--- a/src/HOL/IMP/ACom.thy Sun Jan 01 18:12:11 2012 +0100
+++ b/src/HOL/IMP/ACom.thy Mon Jan 02 10:51:28 2012 +0100
@@ -57,16 +57,33 @@
lemma strip_acom[simp]: "strip (map_acom f c) = strip c"
by (induction c) auto
+lemma map_acom_SKIP:
+ "map_acom f c = SKIP {S'} \<longleftrightarrow> (\<exists>S. c = SKIP {S} \<and> S' = f S)"
+by (cases c) auto
+
+lemma map_acom_Assign:
+ "map_acom f c = x ::= e {S'} \<longleftrightarrow> (\<exists>S. c = x::=e {S} \<and> S' = f S)"
+by (cases c) auto
+
+lemma map_acom_Semi:
+ "map_acom f c = c1';c2' \<longleftrightarrow>
+ (\<exists>c1 c2. c = c1;c2 \<and> map_acom f c1 = c1' \<and> map_acom f c2 = c2')"
+by (cases c) auto
+
+lemma map_acom_If:
+ "map_acom f c = IF b THEN c1' ELSE c2' {S'} \<longleftrightarrow>
+ (\<exists>S c1 c2. c = IF b THEN c1 ELSE c2 {S} \<and> map_acom f c1 = c1' \<and> map_acom f c2 = c2' \<and> S' = f S)"
+by (cases c) auto
+
+lemma map_acom_While:
+ "map_acom f w = {I'} WHILE b DO c' {P'} \<longleftrightarrow>
+ (\<exists>I P c. w = {I} WHILE b DO c {P} \<and> map_acom f c = c' \<and> I' = f I \<and> P' = f P)"
+by (cases w) auto
+
lemma strip_anno[simp]: "strip (anno a c) = c"
by(induct c) simp_all
-lemma strip_eq_SKIP: "strip c = com.SKIP \<longleftrightarrow> (EX P. c = SKIP {P})"
-by (cases c) simp_all
-
-lemma strip_eq_Assign: "strip c = x::=e \<longleftrightarrow> (EX P. c = x::=e {P})"
-by (cases c) simp_all
-
lemma strip_eq_Semi:
"strip c = c1;c2 \<longleftrightarrow> (EX d1 d2. c = d1;d2 & strip d1 = c1 & strip d2 = c2)"
by (cases c) simp_all
--- a/src/HOL/IMP/Abs_Int0.thy Sun Jan 01 18:12:11 2012 +0100
+++ b/src/HOL/IMP/Abs_Int0.thy Mon Jan 02 10:51:28 2012 +0100
@@ -57,49 +57,40 @@
text{* The soundness proofs are textually identical to the ones for the step
function operating on states as functions. *}
-lemma step_preserves_le2:
- "\<lbrakk> S \<subseteq> \<gamma>\<^isub>o S'; cs \<le> \<gamma>\<^isub>c ca; strip cs = c; strip ca = c \<rbrakk>
- \<Longrightarrow> step S cs \<le> \<gamma>\<^isub>c (step' S' ca)"
-proof(induction c arbitrary: cs ca S S')
- case SKIP thus ?case
- by(auto simp:strip_eq_SKIP)
+lemma step_preserves_le:
+ "\<lbrakk> S \<subseteq> \<gamma>\<^isub>o S'; cs \<le> \<gamma>\<^isub>c ca \<rbrakk> \<Longrightarrow> step S cs \<le> \<gamma>\<^isub>c (step' S' ca)"
+proof(induction cs arbitrary: ca S S')
+ case SKIP thus ?case by(auto simp:SKIP_le map_acom_SKIP)
next
case Assign thus ?case
- by (fastforce simp: strip_eq_Assign intro: aval'_sound in_gamma_update
+ by (fastforce simp: Assign_le map_acom_Assign intro: aval'_sound in_gamma_update
split: option.splits del:subsetD)
next
- case Semi thus ?case apply (auto simp: strip_eq_Semi)
+ case Semi thus ?case apply (auto simp: Semi_le map_acom_Semi)
by (metis le_post post_map_acom)
next
- case (If b c1 c2)
- then obtain cs1 cs2 ca1 ca2 P Pa where
- "cs = IF b THEN cs1 ELSE cs2 {P}" "ca = IF b THEN ca1 ELSE ca2 {Pa}"
+ case (If b cs1 cs2 P)
+ then obtain ca1 ca2 Pa where
+ "ca= IF b THEN ca1 ELSE ca2 {Pa}"
"P \<subseteq> \<gamma>\<^isub>o Pa" "cs1 \<le> \<gamma>\<^isub>c ca1" "cs2 \<le> \<gamma>\<^isub>c ca2"
- "strip cs1 = c1" "strip ca1 = c1" "strip cs2 = c2" "strip ca2 = c2"
- by (fastforce simp: strip_eq_If)
+ by (fastforce simp: If_le map_acom_If)
moreover have "post cs1 \<subseteq> \<gamma>\<^isub>o(post ca1 \<squnion> post ca2)"
by (metis (no_types) `cs1 \<le> \<gamma>\<^isub>c ca1` join_ge1 le_post mono_gamma_o order_trans post_map_acom)
moreover have "post cs2 \<subseteq> \<gamma>\<^isub>o(post ca1 \<squnion> post ca2)"
by (metis (no_types) `cs2 \<le> \<gamma>\<^isub>c ca2` join_ge2 le_post mono_gamma_o order_trans post_map_acom)
- ultimately show ?case using If.prems(1) by (simp add: If.IH subset_iff)
+ ultimately show ?case using `S \<subseteq> \<gamma>\<^isub>o S'` by (simp add: If.IH subset_iff)
next
- case (While b c1)
- then obtain cs1 ca1 I P Ia Pa where
- "cs = {I} WHILE b DO cs1 {P}" "ca = {Ia} WHILE b DO ca1 {Pa}"
+ case (While I b cs1 P)
+ then obtain ca1 Ia Pa where
+ "ca = {Ia} WHILE b DO ca1 {Pa}"
"I \<subseteq> \<gamma>\<^isub>o Ia" "P \<subseteq> \<gamma>\<^isub>o Pa" "cs1 \<le> \<gamma>\<^isub>c ca1"
- "strip cs1 = c1" "strip ca1 = c1"
- by (fastforce simp: strip_eq_While)
+ by (fastforce simp: map_acom_While While_le)
moreover have "S \<union> post cs1 \<subseteq> \<gamma>\<^isub>o (S' \<squnion> post ca1)"
using `S \<subseteq> \<gamma>\<^isub>o S'` le_post[OF `cs1 \<le> \<gamma>\<^isub>c ca1`, simplified]
by (metis (no_types) join_ge1 join_ge2 le_sup_iff mono_gamma_o order_trans)
ultimately show ?case by (simp add: While.IH subset_iff)
qed
-lemma step_preserves_le:
- "\<lbrakk> S \<subseteq> \<gamma>\<^isub>o S'; cs \<le> \<gamma>\<^isub>c ca; strip cs = c \<rbrakk>
- \<Longrightarrow> step S cs \<le> \<gamma>\<^isub>c(step' S' ca)"
-by (metis le_strip step_preserves_le2 strip_acom)
-
lemma AI_sound: "AI c = Some c' \<Longrightarrow> CS UNIV c \<le> \<gamma>\<^isub>c c'"
proof(simp add: CS_def AI_def)
assume 1: "lpfp\<^isub>c (step' \<top>) c = Some c'"
@@ -109,7 +100,7 @@
have "lfp (step UNIV) c \<le> \<gamma>\<^isub>c (step' \<top> c')"
proof(rule lfp_lowerbound[simplified,OF 3])
show "step UNIV (\<gamma>\<^isub>c (step' \<top> c')) \<le> \<gamma>\<^isub>c (step' \<top> c')"
- proof(rule step_preserves_le[OF _ _ 3])
+ proof(rule step_preserves_le[OF _ _])
show "UNIV \<subseteq> \<gamma>\<^isub>o \<top>" by simp
show "\<gamma>\<^isub>c (step' \<top> c') \<le> \<gamma>\<^isub>c c'" by(rule mono_gamma_c[OF 2])
qed
--- a/src/HOL/IMP/Abs_Int0_fun.thy Sun Jan 01 18:12:11 2012 +0100
+++ b/src/HOL/IMP/Abs_Int0_fun.thy Mon Jan 02 10:51:28 2012 +0100
@@ -308,49 +308,40 @@
"\<lbrakk> s : \<gamma>\<^isub>f S; i : \<gamma> a \<rbrakk> \<Longrightarrow> s(x := i) : \<gamma>\<^isub>f(S(x := a))"
by(simp add: \<gamma>_fun_def)
-lemma step_preserves_le2:
- "\<lbrakk> S \<subseteq> \<gamma>\<^isub>o S'; cs \<le> \<gamma>\<^isub>c ca; strip cs = c; strip ca = c \<rbrakk>
- \<Longrightarrow> step S cs \<le> \<gamma>\<^isub>c (step' S' ca)"
-proof(induction c arbitrary: cs ca S S')
- case SKIP thus ?case
- by(auto simp:strip_eq_SKIP)
+lemma step_preserves_le:
+ "\<lbrakk> S \<subseteq> \<gamma>\<^isub>o S'; cs \<le> \<gamma>\<^isub>c ca \<rbrakk> \<Longrightarrow> step S cs \<le> \<gamma>\<^isub>c (step' S' ca)"
+proof(induction cs arbitrary: ca S S')
+ case SKIP thus ?case by(auto simp:SKIP_le map_acom_SKIP)
next
case Assign thus ?case
- by (fastforce simp: strip_eq_Assign intro: aval'_sound in_gamma_update
+ by (fastforce simp: Assign_le map_acom_Assign intro: aval'_sound in_gamma_update
split: option.splits del:subsetD)
next
- case Semi thus ?case apply (auto simp: strip_eq_Semi)
+ case Semi thus ?case apply (auto simp: Semi_le map_acom_Semi)
by (metis le_post post_map_acom)
next
- case (If b c1 c2)
- then obtain cs1 cs2 ca1 ca2 P Pa where
- "cs= IF b THEN cs1 ELSE cs2 {P}" "ca= IF b THEN ca1 ELSE ca2 {Pa}"
+ case (If b cs1 cs2 P)
+ then obtain ca1 ca2 Pa where
+ "ca= IF b THEN ca1 ELSE ca2 {Pa}"
"P \<subseteq> \<gamma>\<^isub>o Pa" "cs1 \<le> \<gamma>\<^isub>c ca1" "cs2 \<le> \<gamma>\<^isub>c ca2"
- "strip cs1 = c1" "strip ca1 = c1" "strip cs2 = c2" "strip ca2 = c2"
- by (fastforce simp: strip_eq_If)
+ by (fastforce simp: If_le map_acom_If)
moreover have "post cs1 \<subseteq> \<gamma>\<^isub>o(post ca1 \<squnion> post ca2)"
by (metis (no_types) `cs1 \<le> \<gamma>\<^isub>c ca1` join_ge1 le_post mono_gamma_o order_trans post_map_acom)
moreover have "post cs2 \<subseteq> \<gamma>\<^isub>o(post ca1 \<squnion> post ca2)"
by (metis (no_types) `cs2 \<le> \<gamma>\<^isub>c ca2` join_ge2 le_post mono_gamma_o order_trans post_map_acom)
ultimately show ?case using `S \<subseteq> \<gamma>\<^isub>o S'` by (simp add: If.IH subset_iff)
next
- case (While b c1)
- then obtain cs1 ca1 I P Ia Pa where
- "cs = {I} WHILE b DO cs1 {P}" "ca = {Ia} WHILE b DO ca1 {Pa}"
+ case (While I b cs1 P)
+ then obtain ca1 Ia Pa where
+ "ca = {Ia} WHILE b DO ca1 {Pa}"
"I \<subseteq> \<gamma>\<^isub>o Ia" "P \<subseteq> \<gamma>\<^isub>o Pa" "cs1 \<le> \<gamma>\<^isub>c ca1"
- "strip cs1 = c1" "strip ca1 = c1"
- by (fastforce simp: strip_eq_While)
+ by (fastforce simp: map_acom_While While_le)
moreover have "S \<union> post cs1 \<subseteq> \<gamma>\<^isub>o (S' \<squnion> post ca1)"
using `S \<subseteq> \<gamma>\<^isub>o S'` le_post[OF `cs1 \<le> \<gamma>\<^isub>c ca1`, simplified]
by (metis (no_types) join_ge1 join_ge2 le_sup_iff mono_gamma_o order_trans)
ultimately show ?case by (simp add: While.IH subset_iff)
qed
-lemma step_preserves_le:
- "\<lbrakk> S \<subseteq> \<gamma>\<^isub>o S'; cs \<le> \<gamma>\<^isub>c ca; strip cs = c \<rbrakk>
- \<Longrightarrow> step S cs \<le> \<gamma>\<^isub>c(step' S' ca)"
-by (metis le_strip step_preserves_le2 strip_acom)
-
lemma AI_sound: "AI c = Some c' \<Longrightarrow> CS UNIV c \<le> \<gamma>\<^isub>c c'"
proof(simp add: CS_def AI_def)
assume 1: "lpfp\<^isub>c (step' \<top>) c = Some c'"
@@ -360,12 +351,12 @@
have "lfp (step UNIV) c \<le> \<gamma>\<^isub>c (step' \<top> c')"
proof(rule lfp_lowerbound[simplified,OF 3])
show "step UNIV (\<gamma>\<^isub>c (step' \<top> c')) \<le> \<gamma>\<^isub>c (step' \<top> c')"
- proof(rule step_preserves_le[OF _ _ 3])
+ proof(rule step_preserves_le[OF _ _])
show "UNIV \<subseteq> \<gamma>\<^isub>o \<top>" by simp
show "\<gamma>\<^isub>c (step' \<top> c') \<le> \<gamma>\<^isub>c c'" by(rule mono_gamma_c[OF 2])
qed
qed
- from this 2 show "lfp (step UNIV) c \<le> \<gamma>\<^isub>c c'"
+ with 2 show "lfp (step UNIV) c \<le> \<gamma>\<^isub>c c'"
by (blast intro: mono_gamma_c order_trans)
qed
--- a/src/HOL/IMP/Abs_Int1.thy Sun Jan 01 18:12:11 2012 +0100
+++ b/src/HOL/IMP/Abs_Int1.thy Mon Jan 02 10:51:28 2012 +0100
@@ -168,27 +168,23 @@
"\<lbrakk> s : \<gamma>\<^isub>f S; i : \<gamma> a \<rbrakk> \<Longrightarrow> s(x := i) : \<gamma>\<^isub>f(update S x a)"
by(simp add: \<gamma>_st_def lookup_update)
-
-lemma step_preserves_le2:
- "\<lbrakk> S \<subseteq> \<gamma>\<^isub>o S'; cs \<le> \<gamma>\<^isub>c ca; strip cs = c; strip ca = c \<rbrakk>
- \<Longrightarrow> step S cs \<le> \<gamma>\<^isub>c (step' S' ca)"
-proof(induction c arbitrary: cs ca S S')
- case SKIP thus ?case
- by(auto simp:strip_eq_SKIP)
+lemma step_preserves_le:
+ "\<lbrakk> S \<subseteq> \<gamma>\<^isub>o S'; cs \<le> \<gamma>\<^isub>c ca \<rbrakk> \<Longrightarrow> step S cs \<le> \<gamma>\<^isub>c (step' S' ca)"
+proof(induction cs arbitrary: ca S S')
+ case SKIP thus ?case by(auto simp:SKIP_le map_acom_SKIP)
next
case Assign thus ?case
- by (fastforce simp: strip_eq_Assign intro: aval'_sound in_gamma_update
+ by (fastforce simp: Assign_le map_acom_Assign intro: aval'_sound in_gamma_update
split: option.splits del:subsetD)
next
- case Semi thus ?case apply (auto simp: strip_eq_Semi)
+ case Semi thus ?case apply (auto simp: Semi_le map_acom_Semi)
by (metis le_post post_map_acom)
next
- case (If b c1 c2)
- then obtain cs1 cs2 ca1 ca2 P Pa where
- "cs= IF b THEN cs1 ELSE cs2 {P}" "ca= IF b THEN ca1 ELSE ca2 {Pa}"
+ case (If b cs1 cs2 P)
+ then obtain ca1 ca2 Pa where
+ "ca= IF b THEN ca1 ELSE ca2 {Pa}"
"P \<subseteq> \<gamma>\<^isub>o Pa" "cs1 \<le> \<gamma>\<^isub>c ca1" "cs2 \<le> \<gamma>\<^isub>c ca2"
- "strip cs1 = c1" "strip ca1 = c1" "strip cs2 = c2" "strip ca2 = c2"
- by (fastforce simp: strip_eq_If)
+ by (fastforce simp: If_le map_acom_If)
moreover have "post cs1 \<subseteq> \<gamma>\<^isub>o(post ca1 \<squnion> post ca2)"
by (metis (no_types) `cs1 \<le> \<gamma>\<^isub>c ca1` join_ge1 le_post mono_gamma_o order_trans post_map_acom)
moreover have "post cs2 \<subseteq> \<gamma>\<^isub>o(post ca1 \<squnion> post ca2)"
@@ -196,23 +192,17 @@
ultimately show ?case using `S \<subseteq> \<gamma>\<^isub>o S'`
by (simp add: If.IH subset_iff bfilter_sound)
next
- case (While b c1)
- then obtain cs1 ca1 I P Ia Pa where
- "cs = {I} WHILE b DO cs1 {P}" "ca = {Ia} WHILE b DO ca1 {Pa}"
+ case (While I b cs1 P)
+ then obtain ca1 Ia Pa where
+ "ca = {Ia} WHILE b DO ca1 {Pa}"
"I \<subseteq> \<gamma>\<^isub>o Ia" "P \<subseteq> \<gamma>\<^isub>o Pa" "cs1 \<le> \<gamma>\<^isub>c ca1"
- "strip cs1 = c1" "strip ca1 = c1"
- by (fastforce simp: strip_eq_While)
+ by (fastforce simp: map_acom_While While_le)
moreover have "S \<union> post cs1 \<subseteq> \<gamma>\<^isub>o (S' \<squnion> post ca1)"
using `S \<subseteq> \<gamma>\<^isub>o S'` le_post[OF `cs1 \<le> \<gamma>\<^isub>c ca1`, simplified]
by (metis (no_types) join_ge1 join_ge2 le_sup_iff mono_gamma_o order_trans)
ultimately show ?case by (simp add: While.IH subset_iff bfilter_sound)
qed
-lemma step_preserves_le:
- "\<lbrakk> S \<subseteq> \<gamma>\<^isub>o S'; cs \<le> \<gamma>\<^isub>c ca; strip cs = c \<rbrakk>
- \<Longrightarrow> step S cs \<le> \<gamma>\<^isub>c(step' S' ca)"
-by (metis le_strip step_preserves_le2 strip_acom)
-
lemma AI_sound: "AI c = Some c' \<Longrightarrow> CS UNIV c \<le> \<gamma>\<^isub>c c'"
proof(simp add: CS_def AI_def)
assume 1: "lpfp\<^isub>c (step' \<top>) c = Some c'"
@@ -222,7 +212,7 @@
have "lfp (step UNIV) c \<le> \<gamma>\<^isub>c (step' \<top> c')"
proof(rule lfp_lowerbound[simplified,OF 3])
show "step UNIV (\<gamma>\<^isub>c (step' \<top> c')) \<le> \<gamma>\<^isub>c (step' \<top> c')"
- proof(rule step_preserves_le[OF _ _ 3])
+ proof(rule step_preserves_le[OF _ _])
show "UNIV \<subseteq> \<gamma>\<^isub>o \<top>" by simp
show "\<gamma>\<^isub>c (step' \<top> c') \<le> \<gamma>\<^isub>c c'" by(rule mono_gamma_c[OF 2])
qed
--- a/src/HOL/IMP/Abs_Int2.thy Sun Jan 01 18:12:11 2012 +0100
+++ b/src/HOL/IMP/Abs_Int2.thy Mon Jan 02 10:51:28 2012 +0100
@@ -196,7 +196,7 @@
have "lfp (step UNIV) c \<le> \<gamma>\<^isub>c (step' \<top> c')"
proof(rule lfp_lowerbound[simplified,OF 3])
show "step UNIV (\<gamma>\<^isub>c (step' \<top> c')) \<le> \<gamma>\<^isub>c (step' \<top> c')"
- proof(rule step_preserves_le[OF _ _ 3])
+ proof(rule step_preserves_le[OF _ _])
show "UNIV \<subseteq> \<gamma>\<^isub>o \<top>" by simp
show "\<gamma>\<^isub>c (step' \<top> c') \<le> \<gamma>\<^isub>c c'" by(rule mono_gamma_c[OF 2])
qed