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theory Pointer_Examples(* Title: HOL/Hoare/Pointers.thy ID: $Id: Pointer_Examples.thy,v 1.6 2005/06/17 14:13:07 haftmann Exp $ Author: Tobias Nipkow Copyright 2002 TUM Examples of verifications of pointer programs *) theory Pointer_Examples imports HeapSyntax begin 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 fastsimp apply(fastsimp intro:notin_List_update[THEN iffD2]) (* explicit: apply clarify apply(rename_tac ps b qs) apply clarsimp apply(rename_tac ps') apply(fastsimp 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 fastsimp 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 fastsimp apply fastsimp done text "A longer readable version:" 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 fastsimp 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 fastsimp 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 fastsimp 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 fastsimp 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 fastsimp 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 "Merging two lists" text"This is still a bit rough, especially the proof." constdefs cor :: "bool => bool => bool" "cor P Q == if P then True else Q" cand :: "bool => bool => bool" "cand P Q == if P then Q else False" consts merge :: "'a list * 'a list * ('a => 'a => bool) => 'a list" recdef merge "measure(%(xs,ys,f). size xs + size ys)" "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 (simp_all add: cand_def cor_def) apply (fastsimp) apply clarsimp apply(rule conjI) apply clarsimp apply(rule conjI) apply (fastsimp 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 (fastsimp intro!:Path_snoc intro:Path_upd[THEN iffD2] notin_List_update[THEN iffD2] simp:eq_sym_conv) apply(clarsimp simp add:List_app) 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 (simp_all add: cand_def cor_def) apply (fastsimp) 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) apply(clarsimp simp add:List_app) 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.*} consts ispath:: "('a => 'a ref) => 'a ref => 'a ref => bool" path:: "('a => 'a ref) => 'a ref => 'a ref => 'a list" ML"set quick_and_dirty" text"First some basic lemmas:" lemma [simp]: "ispath f p p" sorry lemma [simp]: "path f p p = []" sorry lemma [simp]: "ispath f p q ==> a ∉ set(path f p q) ==> ispath (f(a := r)) p q" sorry lemma [simp]: "ispath f p q ==> a ∉ set(path f p q) ==> path (f(a := r)) p q = path f p q" sorry text"Some more specific lemmas needed by the example:" lemma [simp]: "ispath (f(a := q)) p (Ref a) ==> ispath (f(a := q)) p q" sorry lemma [simp]: "ispath (f(a := q)) p (Ref a) ==> path (f(a := q)) p q = path (f(a := q)) p (Ref a) @ [a]" sorry lemma [simp]: "ispath f p (Ref a) ==> f a = Ref b ==> b ∉ set (path f p (Ref a))" sorry lemma [simp]: "ispath f p (Ref a) ==> f a = Null ==> islist f p" sorry lemma [simp]: "ispath f p (Ref a) ==> f a = Null ==> list f p = path f p (Ref a) @ [a]" sorry lemma [simp]: "islist f p ==> distinct (list f p)" sorry ML"reset quick_and_dirty" 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 (simp_all add: cand_def cor_def) apply (fastsimp) apply (fastsimp simp: eq_sym_conv) apply(clarsimp) done text"The proof is automatic, but requires a numbet of special lemmas." subsection "Storage allocation" constdefs new :: "'a set => 'a" "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 fastsimp done end
lemma
{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 := tl (addr p); tl := tl(r -> q); q := r OD {List tl q (rev Ps @ Qs)}
lemma
{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 := next (addr p); next := next(r -> q); q := r; qs := hd ps # qs; ps := tl ps OD {List next q (rev Ps @ Qs)}
lemma
{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 := tl (addr p); tl := tl(r -> q); q := r OD {List tl q (rev Ps @ Qs)}
lemma
{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 := tl (addr p); tl := tl(r -> q); q := r OD {islist tl q ∧ list tl q = rev Ps @ Qs}
lemma
{List tl p Ps ∧ X ∈ set Ps} WHILE p ≠ Null ∧ p ≠ Ref X INV {∃ps. List tl p ps ∧ X ∈ set ps} DO p := tl (addr p) OD {p = Ref X}
lemma
{Path tl p Ps X} WHILE p ≠ Null ∧ p ≠ X INV {∃ps. Path tl p ps X} DO p := tl (addr p) OD {p = X}
lemma
{(p, X) ∈ {(Ref x, tl x) |x. True}*} WHILE p ≠ Null ∧ p ≠ X INV {(p, X) ∈ {(Ref x, tl x) |x. True}*} DO p := tl (addr p) OD {p = X}
lemma
{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 := tl (addr p) OD {p = Ref X}
lemma
({} = insert a A ∩ B) = (a ∉ B ∧ {} = A ∩ B)
lemma
({} = A ∩ insert b B) = (b ∉ A ∧ {} = A ∩ B)
lemma
({} = A ∩ (B ∪ C)) = ({} = A ∩ B ∧ {} = A ∩ C)
lemma
{List tl p Ps ∧ List tl q Qs ∧ set Ps ∩ set Qs = {} ∧ (p ≠ Null ∨ q ≠ Null)} IF cor (q = Null) (cand (p ≠ Null) (hd (addr p) ≤ hd (addr q))) THEN r := p; p := tl (addr p) ELSE r := q; q := tl (addr q) FI; s := r; WHILE p ≠ Null ∨ q ≠ Null INV {∃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, %x y. hd x ≤ hd y) = rs @ a # merge (ps, qs, %x y. hd x ≤ hd y) ∧ (tl a = p ∨ tl a = q)} DO IF cor (q = Null) (cand (p ≠ Null) (hd (addr p) ≤ hd (addr q))) THEN tl := tl(s -> p); p := tl (addr p) ELSE tl := tl(s -> q); q := tl (addr q) FI; s := tl (addr s) OD {List tl r (merge (Ps, Qs, %x y. hd x ≤ hd y))}
lemma
{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) (elem (addr p) ≤ elem (addr q))) THEN r := p; p := next (addr p); ps := tl ps ELSE r := q; q := next (addr q); 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, %x y. elem x ≤ elem y) = rs @ a # merge (ps, qs, %x y. elem x ≤ elem y) ∧ (next a = p ∨ next a = q)} DO IF cor (q = Null) (cand (p ≠ Null) (elem (addr p) ≤ elem (addr q))) THEN next := next(s -> p); p := next (addr p); ps := tl ps ELSE next := next(s -> q); q := next (addr q); qs := tl qs FI; rs := rs @ [a]; s := next (addr s); a := addr s OD {List next r (merge (Ps, Qs, %x y. elem x ≤ elem y))}
lemma
ispath f p p
lemma
path f p p = []
lemma
[| ispath f p q; a ∉ set (path f p q) |] ==> ispath (f(a := r)) p q
lemma
[| ispath f p q; a ∉ set (path f p q) |] ==> path (f(a := r)) p q = path f p q
lemma
ispath (f(a := q)) p (Ref a) ==> ispath (f(a := q)) p q
lemma
ispath (f(a := q)) p (Ref a) ==> path (f(a := q)) p q = path (f(a := q)) p (Ref a) @ [a]
lemma
[| ispath f p (Ref a); f a = Ref b |] ==> b ∉ set (path f p (Ref a))
lemma
[| ispath f p (Ref a); f a = Null |] ==> islist f p
lemma
[| ispath f p (Ref a); f a = Null |] ==> list f p = path f p (Ref a) @ [a]
lemma
islist f p ==> distinct (list f p)
lemma
{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) (hd (addr p) ≤ hd (addr q))) THEN r := p; p := tl (addr p) ELSE r := q; q := tl (addr q) FI; s := r; WHILE p ≠ Null ∨ q ≠ Null INV {∃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, %x y. hd x ≤ hd y) = rs @ a # merge (ps, qs, %x y. hd x ≤ hd y) ∧ (tl a = p ∨ tl a = q)} DO IF cor (q = Null) (cand (p ≠ Null) (hd (addr p) ≤ hd (addr q))) THEN tl := tl(s -> p); p := tl (addr p) ELSE tl := tl(s -> q); q := tl (addr q) FI; s := tl (addr s) OD {islist tl r ∧ list tl r = merge (Ps, Qs, %x y. hd x ≤ hd y)} [!]
lemma new_notin:
[| infinite UNIV; finite A; B ⊆ A |] ==> new A ∉ B
lemma
infinite UNIV ==> {Xs = xs ∧ p = Null} 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; next := next(q -> p); elem := elem(q -> hd xs); xs := tl xs; p := q OD {islist next p ∧ map elem (rev (list next p)) = Xs}