tm2: comparison thys/Hoare

equal deleted inserted replaced

-:d826899bc424
+:087d82632852
 theory Hoare_tm2
 imports Hoare_gen
 My_block
 Data_slot
-"~~/src/HOL/Library/FinFun_Syntax"
+FMap
 begin
 ML {*
 fun pretty_terms ctxt trms =
 - TM program (nat \<rightharpoonup> tm_inst)
 - current state (nat)
 - 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 W0 (pos, m) = (pos, m(pos f\<mapsto> Bk))" |
-"next_tape W1 (pos, m) = (pos, m(pos $:= Some Oc))" |
+"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)
+| None \<Rightarrow> (faults + 1, prog, pc, pos, m))"
 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>
+prog f! pc = Some ((action0, pc0), (action1, pc1)) \<longrightarrow>
-m $ pos = Some Bk \<longrightarrow> P (faults, prog, pc0, next_tape action0 (pos, m))) \<and>
+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>
+prog f! pc = Some ((action0, pc0), (action1, pc1)) \<longrightarrow>
-m $ pos = Some Oc \<longrightarrow> P (faults, prog, pc1, next_tape action1 (pos, m))) \<and>
+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>
+prog f! pc = Some ((action0, pc0), (action1, pc1)) \<longrightarrow>
-m $ pos = None \<longrightarrow> P (faults + 1, prog, pc, pos, m)) \<and>
+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)
 datatype tresource =
 TM int Block      (* at the position int of the tape is a Bl or Oc *)
 | TC nat tm_inst    (* at the address nat is a certain instruction *)
 | TAt nat           (* TM is at state nat*)
 | 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}"
 lemmas tpn_set_def = tprog_set_def tpc_set_def tmem_set_def tfaults_set_def tpos_set_def
 "tpg_len (TLabel l) = 0"
 *)
 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 (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') ** (tassemble_to p2 j' j))" |
+"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)>"
 declare tassemble_to.simps [simp del]
 by simp
 thus ?thesis
 by (unfold tpn_set_def, auto)
 qed
-lemma codeD: "(st i ** sg {TC i inst} ** r) (trset_of (ft, prog, i', pos, mem))
+lemma codeD: "(st i \<and>* sg {TC i inst} ** r) (trset_of (ft, prog, i', pos, mem))
-\<Longrightarrow> prog $ i = Some inst"
+\<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>
+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 $ i = Some n} \<union> {TFaults ft}" by simp
+{TPos pos} \<union> {TM i n |i n. mem f! i = Some n} \<union> {TFaults ft}" by simp
 thus ?thesis by auto
 qed
 lemma t_hoare_seq:
 assumes a1: "\<And> i j. \<lbrace>st i \<and>* p\<rbrace> i:[c1]:j \<lbrace>st j \<and>* q\<rbrace>"
 ({TM a v} \<subseteq> tmem_set mem)"
 by (unfold tpn_set_def, auto)
 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 the.simps the_lookup_fmap_upd the_lookup_fmap_upd_other)
-apply (metis finfun_upd_apply_other)
+apply (metis the_lookup_fmap_upd_other)
-by (metis finfun_upd_apply_other option.inject)
+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)"
 by (unfold tpn_set_def, auto)
 assume g: "(tm a v \<and>* r) (trset_of (f, x, y, z, mem))"
 hence h0: "r (tprog_set x \<union> tpc_set y \<union> tpos_set z \<union> (tmem_set mem - {TM a v}) \<union> tfaults_set f)"
 by(sep_drule stimes_sgD, clarify, unfold eq_s, auto)
 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
 from h0 show "r (tprog_set x \<union> tpc_set y \<union> tpos_set z \<union> (tmem_set mem - {TM a v}) \<union> tfaults_set f)" .
 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
 thus ?thesis
 apply (unfold trset_of.simps)
 \<lbrace>st (Suc i) \<and>* ps p \<and>* tm p Bk\<rbrace>"
 proof(fold eq_j, unfold tassemble_to.simps tm.Hoare_gen_def sep_conj_ac, clarify)
 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)
 by(cases v, unfold tm.run_def, auto)
 from h have "i' = p"
 \<lbrace>ps p \<and>* st ?j \<and>* tm p Oc\<rbrace>"
 proof(unfold tassemble_to.simps tm.Hoare_gen_def sep_conj_ac, clarify)
 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)
 by(cases v, unfold tm.run_def, auto)
 from h have "i' = p"
 \<lbrace>st ?j \<and>* tm p v2 \<and>* ps (p - 1)\<rbrace>"
 proof(unfold tassemble_to.simps tm.Hoare_gen_def sep_conj_ac, clarify)
 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:
 "tm.run 1 (ft, prog, cs, i', mem) = (ft, prog, ?j, i' - 1, mem)" (is "?x = ?y")
 apply(sep_frule psD)
 \<lbrace>st ?j \<and>* tm p v1 \<and>* ps (p + 1)\<rbrace>"
 proof(unfold tassemble_to.simps tm.Hoare_gen_def sep_conj_ac, clarify)
 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:
 "tm.run 1 (ft, prog, cs, i', mem) = (ft, prog, ?j, i'+ 1, mem)" (is "?x = ?y")
 apply(sep_frule psD)
 show "\<lbrace>one p \<and>* ps p \<and>* st i\<rbrace>  sg {TC i ((W0, ?j), W1, e)} \<lbrace>one p \<and>* ps p \<and>* st e\<rbrace>"
 proof(unfold tassemble_to.simps tm.Hoare_gen_def sep_conj_ac, clarify)
 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:
 "tm.run 1 (ft, prog, cs, pc, mem) = (ft, prog, e, pc, mem)" (is "?x = ?y")
 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
 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)
 show "\<lbrace>ps p \<and>* st i \<and>* zero p\<rbrace>  sg {TC i ((W0, ?j), W1, e)} \<lbrace>ps p \<and>*  zero p \<and>* st ?j \<rbrace>"
 proof(unfold tassemble_to.simps tm.Hoare_gen_def sep_conj_ac, clarify)
 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:
 "tm.run 1 (ft, prog, cs, pc, mem) = (ft, prog, ?j, pc, mem)" (is "?x = ?y")
 apply(sep_drule stD)
 show "\<lbrace>ps p \<and>* st i \<and>* zero p\<rbrace>  sg {TC i ((W0, e), W1, ?j)} \<lbrace>ps p \<and>* st e \<and>* zero p\<rbrace>"
 proof(unfold tassemble_to.simps tm.Hoare_gen_def sep_conj_ac, clarify)
 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:
 "tm.run 1 (ft, prog, cs, pc, mem) = (ft, prog, e, pc, mem)" (is "?x = ?y")
 apply(sep_drule stD)
 \<lbrace>ps p \<and>* st ?j \<and>* tm p Oc\<rbrace>"
 proof(unfold tassemble_to.simps tm.Hoare_gen_def sep_conj_ac, clarify)
 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")
 apply(sep_drule stD)
 apply(unfold tm.run_def)
 "\<lbrace>st i \<and>* ps p \<and>* tm p v\<rbrace>  i:[jmp e]:j \<lbrace>st e \<and>* ps p \<and>* tm p v\<rbrace>"
 proof(unfold jmp_def tm.Hoare_gen_def tassemble_to.simps sep_conj_ac, clarify)
 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)
 apply (metis finfun_upd_triv)
 by (metis finfun_upd_triv)

changeset 19	087d82632852
parent 17	86918b45b2e6