# Theory Pointer_Examples

theory Pointer_Examples
imports HeapSyntax
```(*  Title:      HOL/Hoare/Pointer_Examples.thy
Author:     Tobias Nipkow

Examples of verifications of pointer programs.
*)

theory Pointer_Examples imports HeapSyntax begin

axiomatization where unproven: "PROP A"

section "Verifications"

subsection "List reversal"

text "A short but unreadable proof:"

lemma "VARS tl p q r
{List tl p Ps ∧ List tl q Qs ∧ set Ps ∩ set Qs = {}}
WHILE p ≠ Null
INV {∃ps qs. List tl p ps ∧ List tl q qs ∧ set ps ∩ set qs = {} ∧
rev ps @ qs = rev Ps @ Qs}
DO r := p; p := p^.tl; r^.tl := q; q := r OD
{List tl q (rev Ps @ Qs)}"
apply vcg_simp
apply fastforce
apply(fastforce intro:notin_List_update[THEN iffD2])
(* explicit:
apply clarify
apply(rename_tac ps b qs)
apply clarsimp
apply(rename_tac ps')
apply(fastforce intro:notin_List_update[THEN iffD2])
apply(rule_tac x = ps' in exI)
apply simp
apply(rule_tac x = "b#qs" in exI)
apply simp
*)
apply fastforce
done

text‹And now with ghost variables @{term ps} and @{term qs}. Even
``more automatic''.›

lemma "VARS next p ps q qs r
{List next p Ps ∧ List next q Qs ∧ set Ps ∩ set Qs = {} ∧
ps = Ps ∧ qs = Qs}
WHILE p ≠ Null
INV {List next p ps ∧ List next q qs ∧ set ps ∩ set qs = {} ∧
rev ps @ qs = rev Ps @ Qs}
DO r := p; p := p^.next; r^.next := q; q := r;
qs := (hd ps) # qs; ps := tl ps OD
{List next q (rev Ps @ Qs)}"
apply vcg_simp
apply fastforce
apply fastforce
done

lemma "VARS tl p q r
{List tl p Ps ∧ List tl q Qs ∧ set Ps ∩ set Qs = {}}
WHILE p ≠ Null
INV {∃ps qs. List tl p ps ∧ List tl q qs ∧ set ps ∩ set qs = {} ∧
rev ps @ qs = rev Ps @ Qs}
DO r := p; p := p^.tl; r^.tl := q; q := r OD
{List tl q (rev Ps @ Qs)}"
proof vcg
fix tl p q r
assume "List tl p Ps ∧ List tl q Qs ∧ set Ps ∩ set Qs = {}"
thus "∃ps qs. List tl p ps ∧ List tl q qs ∧ set ps ∩ set qs = {} ∧
rev ps @ qs = rev Ps @ Qs" by fastforce
next
fix tl p q r
assume "(∃ps qs. List tl p ps ∧ List tl q qs ∧ set ps ∩ set qs = {} ∧
rev ps @ qs = rev Ps @ Qs) ∧ p ≠ Null"
(is "(∃ps qs. ?I ps qs) ∧ _")
then obtain ps qs a where I: "?I ps qs ∧ p = Ref a"
by fast
then obtain ps' where "ps = a # ps'" by fastforce
hence "List (tl(p → q)) (p^.tl) ps' ∧
List (tl(p → q)) p       (a#qs) ∧
set ps' ∩ set (a#qs) = {} ∧
rev ps' @ (a#qs) = rev Ps @ Qs"
using I by fastforce
thus "∃ps' qs'. List (tl(p → q)) (p^.tl) ps' ∧
List (tl(p → q)) p       qs' ∧
set ps' ∩ set qs' = {} ∧
rev ps' @ qs' = rev Ps @ Qs" by fast
next
fix tl p q r
assume "(∃ps qs. List tl p ps ∧ List tl q qs ∧ set ps ∩ set qs = {} ∧
rev ps @ qs = rev Ps @ Qs) ∧ ¬ p ≠ Null"
thus "List tl q (rev Ps @ Qs)" by fastforce
qed

text‹Finaly, the functional version. A bit more verbose, but automatic!›

lemma "VARS tl p q r
{islist tl p ∧ islist tl q ∧
Ps = list tl p ∧ Qs = list tl q ∧ set Ps ∩ set Qs = {}}
WHILE p ≠ Null
INV {islist tl p ∧ islist tl q ∧
set(list tl p) ∩ set(list tl q) = {} ∧
rev(list tl p) @ (list tl q) = rev Ps @ Qs}
DO r := p; p := p^.tl; r^.tl := q; q := r OD
{islist tl q ∧ list tl q = rev Ps @ Qs}"
apply vcg_simp
apply clarsimp
apply clarsimp
apply clarsimp
done

subsection "Searching in a list"

text‹What follows is a sequence of successively more intelligent proofs that
a simple loop finds an element in a linked list.

We start with a proof based on the @{term List} predicate. This means it only
works for acyclic lists.›

lemma "VARS tl p
{List tl p Ps ∧ X ∈ set Ps}
WHILE p ≠ Null ∧ p ≠ Ref X
INV {∃ps. List tl p ps ∧ X ∈ set ps}
DO p := p^.tl OD
{p = Ref X}"
apply vcg_simp
apply blast
apply clarsimp
apply clarsimp
done

text‹Using @{term Path} instead of @{term List} generalizes the correctness
statement to cyclic lists as well:›

lemma "VARS tl p
{Path tl p Ps X}
WHILE p ≠ Null ∧ p ≠ X
INV {∃ps. Path tl p ps X}
DO p := p^.tl OD
{p = X}"
apply vcg_simp
apply blast
apply fastforce
apply clarsimp
done

text‹Now it dawns on us that we do not need the list witness at all --- it
suffices to talk about reachability, i.e.\ we can use relations directly. The
first version uses a relation on @{typ"'a ref"}:›

lemma "VARS tl p
{(p,X) ∈ {(Ref x,tl x) |x. True}^*}
WHILE p ≠ Null ∧ p ≠ X
INV {(p,X) ∈ {(Ref x,tl x) |x. True}^*}
DO p := p^.tl OD
{p = X}"
apply vcg_simp
apply clarsimp
apply(erule converse_rtranclE)
apply simp
apply(clarsimp elim:converse_rtranclE)
apply(fast elim:converse_rtranclE)
done

text‹Finally, a version based on a relation on type @{typ 'a}:›

lemma "VARS tl p
{p ≠ Null ∧ (addr p,X) ∈ {(x,y). tl x = Ref y}^*}
WHILE p ≠ Null ∧ p ≠ Ref X
INV {p ≠ Null ∧ (addr p,X) ∈ {(x,y). tl x = Ref y}^*}
DO p := p^.tl OD
{p = Ref X}"
apply vcg_simp
apply clarsimp
apply(erule converse_rtranclE)
apply simp
apply clarsimp
apply clarsimp
done

subsection "Splicing two lists"

lemma "VARS tl p q pp qq
{List tl p Ps ∧ List tl q Qs ∧ set Ps ∩ set Qs = {} ∧ size Qs ≤ size Ps}
pp := p;
WHILE q ≠ Null
INV {∃as bs qs.
distinct as ∧ Path tl p as pp ∧ List tl pp bs ∧ List tl q qs ∧
set bs ∩ set qs = {} ∧ set as ∩ (set bs ∪ set qs) = {} ∧
size qs ≤ size bs ∧ splice Ps Qs = as @ splice bs qs}
DO qq := q^.tl; q^.tl := pp^.tl; pp^.tl := q; pp := q^.tl; q := qq OD
{List tl p (splice Ps Qs)}"
apply vcg_simp
apply(rule_tac x = "[]" in exI)
apply fastforce
apply clarsimp
apply(rename_tac y bs qqs)
apply(case_tac bs) apply simp
apply clarsimp
apply(rename_tac x bbs)
apply(rule_tac x = "as @ [x,y]" in exI)
apply simp
apply(rule_tac x = "bbs" in exI)
apply simp
apply(rule_tac x = "qqs" in exI)
apply simp
apply (fastforce simp:List_app)
done

subsection "Merging two lists"

text"This is still a bit rough, especially the proof."

definition cor :: "bool ⇒ bool ⇒ bool"
where "cor P Q ⟷ (if P then True else Q)"

definition cand :: "bool ⇒ bool ⇒ bool"
where "cand P Q ⟷ (if P then Q else False)"

fun merge :: "'a list * 'a list * ('a ⇒ 'a ⇒ bool) ⇒ 'a list"
where
"merge(x#xs,y#ys,f) = (if f x y then x # merge(xs,y#ys,f)
else y # merge(x#xs,ys,f))"
| "merge(x#xs,[],f) = x # merge(xs,[],f)"
| "merge([],y#ys,f) = y # merge([],ys,f)"
| "merge([],[],f) = []"

text‹Simplifies the proof a little:›

lemma [simp]: "({} = insert a A ∩ B) = (a ∉ B & {} = A ∩ B)"
by blast
lemma [simp]: "({} = A ∩ insert b B) = (b ∉ A & {} = A ∩ B)"
by blast
lemma [simp]: "({} = A ∩ (B ∪ C)) = ({} = A ∩ B & {} = A ∩ C)"
by blast

lemma "VARS hd tl p q r s
{List tl p Ps ∧ List tl q Qs ∧ set Ps ∩ set Qs = {} ∧
(p ≠ Null ∨ q ≠ Null)}
IF cor (q = Null) (cand (p ≠ Null) (p^.hd ≤ q^.hd))
THEN r := p; p := p^.tl ELSE r := q; q := q^.tl FI;
s := r;
WHILE p ≠ Null ∨ q ≠ Null
INV {EX rs ps qs a. Path tl r rs s ∧ List tl p ps ∧ List tl q qs ∧
distinct(a # ps @ qs @ rs) ∧ s = Ref a ∧
merge(Ps,Qs,λx y. hd x ≤ hd y) =
rs @ a # merge(ps,qs,λx y. hd x ≤ hd y) ∧
(tl a = p ∨ tl a = q)}
DO IF cor (q = Null) (cand (p ≠ Null) (p^.hd ≤ q^.hd))
THEN s^.tl := p; p := p^.tl ELSE s^.tl := q; q := q^.tl FI;
s := s^.tl
OD
{List tl r (merge(Ps,Qs,λx y. hd x ≤ hd y))}"
apply vcg_simp

apply (fastforce)

apply clarsimp
apply(rule conjI)
apply clarsimp
apply(rule conjI)
apply (fastforce intro!:Path_snoc intro:Path_upd[THEN iffD2] notin_List_update[THEN iffD2] simp:eq_sym_conv)
apply clarsimp
apply(rule conjI)
apply (clarsimp)
apply(rule_tac x = "rs @ [a]" in exI)
apply(clarsimp simp:eq_sym_conv)
apply(rule_tac x = "bs" in exI)
apply(clarsimp simp:eq_sym_conv)
apply(rule_tac x = "ya#bsa" in exI)
apply(simp)
apply(clarsimp simp:eq_sym_conv)
apply(rule_tac x = "rs @ [a]" in exI)
apply(clarsimp simp:eq_sym_conv)
apply(rule_tac x = "y#bs" in exI)
apply(clarsimp simp:eq_sym_conv)
apply(rule_tac x = "bsa" in exI)
apply(simp)
apply (fastforce intro!:Path_snoc intro:Path_upd[THEN iffD2] notin_List_update[THEN iffD2] simp:eq_sym_conv)

done

text‹And now with ghost variables:›

lemma "VARS elem next p q r s ps qs rs a
{List next p Ps ∧ List next q Qs ∧ set Ps ∩ set Qs = {} ∧
(p ≠ Null ∨ q ≠ Null) ∧ ps = Ps ∧ qs = Qs}
IF cor (q = Null) (cand (p ≠ Null) (p^.elem ≤ q^.elem))
THEN r := p; p := p^.next; ps := tl ps
ELSE r := q; q := q^.next; qs := tl qs FI;
s := r; rs := []; a := addr s;
WHILE p ≠ Null ∨ q ≠ Null
INV {Path next r rs s ∧ List next p ps ∧ List next q qs ∧
distinct(a # ps @ qs @ rs) ∧ s = Ref a ∧
merge(Ps,Qs,λx y. elem x ≤ elem y) =
rs @ a # merge(ps,qs,λx y. elem x ≤ elem y) ∧
(next a = p ∨ next a = q)}
DO IF cor (q = Null) (cand (p ≠ Null) (p^.elem ≤ q^.elem))
THEN s^.next := p; p := p^.next; ps := tl ps
ELSE s^.next := q; q := q^.next; qs := tl qs FI;
rs := rs @ [a]; s := s^.next; a := addr s
OD
{List next r (merge(Ps,Qs,λx y. elem x ≤ elem y))}"
apply vcg_simp

apply (fastforce)

apply clarsimp
apply(rule conjI)
apply(clarsimp)
apply(rule conjI)
apply(clarsimp simp:neq_commute)
apply(clarsimp simp:neq_commute)
apply(clarsimp simp:neq_commute)

done

text‹The proof is a LOT simpler because it does not need
instantiations anymore, but it is still not quite automatic, probably
because of this wrong orientation business.›

text‹More of the previous proof without ghost variables can be
automated, but the runtime goes up drastically. In general it is
usually more efficient to give the witness directly than to have it
found by proof.

Now we try a functional version of the abstraction relation @{term
Path}. Since the result is not that convincing, we do not prove any of
the lemmas.›

axiomatization
ispath :: "('a ⇒ 'a ref) ⇒ 'a ref ⇒ 'a ref ⇒ bool" and
path :: "('a ⇒ 'a ref) ⇒ 'a ref ⇒ 'a ref ⇒ 'a list"

text"First some basic lemmas:"

lemma [simp]: "ispath f p p"
by (rule unproven)
lemma [simp]: "path f p p = []"
by (rule unproven)
lemma [simp]: "ispath f p q ⟹ a ∉ set(path f p q) ⟹ ispath (f(a := r)) p q"
by (rule unproven)
lemma [simp]: "ispath f p q ⟹ a ∉ set(path f p q) ⟹
path (f(a := r)) p q = path f p q"
by (rule unproven)

text"Some more specific lemmas needed by the example:"

lemma [simp]: "ispath (f(a := q)) p (Ref a) ⟹ ispath (f(a := q)) p q"
by (rule unproven)
lemma [simp]: "ispath (f(a := q)) p (Ref a) ⟹
path (f(a := q)) p q = path (f(a := q)) p (Ref a) @ [a]"
by (rule unproven)
lemma [simp]: "ispath f p (Ref a) ⟹ f a = Ref b ⟹
b ∉ set (path f p (Ref a))"
by (rule unproven)
lemma [simp]: "ispath f p (Ref a) ⟹ f a = Null ⟹ islist f p"
by (rule unproven)
lemma [simp]: "ispath f p (Ref a) ⟹ f a = Null ⟹ list f p = path f p (Ref a) @ [a]"
by (rule unproven)

lemma [simp]: "islist f p ⟹ distinct (list f p)"
by (rule unproven)

lemma "VARS hd tl p q r s
{islist tl p & Ps = list tl p ∧ islist tl q & Qs = list tl q ∧
set Ps ∩ set Qs = {} ∧
(p ≠ Null ∨ q ≠ Null)}
IF cor (q = Null) (cand (p ≠ Null) (p^.hd ≤ q^.hd))
THEN r := p; p := p^.tl ELSE r := q; q := q^.tl FI;
s := r;
WHILE p ≠ Null ∨ q ≠ Null
INV {EX rs ps qs a. ispath tl r s & rs = path tl r s ∧
islist tl p & ps = list tl p ∧ islist tl q & qs = list tl q ∧
distinct(a # ps @ qs @ rs) ∧ s = Ref a ∧
merge(Ps,Qs,λx y. hd x ≤ hd y) =
rs @ a # merge(ps,qs,λx y. hd x ≤ hd y) ∧
(tl a = p ∨ tl a = q)}
DO IF cor (q = Null) (cand (p ≠ Null) (p^.hd ≤ q^.hd))
THEN s^.tl := p; p := p^.tl ELSE s^.tl := q; q := q^.tl FI;
s := s^.tl
OD
{islist tl r & list tl r = (merge(Ps,Qs,λx y. hd x ≤ hd y))}"
apply vcg_simp

apply (fastforce)
apply (fastforce simp: eq_sym_conv)
apply(clarsimp)
done

text"The proof is automatic, but requires a numbet of special lemmas."

subsection "Cyclic list reversal"

text‹We consider two algorithms for the reversal of circular lists.
›

lemma circular_list_rev_I:
"VARS next root p q tmp
{root = Ref r ∧ distPath next root (r#Ps) root}
p := root; q := root^.next;
WHILE q ≠ root
INV {∃ ps qs. distPath next p ps root ∧ distPath next q qs root ∧
root = Ref r ∧ r ∉ set Ps  ∧ set ps ∩ set qs = {} ∧
Ps = (rev ps) @ qs  }
DO tmp := q; q := q^.next; tmp^.next := p; p:=tmp OD;
root^.next := p
{ root = Ref r ∧ distPath next root (r#rev Ps) root}"
apply (simp only:distPath_def)
apply vcg_simp
apply (rule_tac x="[]" in exI)
apply auto
apply (drule (2) neq_dP)
apply clarsimp
apply(rule_tac x="a # ps" in exI)
apply clarsimp
done

text‹In the beginning, we are able to assert @{term"distPath next
root as root"}, with @{term"as"} set to @{term"[]"} or
@{term"[r,a,b,c]"}. Note that @{term"Path next root as root"} would
additionally give us an infinite number of lists with the recurring
sequence @{term"[r,a,b,c]"}.

The precondition states that there exists a non-empty non-repeating
path \mbox{@{term "r # Ps"}} from pointer @{term root} to itself, given that
@{term root} points to location @{term r}. Pointers @{term p} and
@{term q} are then set to @{term root} and the successor of @{term
root} respectively. If @{term "q = root"}, we have circled the loop,
otherwise we set the @{term next} pointer field of @{term q} to point
to @{term p}, and shift @{term p} and @{term q} one step forward. The
invariant thus states that @{term p} and @{term q} point to two
disjoint lists @{term ps} and @{term qs}, such that @{term"Ps = rev ps
@ qs"}. After the loop terminates, one
extra step is needed to close the loop. As expected, the postcondition
states that the @{term distPath} from @{term root} to itself is now
@{term "r # (rev Ps)"}.

It may come as a surprise to the reader that the simple algorithm for
acyclic list reversal, with modified annotations, works for cyclic
lists as well:›

lemma circular_list_rev_II:
"VARS next p q tmp
{p = Ref r ∧ distPath next p (r#Ps) p}
q:=Null;
WHILE p ≠ Null
INV
{ ((q = Null) ⟶ (∃ps. distPath next p (ps) (Ref r) ∧ ps = r#Ps)) ∧
((q ≠ Null) ⟶ (∃ps qs. distPath next q (qs) (Ref r) ∧ List next p ps  ∧
set ps ∩ set qs = {} ∧ rev qs @ ps = Ps@[r])) ∧
¬ (p = Null ∧ q = Null) }
DO tmp := p; p := p^.next; tmp^.next := q; q:=tmp OD
{q = Ref r ∧ distPath next q (r # rev Ps) q}"
apply (simp only:distPath_def)
apply vcg_simp
apply clarsimp
apply clarsimp
apply (case_tac "(q = Null)")
apply (fastforce intro: Path_is_List)
apply clarsimp
apply (rule_tac x= "bs" in exI)
apply (rule_tac x= "y # qs" in exI)
apply clarsimp
apply (auto simp:fun_upd_apply)
done

subsection "Storage allocation"

definition new :: "'a set ⇒ 'a"
where "new A = (SOME a. a ∉ A)"

lemma new_notin:
"⟦ ~finite(UNIV::'a set); finite(A::'a set); B ⊆ A ⟧ ⟹ new (A) ∉ B"
apply(unfold new_def)
apply(rule someI2_ex)
apply (fast intro:ex_new_if_finite)
apply (fast)
done

lemma "~finite(UNIV::'a set) ⟹
VARS xs elem next alloc p q
{Xs = xs ∧ p = (Null::'a ref)}
WHILE xs ≠ []
INV {islist next p ∧ set(list next p) ⊆ set alloc ∧
map elem (rev(list next p)) @ xs = Xs}
DO q := Ref(new(set alloc)); alloc := (addr q)#alloc;
q^.next := p; q^.elem := hd xs; xs := tl xs; p := q
OD
{islist next p ∧ map elem (rev(list next p)) = Xs}"
apply vcg_simp
apply (clarsimp simp: subset_insert_iff neq_Nil_conv fun_upd_apply new_notin)
apply fastforce
done

end
```