
diff -r d826899bc424 -r 087d82632852 thys/Hoare_tm2.thy
--- a/thys/Hoare_tm2.thy	Fri Apr 04 13:15:07 2014 +0100
+++ b/thys/Hoare_tm2.thy	Tue Apr 29 15:26:48 2014 +0100
@@ -6,7 +6,7 @@
 imports Hoare_gen 
         My_block 
         Data_slot
-        "~~/src/HOL/Library/FinFun_Syntax"
+        FMap
 begin
 
 
@@ -64,22 +64,33 @@
    - position of head (int)
    - tape (int \<rightharpoonup> Block)
 *)
-type_synonym tconf = "nat \<times> (nat \<Rightarrow>f tm_inst option) \<times> nat \<times> int \<times> (int \<Rightarrow>f Block option)"
+type_synonym tconf = "nat \<times> (nat f\<rightharpoonup> tm_inst) \<times> nat \<times> int \<times> (int f\<rightharpoonup> Block)"
 
 (* updates the position/tape according to an action *)
 fun 
-  next_tape :: "taction \<Rightarrow> (int \<times>  (int \<Rightarrow>f Block option)) \<Rightarrow> (int \<times>  (int \<Rightarrow>f Block option))"
+  next_tape :: "taction \<Rightarrow> (int \<times>  (int f\<rightharpoonup> Block)) \<Rightarrow> (int \<times>  (int f\<rightharpoonup> Block))"
 where 
-  "next_tape W0 (pos, m) = (pos, m(pos $:= Some Bk))" |
-  "next_tape W1 (pos, m) = (pos, m(pos $:= Some Oc))" |
+  "next_tape W0 (pos, m) = (pos, m(pos f\<mapsto> Bk))" |
+  "next_tape W1 (pos, m) = (pos, m(pos f\<mapsto> Oc))" |
   "next_tape L  (pos, m) = (pos - 1, m)" |
   "next_tape R  (pos, m) = (pos + 1, m)"
 
 fun tstep :: "tconf \<Rightarrow> tconf"
   where "tstep (faults, prog, pc, pos, m) = 
-              (case (prog $ pc) of
+              (case (prog f! pc) of
                   Some ((action0, pc0), (action1, pc1)) \<Rightarrow> 
-                     case (m $ pos) of
+                     case (m f! pos) of
+                       Some Bk \<Rightarrow> (faults, prog, pc0, next_tape action0 (pos, m)) |
+                       Some Oc \<Rightarrow> (faults, prog, pc1, next_tape action1 (pos, m)) |
+                       None \<Rightarrow> (faults + 1, prog, pc, pos, m)
+                | None \<Rightarrow> (faults + 1, prog, pc, pos, m))"
+
+
+fun tstep :: "tconf \<Rightarrow> tconf"
+  where "tstep (faults, prog, pc, pos, m) = 
+              (case (prog f! pc) of
+                  Some ((action0, pc0), (action1, pc1)) \<Rightarrow> 
+                     case (m f! pos) of
                        Some Bk \<Rightarrow> (faults, prog, pc0, next_tape action0 (pos, m)) |
                        Some Oc \<Rightarrow> (faults, prog, pc1, next_tape action1 (pos, m)) |
                        None \<Rightarrow> (faults + 1, prog, pc, pos, m)
@@ -88,19 +99,19 @@
 lemma tstep_splits: 
   "(P (tstep s)) = ((\<forall> faults prog pc pos m action0 pc0 action1 pc1. 
                           s = (faults, prog, pc, pos, m) \<longrightarrow> 
-                          prog $ pc = Some ((action0, pc0), (action1, pc1)) \<longrightarrow> 
-                          m $ pos = Some Bk \<longrightarrow> P (faults, prog, pc0, next_tape action0 (pos, m))) \<and>
+                          prog f! pc = Some ((action0, pc0), (action1, pc1)) \<longrightarrow> 
+                          m f! pos = Some Bk \<longrightarrow> P (faults, prog, pc0, next_tape action0 (pos, m))) \<and>
                     (\<forall> faults prog pc pos m action0 pc0 action1 pc1. 
                           s = (faults, prog, pc, pos, m) \<longrightarrow> 
-                          prog $ pc = Some ((action0, pc0), (action1, pc1)) \<longrightarrow> 
-                          m $ pos = Some Oc \<longrightarrow> P (faults, prog, pc1, next_tape action1 (pos, m))) \<and>
+                          prog f! pc = Some ((action0, pc0), (action1, pc1)) \<longrightarrow> 
+                          m f! pos = Some Oc \<longrightarrow> P (faults, prog, pc1, next_tape action1 (pos, m))) \<and>
                     (\<forall> faults prog pc pos m action0 pc0 action1 pc1. 
                           s = (faults, prog, pc, pos, m) \<longrightarrow> 
-                          prog $ pc = Some ((action0, pc0), (action1, pc1)) \<longrightarrow> 
-                          m $ pos = None \<longrightarrow> P (faults + 1, prog, pc, pos, m)) \<and>
+                          prog f! pc = Some ((action0, pc0), (action1, pc1)) \<longrightarrow> 
+                          m f! pos = None \<longrightarrow> P (faults + 1, prog, pc, pos, m)) \<and>
                     (\<forall> faults prog pc pos m . 
                           s =  (faults, prog, pc, pos, m) \<longrightarrow>
-                          prog $ pc  = None \<longrightarrow> P (faults + 1, prog, pc, pos, m))
+                          prog f! pc  = None \<longrightarrow> P (faults + 1, prog, pc, pos, m))
                    )"
   by (cases s) (auto split: option.splits Block.splits)
 
@@ -111,9 +122,9 @@
   | TPos int          (* head of TM is at position int *)
   | TFaults nat       (* there are nat faults *)
 
-definition "tprog_set prog = {TC i inst | i inst. prog $ i = Some inst}"
+definition "tprog_set prog = {TC i inst | i inst. prog f! i = Some inst}"
 definition "tpc_set pc = {TAt pc}"
-definition "tmem_set m = {TM i n | i n. m $ i = Some n}"
+definition "tmem_set m = {TM i n | i n. m f! i = Some n}"
 definition "tpos_set pos = {TPos pos}"
 definition "tfaults_set faults = {TFaults faults}"
 
@@ -170,8 +181,8 @@
 
 primrec tassemble_to :: "tpg \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> tassert" 
   where 
-  "tassemble_to (TInstr ai) i j = (sg ({TC i ai}) ** <(j = i + 1)>)" |
-  "tassemble_to (TSeq p1 p2) i j = (EXS j'. (tassemble_to p1 i j') ** (tassemble_to p2 j' j))" |
+  "tassemble_to (TInstr ai) i j = (sg ({TC i ai}) \<and>* <(j = i + 1)>)" |
+  "tassemble_to (TSeq p1 p2) i j = (EXS j'. (tassemble_to p1 i j') \<and>* (tassemble_to p2 j' j))" |
   "tassemble_to (TLocal fp) i j  = (EXS l. (tassemble_to (fp l) i j))" | 
   "tassemble_to (TLabel l) i j = <(i = j \<and> j = l)>"
 
@@ -405,21 +416,21 @@
     by (unfold tpn_set_def, auto)
 qed
 
-lemma codeD: "(st i ** sg {TC i inst} ** r) (trset_of (ft, prog, i', pos, mem))
-       \<Longrightarrow> prog $ i = Some inst"
+lemma codeD: "(st i \<and>* sg {TC i inst} ** r) (trset_of (ft, prog, i', pos, mem))
+       \<Longrightarrow> prog f! i = Some inst"
 proof -
-  assume "(st i ** sg {TC i inst} ** r) (trset_of (ft, prog, i', pos, mem))"
+  assume "(st i \<and>* sg {TC i inst} \<and>* 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) 
     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 f! 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 f! i = Some inst} \<union> {TAt i} \<union> 
+    {TPos pos} \<union> {TM i n |i n. mem f! i = Some n} \<union> {TFaults ft}" by simp
   thus ?thesis by auto
 qed
 
@@ -1212,20 +1223,19 @@
 
 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'}"
+        tmem_set (mem(a f\<mapsto> 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)
-
+apply (metis the.simps the_lookup_fmap_upd the_lookup_fmap_upd_other)
+apply (metis the_lookup_fmap_upd_other)
+by (metis option.inject the_lookup_fmap_upd_other)
 
 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 f\<mapsto> 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)"
@@ -1238,11 +1248,11 @@
   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 $:= Some v')) = {TM a v'} \<union> (tmem_set mem - {TM a v})" 
+  have h2: "tmem_set (mem(a f\<mapsto> 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 $:= Some 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 f\<mapsto> 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
@@ -1250,13 +1260,13 @@
     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 f! 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 $:= Some v')) \<union> tprog_set x \<union> tpc_set y \<union> tpos_set z \<union> tfaults_set f =
+      show "tmem_set (mem(a f\<mapsto> 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
@@ -1281,11 +1291,11 @@
       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 f! 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' $:= Some Bk))" (is "?x = ?y")
+        "tm.run 1 (ft, prog, cs, i', mem) = (ft, prog, j, i', mem(i' f\<mapsto> Some Bk))" (is "?x = ?y")
         apply(sep_frule psD)
         apply(sep_frule stD)
         apply(sep_frule memD, simp)
@@ -1331,12 +1341,12 @@
       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 f! 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' $:= Some Oc))" (is "?x = ?y")
+                     (ft, prog, ?j, i', mem(i' f\<mapsto> Some Oc))" (is "?x = ?y")
         apply(sep_frule psD)
         apply(sep_frule stD)
         apply(sep_frule memD, simp)
@@ -1383,7 +1393,7 @@
       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 f! 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:
@@ -1441,7 +1451,7 @@
       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 f! 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:
@@ -1497,12 +1507,12 @@
       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 f! 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 f! pc = Some Oc"
         apply(unfold one_def)
         by(sep_drule memD, simp)
       from h k1 k2 k3 have stp:
@@ -1510,6 +1520,14 @@
         apply(sep_drule stD)
         apply(unfold tm.run_def)
         apply(auto)
+        apply(rule fmap_eqI)
+        apply(simp)
+        apply(subgoal_tac "p \<in> fdom mem")
+        apply(simp add: insert_absorb)
+        apply(simp add: fdomIff)
+        apply(rule_tac x="Some Oc" in exI)
+        apply(auto)[1]
+       apply(simp add: fun_eq_iff)
         by (metis finfun_upd_triv)
 
       from h k2 
@@ -1584,12 +1602,12 @@
       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 f! 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 f! pc = Some Bk"
         apply(unfold zero_def)
         by(sep_drule memD, simp)
       from h k1 k2 k3 have stp:
@@ -1633,12 +1651,12 @@
       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 f! 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 f! pc = Some Bk"
         apply(unfold zero_def)
         by(sep_drule memD, simp)
       from h k1 k2 k3 have stp:
@@ -1712,12 +1730,12 @@
       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 f! 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 f! 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")
@@ -1753,18 +1771,18 @@
   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 f! 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 f! 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)
     apply(unfold tm.run_def)
-    apply(cases "mem $ pos")
+    apply(cases "mem f! pos")
     apply(simp)
     apply(cases v)
     apply(auto)




