Theory "option"

Signature

Definitions

THE_DEF

⊢ ∀x. THE (SOME x) = x

some_def

⊢ ∀P. $some P = if ∃x. P x then SOME (@x. P x) else NONE

SOME_DEF

⊢ ∀x. SOME x = option_ABS (INL x)

OPTREL_def

⊢ ∀R x y.
      OPTREL R x y ⇔
      (x = NONE) ∧ (y = NONE) ∨
      ∃x0 y0. (x = SOME x0) ∧ (y = SOME y0) ∧ R x0 y0

option_TY_DEF

⊢ ∃rep. TYPE_DEFINITION (λx. T) rep

option_REP_ABS_DEF

⊢ (∀a. option_ABS (option_REP a) = a) ∧
  ∀r. (λx. T) r ⇔ (option_REP (option_ABS r) = r)

OPTION_MCOMP_def

⊢ ∀g f m. OPTION_MCOMP g f m = OPTION_BIND (f m) g

OPTION_MAP_DEF

⊢ (∀f x. OPTION_MAP f (SOME x) = SOME (f x)) ∧ ∀f. OPTION_MAP f NONE = NONE

OPTION_MAP2_DEF

⊢ ∀f x y.
      OPTION_MAP2 f x y =
      if IS_SOME x ∧ IS_SOME y then SOME (f (THE x) (THE y)) else NONE

OPTION_JOIN_DEF

⊢ (OPTION_JOIN NONE = NONE) ∧ ∀x. OPTION_JOIN (SOME x) = x

OPTION_IGNORE_BIND_def

⊢ ∀m1 m2. OPTION_IGNORE_BIND m1 m2 = OPTION_BIND m1 (K m2)

OPTION_GUARD_def

⊢ (OPTION_GUARD T = SOME ()) ∧ (OPTION_GUARD F = NONE)

OPTION_CHOICE_def

⊢ (∀m2. OPTION_CHOICE NONE m2 = m2) ∧
  ∀x m2. OPTION_CHOICE (SOME x) m2 = SOME x

option_case_def

⊢ (∀v f. option_CASE NONE v f = v) ∧ ∀x v f. option_CASE (SOME x) v f = f x

OPTION_BIND_def

⊢ (∀f. OPTION_BIND NONE f = NONE) ∧ ∀x f. OPTION_BIND (SOME x) f = f x

OPTION_APPLY_def

⊢ (∀x. NONE <*> x = NONE) ∧ ∀f x. SOME f <*> x = OPTION_MAP f x

OPTION_ALL_def

⊢ (∀P. OPTION_ALL P NONE ⇔ T) ∧ ∀P x. OPTION_ALL P (SOME x) ⇔ P x

NONE_DEF

⊢ NONE = option_ABS (INR ())

IS_SOME_DEF

⊢ (∀x. IS_SOME (SOME x) ⇔ T) ∧ (IS_SOME NONE ⇔ F)

IS_NONE_DEF

⊢ (∀x. IS_NONE (SOME x) ⇔ F) ∧ (IS_NONE NONE ⇔ T)

Theorems

SOME_SOME_APPLY

⊢ SOME f <*> SOME x = SOME (f x)

some_intro

⊢ (∀x. P x ⇒ Q (SOME x)) ∧ ((∀x. ¬P x) ⇒ Q NONE) ⇒ Q ($some P)

some_F

⊢ (some x. F) = NONE

some_EQ

⊢ ((some x. x = y) = SOME y) ∧ ((some x. y = x) = SOME y)

some_elim

⊢ Q ($some P) ⇒ (∃x. P x ∧ Q (SOME x)) ∨ (∀x. ¬P x) ∧ Q NONE

SOME_APPLY_PERMUTE

⊢ f <*> SOME x = SOME (λf. f x) <*> f

SOME_11

⊢ ∀x y. (SOME x = SOME y) ⇔ (x = y)

OPTREL_SOME

⊢ (∀R x y. OPTREL R (SOME x) y ⇔ ∃z. (y = SOME z) ∧ R x z) ∧
  ∀R x y. OPTREL R x (SOME y) ⇔ ∃z. (x = SOME z) ∧ R z y

OPTREL_refl

⊢ (∀x. R x x) ⇒ ∀x. OPTREL R x x

OPTREL_O

⊢ ∀R1 R2. OPTREL (R1 ∘ᵣ R2) = OPTREL R1 ∘ᵣ OPTREL R2

OPTREL_MONO

⊢ (∀x y. P x y ⇒ Q x y) ⇒ OPTREL P x y ⇒ OPTREL Q x y

OPTREL_eq

⊢ OPTREL $= = $=

option_nchotomy

⊢ ∀opt. (opt = NONE) ∨ ∃x. opt = SOME x

OPTION_MCOMP_ID

⊢ (OPTION_MCOMP g SOME = g) ∧ (OPTION_MCOMP SOME f = f)

OPTION_MCOMP_ASSOC

⊢ OPTION_MCOMP f (OPTION_MCOMP g h) = OPTION_MCOMP (OPTION_MCOMP f g) h

OPTION_MAP_EQ_SOME

⊢ ∀f x y. (OPTION_MAP f x = SOME y) ⇔ ∃z. (x = SOME z) ∧ (y = f z)

OPTION_MAP_EQ_NONE_both_ways

⊢ ((OPTION_MAP f x = NONE) ⇔ (x = NONE)) ∧
  ((NONE = OPTION_MAP f x) ⇔ (x = NONE))

OPTION_MAP_EQ_NONE

⊢ ∀f x. (OPTION_MAP f x = NONE) ⇔ (x = NONE)

OPTION_MAP_CONG

⊢ ∀opt1 opt2 f1 f2.
      (opt1 = opt2) ∧ (∀x. (opt2 = SOME x) ⇒ (f1 x = f2 x)) ⇒
      (OPTION_MAP f1 opt1 = OPTION_MAP f2 opt2)

OPTION_MAP_COMPOSE

⊢ OPTION_MAP f (OPTION_MAP g x) = OPTION_MAP (f ∘ g) x

OPTION_MAP_CASE

⊢ OPTION_MAP f x = option_CASE x NONE (SOME ∘ f)

OPTION_MAP2_THM

⊢ (OPTION_MAP2 f (SOME x) (SOME y) = SOME (f x y)) ∧
  (OPTION_MAP2 f (SOME x) NONE = NONE) ∧
  (OPTION_MAP2 f NONE (SOME y) = NONE) ∧ (OPTION_MAP2 f NONE NONE = NONE)

OPTION_MAP2_SOME

⊢ (OPTION_MAP2 f o1 o2 = SOME v) ⇔
  ∃x1 x2. (o1 = SOME x1) ∧ (o2 = SOME x2) ∧ (v = f x1 x2)

OPTION_MAP2_NONE

⊢ (OPTION_MAP2 f o1 o2 = NONE) ⇔ (o1 = NONE) ∨ (o2 = NONE)

OPTION_MAP2_cong

⊢ ∀x1 x2 y1 y2 f1 f2.
      (x1 = x2) ∧ (y1 = y2) ∧
      (∀x y. (x2 = SOME x) ∧ (y2 = SOME y) ⇒ (f1 x y = f2 x y)) ⇒
      (OPTION_MAP2 f1 x1 y1 = OPTION_MAP2 f2 x2 y2)

OPTION_JOIN_EQ_SOME

⊢ ∀x y. (OPTION_JOIN x = SOME y) ⇔ (x = SOME (SOME y))

option_induction

⊢ ∀P. P NONE ∧ (∀a. P (SOME a)) ⇒ ∀x. P x

option_Induct

⊢ ∀P. (∀a. P (SOME a)) ∧ P NONE ⇒ ∀x. P x

OPTION_IGNORE_BIND_thm

⊢ (OPTION_IGNORE_BIND NONE m = NONE) ∧ (OPTION_IGNORE_BIND (SOME v) m = m)

OPTION_IGNORE_BIND_EQUALS_OPTION

⊢ ((OPTION_IGNORE_BIND m1 m2 = NONE) ⇔ (m1 = NONE) ∨ (m2 = NONE)) ∧
  ((OPTION_IGNORE_BIND m1 m2 = SOME y) ⇔ ∃x. (m1 = SOME x) ∧ (m2 = SOME y))

OPTION_GUARD_EQ_THM

⊢ ((OPTION_GUARD b = SOME ()) ⇔ b) ∧ ((OPTION_GUARD b = NONE) ⇔ ¬b)

OPTION_GUARD_COND

⊢ OPTION_GUARD b = if b then SOME () else NONE

option_CLAUSES

⊢ (∀x y. (SOME x = SOME y) ⇔ (x = y)) ∧ (∀x. THE (SOME x) = x) ∧
  (∀x. NONE ≠ SOME x) ∧ (∀x. SOME x ≠ NONE) ∧ (∀x. IS_SOME (SOME x) ⇔ T) ∧
  (IS_SOME NONE ⇔ F) ∧ (∀x. IS_NONE x ⇔ (x = NONE)) ∧
  (∀x. ¬IS_SOME x ⇔ (x = NONE)) ∧ (∀x. IS_SOME x ⇒ (SOME (THE x) = x)) ∧
  (∀x. option_CASE x NONE SOME = x) ∧ (∀x. option_CASE x x SOME = x) ∧
  (∀x. IS_NONE x ⇒ (option_CASE x e f = e)) ∧
  (∀x. IS_SOME x ⇒ (option_CASE x e f = f (THE x))) ∧
  (∀x. IS_SOME x ⇒ (option_CASE x e SOME = x)) ∧
  (∀v f. option_CASE NONE v f = v) ∧
  (∀x v f. option_CASE (SOME x) v f = f x) ∧
  (∀f x. OPTION_MAP f (SOME x) = SOME (f x)) ∧
  (∀f. OPTION_MAP f NONE = NONE) ∧ (OPTION_JOIN NONE = NONE) ∧
  ∀x. OPTION_JOIN (SOME x) = x

OPTION_CHOICE_NONE

⊢ OPTION_CHOICE m1 NONE = m1

OPTION_CHOICE_EQ_NONE

⊢ (OPTION_CHOICE m1 m2 = NONE) ⇔ (m1 = NONE) ∧ (m2 = NONE)

option_CASES

⊢ ∀opt. (∃x. opt = SOME x) ∨ (opt = NONE)

option_case_SOME_ID

⊢ ∀x. option_CASE x x SOME = x

option_case_ID

⊢ ∀x. option_CASE x NONE SOME = x

option_case_eq

⊢ (option_CASE opt nc sc = v) ⇔
  (opt = NONE) ∧ (nc = v) ∨ ∃x. (opt = SOME x) ∧ (sc x = v)

option_case_cong

⊢ ∀M M' v f.
      (M = M') ∧ ((M' = NONE) ⇒ (v = v')) ∧ (∀x. (M' = SOME x) ⇒ (f x = f' x)) ⇒
      (option_CASE M v f = option_CASE M' v' f')

option_case_compute

⊢ option_CASE x e f = if IS_SOME x then f (THE x) else e

OPTION_BIND_SOME

⊢ ∀opt. OPTION_BIND opt SOME = opt

OPTION_BIND_EQUALS_OPTION

⊢ ((OPTION_BIND p f = NONE) ⇔ (p = NONE) ∨ ∃x. (p = SOME x) ∧ (f x = NONE)) ∧
  ((OPTION_BIND p f = SOME y) ⇔ ∃x. (p = SOME x) ∧ (f x = SOME y))

OPTION_BIND_cong

⊢ ∀o1 o2 f1 f2.
      (o1 = o2) ∧ (∀x. (o2 = SOME x) ⇒ (f1 x = f2 x)) ⇒
      (OPTION_BIND o1 f1 = OPTION_BIND o2 f2)

option_Axiom

⊢ ∀e f. ∃fn. (fn NONE = e) ∧ ∀x. fn (SOME x) = f x

OPTION_APPLY_o

⊢ SOME $o <*> f <*> g <*> x = f <*> (g <*> x)

OPTION_APPLY_MAP2

⊢ OPTION_MAP f x <*> y = OPTION_MAP2 f x y

OPTION_ALL_MONO

⊢ (∀x. P x ⇒ P' x) ⇒ OPTION_ALL P opt ⇒ OPTION_ALL P' opt

OPTION_ALL_CONG

⊢ ∀opt opt' P P'.
      (opt = opt') ∧ (∀x. (opt' = SOME x) ⇒ (P x ⇔ P' x)) ⇒
      (OPTION_ALL P opt ⇔ OPTION_ALL P' opt')

NOT_SOME_NONE

⊢ ∀x. SOME x ≠ NONE

NOT_NONE_SOME

⊢ ∀x. NONE ≠ SOME x

NOT_IS_SOME_EQ_NONE

⊢ ∀x. ¬IS_SOME x ⇔ (x = NONE)

IS_SOME_MAP

⊢ IS_SOME (OPTION_MAP f x) ⇔ IS_SOME x

IS_SOME_EXISTS

⊢ ∀opt. IS_SOME opt ⇔ ∃x. opt = SOME x

IS_SOME_BIND

⊢ IS_SOME (OPTION_BIND x g) ⇒ IS_SOME x

IS_NONE_EQ_NONE

⊢ ∀x. IS_NONE x ⇔ (x = NONE)

IF_NONE_EQUALS_OPTION

⊢ (((if P then X else NONE) = NONE) ⇔ P ⇒ IS_NONE X) ∧
  (((if P then NONE else X) = NONE) ⇔ IS_SOME X ⇒ P) ∧
  (((if P then X else NONE) = SOME x) ⇔ P ∧ (X = SOME x)) ∧
  (((if P then NONE else X) = SOME x) ⇔ ¬P ∧ (X = SOME x))

IF_EQUALS_OPTION

⊢ (((if P then SOME x else NONE) = NONE) ⇔ ¬P) ∧
  (((if P then NONE else SOME x) = NONE) ⇔ P) ∧
  (((if P then SOME x else NONE) = SOME y) ⇔ P ∧ (x = y)) ∧
  (((if P then NONE else SOME x) = SOME y) ⇔ ¬P ∧ (x = y))

FORALL_OPTION

⊢ (∀opt. P opt) ⇔ P NONE ∧ ∀x. P (SOME x)

EXISTS_OPTION

⊢ ((∃opt. P opt) ⇔ P NONE) ∨ ∃x. P (SOME x)

datatype_option

⊢ DATATYPE (option NONE SOME)