Theory of maps

Generic Maps

module Map

  type map 'a 'b = 'a -> 'b

  let function get (f: map 'a 'b) (x: 'a) : 'b = f x

  let ghost function set (f: map 'a 'b) (x: 'a) (v: 'b) : map 'a 'b =
    fun y -> if pure {y = x} then v else f y

  let function ([]) (f: map 'a 'b) (x: 'a) : 'b = f x
  let ghost function ([<-]) (f: map 'a 'b) (x: 'a) (v: 'b) : map 'a 'b = set f x v

syntactic sugar

end

module Const

  use Map

  let function const (v: 'b) : map 'a 'b = fun _ -> v

end

module MapExt

  predicate (==) (m1 m2: 'a -> 'b) = forall x: 'a. m1 x = m2 x

  lemma extensionality:
    forall m1 m2: 'a -> 'b. m1 == m2 -> m1 = m2
  (* This lemma is actually provable in Why3, because of how
     eliminate_epsilon handles equality to a lambda-term. *)

end

Sorted Maps (indexed by integers)

module MapSorted

  use int.Int
  use Map

  type elt

  predicate le elt elt

  predicate sorted_sub (a : map int elt) (l u : int) =
    forall i1 i2 : int. l <= i1 <= i2 < u -> le a[i1] a[i2]

sorted_sub a l u is true whenever the array segment a(l..u-1) is sorted w.r.t order relation le

end

Maps Equality (indexed by integers)

module MapEq

  use int.Int
  use Map

  predicate map_eq_sub (a1 a2 : map int 'a) (l u : int) =
    forall i:int. l <= i < u -> a1[i] = a2[i]

end

module MapExchange

  use int.Int
  use Map

  predicate exchange (a1 a2: map int 'a) (l u i j: int) =
    l <= i < u /\ l <= j < u /\
    a1[i] = a2[j] /\ a1[j] = a2[i] /\
    (forall k:int. l <= k < u -> k <> i -> k <> j -> a1[k] = a2[k])

  lemma exchange_set :
    forall a: map int 'a, l u i j: int.
    l <= i < u -> l <= j < u ->
    exchange a a[i <- a[j]][j <- a[i]] l u i j

end

Sum of elements of a map (indexed by integers)

module MapSum

  use int.Int
  use int.Sum as S
  use Map

  function sum (m: map int int) (l h: int) : int = S.sum (get m) l h

sum m l h is the sum of m[i] for l <= i < h

end

Number of occurrences

(* TODO: we could define Occ using theory int.NumOf *)
module Occ

  use int.Int
  use Map

  function occ (v: 'a) (m: map int 'a) (l u: int) : int

number of occurrences of v in m between l included and u excluded

  axiom occ_empty:
    forall v: 'a, m: map int 'a, l u: int.
    u <= l -> occ v m l u = 0

  axiom occ_right_no_add:
    forall v: 'a, m: map int 'a, l u: int.
    l < u -> m[u-1] <> v -> occ v m l u = occ v m l (u-1)

  axiom occ_right_add:
    forall v: 'a, m: map int 'a, l u: int.
    l < u -> m[u-1] = v -> occ v m l u = 1 + occ v m l (u-1)

  lemma occ_left_no_add:
    forall v: 'a, m: map int 'a, l u: int.
    l < u -> m[l] <> v -> occ v m l u = occ v m (l+1) u

  lemma occ_left_add:
    forall v: 'a, m: map int 'a, l u: int.
    l < u -> m[l] = v -> occ v m l u = 1 + occ v m (l+1) u

  lemma occ_bounds:
    forall v: 'a, m: map int 'a, l u: int.
    l <= u -> 0 <= occ v m l u <= u - l
    (* direct when l>=u, by induction on u when l <= u *)

  lemma occ_append:
    forall v: 'a, m: map int 'a, l mid u: int.
    l <= mid <= u -> occ v m l u = occ v m l mid + occ v m mid u
    (* by induction on u *)

  lemma occ_neq:
    forall v: 'a, m: map int 'a, l u: int.
    (forall i: int. l <= i < u -> m[i] <> v) -> occ v m l u = 0
    (* by induction on u *)

  lemma occ_exists:
    forall v: 'a, m: map int 'a, l u: int.
    occ v m l u > 0 -> exists i: int. l <= i < u /\ m[i] = v

  lemma occ_pos:
    forall m: map int 'a, l u i: int.
    l <= i < u -> occ m[i] m l u > 0

  lemma occ_eq:
    forall v: 'a, m1 m2: map int 'a, l u: int.
    (forall i: int. l <= i < u -> m1[i] = m2[i]) -> occ v m1 l u = occ v m2 l u
    (* by induction on u *)

  lemma occ_exchange :
    forall m: map int 'a, l u i j: int, x y z: 'a.
    l <= i < u -> l <= j < u -> i <> j ->
    occ z m[i <- x][j <- y] l u =
    occ z m[i <- y][j <- x] l u

end

module MapPermut

  use int.Int
  use Map
  use Occ

  predicate permut (m1 m2: map int 'a) (l u: int) =
    forall v: 'a. occ v m1 l u = occ v m2 l u

  lemma permut_trans: (* provable, yet useful *)
    forall a1 a2 a3 : map int 'a. forall l u : int.
    permut a1 a2 l u -> permut a2 a3 l u -> permut a1 a3 l u

  lemma permut_exists :
    forall a1 a2: map int 'a, l u i: int.
    permut a1 a2 l u -> l <= i < u ->
    exists j: int. l <= j < u /\ a1[j] = a2[i]

end

Injectivity and surjectivity for maps (indexed by integers)

module MapInjection

  use int.Int
  use export Map

  predicate injective (a: map int int) (n: int) =
    forall i j: int. 0 <= i < n -> 0 <= j < n -> i <> j -> a[i] <> a[j]

injective a n is true when a is an injection on the domain (0..n-1)

  predicate surjective (a: map int int) (n: int) =
    forall i: int. 0 <= i < n -> exists j: int. (0 <= j < n /\ a[j] = i)

surjective a n is true when a is a surjection from (0..n-1) to (0..n-1)

  predicate range (a: map int int) (n: int) =
    forall i: int. 0 <= i < n -> 0 <= a[i] < n

range a n is true when a maps the domain (0..n-1) into (0..n-1)

  lemma injective_surjective:
    forall a: map int int, n: int.
    injective a n -> range a n -> surjective a n

main lemma: an injection on (0..n-1) that ranges into (0..n-1) is also a surjection

  use Occ

  lemma injection_occ:
    forall m: map int int, n: int.
    injective m n <-> forall v:int. (occ v m 0 n <= 1)

end