Theory Fun3

section "Proof of Relative Security for fun3"
theory Fun3
imports "../Instance_IMP/Instance_Secret_IMem" 
  "Relative_Security.Unwinding_fin"
begin 


subsection "Function definition and Boilerplate"
no_notation bot (‹⊥›)

consts NN::"nat"

lemma NN:"int NN ≥ 0" by auto
consts size_aa1 :: nat
consts size_aa2 :: nat
consts mispred :: "predState ⇒ pcounter list ⇒ bool"
consts update :: "predState ⇒ pcounter list ⇒ predState"
consts initPstate :: predState

definition aa1 :: "avname" where "aa1 = ''a1''" 
definition aa2 :: "avname" where "aa2 = ''a2''" 
definition vv :: "avname" where "vv = ''v''" 
definition xx :: "avname" where "xx = ''x''" 
definition tt :: "avname" where "tt = ''t''" 


lemmas vvars_defs = aa1_def aa2_def vv_def xx_def tt_def

lemma vvars_dff[simp]: 
"aa1 ≠ aa2" "aa1 ≠ vv" "aa1 ≠ xx" "aa1 ≠ tt"
"aa2 ≠ aa1" "aa2 ≠ vv" "aa2 ≠ xx" "aa2 ≠ tt"
"vv ≠ aa1" "vv ≠ aa2"  "vv ≠ xx" "vv ≠ tt"
"xx ≠ aa1" "xx ≠ aa2" "xx ≠ vv" "xx ≠ tt"
"tt ≠ aa1" "tt ≠ aa2" "tt ≠ vv" "tt ≠ xx"
unfolding vvars_defs by auto

(* The initial avstore contains two arrays named aa1 and aa2 located one after the other *)
fun initAvstore :: "avstore ⇒ bool" where
"initAvstore (Avstore as)  = (as aa1 = (0, size_aa1) ∧ as aa2 = (size_aa1, size_aa2))"
fun istate ::"state ⇒ bool" where
"istate s = (initAvstore (getAvstore s))"

definition "prog ≡ 
[
 ⌦‹0 › Start , 
 ⌦‹1 › Input U xx ,
 ⌦‹2 › tt ::= (N 0) ,
 ⌦‹3 › IfJump (Less (V xx) (N NN)) 4 7 ,
 ⌦‹4 ›   vv ::= VA aa1 (V xx) ,
 ⌦‹5 ›   Fence ,
 ⌦‹6 ›   tt ::= (VA aa2 (Times (V vv) (N 512))) ,
 ⌦‹7 › Output U (V tt)
]"

(* *)

lemma cases_7: "(i::pcounter) = 0 ∨ i = 1 ∨ i = 2 ∨ i = 3 ∨ i = 4 ∨ i = 5 ∨ 
 i = 6 ∨ i = 7 ∨ i > 7"
apply(cases i, simp_all) 
subgoal for i apply(cases i, simp_all)
subgoal for i apply(cases i, simp_all)
subgoal for i apply(cases i, simp_all)
subgoal for i apply(cases i, simp_all)
subgoal for i apply(cases i, simp_all)
subgoal for i apply(cases i, simp_all)
subgoal for i apply(cases i, simp_all)
. . . . . . . .


lemma xx_NN_cases: "vs xx < int NN ∨ vs xx ≥ int NN" by auto

lemma is_If_pcOf[simp]: 
"pcOf cfg < 8 ⟹ is_IfJump (prog ! (pcOf cfg)) ⟷ pcOf cfg = 3"
apply(cases cfg) subgoal for pc s using cases_7[of "pcOf cfg "] 
by (auto simp: prog_def) .

lemma is_If_pc[simp]: 
"pc < 8 ⟹ is_IfJump (prog ! pc) ⟷ pc = 3"
using cases_7[of pc] 
by (auto simp: prog_def) 

lemma eq_Fence_pc[simp]: 
"pc < 8 ⟹ prog ! pc = Fence ⟷ pc = 5"
using cases_7[of pc] 
by (auto simp: prog_def) 


(* *)

consts resolve :: "predState ⇒ pcounter list ⇒ bool" 



(* *)
interpretation Prog_Mispred_Init where 
prog = prog and initPstate = initPstate and 
mispred = mispred and resolve = resolve and update = update and 
istate = istate
  by (standard, simp add: prog_def)
(* *)

(* Restoring the abbreviations: *)
abbreviation
  stepB_abbrev :: "config × val llist × val llist  ⇒ config × val llist × val llist ⇒ bool" (infix ‹→B› 55)
  where "x →B y == stepB x y"

abbreviation
  stepsB_abbrev :: "config × val llist × val llist ⇒ config × val llist × val llist ⇒ bool" (infix ‹→B*› 55)
  where "x →B* y == star stepB x y"

abbreviation
  stepM_abbrev :: "config × val llist × val llist ⇒ config × val llist × val llist ⇒ bool" (infix ‹→M› 55)
  where "x →M y == stepM x y"

abbreviation
  stepN_abbrev :: "config × val llist × val llist × loc set ⇒ config × val llist × val llist × loc set ⇒ bool" (infix ‹→N› 55)
  where "x →N y == stepN x y"

abbreviation
  stepsN_abbrev :: "config × val llist × val llist × loc set ⇒ config × val llist × val llist × loc set ⇒ bool" (infix ‹→N*› 55)
  where "x →N* y == star stepN x y"

abbreviation
  stepS_abbrev :: "configS ⇒ configS ⇒ bool" (infix ‹→S› 55)
  where "x →S y == stepS x y"

abbreviation
  stepsS_abbrev :: "configS ⇒ configS ⇒ bool" (infix ‹→S*› 55)
  where "x →S* y == star stepS x y"

(* *)

lemma endPC[simp]: "endPC = 8"
unfolding endPC_def unfolding prog_def by auto

(* *)

lemma is_getTrustedInput_pcOf[simp]: "pcOf cfg < 8 ⟹ is_getInput (prog!(pcOf cfg)) ⟷ pcOf cfg = 1"
using cases_7[of "pcOf cfg"] by (auto simp: prog_def)
lemma getUntrustedInput_pcOf[simp]: "prog!1 = Input U xx"
  by (auto simp: prog_def)

lemma getInput_not3[simp]: "¬is_getInput (prog ! 3)"
  by (auto simp: prog_def)

lemma getInput_not4[simp]: "¬is_getInput (prog ! 4)"
  by (auto simp: prog_def)
lemma Output_not4[simp]: "¬is_Output (prog ! 4)"
  by (auto simp: prog_def)

lemma is_Output_pcOf[simp]: "pcOf cfg < 8 ⟹ is_Output (prog!(pcOf cfg)) ⟷ pcOf cfg = 7"
using cases_7[of "pcOf cfg"] by (auto simp: prog_def)

lemma is_Output: "is_Output (prog ! 7)"
  unfolding is_Output_def prog_def by auto

lemma is_Fence[simp]: "(prog ! 5) = Fence"
unfolding prog_def by auto


lemma not_is_getTrustedInput[simp]:"cfg = Config 3 (State (Vstore vs) (Avstore as) (Heap h) p) ⟹ ¬ is_getInput (prog ! pcOf cfg)"
  unfolding is_getInput_def prog_def by simp


lemma not_is_Output[simp]:"cfg = Config pc (State (Vstore vs) (Avstore as) (Heap h) p) ⟹
        pc = 3 ⟹ ¬ is_Output (prog ! pcOf cfg)"
  unfolding is_Output prog_def by simp
(* *)

lemma isSecV_pcOf[simp]: 
"isSecV (cfg,ibT,ibUT) ⟷ pcOf cfg = 0"
using isSecV_def by simp

lemma isSecO_pcOf[simp]: 
"isSecO (pstate,cfg,cfgs,ibT,ibUT,ls) ⟷ (pcOf cfg = 0 ∧ cfgs = [])"
using isSecO_def by simp

(* *)

lemma getInputT_not[simp]: "pcOf cfg < 8 ⟹ 
(prog ! pcOf cfg) ≠ Input T inp"
apply(cases cfg) subgoal for pc s using cases_7[of "pcOf cfg "] 
by (auto simp: prog_def) .

lemma getActV_pcOf[simp]: 
"pcOf cfg < 8 ⟹ 
 getActV (cfg,ibT,ibUT,ls) = 
 (if pcOf cfg = 1 then lhd ibUT else ⊥)"
apply(subst getActV_simps) unfolding prog_def 
  apply simp  
  using cases_7[of "pcOf cfg"] apply(elim disjE)
  using getActV_simps not_is_getTrustedInput_getActV by auto
  
lemma getObsV_pcOf[simp]: 
"pcOf cfg < 8 ⟹  
 getObsV (cfg,ibT,ibUT,ls) = 
 (if pcOf cfg = 7 then 
  (outOf (prog!(pcOf cfg)) (stateOf cfg), ls)
  else ⊥ 
 )"
apply(subst getObsV_simps) 
  unfolding prog_def apply simp
  using getObsV_simps not_is_Output_getObsV is_Output_pcOf prog_def by presburger

lemma getActO_pcOf[simp]: 
"pcOf cfg < 8 ⟹ 
 getActO (pstate,cfg,cfgs,ibT,ibUT,ls) = 
 (if pcOf cfg = 1 ∧ cfgs = [] then lhd ibUT else ⊥)"
apply(subst getActO_simps)
apply(cases cfgs, auto)
  unfolding prog_def apply simp 
  using getActV_simps getActV_pcOf prog_def by presburger

lemma getObsO_pcOf[simp]: 
"pcOf cfg < 8 ⟹  
 getObsO (pstate,cfg,cfgs,ibT,ibUT,ls) = 
 (if (pcOf cfg = 7 ∧ cfgs = []) then 
  (outOf (prog!(pcOf cfg)) (stateOf cfg), ls)
  else ⊥ 
 )"
apply(subst getObsO_simps) 
apply(cases cfgs, auto)
unfolding prog_def apply simp
using getObsV_simps is_Output_pcOf not_is_Output_getObsV prog_def by presburger 

(* *)
(* *) 

lemma eqSec_pcOf[simp]: 
"eqSec (cfg1, ibT,ibUT1, ls1) (pstate3, cfg3, cfgs3, ibT,ibUT3, ls3) ⟷ 
 (pcOf cfg1 = 0 ⟷ pcOf cfg3 = 0 ∧ cfgs3 = []) ∧ 
 (pcOf cfg1 = 0 ⟶ stateOf cfg1 = stateOf cfg3)"
unfolding eqSec_def by simp


(* nextB, nextM and readLocs behavior 
(for nextM we are only interested at branching points, here only program counter 4): *)

lemma nextB_pc0[simp]: 
"nextB (Config 0 s, ibT,ibUT) = 
 (Config 1 s, ibT,ibUT)"
apply(subst nextB_Start_Skip_Fence)
unfolding endPC_def unfolding prog_def by auto

lemma readLocs_pc0[simp]: 
"readLocs (Config 0 s) = {}"
unfolding endPC_def readLocs_def unfolding prog_def by auto

(* *)

lemma nextB_pc1[simp]: 
"ibUT ≠ LNil ⟹ nextB (Config 1 (State (Vstore vs) avst h p), ibT,ibUT) = 
 (Config 2 (State (Vstore (vs(xx := lhd ibUT))) avst h p), ibT, ltl ibUT)"
apply(subst nextB_getUntrustedInput')
unfolding endPC_def unfolding prog_def by auto

lemma readLocs_pc1[simp]: 
"readLocs (Config 1 s) = {}"
unfolding endPC_def readLocs_def unfolding prog_def by auto

lemma nextB_pc1'[simp]: 
"ibUT ≠ LNil ⟹ nextB (Config (Suc 0) (State (Vstore vs) avst h p), ibT,ibUT) = 
 (Config 2 (State (Vstore (vs(xx := lhd ibUT))) avst h p), ibT, ltl ibUT)"
apply(subst nextB_getUntrustedInput')
unfolding endPC_def unfolding prog_def by auto

lemma readLocs_pc1'[simp]: 
"readLocs (Config (Suc 0) s) = {}"
unfolding endPC_def readLocs_def unfolding prog_def by auto

(* *)

lemma nextB_pc2[simp]:
"nextB (Config 2 (State (Vstore vs) avst h p), ibT, ibUT) = 
 (Config 3 (State (Vstore (vs(tt := 0))) avst h p), ibT, ibUT)"
apply(subst nextB_Assign)
unfolding endPC_def unfolding prog_def by auto

lemma readLocs_pc2[simp]: 
"readLocs (Config 2 s) = {}"
unfolding endPC_def readLocs_def unfolding prog_def by auto

(* *)

lemma nextB_pc3_then[simp]: 
"vs xx < int NN ⟹
 nextB (Config 3 (State (Vstore vs) avst h p), ibT, ibUT) = 
 (Config 4 (State (Vstore vs) avst h p), ibT, ibUT)"
apply(subst nextB_IfTrue)
unfolding endPC_def unfolding prog_def by auto

lemma nextB_pc3_else[simp]: 
"vs xx ≥ int NN ⟹
 nextB (Config 3 (State (Vstore vs) avst h p), ibT, ibUT) = 
 (Config 7 (State (Vstore vs) avst h p), ibT, ibUT)"
apply(subst nextB_IfFalse)
unfolding endPC_def unfolding prog_def by auto

lemma nextB_pc3: 
"nextB (Config 3 (State (Vstore vs) avst h p), ibT, ibUT) = 
 (Config (if vs xx < NN then 4 else 7) (State (Vstore vs) avst h p), ibT, ibUT)"
  by(cases "vs xx < NN", auto)

lemma nextM_pc3_then[simp]: 
"vs xx ≥ int NN ⟹
 nextM (Config 3 (State (Vstore vs) avst h p), ibT, ibUT) = 
 (Config 4 (State (Vstore vs) avst h p), ibT, ibUT)"
apply(subst nextM_IfTrue)
unfolding endPC_def unfolding prog_def by auto

lemma nextM_pc3_else[simp]: 
"vs xx < int NN ⟹
 nextM (Config 3 (State (Vstore vs) avst h p), ibT, ibUT) = 
 (Config 7 (State (Vstore vs) avst h p), ibT, ibUT)"
apply(subst nextM_IfFalse)
unfolding endPC_def unfolding prog_def by auto

lemma nextM_pc3: 
"nextM (Config 3 (State (Vstore vs) avst h p), ibT, ibUT) = 
 (Config (if vs xx < NN then 7 else 4) (State (Vstore vs) avst h p), ibT, ibUT)"
by(cases "vs xx < NN", auto)

lemma readLocs_pc3[simp]: 
"readLocs (Config 3 s) = {}"
unfolding endPC_def readLocs_def unfolding prog_def by auto


(* *)

lemma nextB_pc4[simp]: 
"nextB (Config 4 (State (Vstore vs) avst (Heap h) p), ibT,ibUT) = 
 (let l = array_loc aa1 (nat (vs xx)) avst 
  in (Config 5 (State (Vstore (vs(vv := h l))) avst (Heap h) p)), ibT,ibUT)"
apply(subst nextB_Assign)
unfolding endPC_def unfolding prog_def by auto

lemma readLocs_pc4[simp]: 
"readLocs (Config 4 (State (Vstore vs) avst h p)) = {array_loc aa1 (nat (vs xx)) avst}"
unfolding endPC_def readLocs_def unfolding prog_def by auto

(* *)

lemma nextB_pc5[simp]: 
"nextB (Config 5 s, ibT,ibUT) = (Config 6 s, ibT,ibUT)"
apply(subst nextB_Start_Skip_Fence)
unfolding endPC_def unfolding prog_def by auto

lemma readLocs_pc5[simp]: 
"readLocs (Config 5 s) = {}"
unfolding endPC_def readLocs_def unfolding prog_def by auto

(* *)

lemma nextB_pc6[simp]:
"nextB (Config 6 (State (Vstore vs) avst (Heap h) p), ibT,ibUT) =  
  (let l = array_loc aa2 (nat (vs vv * 512)) avst
  in (Config 7 (State (Vstore (vs(tt := h l))) avst (Heap h) p)), ibT,ibUT)"
apply(subst nextB_Assign)
unfolding endPC_def unfolding prog_def by auto


lemma readLocs_pc6[simp]: 
"readLocs (Config 6 (State (Vstore vs) avst h p)) = {array_loc aa2 (nat (vs vv * 512)) avst}"
unfolding endPC_def readLocs_def unfolding prog_def by auto

(* *)

lemma nextB_pc7[simp]:
"nextB (Config 7 s, ibT,ibUT) = (Config 8 s, ibT,ibUT)"
apply(subst nextB_Output)
unfolding endPC_def unfolding prog_def by auto 

lemma readLocs_pc7[simp]: 
"readLocs (Config 7 s) = {}"
unfolding endPC_def readLocs_def unfolding prog_def by auto

(* *)

lemma nextB_stepB_pc: 
"pc < 8 ⟹ (pc = 1 ⟶ ibUT ≠ LNil) ⟹ 
(Config pc s, ibT,ibUT) →B nextB (Config pc s, ibT,ibUT)"
apply(cases s) subgoal for vst avst hh p apply(cases vst, cases avst, cases hh)
subgoal for vs as h
using cases_7[of pc] apply safe
  subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
  by (simp add: prog_def)
  subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
  by (simp add: prog_def)
  (* *)
  subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
  by (simp add: prog_def, metis llist.collapse)
  subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
  by (simp add: prog_def)
  subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
  by (simp add: prog_def)
  (* *)
  subgoal apply(cases "vs xx < NN")  
    subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
    by (simp add: prog_def) 
    subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
    by (simp add: prog_def) .
  subgoal apply(cases "vs xx < NN")  
    subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
    by (simp add: prog_def) 
    subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
    by (simp add: prog_def) .
  (* *)
  subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
  by (simp add: prog_def) 
  subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
  by (simp add: prog_def) 
  (* *)
  subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
  by (simp add: prog_def) 
  subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
  by (simp add: prog_def) 
  (* *)
  subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
  by (simp add: prog_def) 
  subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
  by (simp add: prog_def) 
  (* *)
  subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
  by (simp add: prog_def) 
  subgoal apply simp apply(subst stepB.simps) unfolding endPC_def 
  by (simp add: prog_def) 
  subgoal by auto          
  subgoal by auto 
  . . .

lemma not_finalB: 
"pc < 8 ⟹ (pc = 1 ⟶ ibUT ≠ LNil) ⟹ 
 ¬ finalB (Config pc s, ibT,ibUT)"
using nextB_stepB_pc by (simp add: stepB_iff_nextB)

lemma finalB_pc_iff': 
"pc < 8 ⟹
 finalB (Config pc s, ibT,ibUT) ⟷ 
 (pc = 1 ∧ ibUT = LNil)"
  subgoal apply safe
    subgoal using nextB_stepB_pc[of pc] by (auto simp add: stepB_iff_nextB) 
    subgoal using nextB_stepB_pc[of pc] by (auto simp add: stepB_iff_nextB) 
    subgoal using finalB_iff getUntrustedInput_pcOf by auto . .

lemma finalB_pc_iff: 
"pc ≤ 8 ⟹
 finalB (Config pc s, ibT,ibUT) ⟷ 
 (pc = 1 ∧ ibUT = LNil ∨ pc = 8)"
  using cases_7[of pc] apply (elim disjE, simp add: finalB_def)
  subgoal by (meson final_def stebB_0)
  by (simp add: finalB_pc_iff' finalB_endPC)+

lemma finalB_pcOf_iff[simp]:
"pcOf cfg ≤ 8 ⟹ 
 finalB (cfg, ibT,ibUT) ⟷ (pcOf cfg = 1 ∧ ibUT = LNil ∨ pcOf cfg = 8)"
by (metis config.collapse finalB_pc_iff) 




lemma finalS_cond:"pcOf cfg < 8 ⟹ cfgs = [] ⟹ (pcOf cfg = 1 ⟶ ibUT ≠ LNil) ⟹ ¬ finalS (pstate, cfg, cfgs, ibT,ibUT, ls)"
apply(rule notI, cases cfg)
subgoal for pc s apply(cases s)
subgoal for vst avst hh p apply(cases vst, cases avst, cases hh)
  subgoal for vs as h
    using cases_7[of pc] apply(elim disjE) unfolding finalS_defs
    subgoal by(erule allE[of _ "(pstate, Config 1 (State (Vstore vs) avst hh p), [], ibT,ibUT, ls)"], erule notE, rule nonspec_normal, auto)

      subgoal apply(frule nonspec_normal[of cfgs "Config pc (State (Vstore vs) avst hh p)" 
                                    pstate pstate ibT ibUT 
                                    "Config 2 (State (Vstore (vs(xx:= lhd ibUT))) avst hh p)" 
                                    ibT "ltl ibUT" "[]" "ls ∪ readLocs (Config pc (State (Vstore vs) avst hh p))" ls])
              prefer 7 subgoal by metis by simp_all
      subgoal apply(frule nonspec_normal[of cfgs "Config pc (State (Vstore vs) avst hh p)" 
                                    pstate pstate ibT ibUT 
                                    "Config 3 (State (Vstore (vs(tt:= 0))) avst hh p)" 
                                    ibT ibUT "[]" "ls ∪ readLocs (Config pc (State (Vstore vs) avst hh p))" ls])
              prefer 7 subgoal by metis by simp_all

    subgoal apply(cases "mispred pstate [3]")
      subgoal by(erule allE[of _ "(update pstate [pcOf (Config pc (State (Vstore vs) avst hh p))], 
                                   Config (if vs xx < NN then 4 else 7) (State (Vstore vs) avst hh p), 
                                  [Config (if vs xx < NN then 7 else 4) (State (Vstore vs) avst hh p)], 
                                   ibT,ibUT, ls)"], erule notE, rule nonspec_mispred, auto simp: finalM_iff)

      subgoal apply(frule nonspec_normal[of cfgs "Config pc (State (Vstore vs) avst hh p)"
                                    pstate pstate ibT ibUT 
                                    "Config (if vs xx < NN then 4 else 7) (State (Vstore vs) avst hh p)" 
                                    ibT ibUT "[]" "ls ∪ readLocs (Config pc (State (Vstore vs) avst hh p))" ls])
              prefer 7 subgoal by metis apply simp_all by (simp add: nextB_pc3) .

    subgoal by(erule allE[of _ "(pstate, Config 5 (State (Vstore (vs(vv := h (array_loc aa1 (nat (vs xx)) avst)))) avst hh p), 
                                                                [], ibT,ibUT, ls ∪ {array_loc aa1 (nat (vs xx)) avst})"],
               erule notE, rule nonspec_normal, auto)

    subgoal by(erule allE[of _ "(pstate, Config 6 (State (Vstore vs) avst hh p), [], ibT,ibUT, ls)"], erule notE, rule nonspec_normal, auto)

    subgoal by(erule allE[of _ "(pstate, Config 7 (State (Vstore (vs(tt := h (array_loc aa2 (nat (vs vv * 512)) (Avstore as))))) avst hh p), 
                                 [], ibT,ibUT, ls ∪ {array_loc aa2 (nat (vs vv * 512)) (Avstore as)})"],
              erule notE, rule nonspec_normal, auto)

    subgoal by(erule allE[of _ "(pstate, Config 8 (State (Vstore vs) avst hh p), [], ibT,ibUT, ls)"], erule notE, rule nonspec_normal, auto)

    by simp_all . . .



lemma finalS_cond_spec:
      "pcOf cfg < 8 ⟹ 
       (((pcOf (last cfgs) = 4 ∨ pcOf (last cfgs) = 5) ∧ pcOf cfg = 7) ∨ 
        (pcOf (last cfgs) = 7 ∧ pcOf cfg = 4)) ⟹ 
       length cfgs = Suc 0 ⟹
      ¬ finalS (pstate, cfg, cfgs, ibT,ibUT, ls)"
using not_is_getTrustedInput not_is_Output
  apply(cases cfg)
  subgoal for pc s apply(cases s)
  subgoal for vst avst hh p apply(cases vst, cases avst, cases hh)
  subgoal for vs as h apply(cases "last cfgs")
  subgoal for pcs ss apply(cases ss)
    subgoal for vsts avsts hhs ps apply(cases vsts, cases avsts, cases hhs, simp)
      apply(cases "resolve pstate (pcOf cfg # map pcOf cfgs)")
    subgoal unfolding finalS_defs using spec_resolve by (metis list.size(3) n_not_Suc_n)
    subgoal for vss ass hs apply(elim disjE, elim conjE, elim disjE, simp) 
      unfolding finalS_defs
      subgoal apply(rule notI, 
      erule allE[of _ "(pstate,Config 7 (State (Vstore vs) (Avstore as) (Heap h) p),
                        [Config 5 (State (Vstore (vss(vv := hs (array_loc aa1 (nat (vss xx)) avsts)))) avsts hhs ps)],
                        ibT,ibUT,ls ∪ readLocs (last cfgs))"])
        by(erule notE, 
        rule spec_normal[of _ _ _ _ _ _"Config 5 (State (Vstore (vss(vv := hs (array_loc aa1 (nat (vss xx)) avsts)))) avsts hhs ps)"], auto)
      
      subgoal apply(rule notI, 
      erule allE[of _ "(pstate,Config 7 (State (Vstore vs) (Avstore as) (Heap h) p),[],ibT,ibUT,ls)"])
        using spec_Fence by (metis Zero_not_Suc config.sel(1) is_Fence length_0_conv)

      subgoal apply(rule notI, 
      erule allE[of _ "(update pstate (4 # map pcOf cfgs),Config 4 (State (Vstore vs) (Avstore as) (Heap h) p),
                       [],ibT,ibUT,ls)"])
        using spec_resolve 
        by (metis config.sel(1) is_Output length_1_butlast length_greater_0_conv less_Suc0)
      . . . . . . . 
end