Theory "alist"

Signature

Definitions

fmap_to_alist_def

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

alist_to_fmap_def

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

ALOOKUP_tupled_primitive_def

|- ALOOKUP_tupled =
   WFREC (@R. WF R ∧ ∀y t q x. x ≠ q ⇒ R (t,q) ((x,y)::t,q))
     (λALOOKUP_tupled a.
        case a of
          ([],q) => I NONE
        | ((x,y)::t,q) => I (if x = q then SOME y else ALOOKUP_tupled (t,q)))

ALOOKUP_curried_def

|- ∀x x1. ALOOKUP x x1 = ALOOKUP_tupled (x,x1)

alist_range_def

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

Theorems

fmap_to_alist_FEMPTY

|- fmap_to_alist FEMPTY = []

alist_to_fmap_thm

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

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_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_FAILS

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

ALOOKUP_NONE

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

ALOOKUP_TABULATE

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

ALOOKUP_EQ_FLOOKUP

|- (FLOOKUP (alist_to_fmap al) = ALOOKUP al) ∧
   (ALOOKUP (fmap_to_alist fm) = FLOOKUP fm)

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)

LENGTH_fmap_to_alist

|- ∀fm. LENGTH (fmap_to_alist fm) = CARD (FDOM fm)

fmap_to_alist_to_fmap

|- alist_to_fmap (fmap_to_alist fm) = fm

ALOOKUP_MEM

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

ALOOKUP_SOME_FAPPLY_alist_to_fmap

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

alist_to_fmap_FAPPLY_MEM

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

ALOOKUP_MAP

|- ∀f al. ALOOKUP (MAP (λ(x,y). (x,f y)) al) = OPTION_MAP f o ALOOKUP al

FDOM_alist_to_fmap

|- ∀al. FDOM (alist_to_fmap al) = LIST_TO_SET (MAP FST al)

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_APPEND

|- ∀l1 l2. alist_to_fmap (l1 ++ l2) = alist_to_fmap l1 ⊌ alist_to_fmap l2

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))

ALOOKUP_APPEND

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

FUPDATE_LIST_EQ_APPEND_REVERSE

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

FLOOKUP_FUPDATE_LIST_ALOOKUP_SOME

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

FLOOKUP_FUPDATE_LIST_ALOOKUP_NONE

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

FUNION_alist_to_fmap

|- ∀ls fm. alist_to_fmap ls ⊌ fm = fm |++ REVERSE ls

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_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_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_ALL_DISTINCT_MEM

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

ALL_DISTINCT_fmap_to_alist_keys

|- ∀fm. ALL_DISTINCT (MAP FST (fmap_to_alist fm))

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))

PERM_fmap_to_alist

|- PERM (fmap_to_alist fm1) (fmap_to_alist fm2) ⇔ (fm1 = fm2)

alist_to_fmap_PERM

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

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_ZIP_MAP_SND

|- ∀l1 l2 k f.
     (LENGTH l1 = LENGTH l2) ⇒
     (ALOOKUP (ZIP (l1,MAP f l2)) = OPTION_MAP f o ALOOKUP (ZIP (l1,l2)))

ALOOKUP_FILTER

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

ALOOKUP_APPEND_same

|- ∀l1 l2 l.
     (ALOOKUP l1 = ALOOKUP l2) ⇒ (ALOOKUP (l1 ++ l) = ALOOKUP (l2 ++ l))

ALOOKUP_IN_FRANGE

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

FRANGE_alist_to_fmap_SUBSET

|- FRANGE (alist_to_fmap ls) ⊆ IMAGE SND (LIST_TO_SET ls)

IN_FRANGE_alist_to_fmap_suff

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

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)

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)

MAP_KEYS_I

|- ∀fm. MAP_KEYS I fm = fm

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

set_MAP_FST_fmap_to_alist

|- LIST_TO_SET (MAP FST (fmap_to_alist fm)) = FDOM fm

alookup_distinct_reverse

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

flookup_fupdate_list

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

fupdate_list_funion

|- ∀m l. m |++ l = FEMPTY |++ l ⊌ m

mem_to_flookup

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

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)

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)