Commit 1d789093 authored by Guillaume Melquiond's avatar Guillaume Melquiond

Convert some more examples.

parent 4cb75244
......@@ -84,8 +84,8 @@ module Grid
(** {3 Valid Sudoku Solutions} *)
(** [valid_chunk g i start offsets] is true whenever the chunk
denoted by [start,offsets] from cell [i] is "valid" in grid [g], in
(** `valid_chunk g i start offsets` is true whenever the chunk
denoted by `start,offsets` from cell `i` is "valid" in grid `g`, in
the sense that it contains at most one occurence of each number
between 1 and 9 *)
......@@ -107,16 +107,16 @@ module Grid
predicate valid_square (s:sudoku_chunks) (g:grid) (i:int) =
valid_chunk g i s.square_start s.square_offsets
(** [valid g] is true when all chunks are valid *)
(** `valid g` is true when all chunks are valid *)
predicate valid (s:sudoku_chunks) (g : grid) =
forall i : int. is_index i ->
valid_column s g i /\ valid_row s g i /\ valid_square s g i
(** [full g] is true when all cells are filled *)
(** `full g` is true when all cells are filled *)
predicate full (g : grid) =
forall i : int. is_index i -> 1 <= Map.get g i <= 9
(** [included g1 g2] *)
(** `included g1 g2` *)
predicate included (g1 g2 : grid) =
forall i : int. is_index i /\ 1 <= Map.get g1 i <= 9 ->
Map.get g2 i = Map.get g1 i
......@@ -132,8 +132,8 @@ module Grid
forall s g h.
well_formed_sudoku s /\ included g h /\ valid s h -> valid s g
(** A solution of a grid [data] is a full grid [sol]
that is valid and includes [data] *)
(** A solution of a grid `data` is a full grid `sol`
that is valid and includes `data` *)
predicate is_solution_for (s:sudoku_chunks) (sol:grid) (data:grid) =
included data sol /\ full sol /\ valid s sol
......@@ -248,8 +248,8 @@ module Solver
use import array.Array
(** [check_valid_chunk g i start offsets] checks the validity
of the chunk that includes [i] *)
(** `check_valid_chunk g i start offsets` checks the validity
of the chunk that includes `i` *)
let check_valid_chunk (g:array int) (i:int)
(start:array int) (offsets:array int) : unit
requires { g.length = 81 }
......@@ -283,8 +283,8 @@ module Solver
forall j : int. 0 <= j < i ->
valid_column s g j /\ valid_row s g j /\ valid_square s g j
(** [check_valid s g] checks if the (possibly partially filled) grid
[g] is valid. (This function is not needed by the solver) *)
(** `check_valid s g` checks if the (possibly partially filled) grid
`g` is valid. (This function is not needed by the solver) *)
let check_valid (s:sudoku_chunks) (g : array int) : bool
requires { well_formed_sudoku s }
requires { g.length = 81 }
......@@ -302,7 +302,7 @@ module Solver
with Invalid -> False
end
(** [full_up_to g i] is true when all cells [0..i-1] in grid [g] are
(** `full_up_to g i` is true when all cells `0..i-1` in grid `g` are
non empty *)
predicate full_up_to (g : grid) (i : int) = forall j :
int. 0 <= j < i -> 1 <= Map.get g j <= 9
......@@ -358,31 +358,31 @@ module Solver
(** how to prove the "hard" property : if
[valid_up_to s g i]
`valid_up_to s g i`
and
[h = g[i <- k] (with 1 <= k <= 9)]
`h = g[i <- k` (with 1 <= k <= 9)]
and
[valid_column s h i /\ valid_row s h i /\ valid_square s h i]
`valid_column s h i /\ valid_row s h i /\ valid_square s h i`
then
[valid_up_to s h (i+1)]
`valid_up_to s h (i+1)`
then the problem is that one should prove that for each [j] in [0..i-1] :
then the problem is that one should prove that for each `j` in `0..i-1` :
[valid_column s h j /\ valid_row s h j /\ valid_square s h j]
`valid_column s h j /\ valid_row s h j /\ valid_square s h j`
this is true but with 2 different possible reasons:
if [column_start j = column_start i] then
[valid_column s h j] is true because [valid_column s h i] is true
if `column_start j = column_start i` then
`valid_column s h j` is true because `valid_column s h i` is true
else
[valid_column s h j] is true because [valid_column s g j] is true
because [valid_column s h j] does not depend on [h[i]]
`valid_column s h j` is true because `valid_column s g j` is true
because `valid_column s h j` does not depend on `h[i]`
*)
......
......@@ -4,9 +4,9 @@
We are interested in specifying and proving correct
data structures that support efficient computation of the sum of the
values over an arbitrary range of an array.
Concretely, given an array of integers [a], and given a range
delimited by indices [i] (inclusive) and [j] (exclusive), we wish
to compute the value: [\sum_{k=i}^{j-1} a[k]].
Concretely, given an array of integers `a`, and given a range
delimited by indices `i` (inclusive) and `j` (exclusive), we wish
to compute the value: `\sum_{k=i}^{j-1} a[k]`.
In the first part, we consider a simple loop
for computing the sum in linear time.
......@@ -15,7 +15,7 @@ In the second part, we introduce a cumulative sum array
that allows answering arbitrary range queries in constant time.
In the third part, we explore a tree data structure that
supports modification of values from the underlying array [a],
supports modification of values from the underlying array `a`,
with logarithmic time operations.
*)
......@@ -28,23 +28,23 @@ module ArraySum
use export int.Int
use export array.Array
(** [sum a i j] denotes the sum [\sum_{i <= k < j} a[k]].
(** `sum a i j` denotes the sum `\sum_{i <= k < j} a[k]`.
It is axiomatizated by the two following axioms expressing
the recursive definition
if [i <= j] then [sum a i j = 0]
if `i <= j` then `sum a i j = 0`
if [i < j] then [sum a i j = a[i] + sum a (i+1) j]
if `i < j` then `sum a i j = a[i] + sum a (i+1) j`
*)
let rec function sum (a:array int) (i j:int) : int
let rec function sum (a:array int) (i j:int) : int
requires { 0 <= i <= j <= a.length }
variant { j - i }
= if j <= i then 0 else a[i] + sum a (i+1) j
(** lemma for summation from the right:
if [i < j] then [sum a i j = sum a i (j-1) + a[j-1]]
if `i < j` then `sum a i j = sum a i (j-1) + a[j-1]`
*)
lemma sum_right : forall a : array int, i j : int.
......@@ -63,8 +63,8 @@ module Simple
use import ArraySum
use import ref.Ref
(** [query a i j] returns the sum of elements in [a] between
index [i] inclusive and index [j] exclusive *)
(** `query a i j` returns the sum of elements in `a` between
index `i` inclusive and index `j` exclusive *)
let query (a:array int) (i j:int) : int
requires { 0 <= i <= j <= a.length }
ensures { result = sum a i j }
......@@ -80,7 +80,7 @@ end
(** {2 Additional lemmas on [sum]}
(** {2 Additional lemmas on `sum`}
needed in the remaining code *)
module ExtraLemmas
......@@ -93,8 +93,8 @@ module ExtraLemmas
0 <= i <= j <= k <= a.length ->
sum a i k = sum a i j + sum a j k
(** Frame lemma for [sum], that is [sum a i j] depends only
of values of [a[i..j-1]] *)
(** Frame lemma for `sum`, that is `sum a i j` depends only
of values of `a[i..j-1]` *)
lemma sum_frame : forall a1 a2 : array int, i j : int.
0 <= i <= j ->
j <= a1.length ->
......@@ -102,8 +102,8 @@ module ExtraLemmas
(forall k : int. i <= k < j -> a1[k] = a2[k]) ->
sum a1 i j = sum a2 i j
(** Updated lemma for [sum]: how does [sum a i j] changes when
[a[k]] is changed for some [k] in [[i..j-1]] *)
(** Updated lemma for `sum`: how does `sum a i j` changes when
`a[k]` is changed for some `k` in `[i..j-1]` *)
lemma sum_update : forall a:array int, i v l h:int.
0 <= l <= i < h <= a.length ->
sum (a[i<-v]) l h = sum a l h + v - a[i]
......@@ -135,7 +135,7 @@ module CumulativeArray
c.length = a.length + 1 /\
forall i. 0 <= i < c.length -> c[i] = sum a 0 i
(** [create a] builds the cumulative array associated with [a]. *)
(** `create a` builds the cumulative array associated with `a`. *)
let create (a:array int) : array int
ensures { is_cumulative_array_for result a }
= let l = a.length in
......@@ -146,8 +146,8 @@ module CumulativeArray
done;
s
(** [query c i j a] returns the sum of elements in [a] between
index [i] inclusive and index [j] exclusive, in constant time *)
(** `query c i j a` returns the sum of elements in `a` between
index `i` inclusive and index `j` exclusive, in constant time *)
let query (c:array int) (i j:int) (ghost a:array int): int
requires { is_cumulative_array_for c a }
requires { 0 <= i <= j < c.length }
......@@ -155,8 +155,8 @@ module CumulativeArray
= c[j] - c[i]
(** [update c i v a] updates cell [a[i]] to value [v] and updates
the cumulative array [c] accordingly *)
(** `update c i v a` updates cell `a[i]` to value `v` and updates
the cumulative array `c` accordingly *)
let update (c:array int) (i:int) (v:int) (ghost a:array int) : unit
requires { is_cumulative_array_for c a }
requires { 0 <= i < a.length }
......@@ -290,7 +290,7 @@ module CumulativeTree
= if i=j then 0 else query_aux t a i j
(** frame lemma for predicate [is_tree_for] *)
(** frame lemma for predicate `is_tree_for` *)
lemma is_tree_for_frame : forall t:tree, a:array int, k v i j:int.
0 <= k < a.length ->
k < i \/ k >= j ->
......@@ -343,10 +343,10 @@ module CumulativeTree
(** {2 complexity analysis}
We would like to prove that [query] is really logarithmic. This is
We would like to prove that `query` is really logarithmic. This is
non-trivial because there are two recursive calls in some cases.
So far, we are only able to prove that [update] is logarithmic
So far, we are only able to prove that `update` is logarithmic
We express the complexity by passing a ``credit'' as a ghost
parameter. We pose the precondition that the credit is at least
......@@ -384,7 +384,7 @@ module CumulativeTree
end
(** [update_aux] function instrumented with a credit *)
(** `update_aux` function instrumented with a credit *)
use import ref.Ref
......@@ -419,14 +419,14 @@ module CumulativeTree
(Node {ind with isum = ind.isum + delta} l r',delta) (*>*)
end
(** [query_aux] function instrumented with a credit *)
(** `query_aux` function instrumented with a credit *)
let rec query_aux_complexity (t:tree) (ghost a: array int)
(i j:int) (ghost c:ref int) : int
requires { is_tree_for t a t.indexes.low t.indexes.high }
requires { 0 <= t.indexes.low <= i < j <= t.indexes.high <= a.length }
variant { t }
ensures { !c - old !c <=
ensures { !c - old !c <=
if i = t.indexes.low /\ j = t.indexes.high then 1 else
if i = t.indexes.low \/ j = t.indexes.high then 2 * depth t else
4 * depth t }
......
......@@ -160,26 +160,26 @@ module HeightSmallSpace
(** Count number of leaves in a tree. *)
function leaves (t: tree 'a) : int = 1 + size t
(** [height_limited acc depth lim t]:
Compute the [height t] if the number of leaves in [t] is at most [lim],
fails otherwise. [acc] and [depth] are accumulators.
(** `height_limited acc depth lim t`:
Compute the `height t` if the number of leaves in `t` is at most `lim`,
fails otherwise. `acc` and `depth` are accumulators.
For maintaining the limit within the recursion, this routine
also send back the difference between the number of leaves and
the limit in case of success.
Method: find out one child with number of leaves at most [lim/2] using
Method: find out one child with number of leaves at most `lim/2` using
recursive calls. If no such child is found, the tree has at
least [lim+1] leaves, hence fails. Otherwise, accumulate the result
least `lim+1` leaves, hence fails. Otherwise, accumulate the result
of the recursive call for that child and make a recursive tail-call
for the other child, using the computed difference in order to
update [lim]. Since non-tail-recursive calls halve the limit,
the space complexity is logarithmic in [lim].
update `lim`. Since non-tail-recursive calls halve the limit,
the space complexity is logarithmic in `lim`.
Note that as is, this has a degenerate case:
if the small child is extremely small, we may waste a lot
of computing time on the large child to notice it is large,
while in the end processing only the small child until the
tail-recursive call. Analysis shows that this results in
super-polynomial time behavior (recursion T(N) = T(N/2)+T(N-1))
To mitigate this, we perform recursive calls on all [lim/2^k] limits
To mitigate this, we perform recursive calls on all `lim/2^k` limits
in increasing order (see process_small_child subroutine), until
one succeed or maximal limits both fails. This way,
the time spent by a single phase of the algorithm is representative
......
......@@ -166,8 +166,8 @@ module RedBlackTree
(* insertion *)
(** [almost_rbtree n t]: [t] may have one red-red conflict at its root;
it satisfies [rbtree n] everywhere else *)
(** `almost_rbtree n t`: `t` may have one red-red conflict at its root;
it satisfies `rbtree n` everywhere else *)
predicate almost_rbtree (n : int) (t : tree) =
match t with
......@@ -190,8 +190,8 @@ module RedBlackTree
forall x: key, v: value, l r: tree, n: int.
almost_rbtree n (Node Black l x v r) -> rbtree n (Node Black l x v r)
(** [lbalance c x l r] acts as a black node constructor,
solving a possible red-red conflict on [l], using the following
(** `lbalance c x l r` acts as a black node constructor,
solving a possible red-red conflict on `l`, using the following
schema:
B (R (R a x b) y c) z d
......@@ -216,8 +216,8 @@ module RedBlackTree
Node Black l k v r
end
(** [rbalance l x r] is similar to [lbalance], solving a possible red-red
conflict on [r]. The balancing schema is similar:
(** `rbalance l x r` is similar to `lbalance`, solving a possible red-red
conflict on `r`. The balancing schema is similar:
B a x (R (R b y c) z d)
B a x (R b y (R c z d)) -> R (B a x b) y (R c z d)
......@@ -240,8 +240,8 @@ module RedBlackTree
Node Black l k v r
end
(* [insert x s] inserts [x] in tree [s], resulting in a possible top red-red
conflict when [s] is red. *)
(* `insert x s` inserts `x` in tree `s`, resulting in a possible top red-red
conflict when `s` is red. *)
let rec insert (t : tree) (k : key) (v : value) : tree
requires { bst t /\ exists n: int. rbtree n t }
......@@ -265,7 +265,7 @@ module RedBlackTree
else (* k = k' *) Node Black l k' v r
end
(* finally [add x s] calls [insert] and recolors the root as black *)
(* finally `add x s` calls `insert` and recolors the root as black *)
let add (t : tree) (k : key) (v : value) : tree
requires { bst t /\ exists n:int. rbtree n t }
......@@ -320,7 +320,7 @@ module Vacid0
= let (d, t) = !m in
try find t k with Not_found -> d end
(* the easy way: implements [remove] using [replace] *)
(* the easy way: implements `remove` using `replace` *)
let remove (m : ref rbt) k
requires { inv !m }
ensures { inv !m /\
......
......@@ -63,13 +63,13 @@ module DancingLinks
invariant { length prev = length next = n }
by { prev = make 0 0; next = make 0 0; n = 0 }
(** node [i] is a valid node i.e. it has consistent neighbors *)
(** node `i` is a valid node i.e. it has consistent neighbors *)
predicate valid_in (l: dll) (i: int) =
0 <= i < l.n /\ 0 <= l.prev[i] < l.n /\ 0 <= l.next[i] < l.n /\
l.next[l.prev[i]] = i /\
l.prev[l.next[i]] = i
(** node [i] is ready to be put back in a list *)
(** node `i` is ready to be put back in a list *)
predicate valid_out (l: dll) (i: int) =
0 <= i < l.n /\ 0 <= l.prev[i] < l.n /\ 0 <= l.next[i] < l.n /\
l.next[l.prev[i]] = l.next[i] /\
......@@ -78,8 +78,8 @@ module DancingLinks
use seq.Seq as S
function nth (s: S.seq 'a) (i: int) : 'a = S.([]) s i
(** Representation predicate: Sequence [s] is the list of indices of
a valid circular list in [l].
(** Representation predicate: Sequence `s` is the list of indices of
a valid circular list in `l`.
We choose to model circular lists, since this is the way the
data structure is used in Knuth's dancing links algorithm. *)
predicate is_list (l: dll) (s: S.seq int) =
......@@ -90,7 +90,7 @@ module DancingLinks
(forall k': int. 0 <= k' < S.length s -> k <> k' -> nth s k <> nth s k')
(** Note: the code below works fine even when the list has one element
(necessarily [i] in that case). *)
(necessarily `i` in that case). *)
let remove (l: dll) (i: int) (ghost s: S.seq int)
requires { valid_in l i }
requires { is_list l (S.cons i s) }
......@@ -103,10 +103,10 @@ module DancingLinks
nth (S.cons i s) (k + 1) = nth s k } (* to help SMT with triggers *)
let put_back (l: dll) (i: int) (ghost s: S.seq int)
requires { valid_out l i } (* [i] is ready to be reinserted *)
requires { valid_out l i } (* `i` is ready to be reinserted *)
requires { is_list l s }
requires { 0 < S.length s } (* [s] must contain at least one element *)
requires { l.next[i] = nth s 0 <> i } (* do not link [i] to itself *)
requires { 0 < S.length s } (* `s` must contain at least one element *)
requires { l.next[i] = nth s 0 <> i } (* do not link `i` to itself *)
ensures { valid_in l i }
ensures { is_list l (S.cons i s) }
=
......
......@@ -87,7 +87,7 @@ module RelaxedPrefix
type char
val eq (x y : char) : bool ensures { result = True <-> x = y }
(** [a1[ofs1..ofs1+len]] and [a2[ofs2..ofs2+len]] are valid sub-arrays
(** `a1[ofs1..ofs1+len]` and `a2[ofs2..ofs2+len]` are valid sub-arrays
and they are equal *)
predicate eq_array (a1: array char) (ofs1: int)
(a2: array char) (ofs2: int) (len: int) =
......@@ -110,7 +110,7 @@ module RelaxedPrefix
exception NoPrefix
(** Note regarding the code: the suggested pseudo-code is buggy, as it
may access [a] out of bounds. We fix it by strengthening the
may access `a` out of bounds. We fix it by strengthening the
test in the conditional. *)
let is_relaxed_prefix (pat a: array char) : bool
......
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