Theory "alist"

Signature

Definitions

ADELKEY_def

⊢ ∀k alist. ADELKEY k alist = FILTER (λp. FST p ≠ k) alist

alist_range_def

⊢ ∀m. alist_range m = {v | ∃k. ALOOKUP m k = SOME v}

alist_to_fmap_def

⊢ ∀s. alist_to_fmap s = FOLDR (λ(k,v) f. f |+ (k,v)) FEMPTY s

fmap_to_alist_def

⊢ ∀s. fmap_to_alist s = MAP (λk. (k,s ' k)) (SET_TO_LIST (FDOM s))

Theorems

ADELKEY_AFUPDKEY

⊢ ∀ls f x y.
    x ≠ y ⇒ (ADELKEY x (AFUPDKEY y f ls) = AFUPDKEY y f (ADELKEY x ls))

ADELKEY_AFUPDKEY_same

⊢ ∀fd f ls. ADELKEY fd (AFUPDKEY fd f ls) = ADELKEY fd ls

ADELKEY_unchanged

⊢ ∀x ls. (ADELKEY x ls = ls) ⇔ ¬MEM x (MAP FST ls)

AFUPDKEY_ALOOKUP

⊢ ALOOKUP (AFUPDKEY k2 f al) k1 =
  case ALOOKUP al k1 of
    NONE => NONE
  | SOME v => if k1 = k2 then SOME (f v) else SOME v

AFUPDKEY_I

⊢ AFUPDKEY n I = I

AFUPDKEY_comm

⊢ ∀k1 k2 f1 f2 l.
    k1 ≠ k2 ⇒
    (AFUPDKEY k2 f2 (AFUPDKEY k1 f1 l) = AFUPDKEY k1 f1 (AFUPDKEY k2 f2 l))

AFUPDKEY_def

⊢ (∀k f. AFUPDKEY k f [] = []) ∧
  ∀v rest k' k f.
    AFUPDKEY k f ((k',v)::rest) =
    if k = k' then (k,f v)::rest else (k',v)::AFUPDKEY k f rest

AFUPDKEY_eq

⊢ ∀k f1 l f2.
    (∀v. (ALOOKUP l k = SOME v) ⇒ (f1 v = f2 v)) ⇒
    (AFUPDKEY k f1 l = AFUPDKEY k f2 l)

AFUPDKEY_ind

⊢ ∀P. (∀k f. P k f []) ∧
      (∀k f k' v rest. (k ≠ k' ⇒ P k f rest) ⇒ P k f ((k',v)::rest)) ⇒
      ∀v v1 v2. P v v1 v2

AFUPDKEY_o

⊢ AFUPDKEY k f1 (AFUPDKEY k f2 al) = AFUPDKEY k (f1 ∘ f2) al

AFUPDKEY_unchanged

⊢ ∀k f alist.
    (∀v. (ALOOKUP alist k = SOME v) ⇒ (f v = v)) ⇒
    (AFUPDKEY k f alist = alist)

ALL_DISTINCT_FEVERY_alist_to_fmap

⊢ ALL_DISTINCT (MAP FST ls) ⇒ (FEVERY P (alist_to_fmap ls) ⇔ EVERY P ls)

ALL_DISTINCT_fmap_to_alist_keys

⊢ ∀fm. ALL_DISTINCT (MAP FST (fmap_to_alist fm))

ALOOKUP_ADELKEY

⊢ ∀al. ALOOKUP (ADELKEY k1 al) k2 = if k1 = k2 then NONE else ALOOKUP al k2

ALOOKUP_ALL_DISTINCT_EL

⊢ ∀ls n.
    n < LENGTH ls ∧ ALL_DISTINCT (MAP FST ls) ⇒
    (ALOOKUP ls (FST (EL n ls)) = SOME (SND (EL n ls)))

ALOOKUP_ALL_DISTINCT_MEM

⊢ ALL_DISTINCT (MAP FST al) ∧ MEM (k,v) al ⇒ (ALOOKUP al k = SOME v)

ALOOKUP_ALL_DISTINCT_PERM_same

⊢ ∀l1 l2.
    ALL_DISTINCT (MAP FST l1) ∧ PERM (MAP FST l1) (MAP FST l2) ∧
    (LIST_TO_SET l1 = LIST_TO_SET l2) ⇒
    (ALOOKUP l1 = ALOOKUP l2)

ALOOKUP_APPEND

⊢ ∀l1 l2 k.
    ALOOKUP (l1 ++ l2) k =
    case ALOOKUP l1 k of NONE => ALOOKUP l2 k | SOME v => SOME v

ALOOKUP_APPEND_same

⊢ ∀l1 l2 l.
    (ALOOKUP l1 = ALOOKUP l2) ⇒ (ALOOKUP (l1 ++ l) = ALOOKUP (l2 ++ l))

ALOOKUP_EQ_FLOOKUP

⊢ (FLOOKUP (alist_to_fmap al) = ALOOKUP al) ∧
  (ALOOKUP (fmap_to_alist fm) = FLOOKUP fm)

ALOOKUP_FAILS

⊢ (ALOOKUP l x = NONE) ⇔ ∀k v. MEM (k,v) l ⇒ k ≠ x

ALOOKUP_FILTER

⊢ ∀ls x.
    ALOOKUP (FILTER (λ(k,v). P k) ls) x = if P x then ALOOKUP ls x else NONE

ALOOKUP_IN_FRANGE

⊢ ∀ls k v. (ALOOKUP ls k = SOME v) ⇒ v ∈ FRANGE (alist_to_fmap ls)

ALOOKUP_LEAST_EL

⊢ ∀ls k.
    ALOOKUP ls k =
    if MEM k (MAP FST ls) then
      SOME (EL (LEAST n. EL n (MAP FST ls) = k) (MAP SND ls))
    else NONE

ALOOKUP_MAP

⊢ ∀f al. ALOOKUP (MAP (λ(x,y). (x,f y)) al) = OPTION_MAP f ∘ ALOOKUP al

ALOOKUP_MAP_2

⊢ ∀f al x.
    ALOOKUP (MAP (λ(x,y). (x,f x y)) al) x = OPTION_MAP (f x) (ALOOKUP al x)

ALOOKUP_MEM

⊢ ∀al k v. (ALOOKUP al k = SOME v) ⇒ MEM (k,v) al

ALOOKUP_NONE

⊢ ∀l x. (ALOOKUP l x = NONE) ⇔ ¬MEM x (MAP FST l)

ALOOKUP_SOME_FAPPLY_alist_to_fmap

⊢ ∀al k v. (ALOOKUP al k = SOME v) ⇒ (alist_to_fmap al ' k = v)

ALOOKUP_TABULATE

⊢ MEM x l ⇒ (ALOOKUP (MAP (λk. (k,f k)) l) x = SOME (f x))

ALOOKUP_ZIP_MAP_SND

⊢ ∀l1 l2 k f.
    (LENGTH l1 = LENGTH l2) ⇒
    (ALOOKUP (ZIP (l1,MAP f l2)) = OPTION_MAP f ∘ ALOOKUP (ZIP (l1,l2)))

ALOOKUP_def

⊢ (∀q. ALOOKUP [] q = NONE) ∧
  ∀y x t q. ALOOKUP ((x,y)::t) q = if x = q then SOME y else ALOOKUP t q

ALOOKUP_ind

⊢ ∀P. (∀q. P [] q) ∧ (∀x y t q. (x ≠ q ⇒ P t q) ⇒ P ((x,y)::t) q) ⇒
      ∀v v1. P v v1

ALOOKUP_prefix

⊢ ∀ls k ls2.
    ((ALOOKUP ls k = SOME v) ⇒ (ALOOKUP (ls ++ ls2) k = SOME v)) ∧
    ((ALOOKUP ls k = NONE) ⇒ (ALOOKUP (ls ++ ls2) k = ALOOKUP ls2 k))

FDOM_alist_to_fmap

⊢ ∀al. FDOM (alist_to_fmap al) = LIST_TO_SET (MAP FST al)

FEVERY_alist_to_fmap

⊢ EVERY P ls ⇒ FEVERY P (alist_to_fmap ls)

FLOOKUP_FUPDATE_LIST_ALOOKUP_NONE

⊢ (ALOOKUP ls k = NONE) ⇒ (FLOOKUP (fm |++ REVERSE ls) k = FLOOKUP fm k)

FLOOKUP_FUPDATE_LIST_ALOOKUP_SOME

⊢ (ALOOKUP ls k = SOME v) ⇒ (FLOOKUP (fm |++ REVERSE ls) k = SOME v)

FRANGE_alist_to_fmap_SUBSET

⊢ FRANGE (alist_to_fmap ls) ⊆ IMAGE SND (LIST_TO_SET ls)

FUNION_alist_to_fmap

⊢ ∀ls fm. alist_to_fmap ls FUNION fm = fm |++ REVERSE ls

FUPDATE_LIST_EQ_APPEND_REVERSE

⊢ ∀ls fm. fm |++ ls = alist_to_fmap (REVERSE ls ++ fmap_to_alist fm)

IN_FRANGE_alist_to_fmap_suff

⊢ (∀v. MEM v (MAP SND ls) ⇒ P v) ⇒ ∀v. v ∈ FRANGE (alist_to_fmap ls) ⇒ P v

LENGTH_AFUPDKEY

⊢ ∀ls. LENGTH (AFUPDKEY k f ls) = LENGTH ls

LENGTH_fmap_to_alist

⊢ ∀fm. LENGTH (fmap_to_alist fm) = CARD (FDOM fm)

MAP_FST_AFUPDKEY

⊢ MAP FST (AFUPDKEY f k alist) = MAP FST alist

MAP_KEYS_I

⊢ ∀fm. MAP_KEYS I fm = fm

MAP_values_fmap_to_alist

⊢ ∀f fm. MAP (λ(k,v). (k,f v)) (fmap_to_alist fm) = fmap_to_alist (f o_f fm)

MEM_ADELKEY

⊢ ∀al. MEM (k1,v) (ADELKEY k2 al) ⇔ k1 ≠ k2 ∧ MEM (k1,v) al

MEM_fmap_to_alist

⊢ MEM (x,y) (fmap_to_alist fm) ⇔ x ∈ FDOM fm ∧ (fm ' x = y)

MEM_fmap_to_alist_FLOOKUP

⊢ ∀p fm. MEM p (fmap_to_alist fm) ⇔ (FLOOKUP fm (FST p) = SOME (SND p))

MEM_pair_fmap_to_alist_FLOOKUP

⊢ ∀x y fm. MEM (x,y) (fmap_to_alist fm) ⇔ (FLOOKUP fm x = SOME y)

PERM_fmap_to_alist

⊢ PERM (fmap_to_alist fm1) (fmap_to_alist fm2) ⇔ (fm1 = fm2)

alist_to_fmap_APPEND

⊢ ∀l1 l2. alist_to_fmap (l1 ++ l2) = alist_to_fmap l1 FUNION alist_to_fmap l2

alist_to_fmap_FAPPLY_MEM

⊢ ∀al z. z ∈ FDOM (alist_to_fmap al) ⇒ MEM (z,alist_to_fmap al ' z) al

alist_to_fmap_MAP

⊢ ∀f1 f2 al.
    INJ f1 (LIST_TO_SET (MAP FST al)) 𝕌(:β) ⇒
    (alist_to_fmap (MAP (λ(x,y). (f1 x,f2 y)) al) =
     MAP_KEYS f1 (f2 o_f alist_to_fmap al))

alist_to_fmap_MAP_matchable

⊢ ∀f1 f2 al mal v.
    INJ f1 (LIST_TO_SET (MAP FST al)) 𝕌(:β) ∧
    (mal = MAP (λ(x,y). (f1 x,f2 y)) al) ∧
    (v = MAP_KEYS f1 (f2 o_f alist_to_fmap al)) ⇒
    (alist_to_fmap mal = v)

alist_to_fmap_MAP_values

⊢ ∀f al. alist_to_fmap (MAP (λ(k,v). (k,f v)) al) = f o_f alist_to_fmap al

alist_to_fmap_PERM

⊢ ∀l1 l2.
    PERM l1 l2 ∧ ALL_DISTINCT (MAP FST l1) ⇒
    (alist_to_fmap l1 = alist_to_fmap l2)

alist_to_fmap_prefix

⊢ ∀ls l1 l2.
    (alist_to_fmap l1 = alist_to_fmap l2) ⇒
    (alist_to_fmap (ls ++ l1) = alist_to_fmap (ls ++ l2))

alist_to_fmap_thm

⊢ (alist_to_fmap [] = FEMPTY) ∧
  (alist_to_fmap ((k,v)::t) = alist_to_fmap t |+ (k,v))

alist_to_fmap_to_alist

⊢ ∀al.
    fmap_to_alist (alist_to_fmap al) =
    MAP (λk. (k,THE (ALOOKUP al k))) (SET_TO_LIST (LIST_TO_SET (MAP FST al)))

alist_to_fmap_to_alist_PERM

⊢ ∀al. ALL_DISTINCT (MAP FST al) ⇒ PERM (fmap_to_alist (alist_to_fmap al)) al

alookup_distinct_reverse

⊢ ∀l k. ALL_DISTINCT (MAP FST l) ⇒ (ALOOKUP (REVERSE l) k = ALOOKUP l k)

alookup_filter

⊢ ∀f l x. ALOOKUP l x = ALOOKUP (FILTER (λ(x',y). x = x') l) x

alookup_stable_sorted

⊢ ∀R sort x l.
    transitive R ∧ total R ∧ STABLE sort (inv_image R FST) ⇒
    (ALOOKUP (sort (inv_image R FST) l) x = ALOOKUP l x)

flookup_fupdate_list

⊢ ∀l k m.
    FLOOKUP (m |++ l) k =
    case ALOOKUP (REVERSE l) k of NONE => FLOOKUP m k | SOME v => SOME v

fmap_to_alist_FEMPTY

⊢ fmap_to_alist FEMPTY = []

fmap_to_alist_inj

⊢ ∀f1 f2. (fmap_to_alist f1 = fmap_to_alist f2) ⇒ (f1 = f2)

fmap_to_alist_preserves_FDOM

⊢ ∀fm1 fm2.
    (FDOM fm1 = FDOM fm2) ⇒
    (MAP FST (fmap_to_alist fm1) = MAP FST (fmap_to_alist fm2))

fmap_to_alist_to_fmap

⊢ alist_to_fmap (fmap_to_alist fm) = fm

fupdate_list_funion

⊢ ∀m l. m |++ l = FEMPTY |++ l FUNION m

mem_to_flookup

⊢ ∀x y l.
    ALL_DISTINCT (MAP FST l) ∧ MEM (x,y) l ⇒
    (FLOOKUP (FEMPTY |++ l) x = SOME y)

set_MAP_FST_fmap_to_alist

⊢ LIST_TO_SET (MAP FST (fmap_to_alist fm)) = FDOM fm