
diff -r e116eb9e5e17 -r 9c80e62161a5 src/HOL/IMP/Abs_Int2.thy
--- a/src/HOL/IMP/Abs_Int2.thy	Mon May 13 22:49:00 2013 +0200
+++ b/src/HOL/IMP/Abs_Int2.thy	Tue May 14 06:54:31 2013 +0200
@@ -48,12 +48,12 @@
 locale Val_abs1 = Val_abs1_gamma where \<gamma> = \<gamma>
    for \<gamma> :: "'av::bounded_lattice \<Rightarrow> val set" +
 fixes test_num' :: "val \<Rightarrow> 'av \<Rightarrow> bool"
-and constrain_plus' :: "'av \<Rightarrow> 'av \<Rightarrow> 'av \<Rightarrow> 'av * 'av"
-and constrain_less' :: "bool \<Rightarrow> 'av \<Rightarrow> 'av \<Rightarrow> 'av * 'av"
+and inv_plus' :: "'av \<Rightarrow> 'av \<Rightarrow> 'av \<Rightarrow> 'av * 'av"
+and inv_less' :: "bool \<Rightarrow> 'av \<Rightarrow> 'av \<Rightarrow> 'av * 'av"
 assumes test_num': "test_num' i a = (i : \<gamma> a)"
-and constrain_plus': "constrain_plus' a a1 a2 = (a\<^isub>1',a\<^isub>2') \<Longrightarrow>
+and inv_plus': "inv_plus' a a1 a2 = (a\<^isub>1',a\<^isub>2') \<Longrightarrow>
   i1 : \<gamma> a1 \<Longrightarrow> i2 : \<gamma> a2 \<Longrightarrow> i1+i2 : \<gamma> a \<Longrightarrow> i1 : \<gamma> a\<^isub>1' \<and> i2 : \<gamma> a\<^isub>2'"
-and constrain_less': "constrain_less' (i1<i2) a1 a2 = (a\<^isub>1',a\<^isub>2') \<Longrightarrow>
+and inv_less': "inv_less' (i1<i2) a1 a2 = (a\<^isub>1',a\<^isub>2') \<Longrightarrow>
   i1 : \<gamma> a1 \<Longrightarrow> i2 : \<gamma> a2 \<Longrightarrow> i1 : \<gamma> a\<^isub>1' \<and> i2 : \<gamma> a\<^isub>2'"
 
 
@@ -69,19 +69,19 @@
 "aval'' e None = \<bottom>" |
 "aval'' e (Some S) = aval' e S"
 
-lemma aval''_sound: "s : \<gamma>\<^isub>o S \<Longrightarrow> aval a s : \<gamma>(aval'' a S)"
-by(cases S)(auto simp add: aval'_sound split: option.splits)
+lemma aval''_correct: "s : \<gamma>\<^isub>o S \<Longrightarrow> aval a s : \<gamma>(aval'' a S)"
+by(cases S)(auto simp add: aval'_correct split: option.splits)
 
 subsubsection "Backward analysis"
 
-fun aconstrain :: "aexp \<Rightarrow> 'av \<Rightarrow> 'av st option \<Rightarrow> 'av st option" where
-"aconstrain (N n) a S = (if test_num' n a then S else None)" |
-"aconstrain (V x) a S = (case S of None \<Rightarrow> None | Some S \<Rightarrow>
+fun inv_aval'' :: "aexp \<Rightarrow> 'av \<Rightarrow> 'av st option \<Rightarrow> 'av st option" where
+"inv_aval'' (N n) a S = (if test_num' n a then S else None)" |
+"inv_aval'' (V x) a S = (case S of None \<Rightarrow> None | Some S \<Rightarrow>
   let a' = fun S x \<sqinter> a in
   if a' = \<bottom> then None else Some(update S x a'))" |
-"aconstrain (Plus e1 e2) a S =
- (let (a1,a2) = constrain_plus' a (aval'' e1 S) (aval'' e2 S)
-  in aconstrain e1 a1 (aconstrain e2 a2 S))"
+"inv_aval'' (Plus e1 e2) a S =
+ (let (a1,a2) = inv_plus' a (aval'' e1 S) (aval'' e2 S)
+  in inv_aval'' e1 a1 (inv_aval'' e2 a2 S))"
 
 text{* The test for @{const bot} in the @{const V}-case is important: @{const
 bot} indicates that a variable has no possible values, i.e.\ that the current
@@ -93,17 +93,17 @@
 making the analysis less precise. *}
 
 
-fun bconstrain :: "bexp \<Rightarrow> bool \<Rightarrow> 'av st option \<Rightarrow> 'av st option" where
-"bconstrain (Bc v) res S = (if v=res then S else None)" |
-"bconstrain (Not b) res S = bconstrain b (\<not> res) S" |
-"bconstrain (And b1 b2) res S =
-  (if res then bconstrain b1 True (bconstrain b2 True S)
-   else bconstrain b1 False S \<squnion> bconstrain b2 False S)" |
-"bconstrain (Less e1 e2) res S =
-  (let (a1,a2) = constrain_less' res (aval'' e1 S) (aval'' e2 S)
-   in aconstrain e1 a1 (aconstrain e2 a2 S))"
+fun inv_bval'' :: "bexp \<Rightarrow> bool \<Rightarrow> 'av st option \<Rightarrow> 'av st option" where
+"inv_bval'' (Bc v) res S = (if v=res then S else None)" |
+"inv_bval'' (Not b) res S = inv_bval'' b (\<not> res) S" |
+"inv_bval'' (And b1 b2) res S =
+  (if res then inv_bval'' b1 True (inv_bval'' b2 True S)
+   else inv_bval'' b1 False S \<squnion> inv_bval'' b2 False S)" |
+"inv_bval'' (Less e1 e2) res S =
+  (let (a1,a2) = inv_less' res (aval'' e1 S) (aval'' e2 S)
+   in inv_aval'' e1 a1 (inv_aval'' e2 a2 S))"
 
-lemma aconstrain_sound: "s : \<gamma>\<^isub>o S \<Longrightarrow> aval e s : \<gamma> a \<Longrightarrow> s : \<gamma>\<^isub>o (aconstrain e a S)"
+lemma inv_aval''_correct: "s : \<gamma>\<^isub>o S \<Longrightarrow> aval e s : \<gamma> a \<Longrightarrow> s : \<gamma>\<^isub>o (inv_aval'' e a S)"
 proof(induction e arbitrary: a S)
   case N thus ?case by simp (metis test_num')
 next
@@ -118,11 +118,11 @@
       (metis mono_gamma emptyE in_gamma_inf gamma_bot subset_empty)
 next
   case (Plus e1 e2) thus ?case
-    using constrain_plus'[OF _ aval''_sound aval''_sound]
+    using inv_plus'[OF _ aval''_correct aval''_correct]
     by (auto split: prod.split)
 qed
 
-lemma bconstrain_sound: "s : \<gamma>\<^isub>o S \<Longrightarrow> bv = bval b s \<Longrightarrow> s : \<gamma>\<^isub>o(bconstrain b bv S)"
+lemma inv_bval''_correct: "s : \<gamma>\<^isub>o S \<Longrightarrow> bv = bval b s \<Longrightarrow> s : \<gamma>\<^isub>o(inv_bval'' b bv S)"
 proof(induction b arbitrary: S bv)
   case Bc thus ?case by simp
 next
@@ -133,12 +133,12 @@
 next
   case (Less e1 e2) thus ?case
     by(auto split: prod.split)
-      (metis (lifting) aconstrain_sound aval''_sound constrain_less')
+      (metis (lifting) inv_aval''_correct aval''_correct inv_less')
 qed
 
 definition "step' = Step
   (\<lambda>x e S. case S of None \<Rightarrow> None | Some S \<Rightarrow> Some(update S x (aval' e S)))
-  (\<lambda>b S. bconstrain b True S)"
+  (\<lambda>b S. inv_bval'' b True S)"
 
 definition AI :: "com \<Rightarrow> 'av st option acom option" where
 "AI c = pfp (step' \<top>) (bot c)"
@@ -146,25 +146,25 @@
 lemma strip_step'[simp]: "strip(step' S c) = strip c"
 by(simp add: step'_def)
 
-lemma top_on_aconstrain: "\<lbrakk> top_on_opt S X;  vars e \<subseteq> -X \<rbrakk> \<Longrightarrow> top_on_opt (aconstrain e a S) X"
+lemma top_on_inv_aval'': "\<lbrakk> top_on_opt S X;  vars e \<subseteq> -X \<rbrakk> \<Longrightarrow> top_on_opt (inv_aval'' e a S) X"
 by(induction e arbitrary: a S) (auto simp: Let_def split: option.splits prod.split)
 
-lemma top_on_bconstrain: "\<lbrakk>top_on_opt S X; vars b \<subseteq> -X\<rbrakk> \<Longrightarrow> top_on_opt (bconstrain b r S) X"
-by(induction b arbitrary: r S) (auto simp: top_on_aconstrain top_on_sup split: prod.split)
+lemma top_on_inv_bval'': "\<lbrakk>top_on_opt S X; vars b \<subseteq> -X\<rbrakk> \<Longrightarrow> top_on_opt (inv_bval'' b r S) X"
+by(induction b arbitrary: r S) (auto simp: top_on_inv_aval'' top_on_sup split: prod.split)
 
 lemma top_on_step': "top_on_acom C (- vars C) \<Longrightarrow> top_on_acom (step' \<top> C) (- vars C)"
 unfolding step'_def
 by(rule top_on_Step)
-  (auto simp add: top_on_top top_on_bconstrain split: option.split)
+  (auto simp add: top_on_top top_on_inv_bval'' split: option.split)
 
-subsubsection "Soundness"
+subsubsection "Correctness"
 
 lemma step_step': "step (\<gamma>\<^isub>o S) (\<gamma>\<^isub>c C) \<le> \<gamma>\<^isub>c (step' S C)"
 unfolding step_def step'_def
 by(rule gamma_Step_subcomm)
-  (auto simp: intro!: aval'_sound bconstrain_sound in_gamma_update split: option.splits)
+  (auto simp: intro!: aval'_correct inv_bval''_correct in_gamma_update split: option.splits)
 
-lemma AI_sound: "AI c = Some C \<Longrightarrow> CS c \<le> \<gamma>\<^isub>c C"
+lemma AI_correct: "AI c = Some C \<Longrightarrow> CS c \<le> \<gamma>\<^isub>c C"
 proof(simp add: CS_def AI_def)
   assume 1: "pfp (step' \<top>) (bot c) = Some C"
   have pfp': "step' \<top> C \<le> C" by(rule pfp_pfp[OF 1])
@@ -186,10 +186,10 @@
 
 locale Abs_Int1_mono = Abs_Int1 +
 assumes mono_plus': "a1 \<le> b1 \<Longrightarrow> a2 \<le> b2 \<Longrightarrow> plus' a1 a2 \<le> plus' b1 b2"
-and mono_constrain_plus': "a1 \<le> b1 \<Longrightarrow> a2 \<le> b2 \<Longrightarrow> r \<le> r' \<Longrightarrow>
-  constrain_plus' r a1 a2 \<le> constrain_plus' r' b1 b2"
-and mono_constrain_less': "a1 \<le> b1 \<Longrightarrow> a2 \<le> b2 \<Longrightarrow>
-  constrain_less' bv a1 a2 \<le> constrain_less' bv b1 b2"
+and mono_inv_plus': "a1 \<le> b1 \<Longrightarrow> a2 \<le> b2 \<Longrightarrow> r \<le> r' \<Longrightarrow>
+  inv_plus' r a1 a2 \<le> inv_plus' r' b1 b2"
+and mono_inv_less': "a1 \<le> b1 \<Longrightarrow> a2 \<le> b2 \<Longrightarrow>
+  inv_less' bv a1 a2 \<le> inv_less' bv b1 b2"
 begin
 
 lemma mono_aval':
@@ -204,28 +204,28 @@
  apply simp
 by (simp add: mono_aval')
 
-lemma mono_aconstrain: "r1 \<le> r2 \<Longrightarrow> S1 \<le> S2 \<Longrightarrow> aconstrain e r1 S1 \<le> aconstrain e r2 S2"
+lemma mono_inv_aval'': "r1 \<le> r2 \<Longrightarrow> S1 \<le> S2 \<Longrightarrow> inv_aval'' e r1 S1 \<le> inv_aval'' e r2 S2"
 apply(induction e arbitrary: r1 r2 S1 S2)
    apply(auto simp: test_num' Let_def inf_mono split: option.splits prod.splits)
    apply (metis mono_gamma subsetD)
   apply (metis le_bot inf_mono le_st_iff)
  apply (metis inf_mono mono_update le_st_iff)
-apply(metis mono_aval'' mono_constrain_plus'[simplified less_eq_prod_def] fst_conv snd_conv)
+apply(metis mono_aval'' mono_inv_plus'[simplified less_eq_prod_def] fst_conv snd_conv)
 done
 
-lemma mono_bconstrain: "S1 \<le> S2 \<Longrightarrow> bconstrain b bv S1 \<le> bconstrain b bv S2"
+lemma mono_inv_bval'': "S1 \<le> S2 \<Longrightarrow> inv_bval'' b bv S1 \<le> inv_bval'' b bv S2"
 apply(induction b arbitrary: bv S1 S2)
    apply(simp)
   apply(simp)
  apply simp
  apply(metis order_trans[OF _ sup_ge1] order_trans[OF _ sup_ge2])
 apply (simp split: prod.splits)
-apply(metis mono_aval'' mono_aconstrain mono_constrain_less'[simplified less_eq_prod_def] fst_conv snd_conv)
+apply(metis mono_aval'' mono_inv_aval'' mono_inv_less'[simplified less_eq_prod_def] fst_conv snd_conv)
 done
 
 theorem mono_step': "S1 \<le> S2 \<Longrightarrow> C1 \<le> C2 \<Longrightarrow> step' S1 C1 \<le> step' S2 C2"
 unfolding step'_def
-by(rule mono2_Step) (auto simp: mono_aval' mono_bconstrain split: option.split)
+by(rule mono2_Step) (auto simp: mono_aval' mono_inv_bval'' split: option.split)
 
 lemma mono_step'_top: "C1 \<le> C2 \<Longrightarrow> step' \<top> C1 \<le> step' \<top> C2"
 by (metis mono_step' order_refl)




