# HG changeset patch
# User nipkow
# Date 1374699287 -7200
# Node ID ee0bd6bababd3da9440f7ea2b142605bcfb00c5f
# Parent  ba2bbe825a5e8132309202ea878ad11d33659355
merged Def_Init_Sound_X into Def_Init_X

diff -r ba2bbe825a5e -r ee0bd6bababd src/HOL/IMP/Def_Init_Big.thy
--- a/src/HOL/IMP/Def_Init_Big.thy	Wed Jul 24 13:03:53 2013 +0200
+++ b/src/HOL/IMP/Def_Init_Big.thy	Wed Jul 24 22:54:47 2013 +0200
@@ -1,7 +1,7 @@
 (* Author: Tobias Nipkow *)
 
 theory Def_Init_Big
-imports Com Def_Init_Exp
+imports Def_Init_Exp Def_Init
 begin
 
 subsection "Initialization-Sensitive Big Step Semantics"
@@ -29,4 +29,40 @@
 
 lemmas big_step_induct = big_step.induct[split_format(complete)]
 
+
+subsection "Soundness wrt Big Steps"
+
+text{* Note the special form of the induction because one of the arguments
+of the inductive predicate is not a variable but the term @{term"Some s"}: *}
+
+theorem Sound:
+  "\<lbrakk> (c,Some s) \<Rightarrow> s';  D A c A';  A \<subseteq> dom s \<rbrakk>
+  \<Longrightarrow> \<exists> t. s' = Some t \<and> A' \<subseteq> dom t"
+proof (induction c "Some s" s' arbitrary: s A A' rule:big_step_induct)
+  case AssignNone thus ?case
+    by auto (metis aval_Some option.simps(3) subset_trans)
+next
+  case Seq thus ?case by auto metis
+next
+  case IfTrue thus ?case by auto blast
+next
+  case IfFalse thus ?case by auto blast
+next
+  case IfNone thus ?case
+    by auto (metis bval_Some option.simps(3) order_trans)
+next
+  case WhileNone thus ?case
+    by auto (metis bval_Some option.simps(3) order_trans)
+next
+  case (WhileTrue b s c s' s'')
+  from `D A (WHILE b DO c) A'` obtain A' where "D A c A'" by blast
+  then obtain t' where "s' = Some t'" "A \<subseteq> dom t'"
+    by (metis D_incr WhileTrue(3,7) subset_trans)
+  from WhileTrue(5)[OF this(1) WhileTrue(6) this(2)] show ?case .
+qed auto
+
+corollary sound: "\<lbrakk>  D (dom s) c A';  (c,Some s) \<Rightarrow> s' \<rbrakk> \<Longrightarrow> s' \<noteq> None"
+by (metis Sound not_Some_eq subset_refl)
+
 end
+
diff -r ba2bbe825a5e -r ee0bd6bababd src/HOL/IMP/Def_Init_Small.thy
--- a/src/HOL/IMP/Def_Init_Small.thy	Wed Jul 24 13:03:53 2013 +0200
+++ b/src/HOL/IMP/Def_Init_Small.thy	Wed Jul 24 22:54:47 2013 +0200
@@ -1,7 +1,7 @@
 (* Author: Tobias Nipkow *)
 
 theory Def_Init_Small
-imports Star Com Def_Init_Exp
+imports Star Def_Init_Exp Def_Init
 begin
 
 subsection "Initialization-Sensitive Small Step Semantics"
@@ -24,4 +24,55 @@
 abbreviation small_steps :: "com * state \<Rightarrow> com * state \<Rightarrow> bool" (infix "\<rightarrow>*" 55)
 where "x \<rightarrow>* y == star small_step x y"
 
+
+subsection "Soundness wrt Small Steps"
+
+theorem progress:
+  "D (dom s) c A' \<Longrightarrow> c \<noteq> SKIP \<Longrightarrow> EX cs'. (c,s) \<rightarrow> cs'"
+proof (induction c arbitrary: s A')
+  case Assign thus ?case by auto (metis aval_Some small_step.Assign)
+next
+  case (If b c1 c2)
+  then obtain bv where "bval b s = Some bv" by (auto dest!:bval_Some)
+  then show ?case
+    by(cases bv)(auto intro: small_step.IfTrue small_step.IfFalse)
+qed (fastforce intro: small_step.intros)+
+
+lemma D_mono:  "D A c M \<Longrightarrow> A \<subseteq> A' \<Longrightarrow> EX M'. D A' c M' & M <= M'"
+proof (induction c arbitrary: A A' M)
+  case Seq thus ?case by auto (metis D.intros(3))
+next
+  case (If b c1 c2)
+  then obtain M1 M2 where "vars b \<subseteq> A" "D A c1 M1" "D A c2 M2" "M = M1 \<inter> M2"
+    by auto
+  with If.IH `A \<subseteq> A'` obtain M1' M2'
+    where "D A' c1 M1'" "D A' c2 M2'" and "M1 \<subseteq> M1'" "M2 \<subseteq> M2'" by metis
+  hence "D A' (IF b THEN c1 ELSE c2) (M1' \<inter> M2')" and "M \<subseteq> M1' \<inter> M2'"
+    using `vars b \<subseteq> A` `A \<subseteq> A'` `M = M1 \<inter> M2` by(fastforce intro: D.intros)+
+  thus ?case by metis
+next
+  case While thus ?case by auto (metis D.intros(5) subset_trans)
+qed (auto intro: D.intros)
+
+theorem D_preservation:
+  "(c,s) \<rightarrow> (c',s') \<Longrightarrow> D (dom s) c A \<Longrightarrow> EX A'. D (dom s') c' A' & A <= A'"
+proof (induction arbitrary: A rule: small_step_induct)
+  case (While b c s)
+  then obtain A' where "vars b \<subseteq> dom s" "A = dom s" "D (dom s) c A'" by blast
+  moreover
+  then obtain A'' where "D A' c A''" by (metis D_incr D_mono)
+  ultimately have "D (dom s) (IF b THEN c;; WHILE b DO c ELSE SKIP) (dom s)"
+    by (metis D.If[OF `vars b \<subseteq> dom s` D.Seq[OF `D (dom s) c A'` D.While[OF _ `D A' c A''`]] D.Skip] D_incr Int_absorb1 subset_trans)
+  thus ?case by (metis D_incr `A = dom s`)
+next
+  case Seq2 thus ?case by auto (metis D_mono D.intros(3))
+qed (auto intro: D.intros)
+
+theorem D_sound:
+  "(c,s) \<rightarrow>* (c',s') \<Longrightarrow> D (dom s) c A'
+   \<Longrightarrow> (\<exists>cs''. (c',s') \<rightarrow> cs'') \<or> c' = SKIP"
+apply(induction arbitrary: A' rule:star_induct)
+apply (metis progress)
+by (metis D_preservation)
+
 end
diff -r ba2bbe825a5e -r ee0bd6bababd src/HOL/IMP/Def_Init_Sound_Big.thy
--- a/src/HOL/IMP/Def_Init_Sound_Big.thy	Wed Jul 24 13:03:53 2013 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,41 +0,0 @@
-(* Author: Tobias Nipkow *)
-
-theory Def_Init_Sound_Big
-imports Def_Init Def_Init_Big
-begin
-
-subsection "Soundness wrt Big Steps"
-
-text{* Note the special form of the induction because one of the arguments
-of the inductive predicate is not a variable but the term @{term"Some s"}: *}
-
-theorem Sound:
-  "\<lbrakk> (c,Some s) \<Rightarrow> s';  D A c A';  A \<subseteq> dom s \<rbrakk>
-  \<Longrightarrow> \<exists> t. s' = Some t \<and> A' \<subseteq> dom t"
-proof (induction c "Some s" s' arbitrary: s A A' rule:big_step_induct)
-  case AssignNone thus ?case
-    by auto (metis aval_Some option.simps(3) subset_trans)
-next
-  case Seq thus ?case by auto metis
-next
-  case IfTrue thus ?case by auto blast
-next
-  case IfFalse thus ?case by auto blast
-next
-  case IfNone thus ?case
-    by auto (metis bval_Some option.simps(3) order_trans)
-next
-  case WhileNone thus ?case
-    by auto (metis bval_Some option.simps(3) order_trans)
-next
-  case (WhileTrue b s c s' s'')
-  from `D A (WHILE b DO c) A'` obtain A' where "D A c A'" by blast
-  then obtain t' where "s' = Some t'" "A \<subseteq> dom t'"
-    by (metis D_incr WhileTrue(3,7) subset_trans)
-  from WhileTrue(5)[OF this(1) WhileTrue(6) this(2)] show ?case .
-qed auto
-
-corollary sound: "\<lbrakk>  D (dom s) c A';  (c,Some s) \<Rightarrow> s' \<rbrakk> \<Longrightarrow> s' \<noteq> None"
-by (metis Sound not_Some_eq subset_refl)
-
-end
diff -r ba2bbe825a5e -r ee0bd6bababd src/HOL/IMP/Def_Init_Sound_Small.thy
--- a/src/HOL/IMP/Def_Init_Sound_Small.thy	Wed Jul 24 13:03:53 2013 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,57 +0,0 @@
-(* Author: Tobias Nipkow *)
-
-theory Def_Init_Sound_Small
-imports Def_Init Def_Init_Small
-begin
-
-subsection "Soundness wrt Small Steps"
-
-theorem progress:
-  "D (dom s) c A' \<Longrightarrow> c \<noteq> SKIP \<Longrightarrow> EX cs'. (c,s) \<rightarrow> cs'"
-proof (induction c arbitrary: s A')
-  case Assign thus ?case by auto (metis aval_Some small_step.Assign)
-next
-  case (If b c1 c2)
-  then obtain bv where "bval b s = Some bv" by (auto dest!:bval_Some)
-  then show ?case
-    by(cases bv)(auto intro: small_step.IfTrue small_step.IfFalse)
-qed (fastforce intro: small_step.intros)+
-
-lemma D_mono:  "D A c M \<Longrightarrow> A \<subseteq> A' \<Longrightarrow> EX M'. D A' c M' & M <= M'"
-proof (induction c arbitrary: A A' M)
-  case Seq thus ?case by auto (metis D.intros(3))
-next
-  case (If b c1 c2)
-  then obtain M1 M2 where "vars b \<subseteq> A" "D A c1 M1" "D A c2 M2" "M = M1 \<inter> M2"
-    by auto
-  with If.IH `A \<subseteq> A'` obtain M1' M2'
-    where "D A' c1 M1'" "D A' c2 M2'" and "M1 \<subseteq> M1'" "M2 \<subseteq> M2'" by metis
-  hence "D A' (IF b THEN c1 ELSE c2) (M1' \<inter> M2')" and "M \<subseteq> M1' \<inter> M2'"
-    using `vars b \<subseteq> A` `A \<subseteq> A'` `M = M1 \<inter> M2` by(fastforce intro: D.intros)+
-  thus ?case by metis
-next
-  case While thus ?case by auto (metis D.intros(5) subset_trans)
-qed (auto intro: D.intros)
-
-theorem D_preservation:
-  "(c,s) \<rightarrow> (c',s') \<Longrightarrow> D (dom s) c A \<Longrightarrow> EX A'. D (dom s') c' A' & A <= A'"
-proof (induction arbitrary: A rule: small_step_induct)
-  case (While b c s)
-  then obtain A' where "vars b \<subseteq> dom s" "A = dom s" "D (dom s) c A'" by blast
-  moreover
-  then obtain A'' where "D A' c A''" by (metis D_incr D_mono)
-  ultimately have "D (dom s) (IF b THEN c;; WHILE b DO c ELSE SKIP) (dom s)"
-    by (metis D.If[OF `vars b \<subseteq> dom s` D.Seq[OF `D (dom s) c A'` D.While[OF _ `D A' c A''`]] D.Skip] D_incr Int_absorb1 subset_trans)
-  thus ?case by (metis D_incr `A = dom s`)
-next
-  case Seq2 thus ?case by auto (metis D_mono D.intros(3))
-qed (auto intro: D.intros)
-
-theorem D_sound:
-  "(c,s) \<rightarrow>* (c',s') \<Longrightarrow> D (dom s) c A'
-   \<Longrightarrow> (\<exists>cs''. (c',s') \<rightarrow> cs'') \<or> c' = SKIP"
-apply(induction arbitrary: A' rule:star_induct)
-apply (metis progress)
-by (metis D_preservation)
-
-end
diff -r ba2bbe825a5e -r ee0bd6bababd src/HOL/ROOT
--- a/src/HOL/ROOT	Wed Jul 24 13:03:53 2013 +0200
+++ b/src/HOL/ROOT	Wed Jul 24 22:54:47 2013 +0200
@@ -125,8 +125,9 @@
     Poly_Types
     Sec_Typing
     Sec_TypingT
-    Def_Init_Sound_Big
-    Def_Init_Sound_Small
+    Def_Init_Big
+    Def_Init_Small
+    Fold
     Live
     Live_True
     Hoare_Examples
@@ -147,7 +148,6 @@
     Procs_Stat_Vars_Stat
     C_like
     OO
-    Fold
   files "document/root.bib" "document/root.tex"
 
 session "HOL-IMPP" in IMPP = HOL +