Structure optionTheory

signature optionTheory =
sig
  type thm = Thm.thm
  
  (*  Definitions  *)
    val IS_NONE_DEF : thm
    val IS_SOME_DEF : thm
    val NONE_DEF : thm
    val OPTION_ALL_def : thm
    val OPTION_APPLY_def : thm
    val OPTION_BIND_def : thm
    val OPTION_CHOICE_def : thm
    val OPTION_GUARD_def : thm
    val OPTION_IGNORE_BIND_def : thm
    val OPTION_JOIN_DEF : thm
    val OPTION_MAP2_DEF : thm
    val OPTION_MAP_DEF : thm
    val OPTION_MCOMP_def : thm
    val OPTREL_def : thm
    val SOME_DEF : thm
    val THE_DEF : thm
    val option_REP_ABS_DEF : thm
    val option_TY_DEF : thm
    val option_case_def : thm
    val some_def : thm
  
  (*  Theorems  *)
    val EXISTS_OPTION : thm
    val FORALL_OPTION : thm
    val IF_EQUALS_OPTION : thm
    val IF_NONE_EQUALS_OPTION : thm
    val IS_NONE_EQ_NONE : thm
    val IS_SOME_BIND : thm
    val IS_SOME_EXISTS : thm
    val IS_SOME_MAP : thm
    val NOT_IS_SOME_EQ_NONE : thm
    val NOT_NONE_SOME : thm
    val NOT_SOME_NONE : thm
    val OPTION_ALL_CONG : thm
    val OPTION_ALL_MONO : thm
    val OPTION_APPLY_MAP2 : thm
    val OPTION_APPLY_o : thm
    val OPTION_BIND_EQUALS_OPTION : thm
    val OPTION_BIND_cong : thm
    val OPTION_CHOICE_EQ_NONE : thm
    val OPTION_CHOICE_NONE : thm
    val OPTION_GUARD_COND : thm
    val OPTION_GUARD_EQ_THM : thm
    val OPTION_IGNORE_BIND_EQUALS_OPTION : thm
    val OPTION_IGNORE_BIND_thm : thm
    val OPTION_JOIN_EQ_SOME : thm
    val OPTION_MAP2_NONE : thm
    val OPTION_MAP2_SOME : thm
    val OPTION_MAP2_THM : thm
    val OPTION_MAP2_cong : thm
    val OPTION_MAP_CASE : thm
    val OPTION_MAP_COMPOSE : thm
    val OPTION_MAP_CONG : thm
    val OPTION_MAP_EQ_NONE : thm
    val OPTION_MAP_EQ_NONE_both_ways : thm
    val OPTION_MAP_EQ_SOME : thm
    val OPTION_MCOMP_ASSOC : thm
    val OPTION_MCOMP_ID : thm
    val OPTREL_MONO : thm
    val OPTREL_refl : thm
    val SOME_11 : thm
    val SOME_APPLY_PERMUTE : thm
    val SOME_SOME_APPLY : thm
    val datatype_option : thm
    val option_Axiom : thm
    val option_CASES : thm
    val option_CLAUSES : thm
    val option_Induct : thm
    val option_case_ID : thm
    val option_case_SOME_ID : thm
    val option_case_compute : thm
    val option_case_cong : thm
    val option_case_eq : thm
    val option_induction : thm
    val option_nchotomy : thm
    val some_EQ : thm
    val some_F : thm
    val some_elim : thm
    val some_intro : thm
  
  val option_grammars : type_grammar.grammar * term_grammar.grammar
(*
   [normalForms] Parent theory of "option"
   
   [one] Parent theory of "option"
   
   [sum] Parent theory of "option"
   
   [IS_NONE_DEF]  Definition
      
      ⊢ (∀x. IS_NONE (SOME x) ⇔ F) ∧ (IS_NONE NONE ⇔ T)
   
   [IS_SOME_DEF]  Definition
      
      ⊢ (∀x. IS_SOME (SOME x) ⇔ T) ∧ (IS_SOME NONE ⇔ F)
   
   [NONE_DEF]  Definition
      
      ⊢ NONE = option_ABS (INR ())
   
   [OPTION_ALL_def]  Definition
      
      ⊢ (∀P. OPTION_ALL P NONE ⇔ T) ∧ ∀P x. OPTION_ALL P (SOME x) ⇔ P x
   
   [OPTION_APPLY_def]  Definition
      
      ⊢ (∀x. NONE <*> x = NONE) ∧ ∀f x. SOME f <*> x = OPTION_MAP f x
   
   [OPTION_BIND_def]  Definition
      
      ⊢ (∀f. OPTION_BIND NONE f = NONE) ∧
        ∀x f. OPTION_BIND (SOME x) f = f x
   
   [OPTION_CHOICE_def]  Definition
      
      ⊢ (∀m2. OPTION_CHOICE NONE m2 = m2) ∧
        ∀x m2. OPTION_CHOICE (SOME x) m2 = SOME x
   
   [OPTION_GUARD_def]  Definition
      
      ⊢ (OPTION_GUARD T = SOME ()) ∧ (OPTION_GUARD F = NONE)
   
   [OPTION_IGNORE_BIND_def]  Definition
      
      ⊢ ∀m1 m2. OPTION_IGNORE_BIND m1 m2 = OPTION_BIND m1 (K m2)
   
   [OPTION_JOIN_DEF]  Definition
      
      ⊢ (OPTION_JOIN NONE = NONE) ∧ ∀x. OPTION_JOIN (SOME x) = x
   
   [OPTION_MAP2_DEF]  Definition
      
      ⊢ ∀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_MAP_DEF]  Definition
      
      ⊢ (∀f x. OPTION_MAP f (SOME x) = SOME (f x)) ∧
        ∀f. OPTION_MAP f NONE = NONE
   
   [OPTION_MCOMP_def]  Definition
      
      ⊢ ∀g f m. OPTION_MCOMP g f m = OPTION_BIND (f m) g
   
   [OPTREL_def]  Definition
      
      ⊢ ∀R x y.
            OPTREL R x y ⇔
            (x = NONE) ∧ (y = NONE) ∨
            ∃x0 y0. (x = SOME x0) ∧ (y = SOME y0) ∧ R x0 y0
   
   [SOME_DEF]  Definition
      
      ⊢ ∀x. SOME x = option_ABS (INL x)
   
   [THE_DEF]  Definition
      
      ⊢ ∀x. THE (SOME x) = x
   
   [option_REP_ABS_DEF]  Definition
      
      ⊢ (∀a. option_ABS (option_REP a) = a) ∧
        ∀r. (λx. T) r ⇔ (option_REP (option_ABS r) = r)
   
   [option_TY_DEF]  Definition
      
      ⊢ ∃rep. TYPE_DEFINITION (λx. T) rep
   
   [option_case_def]  Definition
      
      ⊢ (∀v f. option_CASE NONE v f = v) ∧
        ∀x v f. option_CASE (SOME x) v f = f x
   
   [some_def]  Definition
      
      ⊢ ∀P. $some P = if ∃x. P x then SOME (@x. P x) else NONE
   
   [EXISTS_OPTION]  Theorem
      
      ⊢ (∃opt. P opt) ⇔ P NONE ∨ ∃x. P (SOME x)
   
   [FORALL_OPTION]  Theorem
      
      ⊢ (∀opt. P opt) ⇔ P NONE ∧ ∀x. P (SOME x)
   
   [IF_EQUALS_OPTION]  Theorem
      
      ⊢ (((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))
   
   [IF_NONE_EQUALS_OPTION]  Theorem
      
      ⊢ (((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))
   
   [IS_NONE_EQ_NONE]  Theorem
      
      ⊢ ∀x. IS_NONE x ⇔ (x = NONE)
   
   [IS_SOME_BIND]  Theorem
      
      ⊢ IS_SOME (OPTION_BIND x g) ⇒ IS_SOME x
   
   [IS_SOME_EXISTS]  Theorem
      
      ⊢ ∀opt. IS_SOME opt ⇔ ∃x. opt = SOME x
   
   [IS_SOME_MAP]  Theorem
      
      ⊢ IS_SOME (OPTION_MAP f x) ⇔ IS_SOME x
   
   [NOT_IS_SOME_EQ_NONE]  Theorem
      
      ⊢ ∀x. ¬IS_SOME x ⇔ (x = NONE)
   
   [NOT_NONE_SOME]  Theorem
      
      ⊢ ∀x. NONE ≠ SOME x
   
   [NOT_SOME_NONE]  Theorem
      
      ⊢ ∀x. SOME x ≠ NONE
   
   [OPTION_ALL_CONG]  Theorem
      
      ⊢ ∀opt opt' P P'.
            (opt = opt') ∧ (∀x. (opt' = SOME x) ⇒ (P x ⇔ P' x)) ⇒
            (OPTION_ALL P opt ⇔ OPTION_ALL P' opt')
   
   [OPTION_ALL_MONO]  Theorem
      
      ⊢ (∀x. P x ⇒ P' x) ⇒ OPTION_ALL P opt ⇒ OPTION_ALL P' opt
   
   [OPTION_APPLY_MAP2]  Theorem
      
      ⊢ OPTION_MAP f x <*> y = OPTION_MAP2 f x y
   
   [OPTION_APPLY_o]  Theorem
      
      ⊢ SOME $o <*> f <*> g <*> x = f <*> (g <*> x)
   
   [OPTION_BIND_EQUALS_OPTION]  Theorem
      
      ⊢ ((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]  Theorem
      
      ⊢ ∀o1 o2 f1 f2.
            (o1 = o2) ∧ (∀x. (o2 = SOME x) ⇒ (f1 x = f2 x)) ⇒
            (OPTION_BIND o1 f1 = OPTION_BIND o2 f2)
   
   [OPTION_CHOICE_EQ_NONE]  Theorem
      
      ⊢ (OPTION_CHOICE m1 m2 = NONE) ⇔ (m1 = NONE) ∧ (m2 = NONE)
   
   [OPTION_CHOICE_NONE]  Theorem
      
      ⊢ OPTION_CHOICE m1 NONE = m1
   
   [OPTION_GUARD_COND]  Theorem
      
      ⊢ OPTION_GUARD b = if b then SOME () else NONE
   
   [OPTION_GUARD_EQ_THM]  Theorem
      
      ⊢ ((OPTION_GUARD b = SOME ()) ⇔ b) ∧ ((OPTION_GUARD b = NONE) ⇔ ¬b)
   
   [OPTION_IGNORE_BIND_EQUALS_OPTION]  Theorem
      
      ⊢ ((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_IGNORE_BIND_thm]  Theorem
      
      ⊢ (OPTION_IGNORE_BIND NONE m = NONE) ∧
        (OPTION_IGNORE_BIND (SOME v) m = m)
   
   [OPTION_JOIN_EQ_SOME]  Theorem
      
      ⊢ ∀x y. (OPTION_JOIN x = SOME y) ⇔ (x = SOME (SOME y))
   
   [OPTION_MAP2_NONE]  Theorem
      
      ⊢ (OPTION_MAP2 f o1 o2 = NONE) ⇔ (o1 = NONE) ∨ (o2 = NONE)
   
   [OPTION_MAP2_SOME]  Theorem
      
      ⊢ (OPTION_MAP2 f o1 o2 = SOME v) ⇔
        ∃x1 x2. (o1 = SOME x1) ∧ (o2 = SOME x2) ∧ (v = f x1 x2)
   
   [OPTION_MAP2_THM]  Theorem
      
      ⊢ (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_cong]  Theorem
      
      ⊢ ∀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_MAP_CASE]  Theorem
      
      ⊢ OPTION_MAP f x = option_CASE x NONE (SOME ∘ f)
   
   [OPTION_MAP_COMPOSE]  Theorem
      
      ⊢ OPTION_MAP f (OPTION_MAP g x) = OPTION_MAP (f ∘ g) x
   
   [OPTION_MAP_CONG]  Theorem
      
      ⊢ ∀opt1 opt2 f1 f2.
            (opt1 = opt2) ∧ (∀x. (opt2 = SOME x) ⇒ (f1 x = f2 x)) ⇒
            (OPTION_MAP f1 opt1 = OPTION_MAP f2 opt2)
   
   [OPTION_MAP_EQ_NONE]  Theorem
      
      ⊢ ∀f x. (OPTION_MAP f x = NONE) ⇔ (x = NONE)
   
   [OPTION_MAP_EQ_NONE_both_ways]  Theorem
      
      ⊢ ((OPTION_MAP f x = NONE) ⇔ (x = NONE)) ∧
        ((NONE = OPTION_MAP f x) ⇔ (x = NONE))
   
   [OPTION_MAP_EQ_SOME]  Theorem
      
      ⊢ ∀f x y. (OPTION_MAP f x = SOME y) ⇔ ∃z. (x = SOME z) ∧ (y = f z)
   
   [OPTION_MCOMP_ASSOC]  Theorem
      
      ⊢ OPTION_MCOMP f (OPTION_MCOMP g h) =
        OPTION_MCOMP (OPTION_MCOMP f g) h
   
   [OPTION_MCOMP_ID]  Theorem
      
      ⊢ (OPTION_MCOMP g SOME = g) ∧ (OPTION_MCOMP SOME f = f)
   
   [OPTREL_MONO]  Theorem
      
      ⊢ (∀x y. P x y ⇒ Q x y) ⇒ OPTREL P x y ⇒ OPTREL Q x y
   
   [OPTREL_refl]  Theorem
      
      ⊢ (∀x. R x x) ⇒ ∀x. OPTREL R x x
   
   [SOME_11]  Theorem
      
      ⊢ ∀x y. (SOME x = SOME y) ⇔ (x = y)
   
   [SOME_APPLY_PERMUTE]  Theorem
      
      ⊢ f <*> SOME x = SOME (λf. f x) <*> f
   
   [SOME_SOME_APPLY]  Theorem
      
      ⊢ SOME f <*> SOME x = SOME (f x)
   
   [datatype_option]  Theorem
      
      ⊢ DATATYPE (option NONE SOME)
   
   [option_Axiom]  Theorem
      
      ⊢ ∀e f. ∃fn. (fn NONE = e) ∧ ∀x. fn (SOME x) = f x
   
   [option_CASES]  Theorem
      
      ⊢ ∀opt. (∃x. opt = SOME x) ∨ (opt = NONE)
   
   [option_CLAUSES]  Theorem
      
      ⊢ (∀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_Induct]  Theorem
      
      ⊢ ∀P. (∀a. P (SOME a)) ∧ P NONE ⇒ ∀x. P x
   
   [option_case_ID]  Theorem
      
      ⊢ ∀x. option_CASE x NONE SOME = x
   
   [option_case_SOME_ID]  Theorem
      
      ⊢ ∀x. option_CASE x x SOME = x
   
   [option_case_compute]  Theorem
      
      ⊢ option_CASE x e f = if IS_SOME x then f (THE x) else e
   
   [option_case_cong]  Theorem
      
      ⊢ ∀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_eq]  Theorem
      
      ⊢ (option_CASE opt nc sc = v) ⇔
        (opt = NONE) ∧ (nc = v) ∨ ∃x. (opt = SOME x) ∧ (sc x = v)
   
   [option_induction]  Theorem
      
      ⊢ ∀P. P NONE ∧ (∀a. P (SOME a)) ⇒ ∀x. P x
   
   [option_nchotomy]  Theorem
      
      ⊢ ∀opt. (opt = NONE) ∨ ∃x. opt = SOME x
   
   [some_EQ]  Theorem
      
      ⊢ ((some x. x = y) = SOME y) ∧ ((some x. y = x) = SOME y)
   
   [some_F]  Theorem
      
      ⊢ (some x. F) = NONE
   
   [some_elim]  Theorem
      
      ⊢ Q ($some P) ⇒ (∃x. P x ∧ Q (SOME x)) ∨ (∀x. ¬P x) ∧ Q NONE
   
   [some_intro]  Theorem
      
      ⊢ (∀x. P x ⇒ Q (SOME x)) ∧ ((∀x. ¬P x) ⇒ Q NONE) ⇒ Q ($some P)
   
   
*)
end

HOL 4, Kananaskis-11