
diff -r 0352ad5ee9c5 -r 86918b45b2e6 thys/Hoare_tm2.thy
--- a/thys/Hoare_tm2.thy	Thu Apr 03 14:02:58 2014 +0100
+++ b/thys/Hoare_tm2.thy	Thu Apr 03 15:28:01 2014 +0100
@@ -2,7 +2,7 @@
   Separation logic for Turing Machines
 *}
 
-theory Hoare_tm
+theory Hoare_tm2
 imports Hoare_gen 
         My_block 
         Data_slot
@@ -126,7 +126,6 @@
 interpretation tm: sep_exec tstep trset_of .
 
 
-
 section {* Assembly language for TMs and its program logic *}
 
 subsection {* The assembly language *}
@@ -411,17 +410,16 @@
 proof -
   assume "(st i ** sg {TC i inst} ** r) (trset_of (ft, prog, i', pos, mem))"
   thus ?thesis
-    apply(unfold sep_conj_def set_ins_def sg_def trset_of.simps tpn_set_def)
-    
+    apply(unfold sep_conj_def set_ins_def sg_def trset_of.simps tpn_set_def) 
     by auto
 qed
 
-lemma memD: "((tm a v) ** r) (trset_of (ft, prog, i, pos, mem))  \<Longrightarrow> mem a = Some v"
+lemma memD: "((tm a v) ** r) (trset_of (ft, prog, i, pos, mem))  \<Longrightarrow> mem $ a = Some v"
 proof -
   assume "((tm a v) ** r) (trset_of (ft, prog, i, pos, mem))"
   from stimes_sgD[OF this[unfolded trset_of.simps tpn_set_def tm_def]]
-  have "{TM a v} \<subseteq> {TC i inst |i inst. prog i = Some inst} \<union> {TAt i} \<union> 
-    {TPos pos} \<union> {TM i n |i n. mem i = Some n} \<union> {TFaults ft}" by simp
+  have "{TM a v} \<subseteq> {TC i inst |i inst. prog $ i = Some inst} \<union> {TAt i} \<union> 
+    {TPos pos} \<union> {TM i n |i n. mem $ i = Some n} \<union> {TFaults ft}" by simp
   thus ?thesis by auto
 qed
 
@@ -1214,15 +1212,20 @@
 
 lemma tmem_set_upd: 
   "{TM a v} \<subseteq> tmem_set mem \<Longrightarrow> 
-        tmem_set (mem(a:=Some v')) = ((tmem_set mem) - {TM a v}) \<union> {TM a v'}"
-  by (unfold tpn_set_def, auto)
+        tmem_set (mem(a $:=Some v')) = ((tmem_set mem) - {TM a v}) \<union> {TM a v'}"
+apply(unfold tpn_set_def) 
+apply(auto)
+apply (metis finfun_upd_apply option.inject)
+apply (metis finfun_upd_apply_other)
+by (metis finfun_upd_apply_other option.inject)
+
 
 lemma tmem_set_disj: "{TM a v} \<subseteq> tmem_set mem \<Longrightarrow> 
                             {TM a v'} \<inter>  (tmem_set mem - {TM a v}) = {}"
   by (unfold tpn_set_def, auto)
 
 lemma smem_upd: "((tm a v) ** r) (trset_of (f, x, y, z, mem))  \<Longrightarrow> 
-                    ((tm a v') ** r) (trset_of (f, x, y, z, mem(a := Some v')))"
+                    ((tm a v') ** r) (trset_of (f, x, y, z, mem(a $:= Some v')))"
 proof -
   have eq_s: "(tprog_set x \<union> tpc_set y \<union> tpos_set z \<union> tmem_set mem \<union> tfaults_set f - {TM a v}) =
     (tprog_set x \<union> tpc_set y \<union> tpos_set z \<union> (tmem_set mem - {TM a v}) \<union> tfaults_set f)"
@@ -1235,11 +1238,11 @@
   from h TM_in_simp have "{TM a v} \<subseteq> tmem_set mem"
     by(sep_drule stimes_sgD, auto)
   from tmem_set_upd [OF this] tmem_set_disj[OF this]
-  have h2: "tmem_set (mem(a \<mapsto> v')) = {TM a v'} \<union> (tmem_set mem - {TM a v})" 
+  have h2: "tmem_set (mem(a $:= Some v')) = {TM a v'} \<union> (tmem_set mem - {TM a v})" 
            "{TM a v'} \<inter> (tmem_set mem - {TM a v}) = {}" by auto
   show ?thesis
   proof -
-    have "(tm a v' ** r) (tmem_set (mem(a \<mapsto> v')) \<union> tprog_set x \<union> tpc_set y \<union> tpos_set z \<union> tfaults_set f)"
+    have "(tm a v' ** r) (tmem_set (mem(a $:= Some v')) \<union> tprog_set x \<union> tpc_set y \<union> tpos_set z \<union> tfaults_set f)"
     proof(rule sep_conjI)
       show "tm a v' ({TM a v'})" by (unfold tm_def sg_def, simp)
     next
@@ -1247,13 +1250,13 @@
     next
       show "{TM a v'} ## tprog_set x \<union> tpc_set y \<union> tpos_set z \<union> (tmem_set mem - {TM a v}) \<union> tfaults_set f"
       proof -
-        from g have " mem a = Some v"
+        from g have " mem $ a = Some v"
           by(sep_frule memD, simp)
         thus "?thesis"
           by(unfold tpn_set_def set_ins_def, auto)
       qed
     next
-      show "tmem_set (mem(a \<mapsto> v')) \<union> tprog_set x \<union> tpc_set y \<union> tpos_set z \<union> tfaults_set f =
+      show "tmem_set (mem(a $:= Some v')) \<union> tprog_set x \<union> tpc_set y \<union> tpos_set z \<union> tfaults_set f =
     {TM a v'} + (tprog_set x \<union> tpc_set y \<union> tpos_set z \<union> (tmem_set mem - {TM a v}) \<union> tfaults_set f)"
         by (unfold h2(1) set_ins_def eq_s, auto)
     qed
@@ -1278,11 +1281,11 @@
       fix ft prog cs i' mem r
       assume h: "(r \<and>* ps p \<and>* st i \<and>* tm p v \<and>* sg {TC i ((W0, j), W0, j)})
               (trset_of (ft, prog, cs, i', mem))"
-      from h have "prog i = Some ((W0, j), W0, j)"
+      from h have "prog $ i = Some ((W0, j), W0, j)"
         apply(rule_tac r = "r \<and>* ps p \<and>* tm p v" in codeD)
         by(simp add: sep_conj_ac)
       from h and this have stp:
-        "tm.run 1 (ft, prog, cs, i', mem) = (ft, prog, j, i', mem(i'\<mapsto> Bk))" (is "?x = ?y")
+        "tm.run 1 (ft, prog, cs, i', mem) = (ft, prog, j, i', mem(i' $:= Some Bk))" (is "?x = ?y")
         apply(sep_frule psD)
         apply(sep_frule stD)
         apply(sep_frule memD, simp)
@@ -1328,12 +1331,12 @@
       fix ft prog cs i' mem r
       assume h: "(r \<and>* ps p \<and>* st i \<and>* tm p v \<and>* sg {TC i ((W1, ?j), W1, ?j)})
               (trset_of (ft, prog, cs, i', mem))"
-      from h have "prog i = Some ((W1, ?j), W1, ?j)"
+      from h have "prog $ i = Some ((W1, ?j), W1, ?j)"
         apply(rule_tac r = "r \<and>* ps p \<and>* tm p v" in codeD)
         by(simp add: sep_conj_ac)
       from h and this have stp:
         "tm.run 1 (ft, prog, cs, i', mem) = 
-                     (ft, prog, ?j, i', mem(i'\<mapsto> Oc))" (is "?x = ?y")
+                     (ft, prog, ?j, i', mem(i' $:= Some Oc))" (is "?x = ?y")
         apply(sep_frule psD)
         apply(sep_frule stD)
         apply(sep_frule memD, simp)
@@ -1380,7 +1383,7 @@
       fix ft prog cs i' mem r
       assume h: "(r \<and>* ps p \<and>* st i \<and>* tm p v2 \<and>* sg {TC i ((L, ?j), L, ?j)}) 
                        (trset_of (ft, prog, cs, i',  mem))"
-      from h have "prog i = Some ((L, ?j), L, ?j)"
+      from h have "prog $ i = Some ((L, ?j), L, ?j)"
         apply(rule_tac r = "r \<and>* ps p \<and>* tm p v2" in codeD)
         by(simp add: sep_conj_ac)
       from h and this have stp:
@@ -1438,7 +1441,7 @@
       fix ft prog cs i' mem r
       assume h: "(r \<and>* ps p \<and>* st i \<and>* tm p v1 \<and>* sg {TC i ((R, ?j), R, ?j)}) 
                        (trset_of (ft, prog, cs, i',  mem))"
-      from h have "prog i = Some ((R, ?j), R, ?j)"
+      from h have "prog $ i = Some ((R, ?j), R, ?j)"
         apply(rule_tac r = "r \<and>* ps p \<and>* tm p v1" in codeD)
         by(simp add: sep_conj_ac)
       from h and this have stp:
@@ -1494,18 +1497,21 @@
       fix ft prog cs pc mem r
       assume h: "(r \<and>* one p \<and>* ps p \<and>* st i \<and>* sg {TC i ((W0, ?j), W1, e)}) 
         (trset_of (ft, prog, cs, pc, mem))"
-      from h have k1: "prog i = Some ((W0, ?j), W1, e)"
+      from h have k1: "prog $ i = Some ((W0, ?j), W1, e)"
         apply(rule_tac r = "r \<and>* one p \<and>* ps p" in codeD)
         by(simp add: sep_conj_ac)
       from h have k2: "pc = p"
         by(sep_drule psD, simp)
-      from h and this have k3: "mem pc = Some Oc"
+      from h and this have k3: "mem $ pc = Some Oc"
         apply(unfold one_def)
         by(sep_drule memD, simp)
       from h k1 k2 k3 have stp:
         "tm.run 1 (ft, prog, cs, pc, mem) = (ft, prog, e, pc, mem)" (is "?x = ?y")
         apply(sep_drule stD)
-        by(unfold tm.run_def, auto)
+        apply(unfold tm.run_def)
+        apply(auto)
+        by (metis finfun_upd_triv)
+
       from h k2 
       have "(r \<and>* one p \<and>* ps p \<and>* st e \<and>* sg {TC i ((W0, ?j), W1, e)})  (trset_of ?x)"
         apply(unfold stp)
@@ -1578,18 +1584,20 @@
       fix ft prog cs pc mem r
       assume h: "(r \<and>* ps p \<and>* st i \<and>* zero p \<and>* sg {TC i ((W0, ?j), W1, e)})
         (trset_of (ft, prog, cs, pc, mem))"
-      from h have k1: "prog i = Some ((W0, ?j), W1, e)"
+      from h have k1: "prog $ i = Some ((W0, ?j), W1, e)"
         apply(rule_tac r = "r \<and>* zero p \<and>* ps p" in codeD)
         by(simp add: sep_conj_ac)
       from h have k2: "pc = p"
         by(sep_drule psD, simp)
-      from h and this have k3: "mem pc = Some Bk"
+      from h and this have k3: "mem $ pc = Some Bk"
         apply(unfold zero_def)
         by(sep_drule memD, simp)
       from h k1 k2 k3 have stp:
         "tm.run 1 (ft, prog, cs, pc, mem) = (ft, prog, ?j, pc, mem)" (is "?x = ?y")
         apply(sep_drule stD)
-        by(unfold tm.run_def, auto)
+        apply(unfold tm.run_def)
+        apply(auto)
+        by (metis finfun_upd_triv)
       from h k2 
       have "(r \<and>* zero p \<and>* ps p \<and>* st ?j \<and>* sg {TC i ((W0, ?j), W1, e)})  (trset_of ?x)"
         apply (unfold stp)
@@ -1625,18 +1633,20 @@
       fix ft prog cs pc mem r
       assume h: "(r \<and>* ps p \<and>* st i \<and>* zero p \<and>* sg {TC i ((W0, e), W1, ?j)})
         (trset_of (ft, prog, cs, pc, mem))"
-      from h have k1: "prog i = Some ((W0, e), W1, ?j)"
+      from h have k1: "prog $ i = Some ((W0, e), W1, ?j)"
         apply(rule_tac r = "r \<and>* zero p \<and>* ps p" in codeD)
         by(simp add: sep_conj_ac)
       from h have k2: "pc = p"
         by(sep_drule psD, simp)
-      from h and this have k3: "mem pc = Some Bk"
+      from h and this have k3: "mem $ pc = Some Bk"
         apply(unfold zero_def)
         by(sep_drule memD, simp)
       from h k1 k2 k3 have stp:
         "tm.run 1 (ft, prog, cs, pc, mem) = (ft, prog, e, pc, mem)" (is "?x = ?y")
         apply(sep_drule stD)
-        by(unfold tm.run_def, auto)
+        apply(unfold tm.run_def)
+        apply(auto)
+        by (metis finfun_upd_triv)  
       from h k2 
       have "(r \<and>* zero p \<and>* ps p \<and>* st e \<and>* sg {TC i ((W0, e), W1, ?j)})  (trset_of ?x)"
         apply(unfold stp)
@@ -1702,17 +1712,19 @@
       fix ft prog cs pc mem r
       assume h: "(r \<and>* ps p \<and>* st i \<and>* tm p Oc \<and>* sg {TC i ((W0, e), W1, ?j)})
         (trset_of (ft, prog, cs, pc, mem))"
-      from h have k1: "prog i = Some ((W0, e), W1, ?j)"
+      from h have k1: "prog $ i = Some ((W0, e), W1, ?j)"
         apply(rule_tac r = "r \<and>* tm p Oc \<and>* ps p" in codeD)
         by(simp add: sep_conj_ac)
       from h have k2: "pc = p"
         by(sep_drule psD, simp)
-      from h and this have k3: "mem pc = Some Oc"
+      from h and this have k3: "mem $ pc = Some Oc"
         by(sep_drule memD, simp)
       from h k1 k2 k3 have stp:
         "tm.run 1 (ft, prog, cs, pc, mem) = (ft, prog, ?j, pc, mem)" (is "?x = ?y")
         apply(sep_drule stD)
-        by(unfold tm.run_def, auto)
+        apply(unfold tm.run_def)
+        apply(auto)
+        by (metis finfun_upd_triv)
       from h k2 
       have "(r \<and>* tm p Oc \<and>* ps p \<and>* st ?j \<and>* sg {TC i ((W0, e), W1, ?j)})  (trset_of ?x)"
         apply(unfold stp)
@@ -1741,17 +1753,23 @@
   fix ft prog cs pos mem r
   assume h: "(r \<and>* ps p \<and>* st i \<and>* tm p v \<and>* <(j = i + 1)> \<and>* sg {TC i ((W0, e), W1, e)})
     (trset_of (ft, prog, cs, pos, mem))"
-  from h have k1: "prog i = Some ((W0, e), W1, e)"
+  from h have k1: "prog $ i = Some ((W0, e), W1, e)"
     apply(rule_tac r = "r \<and>* <(j = i + 1)> \<and>* tm p v \<and>* ps p" in codeD)
     by(simp add: sep_conj_ac)
   from h have k2: "p = pos"
     by(sep_drule psD, simp)
-  from h k2 have k3: "mem pos = Some v"
+  from h k2 have k3: "mem $ pos = Some v"
     by(sep_drule memD, simp)
   from h k1 k2 k3 have 
     stp: "tm.run 1 (ft, prog, cs, pos, mem) = (ft, prog, e, pos, mem)" (is "?x = ?y")
     apply(sep_drule stD)
-    by(unfold tm.run_def, cases "mem pos", simp, cases v, auto)
+    apply(unfold tm.run_def)
+    apply(cases "mem $ pos")
+    apply(simp)
+    apply(cases v)
+    apply(auto)
+    apply (metis finfun_upd_triv)
+    by (metis finfun_upd_triv)
   from h k2 
   have "(r \<and>* ps p \<and>* st e \<and>* tm p v \<and>* <(j = i + 1)> \<and>* 
            sg {TC i ((W0, e), W1, e)}) (trset_of ?x)"
@@ -4213,7 +4231,7 @@
   by (smt cond_eqI cond_true_eq sep_conj_commute sep_conj_empty)
 
 lemma sep_conj_cond4 : "(<cond1> \<and>* <cond2> \<and>* r) = (<(cond1 \<and> cond2)> \<and>* r)"
-  by (metis Hoare_gen.sep_conj_cond3 Hoare_tm.sep_conj_cond3)
+by (metis Hoare_tm2.sep_conj_cond3 sep_conj_assoc)
 
 lemmas sep_conj_cond = sep_conj_cond3 sep_conj_cond4 sep_conj_cond 
 




