Quelle  SchorrWaite.thy

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

section ‹Proof of the Schorr-Waite graph marking algorithm›

theory SchorrWaite
  imports HeapSyntax
begin

subsection ‹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 = (∪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"
    (‹(‹notation=‹mixfix relation restriction›\›_/ | _)› [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


subsection ‹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 "Valid
  {(c, m, l, r, t, p, q, root). ?Pre c m l r root}
  (Seq _ (Seq _ (While {(c, m, l, r, t, p, q, root). ?whileB m t p} _)))
  (Aseq _ (Aseq _ (Awhile {(c, m, l, r, t, p, q, root). ?inv c m l r t p} _ _))) _")
 proof (vcg)
  {
  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) ∧ ?whileB m t p"
  then obtain stack where inv: "?Inv stack" and whileB: "?whileB m t p" 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 ‹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 ‹∀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 🍋‹p^.r› used later in the proof›
  have pDisj:"p^.r = Null ∨ (p^.r ≠ Null ∧ p^.r^.m)" using poI1 poI2
  by auto
  🍋 ‹🍋‹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 🍋‹l› and 🍋‹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 🍋‹l› and 🍋‹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 🍋‹l› and 🍋‹r› fields are unchanged›
  from i6 stack_eq
  have "?swI6"
  by clarsimp
  moreover
 
  🍋 ‹If it is on the stack, then its 🍋‹l› and 🍋‹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 🍋‹l› and 🍋‹r› fields are unchanged›
  from i6
  have "?puI6"
  by (simp add:new_stack_eq)
  moreover
 
  🍋 ‹If it is on the stack, then its 🍋‹l› and 🍋‹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