Theory "finite_map"

Signature

Definitions

fmap_inverse_def

|- ∀m1 m2.
     fmap_inverse m1 m2 ⇔
     ∀k. k ∈ FDOM m1 ⇒ ∃v. (FLOOKUP m1 k = SOME v) ∧ (FLOOKUP m2 v = SOME k)

FMAP_MAP2_def

|- ∀f m. FMAP_MAP2 f m = FUN_FMAP (λx. f (x,m ' x)) (FDOM m)

FUPDATE_LIST

|- $|++ = FOLDL $|+

SUBMAP_DEF

|- ∀f g. f ⊑ g ⇔ ∀x. x ∈ FDOM f ⇒ x ∈ FDOM g ∧ (f ' x = g ' x)

FCARD_DEF

|- ∀fm. FCARD fm = CARD (FDOM fm)

FDOM_DEF

|- ∀f x. FDOM f x ⇔ ISL (fmap_REP f x)

FAPPLY_DEF

|- ∀f x. f ' x = OUTL (fmap_REP f x)

FEMPTY_DEF

|- FEMPTY = fmap_ABS (λa. INR ())

FUPDATE_DEF

|- ∀f x y. f |+ (x,y) = fmap_ABS (λa. if a = x then INL y else fmap_REP f a)

fmap_ISO_DEF

|- (∀a. fmap_ABS (fmap_REP a) = a) ∧
   ∀r. is_fmap r ⇔ (fmap_REP (fmap_ABS r) = r)

fmap_TY_DEF

|- ∃rep. TYPE_DEFINITION is_fmap rep

is_fmap_def

|- is_fmap =
   (λa0.
      ∀is_fmap'.
        (∀a0.
           (a0 = (λa. INR ())) ∨
           (∃f a b. (a0 = (λx. if x = a then INL b else f x)) ∧ is_fmap' f) ⇒
           is_fmap' a0) ⇒
        is_fmap' a0)

DRESTRICT_DEF

|- ∀f r.
     (FDOM (DRESTRICT f r) = FDOM f ∩ r) ∧
     ∀x. DRESTRICT f r ' x = if x ∈ FDOM f ∩ r then f ' x else FEMPTY ' x

FUNION_DEF

|- ∀f g.
     (FDOM (f ⊌ g) = FDOM f ∪ FDOM g) ∧
     ∀x. (f ⊌ g) ' x = if x ∈ FDOM f then f ' x else g ' x

fmap_domsub

|- ∀fm k. fm \\ k = DRESTRICT fm (COMPL {k})

f_o_DEF

|- ∀f g. f f_o g = f f_o_f FUN_FMAP g {x | g x ∈ FDOM f}

FUN_FMAP_DEF

|- ∀f P.
     FINITE P ⇒
     (FDOM (FUN_FMAP f P) = P) ∧ ∀x. x ∈ P ⇒ (FUN_FMAP f P ' x = f x)

RRESTRICT_DEF

|- ∀f r.
     (FDOM (RRESTRICT f r) = {x | x ∈ FDOM f ∧ f ' x ∈ r}) ∧
     ∀x.
       RRESTRICT f r ' x =
       if x ∈ FDOM f ∧ f ' x ∈ r then f ' x else FEMPTY ' x

FRANGE_DEF

|- ∀f. FRANGE f = {y | ∃x. x ∈ FDOM f ∧ (f ' x = y)}

o_f_DEF

|- ∀f g.
     (FDOM (f o_f g) = FDOM g) ∧
     ∀x. x ∈ FDOM (f o_f g) ⇒ ((f o_f g) ' x = f (g ' x))

f_o_f_DEF

|- ∀f g.
     (FDOM (f f_o_f g) = FDOM g ∩ {x | g ' x ∈ FDOM f}) ∧
     ∀x. x ∈ FDOM (f f_o_f g) ⇒ ((f f_o_f g) ' x = f ' (g ' x))

FEVERY_DEF

|- ∀P f. FEVERY P f ⇔ ∀x. x ∈ FDOM f ⇒ P (x,f ' x)

FLOOKUP_DEF

|- ∀f x. FLOOKUP f x = if x ∈ FDOM f then SOME (f ' x) else NONE

FMERGE_DEF

|- ∀m f g.
     (FDOM (FMERGE m f g) = FDOM f ∪ FDOM g) ∧
     ∀x.
       FMERGE m f g ' x =
       if x ∉ FDOM f then g ' x
       else if x ∉ FDOM g then f ' x
       else m (f ' x) (g ' x)

MAP_KEYS_def

|- ∀f fm.
     (FDOM (MAP_KEYS f fm) = IMAGE f (FDOM fm)) ∧
     (INJ f (FDOM fm) 𝕌(:β) ⇒
      ∀x. x ∈ FDOM fm ⇒ (MAP_KEYS f fm ' (f x) = fm ' x))

fmap_rel_def

|- ∀R f1 f2.
     fmap_rel R f1 f2 ⇔
     (FDOM f2 = FDOM f1) ∧ ∀x. x ∈ FDOM f1 ⇒ R (f1 ' x) (f2 ' x)

fmap_EQ_UPTO_def

|- ∀f1 f2 vs.
     fmap_EQ_UPTO f1 f2 vs ⇔
     (FDOM f1 ∩ COMPL vs = FDOM f2 ∩ COMPL vs) ∧
     ∀x. x ∈ FDOM f1 ∩ COMPL vs ⇒ (f1 ' x = f2 ' x)

fmap_size_def

|- ∀kz vz fm. fmap_size kz vz fm = ∑ (λk. kz k + vz (fm ' k)) (FDOM fm)

Theorems

fevery_funion

|- ∀P m1 m2. FEVERY P m1 ∧ FEVERY P m2 ⇒ FEVERY P (m1 ⊌ m2)

drestrict_iter_list

|- ∀m l. FOLDR (λk m. m \\ k) m l = DRESTRICT m (COMPL (LIST_TO_SET l))

disjoint_drestrict

|- ∀s m. DISJOINT s (FDOM m) ⇒ (DRESTRICT m (COMPL s) = m)

fmap_to_list

|- ∀m. ∃l. ALL_DISTINCT (MAP FST l) ∧ (m = FEMPTY |++ l)

fupdate_list_foldl

|- ∀m l. FOLDL (λenv (k,v). env |+ (k,v)) m l = m |++ l

fupdate_list_foldr

|- ∀m l. FOLDR (λ(k,v) env. env |+ (k,v)) m l = m |++ REVERSE l

FUPDATE_EQ_FUPDATE_LIST

|- ∀fm kv. fm |+ kv = fm |++ [kv]

flookup_thm

|- ∀f x v.
     ((FLOOKUP f x = NONE) ⇔ x ∉ FDOM f) ∧
     ((FLOOKUP f x = SOME v) ⇔ x ∈ FDOM f ∧ (f ' x = v))

fdom_fupdate_list2

|- ∀kvl fm.
     FDOM (fm |++ kvl) =
     FDOM fm DIFF LIST_TO_SET (MAP FST kvl) ∪ LIST_TO_SET (MAP FST kvl)

fupdate_list_map

|- ∀l f x y.
     x ∈ FDOM (FEMPTY |++ l) ⇒
     ((FEMPTY |++ MAP (λ(a,b). (a,f b)) l) ' x = f ((FEMPTY |++ l) ' x))

fmap_eq_flookup

|- ∀m1 m2. (m1 = m2) ⇔ ∀k. FLOOKUP m1 k = FLOOKUP m2 k

fmap_rel_sym

|- (∀x y. R x y ⇒ R y x) ⇒ ∀x y. fmap_rel R x y ⇒ fmap_rel R y x

fmap_rel_trans

|- (∀x y z. R x y ∧ R y z ⇒ R x z) ⇒
   ∀x y z. fmap_rel R x y ∧ fmap_rel R y z ⇒ fmap_rel R x z

f_o_f_FUPDATE_compose

|- ∀f1 f2 k x v.
     x ∉ FDOM f1 ∧ x ∉ FRANGE f2 ⇒
     (f1 |+ (x,v) f_o_f f2 |+ (k,x) = (f1 f_o_f f2) |+ (k,v))

DRESTRICT_SUBSET

|- ∀f1 f2 s t.
     (DRESTRICT f1 s = DRESTRICT f2 s) ∧ t ⊆ s ⇒
     (DRESTRICT f1 t = DRESTRICT f2 t)

DRESTRICT_FDOM

|- ∀f. DRESTRICT f (FDOM f) = f

FRANGE_DRESTRICT_SUBSET

|- FRANGE (DRESTRICT fm s) ⊆ FRANGE fm

IN_FRANGE_DOMSUB_suff

|- (∀v. v ∈ FRANGE fm ⇒ P v) ⇒ ∀v. v ∈ FRANGE (fm \\ k) ⇒ P v

FRANGE_DOMSUB_SUBSET

|- FRANGE (fm \\ k) ⊆ FRANGE fm

IN_FRANGE_FUNION_suff

|- (∀v. v ∈ FRANGE f1 ⇒ P v) ∧ (∀v. v ∈ FRANGE f2 ⇒ P v) ⇒
   ∀v. v ∈ FRANGE (f1 ⊌ f2) ⇒ P v

FRANGE_FUNION_SUBSET

|- FRANGE (f1 ⊌ f2) ⊆ FRANGE f1 ∪ FRANGE f2

IN_FRANGE_FUPDATE_LIST_suff

|- (∀v. v ∈ FRANGE fm ⇒ P v) ∧ (∀v. MEM v (MAP SND ls) ⇒ P v) ⇒
   ∀v. v ∈ FRANGE (fm |++ ls) ⇒ P v

FRANGE_FUPDATE_LIST_SUBSET

|- ∀ls fm. FRANGE (fm |++ ls) ⊆ FRANGE fm ∪ LIST_TO_SET (MAP SND ls)

IN_FRANGE_FLOOKUP

|- ∀f v. v ∈ FRANGE f ⇔ ∃k. FLOOKUP f k = SOME v

IN_FRANGE

|- ∀f v. v ∈ FRANGE f ⇔ ∃k. k ∈ FDOM f ∧ (f ' k = v)

FUPDATE_LIST_ALL_DISTINCT_REVERSE

|- ∀ls. ALL_DISTINCT (MAP FST ls) ⇒ ∀fm. fm |++ REVERSE ls = fm |++ ls

FUPDATE_LIST_ALL_DISTINCT_APPLY_MEM

|- ∀P ls k v fm.
     ALL_DISTINCT (MAP FST ls) ∧ MEM (k,v) ls ∧ P v ⇒ P ((fm |++ ls) ' k)

FUPDATE_SAME_LIST_APPLY

|- ∀kvl fm1 fm2 x.
     MEM x (MAP FST kvl) ⇒ ((fm1 |++ kvl) ' x = (fm2 |++ kvl) ' x)

FUPDATE_SAME_APPLY

|- (x = FST kv) ∨ (fm1 ' x = fm2 ' x) ⇒ ((fm1 |+ kv) ' x = (fm2 |+ kv) ' x)

FUPDATE_LIST_APPLY_HO_THM

|- ∀P f kvl k.
     (∃n.
        n < LENGTH kvl ∧ (k = EL n (MAP FST kvl)) ∧ P (EL n (MAP SND kvl)) ∧
        ∀m. n < m ∧ m < LENGTH kvl ⇒ EL m (MAP FST kvl) ≠ k) ∨
     ¬MEM k (MAP FST kvl) ∧ P (f ' k) ⇒
     P ((f |++ kvl) ' k)

FUPDATE_LIST_APPLY_NOT_MEM_matchable

|- ∀kvl f k v. ¬MEM k (MAP FST kvl) ∧ (v = f ' k) ⇒ ((f |++ kvl) ' k = v)

DRESTRICT_FUNION_SUBSET

|- s2 ⊆ s1 ⇒ ∃h. DRESTRICT f s1 ⊌ g = DRESTRICT f s2 ⊌ h

FOLDL2_FUPDATE_LIST_paired

|- ∀f1 f2 bs cs a.
     (LENGTH bs = LENGTH cs) ⇒
     (FOLDL2 (λfm b (c,d). fm |+ (f1 b c d,f2 b c d)) a bs cs =
      a |++
      ZIP (MAP2 (λb. UNCURRY (f1 b)) bs cs,MAP2 (λb. UNCURRY (f2 b)) bs cs))

FOLDL2_FUPDATE_LIST

|- ∀f1 f2 bs cs a.
     (LENGTH bs = LENGTH cs) ⇒
     (FOLDL2 (λfm b c. fm |+ (f1 b c,f2 b c)) a bs cs =
      a |++ ZIP (MAP2 f1 bs cs,MAP2 f2 bs cs))

DRESTRICT_EQ_DRESTRICT_SAME

|- (DRESTRICT f1 s = DRESTRICT f2 s) ⇔
   (s ∩ FDOM f1 = s ∩ FDOM f2) ∧ ∀x. x ∈ FDOM f1 ∧ x ∈ s ⇒ (f1 ' x = f2 ' x)

DRESTRICT_FUNION_SAME

|- ∀fm s. DRESTRICT fm s ⊌ fm = fm

DRESTRICT_SUBSET_SUBMAP_gen

|- ∀f1 f2 s t.
     DRESTRICT f1 s ⊑ DRESTRICT f2 s ∧ t ⊆ s ⇒ DRESTRICT f1 t ⊑ DRESTRICT f2 t

DRESTRICT_DOMSUB

|- ∀f s k. DRESTRICT f s \\ k = DRESTRICT f (s DELETE k)

DOMSUB_SUBMAP

|- ∀f g x. f ⊑ g ∧ x ∉ FDOM f ⇒ f ⊑ g \\ x

SUBMAP_DOMSUB_gen

|- ∀f g k. f \\ k ⊑ g ⇔ f \\ k ⊑ g \\ k

SUBMAP_mono_FUPDATE

|- ∀f g x y. f \\ x ⊑ g \\ x ⇒ f |+ (x,y) ⊑ g |+ (x,y)

IMAGE_FRANGE

|- ∀f fm. IMAGE f (FRANGE fm) = FRANGE (f o_f fm)

DRESTRICTED_FUNION

|- ∀f1 f2 s.
     DRESTRICT (f1 ⊌ f2) s = DRESTRICT f1 s ⊌ DRESTRICT f2 (s DIFF FDOM f1)

DRESTRICT_SUBSET_SUBMAP

|- s1 ⊆ s2 ⇒ DRESTRICT f s1 ⊑ DRESTRICT f s2

DRESTRICT_SUBMAP_gen

|- f ⊑ g ⇒ DRESTRICT f P ⊑ g

IN_FRANGE_o_f_suff

|- (∀v. v ∈ FRANGE fm ⇒ P (f v)) ⇒ ∀v. v ∈ FRANGE (f o_f fm) ⇒ P v

IN_FRANGE_FUPDATE_suff

|- (∀v. v ∈ FRANGE fm ⇒ P v) ∧ P (SND kv) ⇒ ∀v. v ∈ FRANGE (fm |+ kv) ⇒ P v

FRANGE_FUPDATE_SUBSET

|- FRANGE (fm |+ kv) ⊆ FRANGE fm ∪ {SND kv}

IN_FRANGE_DRESTRICT_suff

|- (∀v. v ∈ FRANGE fm ⇒ P v) ⇒ ∀v. v ∈ FRANGE (DRESTRICT fm s) ⇒ P v

FUNION_ASSOC

|- ∀f g h. f ⊌ (g ⊌ h) = f ⊌ g ⊌ h

FUNION_COMM

|- ∀f g. DISJOINT (FDOM f) (FDOM g) ⇒ (f ⊌ g = g ⊌ f)

DOMSUB_FUNION

|- (f ⊌ g) \\ k = f \\ k ⊌ g \\ k

FUNION_EQ_IMPL

|- ∀f1 f2 f3.
     DISJOINT (FDOM f1) (FDOM f2) ∧ DISJOINT (FDOM f1) (FDOM f3) ∧ (f2 = f3) ⇒
     (f1 ⊌ f2 = f1 ⊌ f3)

FUNION_EQ

|- ∀f1 f2 f3.
     DISJOINT (FDOM f1) (FDOM f2) ∧ DISJOINT (FDOM f1) (FDOM f3) ⇒
     ((f1 ⊌ f2 = f1 ⊌ f3) ⇔ (f2 = f3))

FEMPTY_SUBMAP

|- ∀h. h ⊑ FEMPTY ⇔ (h = FEMPTY)

SUBMAP_FUNION_ID

|- (∀f1 f2. f1 ⊑ f1 ⊌ f2) ∧
   ∀f1 f2. DISJOINT (FDOM f1) (FDOM f2) ⇒ f2 ⊑ f1 ⊌ f2

SUBMAP_FUNION

|- ∀f1 f2 f3. f1 ⊑ f2 ∨ DISJOINT (FDOM f1) (FDOM f2) ∧ f1 ⊑ f3 ⇒ f1 ⊑ f2 ⊌ f3

SUBMAP_FUNION_EQ

|- (∀f1 f2 f3. DISJOINT (FDOM f1) (FDOM f2) ⇒ (f1 ⊑ f2 ⊌ f3 ⇔ f1 ⊑ f3)) ∧
   ∀f1 f2 f3.
     DISJOINT (FDOM f1) (FDOM f3 DIFF FDOM f2) ⇒ (f1 ⊑ f2 ⊌ f3 ⇔ f1 ⊑ f2)

FUNION_EQ_FEMPTY

|- ∀h1 h2. (h1 ⊌ h2 = FEMPTY) ⇔ (h1 = FEMPTY) ∧ (h2 = FEMPTY)

FEVERY_DRESTRICT_COMPL

|- FEVERY P (DRESTRICT (f |+ (k,v)) (COMPL s)) ⇔
   (k ∉ s ⇒ P (k,v)) ∧ FEVERY P (DRESTRICT f (COMPL (k INSERT s)))

FUPDATE_ELIM

|- ∀k v f. k ∈ FDOM f ∧ (f ' k = v) ⇒ (f |+ (k,v) = f)

FEVERY_STRENGTHEN_THM

|- FEVERY P FEMPTY ∧ (FEVERY P f ∧ P (x,y) ⇒ FEVERY P (f |+ (x,y)))

FMAP_MAP2_FUPDATE

|- FMAP_MAP2 f (m |+ (x,v)) = FMAP_MAP2 f m |+ (x,f (x,v))

FMAP_MAP2_FEMPTY

|- FMAP_MAP2 f FEMPTY = FEMPTY

FMAP_MAP2_THM

|- (FDOM (FMAP_MAP2 f m) = FDOM m) ∧
   ∀x. x ∈ FDOM m ⇒ (FMAP_MAP2 f m ' x = f (x,m ' x))

FUPDATE_PURGE

|- ∀f x y. f |+ (x,y) = f \\ x |+ (x,y)

FMEQ_SINGLE_SIMPLE_DISJ_ELIM

|- ∀fm k v ck cv.
     (fm |+ (k,v) = FEMPTY |+ (ck,cv)) ⇔
     (k = ck) ∧ (v = cv) ∧ ((fm = FEMPTY) ∨ ∃v'. fm = FEMPTY |+ (k,v'))

FMEQ_SINGLE_SIMPLE_ELIM

|- ∀P k v ck cv nv.
     (∃fm. (fm |+ (k,v) = FEMPTY |+ (ck,cv)) ∧ P (fm |+ (k,nv))) ⇔
     (k = ck) ∧ (v = cv) ∧ P (FEMPTY |+ (ck,nv))

FMEQ_ENUMERATE_CASES

|- ∀f1 kvl p. (f1 |+ p = FEMPTY |++ kvl) ⇒ MEM p kvl

FUPDATE_LIST_SAME_KEYS_UNWIND

|- ∀f1 f2 kvl1 kvl2.
     (f1 |++ kvl1 = f2 |++ kvl2) ∧ (MAP FST kvl1 = MAP FST kvl2) ∧
     ALL_DISTINCT (MAP FST kvl1) ⇒
     (kvl1 = kvl2) ∧
     ∀kvl. (MAP FST kvl = MAP FST kvl1) ⇒ (f1 |++ kvl = f2 |++ kvl)

FUPDATE_LIST_SAME_UPDATE

|- ∀kvl f1 f2.
     (f1 |++ kvl = f2 |++ kvl) ⇔
     (DRESTRICT f1 (COMPL (LIST_TO_SET (MAP FST kvl))) =
      DRESTRICT f2 (COMPL (LIST_TO_SET (MAP FST kvl))))

FDOM_FUPDATE_LIST

|- ∀kvl fm. FDOM (fm |++ kvl) = FDOM fm ∪ LIST_TO_SET (MAP FST kvl)

FUPD11_SAME_UPDATE

|- ∀f1 f2 k v.
     (f1 |+ (k,v) = f2 |+ (k,v)) ⇔
     (DRESTRICT f1 (COMPL {k}) = DRESTRICT f2 (COMPL {k}))

FUPD_SAME_KEY_UNWIND

|- ∀f1 f2 k v1 v2.
     (f1 |+ (k,v1) = f2 |+ (k,v2)) ⇒ (v1 = v2) ∧ ∀v. f1 |+ (k,v) = f2 |+ (k,v)

FUPD11_SAME_BASE

|- ∀f k1 v1 k2 v2.
     (f |+ (k1,v1) = f |+ (k2,v2)) ⇔
     (k1 = k2) ∧ (v1 = v2) ∨
     k1 ≠ k2 ∧ k1 ∈ FDOM f ∧ k2 ∈ FDOM f ∧ (f |+ (k1,v1) = f) ∧
     (f |+ (k2,v2) = f)

SAME_KEY_UPDATES_DIFFER

|- ∀f1 f2 k v1 v2. v1 ≠ v2 ⇒ f1 |+ (k,v1) ≠ f2 |+ (k,v2)

FUPD11_SAME_NEW_KEY

|- ∀f1 f2 k v1 v2.
     k ∉ FDOM f1 ∧ k ∉ FDOM f2 ⇒
     ((f1 |+ (k,v1) = f2 |+ (k,v2)) ⇔ (f1 = f2) ∧ (v1 = v2))

FUPD11_SAME_KEY_AND_BASE

|- ∀f k v1 v2. (f |+ (k,v1) = f |+ (k,v2)) ⇔ (v1 = v2)

FUPDATE_LIST_SNOC

|- ∀xs x fm. fm |++ SNOC x xs = (fm |++ xs) |+ x

FOLDL_FUPDATE_LIST

|- ∀f1 f2 ls a.
     FOLDL (λfm k. fm |+ (f1 k,f2 k)) a ls = a |++ MAP (λk. (f1 k,f2 k)) ls

FUPDATE_LIST_APPLY_MEM

|- ∀kvl f k v n.
     n < LENGTH kvl ∧ (k = EL n (MAP FST kvl)) ∧ (v = EL n (MAP SND kvl)) ∧
     (∀m. n < m ∧ m < LENGTH kvl ⇒ EL m (MAP FST kvl) ≠ k) ⇒
     ((f |++ kvl) ' k = v)

FEVERY_FUPDATE_LIST

|- ALL_DISTINCT (MAP FST kvl) ⇒
   (FEVERY P (fm |++ kvl) ⇔
    EVERY P kvl ∧ FEVERY P (DRESTRICT fm (COMPL (LIST_TO_SET (MAP FST kvl)))))

FUPDATE_FUPDATE_LIST_MEM

|- MEM k (MAP FST kvl) ⇒ (fm |+ (k,v) |++ kvl = fm |++ kvl)

FUPDATE_FUPDATE_LIST_COMMUTES

|- ¬MEM k (MAP FST kvl) ⇒ (fm |+ (k,v) |++ kvl = (fm |++ kvl) |+ (k,v))

FUPDATE_LIST_APPEND

|- fm |++ (kvl1 ++ kvl2) = fm |++ kvl1 |++ kvl2

FUPDATE_LIST_APPLY_NOT_MEM

|- ∀kvl f k. ¬MEM k (MAP FST kvl) ⇒ ((f |++ kvl) ' k = f ' k)

FUPDATE_LIST_THM

|- ∀f. (f |++ [] = f) ∧ ∀h t. f |++ (h::t) = f |+ h |++ t

SUBMAP_FUPDATE

|- ∀f g x y. f |+ (x,y) ⊑ g ⇔ x ∈ FDOM g ∧ (g ' x = y) ∧ f \\ x ⊑ g \\ x

SUBMAP_TRANS

|- ∀f g h. f ⊑ g ∧ g ⊑ h ⇒ f ⊑ h

SUBMAP_ANTISYM

|- ∀f g. f ⊑ g ∧ g ⊑ f ⇔ (f = g)

SUBMAP_REFL

|- ∀f. f ⊑ f

SUBMAP_FEMPTY

|- ∀f. FEMPTY ⊑ f

fmap_EXT

|- ∀f g. (f = g) ⇔ (FDOM f = FDOM g) ∧ ∀x. x ∈ FDOM f ⇒ (f ' x = g ' x)

fmap_EQ_THM

|- ∀f g. (FDOM f = FDOM g) ∧ (∀x. x ∈ FDOM f ⇒ (f ' x = g ' x)) ⇔ (f = g)

fmap_EQ

|- ∀f g. (FDOM f = FDOM g) ∧ ($' f = $' g) ⇔ (f = g)

NOT_FDOM_FAPPLY_FEMPTY

|- ∀f x. x ∉ FDOM f ⇒ (f ' x = FEMPTY ' x)

FM_PULL_APART

|- ∀fm k. k ∈ FDOM fm ⇒ ∃fm0 v. (fm = fm0 |+ (k,v)) ∧ k ∉ FDOM fm0

fmap_INDUCT

|- ∀P. P FEMPTY ∧ (∀f. P f ⇒ ∀x y. x ∉ FDOM f ⇒ P (f |+ (x,y))) ⇒ ∀f. P f

FCARD_SUC

|- ∀f n.
     (FCARD f = SUC n) ⇔
     ∃f' x y. x ∉ FDOM f' ∧ (FCARD f' = n) ∧ (f = f' |+ (x,y))

FCARD_0_FEMPTY

|- ∀f. (FCARD f = 0) ⇔ (f = FEMPTY)

FCARD_FUPDATE

|- ∀fm a b.
     FCARD (fm |+ (a,b)) = if a ∈ FDOM fm then FCARD fm else 1 + FCARD fm

FCARD_FEMPTY

|- FCARD FEMPTY = 0

FDOM_FINITE

|- ∀fm. FINITE (FDOM fm)

FDOM_F_FEMPTY1

|- ∀f. (∀a. a ∉ FDOM f) ⇔ (f = FEMPTY)

FDOM_EQ_EMPTY_SYM

|- ∀f. (∅ = FDOM f) ⇔ (f = FEMPTY)

FDOM_EQ_EMPTY

|- ∀f. (FDOM f = ∅) ⇔ (f = FEMPTY)

fmap_SIMPLE_INDUCT

|- ∀P. P FEMPTY ∧ (∀f. P f ⇒ ∀x y. P (f |+ (x,y))) ⇒ ∀f. P f

FDOM_EQ_FDOM_FUPDATE

|- ∀f x. x ∈ FDOM f ⇒ ∀y. FDOM (f |+ (x,y)) = FDOM f

NOT_EQ_FEMPTY_FUPDATE

|- ∀f a b. FEMPTY ≠ f |+ (a,b)

FAPPLY_FUPDATE_THM

|- ∀f a b x. (f |+ (a,b)) ' x = if x = a then b else f ' x

FDOM_FUPDATE

|- ∀f a b. FDOM (f |+ (a,b)) = a INSERT FDOM f

FDOM_FEMPTY

|- FDOM FEMPTY = ∅

FUPDATE_EQ

|- ∀f a b c. f |+ (a,b) |+ (a,c) = f |+ (a,c)

FUPDATE_COMMUTES

|- ∀f a b c d. a ≠ c ⇒ (f |+ (a,b) |+ (c,d) = f |+ (c,d) |+ (a,b))

NOT_EQ_FAPPLY

|- ∀f a x y. a ≠ x ⇒ ((f |+ (x,y)) ' a = f ' a)

FAPPLY_FUPDATE

|- ∀f x y. (f |+ (x,y)) ' x = y

is_fmap_cases

|- ∀a0.
     is_fmap a0 ⇔
     (a0 = (λa. INR ())) ∨
     ∃f a b. (a0 = (λx. if x = a then INL b else f x)) ∧ is_fmap f

is_fmap_strongind

|- ∀is_fmap'.
     is_fmap' (λa. INR ()) ∧
     (∀f a b.
        is_fmap f ∧ is_fmap' f ⇒
        is_fmap' (λx. if x = a then INL b else f x)) ⇒
     ∀a0. is_fmap a0 ⇒ is_fmap' a0

is_fmap_ind

|- ∀is_fmap'.
     is_fmap' (λa. INR ()) ∧
     (∀f a b. is_fmap' f ⇒ is_fmap' (λx. if x = a then INL b else f x)) ⇒
     ∀a0. is_fmap a0 ⇒ is_fmap' a0

is_fmap_rules

|- is_fmap (λa. INR ()) ∧
   ∀f a b. is_fmap f ⇒ is_fmap (λx. if x = a then INL b else f x)

FUNION_FEMPTY_1

|- ∀g. FEMPTY ⊌ g = g

FUNION_FEMPTY_2

|- ∀f. f ⊌ FEMPTY = f

FUNION_FUPDATE_1

|- ∀f g x y. f |+ (x,y) ⊌ g = (f ⊌ g) |+ (x,y)

FUNION_FUPDATE_2

|- ∀f g x y. f ⊌ g |+ (x,y) = if x ∈ FDOM f then f ⊌ g else (f ⊌ g) |+ (x,y)

EQ_FDOM_SUBMAP

|- (f = g) ⇔ f ⊑ g ∧ (FDOM f = FDOM g)

SUBMAP_FUPDATE_EQN

|- f ⊑ f |+ (x,y) ⇔ x ∉ FDOM f ∨ (f ' x = y) ∧ x ∈ FDOM f

DRESTRICT_FEMPTY

|- ∀r. DRESTRICT FEMPTY r = FEMPTY

DRESTRICT_FUPDATE

|- ∀f r x y.
     DRESTRICT (f |+ (x,y)) r =
     if x ∈ r then DRESTRICT f r |+ (x,y) else DRESTRICT f r

STRONG_DRESTRICT_FUPDATE

|- ∀f r x y.
     x ∈ r ⇒ (DRESTRICT (f |+ (x,y)) r = DRESTRICT f (r DELETE x) |+ (x,y))

FDOM_DRESTRICT

|- ∀f r x. FDOM (DRESTRICT f r) = FDOM f ∩ r

NOT_FDOM_DRESTRICT

|- ∀f x. x ∉ FDOM f ⇒ (DRESTRICT f (COMPL {x}) = f)

DRESTRICT_SUBMAP

|- ∀f r. DRESTRICT f r ⊑ f

DRESTRICT_DRESTRICT

|- ∀f P Q. DRESTRICT (DRESTRICT f P) Q = DRESTRICT f (P ∩ Q)

DRESTRICT_IS_FEMPTY

|- ∀f. DRESTRICT f ∅ = FEMPTY

FUPDATE_DRESTRICT

|- ∀f x y. f |+ (x,y) = DRESTRICT f (COMPL {x}) |+ (x,y)

STRONG_DRESTRICT_FUPDATE_THM

|- ∀f r x y.
     DRESTRICT (f |+ (x,y)) r =
     if x ∈ r then DRESTRICT f (COMPL {x} ∩ r) |+ (x,y)
     else DRESTRICT f (COMPL {x} ∩ r)

DRESTRICT_UNIV

|- ∀f. DRESTRICT f 𝕌(:α) = f

SUBMAP_DRESTRICT

|- DRESTRICT f P ⊑ f

DRESTRICT_EQ_DRESTRICT

|- ∀f1 f2 s1 s2.
     (DRESTRICT f1 s1 = DRESTRICT f2 s2) ⇔
     DRESTRICT f1 s1 ⊑ f2 ∧ DRESTRICT f2 s2 ⊑ f1 ∧
     (s1 ∩ FDOM f1 = s2 ∩ FDOM f2)

FDOM_FUNION

|- ∀f g x. FDOM (f ⊌ g) = FDOM f ∪ FDOM g

FMERGE_DOMSUB

|- ∀m m1 m2 k. FMERGE m m1 m2 \\ k = FMERGE m (m1 \\ k) (m2 \\ k)

SUBMAP_DOMSUB

|- f \\ k ⊑ f

fmap_CASES

|- ∀f. (f = FEMPTY) ∨ ∃g x y. f = g |+ (x,y)

DOMSUB_NOT_IN_DOM

|- k ∉ FDOM fm ⇒ (fm \\ k = fm)

o_f_FUPDATE

|- f o_f fm |+ (k,v) = (f o_f fm \\ k) |+ (k,f v)

DOMSUB_COMMUTES

|- fm \\ k1 \\ k2 = fm \\ k2 \\ k1

DOMSUB_IDEM

|- fm \\ k \\ k = fm \\ k

o_f_DOMSUB

|- (g o_f fm) \\ k = g o_f fm \\ k

FRANGE_FUPDATE_DOMSUB

|- ∀fm k v. FRANGE (fm |+ (k,v)) = v INSERT FRANGE (fm \\ k)

DOMSUB_FLOOKUP_THM

|- ∀fm k1 k2. FLOOKUP (fm \\ k1) k2 = if k1 = k2 then NONE else FLOOKUP fm k2

DOMSUB_FLOOKUP_NEQ

|- ∀fm k1 k2. k1 ≠ k2 ⇒ (FLOOKUP (fm \\ k1) k2 = FLOOKUP fm k2)

DOMSUB_FLOOKUP

|- ∀fm k. FLOOKUP (fm \\ k) k = NONE

DOMSUB_FAPPLY_THM

|- ∀fm k1 k2. (fm \\ k1) ' k2 = if k1 = k2 then FEMPTY ' k2 else fm ' k2

DOMSUB_FAPPLY_NEQ

|- ∀fm k1 k2. k1 ≠ k2 ⇒ ((fm \\ k1) ' k2 = fm ' k2)

DOMSUB_FAPPLY

|- ∀fm k. (fm \\ k) ' k = FEMPTY ' k

FDOM_DOMSUB

|- ∀fm k. FDOM (fm \\ k) = FDOM fm DELETE k

DOMSUB_FUPDATE_THM

|- ∀fm k1 k2 v.
     fm |+ (k1,v) \\ k2 = if k1 = k2 then fm \\ k2 else fm \\ k2 |+ (k1,v)

DOMSUB_FUPDATE_NEQ

|- ∀fm k1 k2 v. k1 ≠ k2 ⇒ (fm |+ (k1,v) \\ k2 = fm \\ k2 |+ (k1,v))

DOMSUB_FUPDATE

|- ∀fm k v. fm |+ (k,v) \\ k = fm \\ k

DOMSUB_FEMPTY

|- ∀k. FEMPTY \\ k = FEMPTY

FINITE_PRED_11

|- ∀g. (∀x y. (g x = g y) ⇔ (x = y)) ⇒ ∀f. FINITE {x | g x ∈ FDOM f}

FAPPLY_f_o

|- ∀f g.
     FINITE {x | g x ∈ FDOM f} ⇒
     ∀x. x ∈ FDOM (f f_o g) ⇒ ((f f_o g) ' x = f ' (g x))

f_o_FUPDATE

|- ∀fm k v g.
     FINITE {x | g x ∈ FDOM fm} ∧ FINITE {x | g x = k} ⇒
     (fm |+ (k,v) f_o g =
      FMERGE (combin$C K) (fm f_o g) (FUN_FMAP (K v) {x | g x = k}))

f_o_FEMPTY

|- ∀g. FEMPTY f_o g = FEMPTY

FDOM_f_o

|- ∀f g. FINITE {x | g x ∈ FDOM f} ⇒ (FDOM (f f_o g) = {x | g x ∈ FDOM f})

FLOOKUP_FUN_FMAP

|- FINITE P ⇒ (FLOOKUP (FUN_FMAP f P) k = if k ∈ P then SOME (f k) else NONE)

FDOM_FMAP

|- ∀f s. FINITE s ⇒ (FDOM (FUN_FMAP f s) = s)

FRANGE_FMAP

|- FINITE P ⇒ (FRANGE (FUN_FMAP f P) = IMAGE f P)

FUN_FMAP_EMPTY

|- FUN_FMAP f ∅ = FEMPTY

RRESTRICT_FUPDATE

|- ∀f r x y.
     RRESTRICT (f |+ (x,y)) r =
     if y ∈ r then RRESTRICT f r |+ (x,y)
     else RRESTRICT (DRESTRICT f (COMPL {x})) r

RRESTRICT_FEMPTY

|- ∀r. RRESTRICT FEMPTY r = FEMPTY

FRANGE_FUNION

|- DISJOINT (FDOM fm1) (FDOM fm2) ⇒
   (FRANGE (fm1 ⊌ fm2) = FRANGE fm1 ∪ FRANGE fm2)

FRANGE_FLOOKUP

|- v ∈ FRANGE f ⇔ ∃k. FLOOKUP f k = SOME v

o_f_FRANGE

|- x ∈ FRANGE g ⇒ f x ∈ FRANGE (f o_f g)

FINITE_FRANGE

|- ∀fm. FINITE (FRANGE fm)

SUBMAP_FRANGE

|- ∀f g. f ⊑ g ⇒ FRANGE f ⊆ FRANGE g

FRANGE_FUPDATE

|- ∀f x y. FRANGE (f |+ (x,y)) = y INSERT FRANGE (DRESTRICT f (COMPL {x}))

FRANGE_FEMPTY

|- FRANGE FEMPTY = ∅

FLOOKUP_o_f

|- FLOOKUP (f o_f fm) k =
   case FLOOKUP fm k of NONE => NONE | SOME v => SOME (f v)

o_f_o_f

|- f o_f (g o_f h) = f o g o_f h

FEVERY_o_f

|- ∀m P f. FEVERY P (f o_f m) ⇔ FEVERY (λx. P (FST x,f (SND x))) m

o_f_FEMPTY

|- f o_f FEMPTY = FEMPTY

o_f_FAPPLY

|- ∀f g x. x ∈ FDOM g ⇒ ((f o_f g) ' x = f (g ' x))

FDOM_o_f

|- ∀f g. FDOM (f o_f g) = FDOM g

o_f_FDOM

|- ∀f g. FDOM g = FDOM (f o_f g)

f_o_f_FEMPTY_2

|- ∀f. f f_o_f FEMPTY = FEMPTY

f_o_f_FEMPTY_1

|- ∀f. FEMPTY f_o_f f = FEMPTY

FEVERY_FLOOKUP

|- FEVERY P f ∧ (FLOOKUP f k = SOME v) ⇒ P (k,v)

FEVERY_FUPDATE

|- ∀P f x y.
     FEVERY P (f |+ (x,y)) ⇔ P (x,y) ∧ FEVERY P (DRESTRICT f (COMPL {x}))

FEVERY_FEMPTY

|- ∀P. FEVERY P FEMPTY

FLOOKUP_EXT

|- (f1 = f2) ⇔ (FLOOKUP f1 = FLOOKUP f2)

FLOOKUP_FUNION

|- FLOOKUP (f1 ⊌ f2) k =
   case FLOOKUP f1 k of NONE => FLOOKUP f2 k | SOME v => SOME v

SUBMAP_FUPDATE_FLOOKUP

|- f ⊑ f |+ (x,y) ⇔ (FLOOKUP f x = NONE) ∨ (FLOOKUP f x = SOME y)

FLOOKUP_SUBMAP

|- f ⊑ g ∧ (FLOOKUP f k = SOME v) ⇒ (FLOOKUP g k = SOME v)

FLOOKUP_UPDATE

|- FLOOKUP (fm |+ (k1,v)) k2 = if k1 = k2 then SOME v else FLOOKUP fm k2

FLOOKUP_EMPTY

|- FLOOKUP FEMPTY k = NONE

FMERGE_EQ_FEMPTY

|- (FMERGE m f g = FEMPTY) ⇔ (f = FEMPTY) ∧ (g = FEMPTY)

FMERGE_DRESTRICT

|- DRESTRICT (FMERGE f st1 st2) vs =
   FMERGE f (DRESTRICT st1 vs) (DRESTRICT st2 vs)

FMERGE_ASSOC

|- ASSOC (FMERGE m) ⇔ ASSOC m

FMERGE_COMM

|- COMM (FMERGE m) ⇔ COMM m

FMERGE_NO_CHANGE

|- ((FMERGE m f1 f2 = f1) ⇔
    ∀x. x ∈ FDOM f2 ⇒ x ∈ FDOM f1 ∧ (m (f1 ' x) (f2 ' x) = f1 ' x)) ∧
   ((FMERGE m f1 f2 = f2) ⇔
    ∀x. x ∈ FDOM f1 ⇒ x ∈ FDOM f2 ∧ (m (f1 ' x) (f2 ' x) = f2 ' x))

FUNION_FMERGE

|- ∀f1 f2 m. DISJOINT (FDOM f1) (FDOM f2) ⇒ (FMERGE m f1 f2 = f1 ⊌ f2)

FMERGE_FUNION

|- $⊌ = FMERGE (λx y. x)

FDOM_FMERGE

|- ∀m f g. FDOM (FMERGE m f g) = FDOM f ∪ FDOM g

FMERGE_FEMPTY

|- (FMERGE m f FEMPTY = f) ∧ (FMERGE m FEMPTY f = f)

FUNION_IDEMPOT

|- fm ⊌ fm = fm

DRESTRICT_FUNION

|- ∀h s1 s2. DRESTRICT h s1 ⊌ DRESTRICT h s2 = DRESTRICT h (s1 ∪ s2)

DRESTRICT_EQ_FUNION

|- ∀h h1 h2.
     DISJOINT (FDOM h1) (FDOM h2) ∧ (h1 ⊌ h2 = h) ⇒
     (h2 = DRESTRICT h (COMPL (FDOM h1)))

IN_FDOM_FOLDR_UNION

|- ∀x hL. x ∈ FDOM (FOLDR $⊌ FEMPTY hL) ⇔ ∃h. MEM h hL ∧ x ∈ FDOM h

DRESTRICT_FUNION_DRESTRICT_COMPL

|- DRESTRICT f s ⊌ DRESTRICT f (COMPL s) = f

DRESTRICT_IDEMPOT

|- ∀s vs. DRESTRICT (DRESTRICT s vs) vs = DRESTRICT s vs

SUBMAP_FUNION_ABSORPTION

|- ∀f g. f ⊑ g ⇔ (f ⊌ g = g)

MAP_KEYS_witness

|- let m f fm =
         if INJ f (FDOM fm) 𝕌(:β) then
           fm f_o_f FUN_FMAP (LINV f (FDOM fm)) (IMAGE f (FDOM fm))
         else FUN_FMAP ARB (IMAGE f (FDOM fm))
   in
     ∀f fm.
       (FDOM (m f fm) = IMAGE f (FDOM fm)) ∧
       (INJ f (FDOM fm) 𝕌(:β) ⇒ ∀x. x ∈ FDOM fm ⇒ (m f fm ' (f x) = fm ' x))

MAP_KEYS_FEMPTY

|- ∀f. MAP_KEYS f FEMPTY = FEMPTY

MAP_KEYS_FUPDATE

|- ∀f fm k v.
     INJ f (k INSERT FDOM fm) 𝕌(:β) ⇒
     (MAP_KEYS f (fm |+ (k,v)) = MAP_KEYS f fm |+ (f k,v))

MAP_KEYS_using_LINV

|- ∀f fm.
     INJ f (FDOM fm) 𝕌(:β) ⇒
     (MAP_KEYS f fm =
      fm f_o_f FUN_FMAP (LINV f (FDOM fm)) (IMAGE f (FDOM fm)))

fmap_rel_FUPDATE_same

|- fmap_rel R f1 f2 ∧ R v1 v2 ⇒ fmap_rel R (f1 |+ (k,v1)) (f2 |+ (k,v2))

fmap_rel_FUPDATE_LIST_same

|- ∀R ls1 ls2 f1 f2.
     fmap_rel R f1 f2 ∧ (MAP FST ls1 = MAP FST ls2) ∧
     LIST_REL R (MAP SND ls1) (MAP SND ls2) ⇒
     fmap_rel R (f1 |++ ls1) (f2 |++ ls2)

fmap_rel_FEMPTY

|- fmap_rel R FEMPTY FEMPTY

fmap_rel_FEMPTY2

|- (fmap_rel R FEMPTY f ⇔ (f = FEMPTY)) ∧ (fmap_rel R f FEMPTY ⇔ (f = FEMPTY))

fmap_rel_refl

|- (∀x. R x x) ⇒ fmap_rel R x x

fmap_rel_FUNION_rels

|- fmap_rel R f1 f2 ∧ fmap_rel R f3 f4 ⇒ fmap_rel R (f1 ⊌ f3) (f2 ⊌ f4)

fmap_rel_FUPDATE_I

|- fmap_rel R (f1 \\ k) (f2 \\ k) ∧ R v1 v2 ⇒
   fmap_rel R (f1 |+ (k,v1)) (f2 |+ (k,v2))

fmap_rel_mono

|- (∀x y. R1 x y ⇒ R2 x y) ⇒ fmap_rel R1 f1 f2 ⇒ fmap_rel R2 f1 f2

fmap_EQ_UPTO___EMPTY

|- ∀f1 f2. fmap_EQ_UPTO f1 f2 ∅ ⇔ (f1 = f2)

fmap_EQ_UPTO___EQ

|- ∀vs f. fmap_EQ_UPTO f f vs

fmap_EQ_UPTO___FUPDATE_BOTH

|- ∀f1 f2 ks k v.
     fmap_EQ_UPTO f1 f2 ks ⇒
     fmap_EQ_UPTO (f1 |+ (k,v)) (f2 |+ (k,v)) (ks DELETE k)

fmap_EQ_UPTO___FUPDATE_BOTH___NO_DELETE

|- ∀f1 f2 ks k v.
     fmap_EQ_UPTO f1 f2 ks ⇒ fmap_EQ_UPTO (f1 |+ (k,v)) (f2 |+ (k,v)) ks

fmap_EQ_UPTO___FUPDATE_SING

|- ∀f1 f2 ks k v.
     fmap_EQ_UPTO f1 f2 ks ⇒ fmap_EQ_UPTO (f1 |+ (k,v)) f2 (k INSERT ks)

o_f_FUNION

|- f o_f (f1 ⊌ f2) = f o_f f1 ⊌ (f o_f f2)

FDOM_FOLDR_DOMSUB

|- ∀ls fm. FDOM (FOLDR (λk m. m \\ k) fm ls) = FDOM fm DIFF LIST_TO_SET ls

FEVERY_SUBMAP

|- FEVERY P fm ∧ fm0 ⊑ fm ⇒ FEVERY P fm0

FEVERY_ALL_FLOOKUP

|- ∀P f. FEVERY P f ⇔ ∀k v. (FLOOKUP f k = SOME v) ⇒ P (k,v)

FUPDATE_LIST_CANCEL

|- ∀ls1 fm ls2.
     (∀k. MEM k (MAP FST ls1) ⇒ MEM k (MAP FST ls2)) ⇒
     (fm |++ ls1 |++ ls2 = fm |++ ls2)

FUPDATE_EQ_FUNION

|- ∀fm kv. fm |+ kv = FEMPTY |+ kv ⊌ fm

FUPDATE_LIST_APPEND_COMMUTES

|- ∀l1 l2 fm.
     DISJOINT (LIST_TO_SET (MAP FST l1)) (LIST_TO_SET (MAP FST l2)) ⇒
     (fm |++ l1 |++ l2 = fm |++ l2 |++ l1)

fmap_rel_OPTREL_FLOOKUP

|- fmap_rel R f1 f2 ⇔ ∀k. OPTREL R (FLOOKUP f1 k) (FLOOKUP f2 k)

FLOOKUP_DRESTRICT

|- ∀fm s k. FLOOKUP (DRESTRICT fm s) k = if k ∈ s then FLOOKUP fm k else NONE

FUPDATE_LIST_ALL_DISTINCT_PERM

|- ∀ls ls' fm.
     ALL_DISTINCT (MAP FST ls) ∧ PERM ls ls' ⇒ (fm |++ ls = fm |++ ls')