Theory SchorrWaite

theory SchorrWaite
imports HeapSyntax
(*  Title:      HOL/Hoare/SchorrWaite.thy
    Author:     Farhad Mehta
    Copyright   2003 TUM

Proof of the Schorr-Waite graph marking algorithm.
*)


theory SchorrWaite imports HeapSyntax begin

section {* Machinery for the Schorr-Waite proof*}

definition
  -- "Relations induced by a mapping"
  rel :: "('a => 'a ref) => ('a × 'a) set"
  where "rel m = {(x,y). m x = Ref y}"

definition
  relS :: "('a => 'a ref) set => ('a × 'a) set"
  where "relS M = (\<Union> m ∈ M. rel m)"

definition
  addrs :: "'a ref set => 'a set"
  where "addrs P = {a. Ref a ∈ P}"

definition
  reachable :: "('a × 'a) set => 'a ref set => 'a set"
  where "reachable r P = (r* `` addrs P)"

lemmas rel_defs = relS_def rel_def

text {* Rewrite rules for relations induced by a mapping*}

lemma self_reachable: "b ∈ B ==> b ∈ R* `` B"
apply blast
done

lemma oneStep_reachable: "b ∈ R``B ==> b ∈ R* `` B"
apply blast
done

lemma still_reachable: "[|B⊆Ra*``A; ∀ (x,y) ∈ Rb-Ra. y∈ (Ra*``A)|] ==> Rb* `` B ⊆ Ra* `` A "
apply (clarsimp simp only:Image_iff)
apply (erule rtrancl_induct)
 apply blast
apply (subgoal_tac "(y, z) ∈ Ra∪(Rb-Ra)")
 apply (erule UnE)
 apply (auto intro:rtrancl_into_rtrancl)
apply blast
done

lemma still_reachable_eq: "[| A⊆Rb*``B; B⊆Ra*``A; ∀ (x,y) ∈ Ra-Rb. y ∈(Rb*``B); ∀ (x,y) ∈ Rb-Ra. y∈ (Ra*``A)|] ==> Ra*``A =  Rb*``B "
apply (rule equalityI)
 apply (erule still_reachable ,assumption)+
done

lemma reachable_null: "reachable mS {Null} = {}"
apply (simp add: reachable_def addrs_def)
done

lemma reachable_empty: "reachable mS {} = {}"
apply (simp add: reachable_def addrs_def)
done

lemma reachable_union: "(reachable mS aS ∪ reachable mS bS) = reachable mS (aS ∪ bS)"
apply (simp add: reachable_def rel_defs addrs_def)
apply blast
done

lemma reachable_union_sym: "reachable r (insert a aS) = (r* `` addrs {a}) ∪ reachable r aS"
apply (simp add: reachable_def rel_defs addrs_def)
apply blast
done

lemma rel_upd1: "(a,b) ∉ rel (r(q:=t)) ==> (a,b) ∈ rel r ==> a=q"
apply (rule classical)
apply (simp add:rel_defs fun_upd_apply)
done

lemma rel_upd2: "(a,b)  ∉ rel r ==> (a,b) ∈ rel (r(q:=t)) ==> a=q"
apply (rule classical)
apply (simp add:rel_defs fun_upd_apply)
done

definition
  -- "Restriction of a relation"
  restr ::"('a × 'a) set => ('a => bool) => ('a × 'a) set"       ("(_/ | _)" [50, 51] 50)
  where "restr r m = {(x,y). (x,y) ∈ r ∧ ¬ m x}"

text {* Rewrite rules for the restriction of a relation *}

lemma restr_identity[simp]:
 " (∀x. ¬ m x) ==> (R |m) = R"
by (auto simp add:restr_def)

lemma restr_rtrancl[simp]: " [|m l|] ==> (R | m)* `` {l} = {l}"
by (auto simp add:restr_def elim:converse_rtranclE)

lemma [simp]: " [|m l|] ==> (l,x) ∈ (R | m)* = (l=x)"
by (auto simp add:restr_def elim:converse_rtranclE)

lemma restr_upd: "((rel (r (q := t)))|(m(q := True))) = ((rel (r))|(m(q := True))) "
apply (auto simp:restr_def rel_def fun_upd_apply)
apply (rename_tac a b)
apply (case_tac "a=q")
 apply auto
done

lemma restr_un: "((r ∪ s)|m) = (r|m) ∪ (s|m)"
  by (auto simp add:restr_def)

lemma rel_upd3: "(a, b) ∉ (r|(m(q := t))) ==> (a,b) ∈ (r|m) ==> a = q "
apply (rule classical)
apply (simp add:restr_def fun_upd_apply)
done

definition
  -- "A short form for the stack mapping function for List"
  S :: "('a => bool) => ('a => 'a ref) => ('a => 'a ref) => ('a => 'a ref)"
  where "S c l r = (λx. if c x then r x else l x)"

text {* Rewrite rules for Lists using S as their mapping *}

lemma [rule_format,simp]:
 "∀p. a ∉ set stack --> List (S c l r) p stack = List (S (c(a:=x)) (l(a:=y)) (r(a:=z))) p stack"
apply(induct_tac stack)
 apply(simp add:fun_upd_apply S_def)+
done

lemma [rule_format,simp]:
 "∀p. a ∉ set stack --> List (S c l (r(a:=z))) p stack = List (S c l r) p stack"
apply(induct_tac stack)
 apply(simp add:fun_upd_apply S_def)+
done

lemma [rule_format,simp]:
 "∀p. a ∉ set stack --> List (S c (l(a:=z)) r) p stack = List (S c l r) p stack"
apply(induct_tac stack)
 apply(simp add:fun_upd_apply S_def)+
done

lemma [rule_format,simp]:
 "∀p. a ∉ set stack --> List (S (c(a:=z)) l r) p stack = List (S c l r) p stack"
apply(induct_tac stack)
 apply(simp add:fun_upd_apply S_def)+
done

primrec
  --"Recursive definition of what is means for a the graph/stack structure to be reconstructible"
  stkOk :: "('a => bool) => ('a => 'a ref) => ('a => 'a ref) => ('a => 'a ref) => ('a => 'a ref) => 'a ref =>'a list =>  bool"
where
  stkOk_nil:  "stkOk c l r iL iR t [] = True"
| stkOk_cons:
    "stkOk c l r iL iR t (p#stk) = (stkOk c l r iL iR (Ref p) (stk) ∧ 
      iL p = (if c p then l p else t) ∧
      iR p = (if c p then t else r p))"

text {* Rewrite rules for stkOk *}

lemma [simp]: "!!t. [| x ∉ set xs; Ref x≠t |] ==>
  stkOk (c(x := f)) l r iL iR t xs = stkOk c l r iL iR t xs"
apply (induct xs)
 apply (auto simp:eq_sym_conv)
done

lemma [simp]: "!!t. [| x ∉ set xs; Ref x≠t |] ==>
 stkOk c (l(x := g)) r iL iR t xs = stkOk c l r iL iR t xs"
apply (induct xs)
 apply (auto simp:eq_sym_conv)
done

lemma [simp]: "!!t. [| x ∉ set xs; Ref x≠t |] ==>
 stkOk c l (r(x := g)) iL iR t xs = stkOk c l r iL iR t xs"
apply (induct xs)
 apply (auto simp:eq_sym_conv)
done

lemma stkOk_r_rewrite [simp]: "!!x. x ∉ set xs ==>
  stkOk c l (r(x := g)) iL iR (Ref x) xs = stkOk c l r iL iR (Ref x) xs"
apply (induct xs)
 apply (auto simp:eq_sym_conv)
done

lemma [simp]: "!!x. x ∉ set xs ==>
 stkOk c (l(x := g)) r iL iR (Ref x) xs = stkOk c l r iL iR (Ref x) xs"
apply (induct xs)
 apply (auto simp:eq_sym_conv)
done

lemma [simp]: "!!x. x ∉ set xs ==>
 stkOk (c(x := g)) l r iL iR (Ref x) xs = stkOk c l r iL iR (Ref x) xs"
apply (induct xs)
 apply (auto simp:eq_sym_conv)
done


section{*The Schorr-Waite algorithm*}


theorem SchorrWaiteAlgorithm: 
"VARS c m l r t p q root
 {R = reachable (relS {l, r}) {root} ∧ (∀x. ¬ m x) ∧ iR = r ∧ iL = l} 
 t := root; p := Null;
 WHILE p ≠ Null ∨ t ≠ Null ∧ ¬ t^.m
 INV {∃stack.
          List (S c l r) p stack ∧                                         (*i1*)
          (∀x ∈ set stack. m x) ∧                                        (*i2*)
          R = reachable (relS{l, r}) {t,p} ∧                           (*i3*)
          (∀x. x ∈ R ∧ ¬m x -->                                        (*i4*)
                 x ∈ reachable (relS{l,r}|m) ({t}∪set(map r stack))) ∧
          (∀x. m x --> x ∈ R) ∧                                         (*i5*)
          (∀x. x ∉ set stack --> r x = iR x ∧ l x = iL x) ∧       (*i6*)
          (stkOk c l r iL iR t stack)                                    (*i7*) }
 DO IF t = Null ∨ t^.m
      THEN IF p^.c
               THEN q := t; t := p; p := p^.r; t^.r := q               (*pop*)
               ELSE q := t; t := p^.r; p^.r := p^.l;                      (*swing*)
                        p^.l := q; p^.c := True          FI    
      ELSE q := p; p := t; t := t^.l; p^.l := q;                         (*push*)
               p^.m := True; p^.c := False            FI       OD
 {(∀x. (x ∈ R) = m x) ∧ (r = iR ∧ l = iL) }"
  (is "VARS c m l r t p q root {?Pre c m l r root} (?c1; ?c2; ?c3) {?Post c m l r}")
proof (vcg)
  let "While {(c, m, l, r, t, p, q, root). ?whileB m t p}
    {(c, m, l, r, t, p, q, root). ?inv c m l r t p} ?body" = ?c3
  {

    fix c m l r t p q root
    assume "?Pre c m l r root"
    thus "?inv c m l r root Null"  by (auto simp add: reachable_def addrs_def)
  next

    fix c m l r t p q
    let "∃stack. ?Inv stack"  =  "?inv c m l r t p"
    assume a: "?inv c m l r t p ∧ ¬(p ≠ Null ∨ t ≠ Null ∧ ¬ t^.m)"  
    then obtain stack where inv: "?Inv stack" by blast
    from a have pNull: "p = Null" and tDisj: "t=Null ∨ (t≠Null ∧ t^.m )" by auto
    let "?I1 ∧ _ ∧ _ ∧ ?I4 ∧ ?I5 ∧ ?I6 ∧ _"  =  "?Inv stack"
    from inv have i1: "?I1" and i4: "?I4" and i5: "?I5" and i6: "?I6" by simp+
    from pNull i1 have stackEmpty: "stack = []" by simp
    from tDisj i4 have RisMarked[rule_format]: "∀x.  x ∈ R --> m x"  by(auto simp: reachable_def addrs_def stackEmpty)
    from i5 i6 show "(∀x.(x ∈ R) = m x) ∧ r = iR ∧ l = iL"  by(auto simp: stackEmpty fun_eq_iff intro:RisMarked)

  next   
      fix c m l r t p q root
      let "∃stack. ?Inv stack"  =  "?inv c m l r t p"
      let "∃stack. ?popInv stack"  =  "?inv c m l (r(p -> t)) p (p^.r)"
      let "∃stack. ?swInv stack"  =
        "?inv (c(p -> True)) m (l(p -> t)) (r(p -> p^.l)) (p^.r) p"
      let "∃stack. ?puInv stack"  =
        "?inv (c(t -> False)) (m(t -> True)) (l(t -> p)) r (t^.l) t"
      let "?ifB1"  =  "(t = Null ∨ t^.m)"
      let "?ifB2"  =  "p^.c" 

      assume "(∃stack.?Inv stack) ∧ (p ≠ Null ∨ t ≠ Null ∧ ¬ t^.m)" (is "_ ∧ ?whileB")
      then obtain stack where inv: "?Inv stack" and whileB: "?whileB" by blast
      let "?I1 ∧ ?I2 ∧ ?I3 ∧ ?I4 ∧ ?I5 ∧ ?I6 ∧ ?I7" = "?Inv stack"
      from inv have i1: "?I1" and i2: "?I2" and i3: "?I3" and i4: "?I4"
                  and i5: "?I5" and i6: "?I6" and i7: "?I7" by simp+        
      have stackDist: "distinct (stack)" using i1 by (rule List_distinct)

      show "(?ifB1 --> (?ifB2 --> (∃stack.?popInv stack)) ∧ 
                            (¬?ifB2 --> (∃stack.?swInv stack)) ) ∧
              (¬?ifB1 --> (∃stack.?puInv stack))"
      proof - 
        {
          assume ifB1: "t = Null ∨ t^.m" and ifB2: "p^.c"
          from ifB1 whileB have pNotNull: "p ≠ Null" by auto
          then obtain addr_p where addr_p_eq: "p = Ref addr_p" by auto
          with i1 obtain stack_tl where stack_eq: "stack = (addr p) # stack_tl"
            by auto
          with i2 have m_addr_p: "p^.m" by auto
          have stackDist: "distinct (stack)" using i1 by (rule List_distinct)
          from stack_eq stackDist have p_notin_stack_tl: "addr p ∉ set stack_tl" by simp
          let "?poI1∧ ?poI2∧ ?poI3∧ ?poI4∧ ?poI5∧ ?poI6∧ ?poI7" = "?popInv stack_tl"
          have "?popInv stack_tl"
          proof -

            -- {*List property is maintained:*}
            from i1 p_notin_stack_tl ifB2
            have poI1: "List (S c l (r(p -> t))) (p^.r) stack_tl" 
              by(simp add: addr_p_eq stack_eq, simp add: S_def)

            moreover
            -- {*Everything on the stack is marked:*}
            from i2 have poI2: "∀ x ∈ set stack_tl. m x" by (simp add:stack_eq)
            moreover

            -- {*Everything is still reachable:*}
            let "(R = reachable ?Ra ?A)" = "?I3"
            let "?Rb" = "(relS {l, r(p -> t)})"
            let "?B" = "{p, p^.r}"
            -- {*Our goal is @{text"R = reachable ?Rb ?B"}.*}
            have "?Ra* `` addrs ?A = ?Rb* `` addrs ?B" (is "?L = ?R")
            proof
              show "?L ⊆ ?R"
              proof (rule still_reachable)
                show "addrs ?A ⊆ ?Rb* `` addrs ?B" by(fastforce simp:addrs_def relS_def rel_def addr_p_eq 
                     intro:oneStep_reachable Image_iff[THEN iffD2])
                show "∀(x,y) ∈ ?Ra-?Rb. y ∈ (?Rb* `` addrs ?B)" by (clarsimp simp:relS_def) 
                     (fastforce simp add:rel_def Image_iff addrs_def dest:rel_upd1)
              qed
              show "?R ⊆ ?L"
              proof (rule still_reachable)
                show "addrs ?B ⊆ ?Ra* `` addrs ?A"
                  by(fastforce simp:addrs_def rel_defs addr_p_eq 
                      intro:oneStep_reachable Image_iff[THEN iffD2])
              next
                show "∀(x, y)∈?Rb-?Ra. y∈(?Ra*``addrs ?A)"
                  by (clarsimp simp:relS_def) 
                     (fastforce simp add:rel_def Image_iff addrs_def dest:rel_upd2)
              qed
            qed
            with i3 have poI3: "R = reachable ?Rb ?B"  by (simp add:reachable_def) 
            moreover

            -- "If it is reachable and not marked, it is still reachable using..."
            let "∀x. x ∈ R ∧ ¬ m x --> x ∈ reachable ?Ra ?A"  =  ?I4        
            let "?Rb" = "relS {l, r(p -> t)} | m"
            let "?B" = "{p} ∪ set (map (r(p -> t)) stack_tl)"
            -- {*Our goal is @{text"∀x. x ∈ R ∧ ¬ m x --> x ∈ reachable ?Rb ?B"}.*}
            let ?T = "{t, p^.r}"

            have "?Ra* `` addrs ?A ⊆ ?Rb* `` (addrs ?B ∪ addrs ?T)"
            proof (rule still_reachable)
              have rewrite: "∀s∈set stack_tl. (r(p -> t)) s = r s"
                by (auto simp add:p_notin_stack_tl intro:fun_upd_other) 
              show "addrs ?A ⊆ ?Rb* `` (addrs ?B ∪ addrs ?T)"
                by (fastforce cong:map_cong simp:stack_eq addrs_def rewrite intro:self_reachable)
              show "∀(x, y)∈?Ra-?Rb. y∈(?Rb*``(addrs ?B ∪ addrs ?T))"
                by (clarsimp simp:restr_def relS_def) 
                  (fastforce simp add:rel_def Image_iff addrs_def dest:rel_upd1)
            qed
            -- "We now bring a term from the right to the left of the subset relation."
            hence subset: "?Ra* `` addrs ?A - ?Rb* `` addrs ?T ⊆ ?Rb* `` addrs ?B"
              by blast
            have poI4: "∀x. x ∈ R ∧ ¬ m x --> x ∈ reachable ?Rb ?B"
            proof (rule allI, rule impI)
              fix x
              assume a: "x ∈ R ∧ ¬ m x"
              -- {*First, a disjunction on @{term"p^.r"} used later in the proof*}
              have pDisj:"p^.r = Null ∨ (p^.r ≠ Null ∧ p^.r^.m)" using poI1 poI2 
                by auto
              -- {*@{term x} belongs to the left hand side of @{thm[source] subset}:*}
              have incl: "x ∈ ?Ra*``addrs ?A" using  a i4 by (simp only:reachable_def, clarsimp)
              have excl: "x ∉ ?Rb*`` addrs ?T" using pDisj ifB1 a by (auto simp add:addrs_def)
              -- {*And therefore also belongs to the right hand side of @{thm[source]subset},*}
              -- {*which corresponds to our goal.*}
              from incl excl subset  show "x ∈ reachable ?Rb ?B" by (auto simp add:reachable_def)
            qed
            moreover

            -- "If it is marked, then it is reachable"
            from i5 have poI5: "∀x. m x --> x ∈ R" .
            moreover

            -- {*If it is not on the stack, then its @{term l} and @{term r} fields are unchanged*}
            from i7 i6 ifB2 
            have poI6: "∀x. x ∉ set stack_tl --> (r(p -> t)) x = iR x ∧ l x = iL x" 
              by(auto simp: addr_p_eq stack_eq fun_upd_apply)

            moreover

            -- {*If it is on the stack, then its @{term l} and @{term r} fields can be reconstructed*}
            from p_notin_stack_tl i7 have poI7: "stkOk c l (r(p -> t)) iL iR p stack_tl"
              by (clarsimp simp:stack_eq addr_p_eq)

            ultimately show "?popInv stack_tl" by simp
          qed
          hence "∃stack. ?popInv stack" ..
        }
        moreover

        -- "Proofs of the Swing and Push arm follow."
        -- "Since they are in principle simmilar to the Pop arm proof,"
        -- "we show fewer comments and use frequent pattern matching."
        {
          -- "Swing arm"
          assume ifB1: "?ifB1" and nifB2: "¬?ifB2"
          from ifB1 whileB have pNotNull: "p ≠ Null" by clarsimp
          then obtain addr_p where addr_p_eq: "p = Ref addr_p" by clarsimp
          with i1 obtain stack_tl where stack_eq: "stack = (addr p) # stack_tl" by clarsimp
          with i2 have m_addr_p: "p^.m" by clarsimp
          from stack_eq stackDist have p_notin_stack_tl: "(addr p) ∉ set stack_tl"
            by simp
          let "?swI1∧?swI2∧?swI3∧?swI4∧?swI5∧?swI6∧?swI7" = "?swInv stack"
          have "?swInv stack"
          proof -
            
            -- {*List property is maintained:*}
            from i1 p_notin_stack_tl nifB2
            have swI1: "?swI1"
              by (simp add:addr_p_eq stack_eq, simp add:S_def)
            moreover
            
            -- {*Everything on the stack is marked:*}
            from i2
            have swI2: "?swI2" .
            moreover
            
            -- {*Everything is still reachable:*}
            let "R = reachable ?Ra ?A" = "?I3"
            let "R = reachable ?Rb ?B" = "?swI3"
            have "?Ra* `` addrs ?A = ?Rb* `` addrs ?B"
            proof (rule still_reachable_eq)
              show "addrs ?A ⊆ ?Rb* `` addrs ?B"
                by(fastforce simp:addrs_def rel_defs addr_p_eq intro:oneStep_reachable Image_iff[THEN iffD2])
            next
              show "addrs ?B ⊆ ?Ra* `` addrs ?A"
                by(fastforce simp:addrs_def rel_defs addr_p_eq intro:oneStep_reachable Image_iff[THEN iffD2])
            next
              show "∀(x, y)∈?Ra-?Rb. y∈(?Rb*``addrs ?B)"
                by (clarsimp simp:relS_def) (fastforce simp add:rel_def Image_iff addrs_def fun_upd_apply dest:rel_upd1)
            next
              show "∀(x, y)∈?Rb-?Ra. y∈(?Ra*``addrs ?A)"
                by (clarsimp simp:relS_def) (fastforce simp add:rel_def Image_iff addrs_def fun_upd_apply dest:rel_upd2)
            qed
            with i3
            have swI3: "?swI3" by (simp add:reachable_def) 
            moreover

            -- "If it is reachable and not marked, it is still reachable using..."
            let "∀x. x ∈ R ∧ ¬ m x --> x ∈ reachable ?Ra ?A" = ?I4
            let "∀x. x ∈ R ∧ ¬ m x --> x ∈ reachable ?Rb ?B" = ?swI4
            let ?T = "{t}"
            have "?Ra*``addrs ?A ⊆ ?Rb*``(addrs ?B ∪ addrs ?T)"
            proof (rule still_reachable)
              have rewrite: "(∀s∈set stack_tl. (r(addr p := l(addr p))) s = r s)"
                by (auto simp add:p_notin_stack_tl intro:fun_upd_other)
              show "addrs ?A ⊆ ?Rb* `` (addrs ?B ∪ addrs ?T)"
                by (fastforce cong:map_cong simp:stack_eq addrs_def rewrite intro:self_reachable)
            next
              show "∀(x, y)∈?Ra-?Rb. y∈(?Rb*``(addrs ?B ∪ addrs ?T))"
                by (clarsimp simp:relS_def restr_def) (fastforce simp add:rel_def Image_iff addrs_def fun_upd_apply dest:rel_upd1)
            qed
            then have subset: "?Ra*``addrs ?A - ?Rb*``addrs ?T ⊆ ?Rb*``addrs ?B"
              by blast
            have ?swI4
            proof (rule allI, rule impI)
              fix x
              assume a: "x ∈ R ∧¬ m x"
              with i4 addr_p_eq stack_eq  have inc: "x ∈ ?Ra*``addrs ?A" 
                by (simp only:reachable_def, clarsimp)
              with ifB1 a 
              have exc: "x ∉ ?Rb*`` addrs ?T" 
                by (auto simp add:addrs_def)
              from inc exc subset  show "x ∈ reachable ?Rb ?B" 
                by (auto simp add:reachable_def)
            qed
            moreover
            
            -- "If it is marked, then it is reachable"
            from i5
            have "?swI5" .
            moreover

            -- {*If it is not on the stack, then its @{term l} and @{term r} fields are unchanged*}
            from i6 stack_eq
            have "?swI6"
              by clarsimp           
            moreover

            -- {*If it is on the stack, then its @{term l} and @{term r} fields can be reconstructed*}
            from stackDist i7 nifB2 
            have "?swI7"
              by (clarsimp simp:addr_p_eq stack_eq)

            ultimately show ?thesis by auto
          qed
          then have "∃stack. ?swInv stack" by blast
        }
        moreover

        {
          -- "Push arm"
          assume nifB1: "¬?ifB1"
          from nifB1 whileB have tNotNull: "t ≠ Null" by clarsimp
          then obtain addr_t where addr_t_eq: "t = Ref addr_t" by clarsimp
          with i1 obtain new_stack where new_stack_eq: "new_stack = (addr t) # stack" by clarsimp
          from tNotNull nifB1 have n_m_addr_t: "¬ (t^.m)" by clarsimp
          with i2 have t_notin_stack: "(addr t) ∉ set stack" by blast
          let "?puI1∧?puI2∧?puI3∧?puI4∧?puI5∧?puI6∧?puI7" = "?puInv new_stack"
          have "?puInv new_stack"
          proof -
            
            -- {*List property is maintained:*}
            from i1 t_notin_stack
            have puI1: "?puI1"
              by (simp add:addr_t_eq new_stack_eq, simp add:S_def)
            moreover
            
            -- {*Everything on the stack is marked:*}
            from i2
            have puI2: "?puI2" 
              by (simp add:new_stack_eq fun_upd_apply)
            moreover
            
            -- {*Everything is still reachable:*}
            let "R = reachable ?Ra ?A" = "?I3"
            let "R = reachable ?Rb ?B" = "?puI3"
            have "?Ra* `` addrs ?A = ?Rb* `` addrs ?B"
            proof (rule still_reachable_eq)
              show "addrs ?A ⊆ ?Rb* `` addrs ?B"
                by(fastforce simp:addrs_def rel_defs addr_t_eq intro:oneStep_reachable Image_iff[THEN iffD2])
            next
              show "addrs ?B ⊆ ?Ra* `` addrs ?A"
                by(fastforce simp:addrs_def rel_defs addr_t_eq intro:oneStep_reachable Image_iff[THEN iffD2])
            next
              show "∀(x, y)∈?Ra-?Rb. y∈(?Rb*``addrs ?B)"
                by (clarsimp simp:relS_def) (fastforce simp add:rel_def Image_iff addrs_def dest:rel_upd1)
            next
              show "∀(x, y)∈?Rb-?Ra. y∈(?Ra*``addrs ?A)"
                by (clarsimp simp:relS_def) (fastforce simp add:rel_def Image_iff addrs_def fun_upd_apply dest:rel_upd2)
            qed
            with i3
            have puI3: "?puI3" by (simp add:reachable_def) 
            moreover
            
            -- "If it is reachable and not marked, it is still reachable using..."
            let "∀x. x ∈ R ∧ ¬ m x --> x ∈ reachable ?Ra ?A" = ?I4
            let "∀x. x ∈ R ∧ ¬ ?new_m x --> x ∈ reachable ?Rb ?B" = ?puI4
            let ?T = "{t}"
            have "?Ra*``addrs ?A ⊆ ?Rb*``(addrs ?B ∪ addrs ?T)"
            proof (rule still_reachable)
              show "addrs ?A ⊆ ?Rb* `` (addrs ?B ∪ addrs ?T)"
                by (fastforce simp:new_stack_eq addrs_def intro:self_reachable)
            next
              show "∀(x, y)∈?Ra-?Rb. y∈(?Rb*``(addrs ?B ∪ addrs ?T))"
                by (clarsimp simp:relS_def new_stack_eq restr_un restr_upd) 
                   (fastforce simp add:rel_def Image_iff restr_def addrs_def fun_upd_apply addr_t_eq dest:rel_upd3)
            qed
            then have subset: "?Ra*``addrs ?A - ?Rb*``addrs ?T ⊆ ?Rb*``addrs ?B"
              by blast
            have ?puI4
            proof (rule allI, rule impI)
              fix x
              assume a: "x ∈ R ∧ ¬ ?new_m x"
              have xDisj: "x=(addr t) ∨ x≠(addr t)" by simp
              with i4 a have inc: "x ∈ ?Ra*``addrs ?A"
                by (fastforce simp:addr_t_eq addrs_def reachable_def intro:self_reachable)
              have exc: "x ∉ ?Rb*`` addrs ?T"
                using xDisj a n_m_addr_t
                by (clarsimp simp add:addrs_def addr_t_eq) 
              from inc exc subset  show "x ∈ reachable ?Rb ?B" 
                by (auto simp add:reachable_def)
            qed  
            moreover
            
            -- "If it is marked, then it is reachable"
            from i5
            have "?puI5"
              by (auto simp:addrs_def i3 reachable_def addr_t_eq fun_upd_apply intro:self_reachable)
            moreover
            
            -- {*If it is not on the stack, then its @{term l} and @{term r} fields are unchanged*}
            from i6 
            have "?puI6"
              by (simp add:new_stack_eq)
            moreover

            -- {*If it is on the stack, then its @{term l} and @{term r} fields can be reconstructed*}
            from stackDist i6 t_notin_stack i7
            have "?puI7" by (clarsimp simp:addr_t_eq new_stack_eq)

            ultimately show ?thesis by auto
          qed
          then have "∃stack. ?puInv stack" by blast

        }
        ultimately show ?thesis by blast
      qed
    }
  qed

end