_tac) *}) (* Skip, Ass, Local *)
+prefer 3 apply   (force) (* Call *)
+apply  (erule_tac [2] evaln_elim_cases) (* If *)
+apply   blast+
+done
+
+
+section "completeness"
+
+(* Both versions *)
+
+(*unused*)
+lemma MGT_alternI: "G|-MGT c ==>  
+  G|-{%Z s0. !s1. <c,s0> -c-> s1 --> Z=s1}. c .{%Z s1. Z=s1}"
+apply (unfold MGT_def)
+apply (erule conseq12)
+apply auto
+done
+
+(* requires com_det *)
+lemma MGT_alternD: "state_not_singleton ==>  
+  G|-{%Z s0. !s1. <c,s0> -c-> s1 --> Z=s1}. c .{%Z s1. Z=s1} ==> G|-MGT c"
+apply (unfold MGT_def)
+apply (erule conseq12)
+apply auto
+apply (case_tac "? t. <c,?s> -c-> t")
+apply  (fast elim: com_det)
+apply clarsimp
+apply (drule single_stateE)
+apply blast
+done
+
+lemma MGF_complete: 
+ "{}|-(MGT c::state triple) ==> {}|={P}.c.{Q} ==> {}|-{P}.c.{Q::state assn}"
+apply (unfold MGT_def)
+apply (erule conseq12)
+apply (clarsimp simp add: hoare_valids_def eval_eq triple_valid_def2)
+done
+
+declare WTs_elim_cases [elim!]
+declare not_None_eq [iff]
+(* requires com_det, escape (i.e. hoare_derivs.conseq) *)
+lemma MGF_lemma1 [rule_format (no_asm)]: "state_not_singleton ==>  
+  !pn:dom body. G|-{=}.BODY pn.{->} ==> WT c --> G|-{=}.c.{->}"
+apply (induct_tac c)
+apply        (tactic {* ALLGOALS (CLASIMPSET' clarsimp_tac) *})
+prefer 7 apply        (fast intro: domI)
+apply (erule_tac [6] MGT_alternD)
+apply       (unfold MGT_def)
+apply       (drule_tac [7] bspec, erule_tac [7] domI)
+apply       (rule_tac [7] escape, tactic {* CLASIMPSET' clarsimp_tac 7 *},
+  rule_tac [7] P1 = "%Z' s. s= (setlocs Z newlocs) [Loc Arg ::= fun Z]" in hoare_derivs.Call [THEN conseq1], erule_tac [7] conseq12)
+apply        (erule_tac [!] thin_rl)
+apply (rule hoare_derivs.Skip [THEN conseq2])
+apply (rule_tac [2] hoare_derivs.Ass [THEN conseq1])
+apply        (rule_tac [3] escape, tactic {* CLASIMPSET' clarsimp_tac 3 *},
+  rule_tac [3] P1 = "%Z' s. s= (Z[Loc loc::=fun Z])" in hoare_derivs.Local [THEN conseq1],
+  erule_tac [3] conseq12)
+apply         (erule_tac [5] hoare_derivs.Comp, erule_tac [5] conseq12)
+apply         (tactic {* (rtac (thm "hoare_derivs.If") THEN_ALL_NEW etac (thm "conseq12")) 6 *})
+apply          (rule_tac [8] hoare_derivs.Loop [THEN conseq2], erule_tac [8] conseq12)
+apply           auto
+done
+
+(* Version: nested single recursion *)
+
+lemma nesting_lemma [rule_format]:
+  assumes "!!G ts. ts <= G ==> P G ts"
+    and "!!G pn. P (insert (mgt_call pn) G) {mgt(the(body pn))} ==> P G {mgt_call pn}"
+    and "!!G c. [| wt c; !pn:U. P G {mgt_call pn} |] ==> P G {mgt c}"
+    and "!!pn. pn : U ==> wt (the (body pn))"
+  shows "finite U ==> uG = mgt_call`U ==>  
+  !G. G <= uG --> n <= card uG --> card G = card uG - n --> (!c. wt c --> P G {mgt c})"
+apply (induct_tac n)
+apply  (tactic {* ALLGOALS (CLASIMPSET' clarsimp_tac) *})
+apply  (subgoal_tac "G = mgt_call ` U")
+prefer 2
+apply   (simp add: card_seteq finite_imageI)
+apply  simp
+apply  (erule prems(3-)) (*MGF_lemma1*)
+apply (rule ballI)
+apply  (rule prems) (*hoare_derivs.asm*)
+apply  fast
+apply (erule prems(3-)) (*MGF_lemma1*)
+apply (rule ballI)
+apply (case_tac "mgt_call pn : G")
+apply  (rule prems) (*hoare_derivs.asm*)
+apply  fast
+apply (rule prems(2-)) (*MGT_BodyN*)
+apply (drule spec, erule impE, erule_tac [2] impE, drule_tac [3] spec, erule_tac [3] mp)
+apply   (erule_tac [3] prems(4-))
+apply   fast
+apply (drule finite_subset)
+apply (erule finite_imageI)
+apply (simp (no_asm_simp))
+apply arith
+done
+
+lemma MGT_BodyN: "insert ({=}.BODY pn.{->}) G|-{=}. the (body pn) .{->} ==>  
+  G|-{=}.BODY pn.{->}"
+apply (unfold MGT_def)
+apply (rule BodyN)
+apply (erule conseq2)
+apply force
+done
+
+(* requires BodyN, com_det *)
+lemma MGF: "[| state_not_singleton; WT_bodies; WT c |] ==> {}|-MGT c"
+apply (rule_tac P = "%G ts. G||-ts" and U = "dom body" in nesting_lemma)
+apply (erule hoare_derivs.asm)
+apply (erule MGT_BodyN)
+apply (rule_tac [3] finite_dom_body)
+apply (erule MGF_lemma1)
+prefer 2 apply (assumption)
+apply       blast
+apply      clarsimp
+apply     (erule (1) WT_bodiesD)
+apply (rule_tac [3] le_refl)
+apply    auto
+done
+
+
+(* Version: simultaneous recursion in call rule *)
+
+(* finiteness not really necessary here *)
+lemma MGT_Body: "[| G Un (%pn. {=}.      BODY pn  .{->})`Procs  
+                          ||-(%pn. {=}. the (body pn) .{->})`Procs;  
+  finite Procs |] ==>   G ||-(%pn. {=}.      BODY pn  .{->})`Procs"
+apply (unfold MGT_def)
+apply (rule hoare_derivs.Body)
+apply (erule finite_pointwise)
+prefer 2 apply (assumption)
+apply clarify
+apply (erule conseq2)
+apply auto
+done
+
+(* requires empty, insert, com_det *)
+lemma MGF_lemma2_simult [rule_format (no_asm)]: "[| state_not_singleton; WT_bodies;  
+  F<=(%pn. {=}.the (body pn).{->})`dom body |] ==>  
+     (%pn. {=}.     BODY pn .{->})`dom body||-F"
+apply (frule finite_subset)
+apply (rule finite_dom_body [THEN finite_imageI])
+apply (rotate_tac 2)
+apply (tactic "make_imp_tac 1")
+apply (erule finite_induct)
+apply  (clarsimp intro!: hoare_derivs.empty)
+apply (clarsimp intro!: hoare_derivs.insert simp del: range_composition)
+apply (erule MGF_lemma1)
+prefer 2 apply  (fast dest: WT_bodiesD)
+apply clarsimp
+apply (rule hoare_derivs.asm)
+apply (fast intro: domI)
+done
+
+(* requires Body, empty, insert, com_det *)
+lemma MGF': "[| state_not_singleton; WT_bodies; WT c |] ==> {}|-MGT c"
+apply (rule MGF_lemma1)
+apply assumption
+prefer 2 apply (assumption)
+apply clarsimp
+apply (subgoal_tac "{}||- (%pn. {=}. BODY pn .{->}) `dom body")
+apply (erule hoare_derivs.weaken)
+apply  (fast intro: domI)
+apply (rule finite_dom_body [THEN [2] MGT_Body])
+apply (simp (no_asm))
+apply (erule (1) MGF_lemma2_simult)
+apply (rule subset_refl)
+done
+
+(* requires Body+empty+insert / BodyN, com_det *)
+lemmas hoare_complete = MGF' [THEN MGF_complete, standard]
+
+
+subsection "unused derived rules"
+
+lemma falseE: "G|-{%Z s. False}.c.{Q}"
+apply (rule hoare_derivs.conseq)
+apply fast
+done
+
+lemma trueI: "G|-{P}.c.{%Z s. True}"
+apply (rule hoare_derivs.conseq)
+apply (fast intro!: falseE)
+done
+
+lemma disj: "[| G|-{P}.c.{Q}; G|-{P'}.c.{Q'} |]  
+        ==> G|-{%Z s. P Z s | P' Z s}.c.{%Z s. Q Z s | Q' Z s}"
+apply (rule hoare_derivs.conseq)
+apply (fast elim: conseq12)
+done (* analogue conj non-derivable *)
+
+lemma hoare_SkipI: "(!Z s. P Z s --> Q Z s) ==> G|-{P}. SKIP .{Q}"
+apply (rule conseq12)
+apply (rule hoare_derivs.Skip)
+apply fast
+done
+
+
+subsection "useful derived rules"
+
+lemma single_asm: "{t}|-t"
+apply (rule hoare_derivs.asm)
+apply (rule subset_refl)
+done
+
+lemma export_s: "[| !!s'. G|-{%Z s. s'=s & P Z s}.c.{Q} |] ==> G|-{P}.c.{Q}"
+apply (rule hoare_derivs.conseq)
+apply auto
+done
+
+
+lemma weak_Local: "[| G|-{P}. c .{Q}; !k Z s. Q Z s --> Q Z (s[Loc Y::=k]) |] ==>  
+    G|-{%Z s. P Z (s[Loc Y::=a s])}. LOCAL Y:=a IN c .{Q}"
+apply (rule export_s)
+apply (rule hoare_derivs.Local)
+apply (erule conseq2)
+apply (erule spec)
+done
+
+(*
+Goal "!Q. G |-{%Z s. ~(? s'. <c,s> -c-> s')}. c .{Q}"
+by (induct_tac "c" 1);
+by Auto_tac;
+by (rtac conseq1 1);
+by (rtac hoare_derivs.Skip 1);
+force 1;
+by (rtac conseq1 1);
+by (rtac hoare_derivs.Ass 1);
+force 1;
+by (defer_tac 1);
+###
+by (rtac hoare_derivs.Comp 1);
+by (dtac spec 2);
+by (dtac spec 2);
+by (assume_tac 2);
+by (etac conseq1 2);
+by (Clarsimp_tac 2);
+force 1;
+*)
 
 end




