Structure boolTheory

signature boolTheory =
sig
  type thm = Thm.thm
  
  (*  Axioms  *)
    val BOOL_CASES_AX : thm
    val ETA_AX : thm
    val INFINITY_AX : thm
    val SELECT_AX : thm
  
  (*  Definitions  *)
    val AND_DEF : thm
    val BOUNDED_DEF : thm
    val COND_DEF : thm
    val DATATYPE_TAG_DEF : thm
    val EXISTS_DEF : thm
    val EXISTS_UNIQUE_DEF : thm
    val FORALL_DEF : thm
    val F_DEF : thm
    val IN_DEF : thm
    val LET_DEF : thm
    val NOT_DEF : thm
    val ONE_ONE_DEF : thm
    val ONTO_DEF : thm
    val OR_DEF : thm
    val RES_ABSTRACT_DEF : thm
    val RES_EXISTS_DEF : thm
    val RES_EXISTS_UNIQUE_DEF : thm
    val RES_FORALL_DEF : thm
    val RES_SELECT_DEF : thm
    val TYPE_DEFINITION : thm
    val T_DEF : thm
    val itself_TY_DEF : thm
    val itself_case_thm : thm
    val literal_case_DEF : thm
  
  (*  Theorems  *)
    val ABS_REP_THM : thm
    val ABS_SIMP : thm
    val AND1_THM : thm
    val AND2_THM : thm
    val AND_CLAUSES : thm
    val AND_CONG : thm
    val AND_IMP_INTRO : thm
    val AND_INTRO_THM : thm
    val BETA_THM : thm
    val BOOL_EQ_DISTINCT : thm
    val BOOL_FUN_CASES_THM : thm
    val BOOL_FUN_INDUCT : thm
    val BOTH_EXISTS_AND_THM : thm
    val BOTH_EXISTS_IMP_THM : thm
    val BOTH_FORALL_IMP_THM : thm
    val BOTH_FORALL_OR_THM : thm
    val BOUNDED_THM : thm
    val COND_ABS : thm
    val COND_CLAUSES : thm
    val COND_CONG : thm
    val COND_EXPAND : thm
    val COND_EXPAND_IMP : thm
    val COND_EXPAND_OR : thm
    val COND_ID : thm
    val COND_RAND : thm
    val COND_RATOR : thm
    val CONJ_ASSOC : thm
    val CONJ_COMM : thm
    val CONJ_SYM : thm
    val DATATYPE_BOOL : thm
    val DATATYPE_TAG_THM : thm
    val DE_MORGAN_THM : thm
    val DISJ_ASSOC : thm
    val DISJ_COMM : thm
    val DISJ_EQ_IMP : thm
    val DISJ_IMP_THM : thm
    val DISJ_SYM : thm
    val EQ_CLAUSES : thm
    val EQ_EXPAND : thm
    val EQ_EXT : thm
    val EQ_IMP_THM : thm
    val EQ_REFL : thm
    val EQ_SYM : thm
    val EQ_SYM_EQ : thm
    val EQ_TRANS : thm
    val ETA_THM : thm
    val EXCLUDED_MIDDLE : thm
    val EXISTS_OR_THM : thm
    val EXISTS_REFL : thm
    val EXISTS_SIMP : thm
    val EXISTS_THM : thm
    val EXISTS_UNIQUE_ALT' : thm
    val EXISTS_UNIQUE_REFL : thm
    val EXISTS_UNIQUE_THM : thm
    val EXISTS_itself : thm
    val FALSITY : thm
    val FORALL_AND_THM : thm
    val FORALL_BOOL : thm
    val FORALL_SIMP : thm
    val FORALL_THM : thm
    val FORALL_itself : thm
    val FUN_EQ_THM : thm
    val F_IMP : thm
    val IMP_ANTISYM_AX : thm
    val IMP_CLAUSES : thm
    val IMP_CONG : thm
    val IMP_CONJ_THM : thm
    val IMP_DISJ_THM : thm
    val IMP_F : thm
    val IMP_F_EQ_F : thm
    val ITSELF_UNIQUE : thm
    val JRH_INDUCT_UTIL : thm
    val LCOMM_THM : thm
    val LEFT_AND_CONG : thm
    val LEFT_AND_FORALL_THM : thm
    val LEFT_AND_OVER_OR : thm
    val LEFT_EXISTS_AND_THM : thm
    val LEFT_EXISTS_IMP_THM : thm
    val LEFT_FORALL_IMP_THM : thm
    val LEFT_FORALL_OR_THM : thm
    val LEFT_OR_CONG : thm
    val LEFT_OR_EXISTS_THM : thm
    val LEFT_OR_OVER_AND : thm
    val LET_CONG : thm
    val LET_RAND : thm
    val LET_RATOR : thm
    val LET_THM : thm
    val MONO_ALL : thm
    val MONO_AND : thm
    val MONO_COND : thm
    val MONO_EXISTS : thm
    val MONO_IMP : thm
    val MONO_NOT : thm
    val MONO_NOT_EQ : thm
    val MONO_OR : thm
    val NOT_AND : thm
    val NOT_CLAUSES : thm
    val NOT_EXISTS_THM : thm
    val NOT_F : thm
    val NOT_FORALL_THM : thm
    val NOT_IMP : thm
    val ONE_ONE_THM : thm
    val ONTO_THM : thm
    val OR_CLAUSES : thm
    val OR_CONG : thm
    val OR_ELIM_THM : thm
    val OR_IMP_THM : thm
    val OR_INTRO_THM1 : thm
    val OR_INTRO_THM2 : thm
    val PEIRCE : thm
    val PULL_EXISTS : thm
    val PULL_FORALL : thm
    val REFL_CLAUSE : thm
    val RES_EXISTS_CONG : thm
    val RES_EXISTS_FALSE : thm
    val RES_EXISTS_THM : thm
    val RES_EXISTS_UNIQUE_THM : thm
    val RES_FORALL_CONG : thm
    val RES_FORALL_THM : thm
    val RES_FORALL_TRUE : thm
    val RES_SELECT_THM : thm
    val RIGHT_AND_FORALL_THM : thm
    val RIGHT_AND_OVER_OR : thm
    val RIGHT_EXISTS_AND_THM : thm
    val RIGHT_EXISTS_IMP_THM : thm
    val RIGHT_FORALL_IMP_THM : thm
    val RIGHT_FORALL_OR_THM : thm
    val RIGHT_OR_EXISTS_THM : thm
    val RIGHT_OR_OVER_AND : thm
    val SELECT_ELIM_THM : thm
    val SELECT_REFL : thm
    val SELECT_REFL_2 : thm
    val SELECT_THM : thm
    val SELECT_UNIQUE : thm
    val SKOLEM_THM : thm
    val SWAP_EXISTS_THM : thm
    val SWAP_FORALL_THM : thm
    val TRUTH : thm
    val TYPE_DEFINITION_THM : thm
    val UEXISTS_OR_THM : thm
    val UEXISTS_SIMP : thm
    val UNWIND_FORALL_THM1 : thm
    val UNWIND_FORALL_THM2 : thm
    val UNWIND_THM1 : thm
    val UNWIND_THM2 : thm
    val boolAxiom : thm
    val bool_INDUCT : thm
    val bool_case_CONG : thm
    val bool_case_ID : thm
    val bool_case_thm : thm
    val itself_Axiom : thm
    val itself_induction : thm
    val literal_case_CONG : thm
    val literal_case_RAND : thm
    val literal_case_RATOR : thm
    val literal_case_THM : thm
    val literal_case_id : thm
  
  val bool_grammars : type_grammar.grammar * term_grammar.grammar
(*
   [min] Parent theory of "bool"
   
   [BOOL_CASES_AX]  Axiom
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀t. (t ⇔ T) ∨ (t ⇔ F)
   
   [ETA_AX]  Axiom
      
      [oracles: ] [axioms: ETA_AX] [] ⊢ ∀t. (λx. t x) = t
   
   [INFINITY_AX]  Axiom
      
      [oracles: ] [axioms: INFINITY_AX] [] ⊢ ∃f. ONE_ONE f ∧ ¬ONTO f
   
   [SELECT_AX]  Axiom
      
      [oracles: ] [axioms: SELECT_AX] [] ⊢ ∀P x. P x ⇒ P ($@ P)
   
   [AND_DEF]  Definition
      
      ⊢ $/\ = (λt1 t2. ∀t. (t1 ⇒ t2 ⇒ t) ⇒ t)
   
   [BOUNDED_DEF]  Definition
      
      ⊢ BOUNDED = (λv. T)
   
   [COND_DEF]  Definition
      
      ⊢ COND = (λt t1 t2. @x. ((t ⇔ T) ⇒ x = t1) ∧ ((t ⇔ F) ⇒ x = t2))
   
   [DATATYPE_TAG_DEF]  Definition
      
      ⊢ DATATYPE = (λx. T)
   
   [EXISTS_DEF]  Definition
      
      ⊢ $? = (λP. P ($@ P))
   
   [EXISTS_UNIQUE_DEF]  Definition
      
      ⊢ $?! = (λP. $? P ∧ ∀x y. P x ∧ P y ⇒ x = y)
   
   [FORALL_DEF]  Definition
      
      ⊢ $! = (λP. P = (λx. T))
   
   [F_DEF]  Definition
      
      ⊢ F ⇔ ∀t. t
   
   [IN_DEF]  Definition
      
      ⊢ $IN = (λx f. f x)
   
   [LET_DEF]  Definition
      
      ⊢ LET = (λf x. f x)
   
   [NOT_DEF]  Definition
      
      ⊢ $¬ = (λt. t ⇒ F)
   
   [ONE_ONE_DEF]  Definition
      
      ⊢ ONE_ONE = (λf. ∀x1 x2. f x1 = f x2 ⇒ x1 = x2)
   
   [ONTO_DEF]  Definition
      
      ⊢ ONTO = (λf. ∀y. ∃x. y = f x)
   
   [OR_DEF]  Definition
      
      ⊢ $\/ = (λt1 t2. ∀t. (t1 ⇒ t) ⇒ (t2 ⇒ t) ⇒ t)
   
   [RES_ABSTRACT_DEF]  Definition
      
      [oracles: ] [axioms: ETA_AX, BOOL_CASES_AX, SELECT_AX] []
      ⊢ (∀p m x. x ∈ p ⇒ RES_ABSTRACT p m x = m x) ∧
        ∀p m1 m2.
          (∀x. x ∈ p ⇒ m1 x = m2 x) ⇒ RES_ABSTRACT p m1 = RES_ABSTRACT p m2
   
   [RES_EXISTS_DEF]  Definition
      
      ⊢ RES_EXISTS = (λp m. ∃x. x ∈ p ∧ m x)
   
   [RES_EXISTS_UNIQUE_DEF]  Definition
      
      ⊢ RES_EXISTS_UNIQUE =
        (λp m. (∃x::p. m x) ∧ ∀x y::p. m x ∧ m y ⇒ x = y)
   
   [RES_FORALL_DEF]  Definition
      
      ⊢ RES_FORALL = (λp m. ∀x. x ∈ p ⇒ m x)
   
   [RES_SELECT_DEF]  Definition
      
      ⊢ RES_SELECT = (λp m. @x. x ∈ p ∧ m x)
   
   [TYPE_DEFINITION]  Definition
      
      ⊢ TYPE_DEFINITION =
        (λP rep.
             (∀x' x''. rep x' = rep x'' ⇒ x' = x'') ∧
             ∀x. P x ⇔ ∃x'. x = rep x')
   
   [T_DEF]  Definition
      
      ⊢ T ⇔ (λx. x) = (λx. x)
   
   [itself_TY_DEF]  Definition
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∃rep. TYPE_DEFINITION ($= ARB) rep
   
   [itself_case_thm]  Definition
      
      ⊢ ∀b. itself_case (:α) b = b
   
   [literal_case_DEF]  Definition
      
      ⊢ literal_case = (λf x. f x)
   
   [ABS_REP_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX, SELECT_AX] []
      ⊢ ∀P. (∃rep. TYPE_DEFINITION P rep) ⇒
            ∃rep abs. (∀a. abs (rep a) = a) ∧ ∀r. P r ⇔ rep (abs r) = r
   
   [ABS_SIMP]  Theorem
      
      ⊢ ∀t1 t2. (λx. t1) t2 = t1
   
   [AND1_THM]  Theorem
      
      ⊢ ∀t1 t2. t1 ∧ t2 ⇒ t1
   
   [AND2_THM]  Theorem
      
      ⊢ ∀t1 t2. t1 ∧ t2 ⇒ t2
   
   [AND_CLAUSES]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀t. (T ∧ t ⇔ t) ∧ (t ∧ T ⇔ t) ∧ (F ∧ t ⇔ F) ∧ (t ∧ F ⇔ F) ∧
            (t ∧ t ⇔ t)
   
   [AND_CONG]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P P' Q Q'. (Q ⇒ (P ⇔ P')) ∧ (P' ⇒ (Q ⇔ Q')) ⇒ (P ∧ Q ⇔ P' ∧ Q')
   
   [AND_IMP_INTRO]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀t1 t2 t3. t1 ⇒ t2 ⇒ t3 ⇔ t1 ∧ t2 ⇒ t3
   
   [AND_INTRO_THM]  Theorem
      
      ⊢ ∀t1 t2. t1 ⇒ t2 ⇒ t1 ∧ t2
   
   [BETA_THM]  Theorem
      
      ⊢ ∀f y. (λx. f x) y = f y
   
   [BOOL_EQ_DISTINCT]  Theorem
      
      ⊢ (T ⇎ F) ∧ (F ⇎ T)
   
   [BOOL_FUN_CASES_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX, ETA_AX] []
      ⊢ ∀f. f = (λb. T) ∨ f = (λb. F) ∨ f = (λb. b) ∨ f = (λb. ¬b)
   
   [BOOL_FUN_INDUCT]  Theorem
      
      [oracles: ] [axioms: ETA_AX, BOOL_CASES_AX] []
      ⊢ ∀P. P (λb. T) ∧ P (λb. F) ∧ P (λb. b) ∧ P (λb. ¬b) ⇒ ∀f. P f
   
   [BOTH_EXISTS_AND_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∃x. P ∧ Q) ⇔ (∃x. P) ∧ ∃x. Q
   
   [BOTH_EXISTS_IMP_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∃x. P ⇒ Q) ⇔ (∀x. P) ⇒ ∃x. Q
   
   [BOTH_FORALL_IMP_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∀x. P ⇒ Q) ⇔ (∃x. P) ⇒ ∀x. Q
   
   [BOTH_FORALL_OR_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∀x. P ∨ Q) ⇔ (∀x. P) ∨ ∀x. Q
   
   [BOUNDED_THM]  Theorem
      
      ⊢ ∀v. BOUNDED v ⇔ T
   
   [COND_ABS]  Theorem
      
      [oracles: ] [axioms: SELECT_AX, BOOL_CASES_AX, ETA_AX] []
      ⊢ ∀b f g. (λx. if b then f x else g x) = if b then f else g
   
   [COND_CLAUSES]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX, SELECT_AX] []
      ⊢ ∀t1 t2. (if T then t1 else t2) = t1 ∧ (if F then t1 else t2) = t2
   
   [COND_CONG]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX, SELECT_AX] []
      ⊢ ∀P Q x x' y y'.
          (P ⇔ Q) ∧ (Q ⇒ x = x') ∧ (¬Q ⇒ y = y') ⇒
          (if P then x else y) = if Q then x' else y'
   
   [COND_EXPAND]  Theorem
      
      [oracles: ] [axioms: SELECT_AX, BOOL_CASES_AX] []
      ⊢ ∀b t1 t2. (if b then t1 else t2) ⇔ (¬b ∨ t1) ∧ (b ∨ t2)
   
   [COND_EXPAND_IMP]  Theorem
      
      [oracles: ] [axioms: SELECT_AX, BOOL_CASES_AX] []
      ⊢ ∀b t1 t2. (if b then t1 else t2) ⇔ (b ⇒ t1) ∧ (¬b ⇒ t2)
   
   [COND_EXPAND_OR]  Theorem
      
      [oracles: ] [axioms: SELECT_AX, BOOL_CASES_AX] []
      ⊢ ∀b t1 t2. (if b then t1 else t2) ⇔ b ∧ t1 ∨ ¬b ∧ t2
   
   [COND_ID]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX, SELECT_AX] []
      ⊢ ∀b t. (if b then t else t) = t
   
   [COND_RAND]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX, SELECT_AX] []
      ⊢ ∀f b x y. f (if b then x else y) = if b then f x else f y
   
   [COND_RATOR]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX, SELECT_AX] []
      ⊢ ∀b f g x. (if b then f else g) x = if b then f x else g x
   
   [CONJ_ASSOC]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀t1 t2 t3. t1 ∧ t2 ∧ t3 ⇔ (t1 ∧ t2) ∧ t3
   
   [CONJ_COMM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀t1 t2. t1 ∧ t2 ⇔ t2 ∧ t1
   
   [CONJ_SYM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀t1 t2. t1 ∧ t2 ⇔ t2 ∧ t1
   
   [DATATYPE_BOOL]  Theorem
      
      ⊢ DATATYPE (bool T F) ⇔ T
   
   [DATATYPE_TAG_THM]  Theorem
      
      ⊢ ∀x. DATATYPE x ⇔ T
   
   [DE_MORGAN_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀A B. (¬(A ∧ B) ⇔ ¬A ∨ ¬B) ∧ (¬(A ∨ B) ⇔ ¬A ∧ ¬B)
   
   [DISJ_ASSOC]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀A B C. A ∨ B ∨ C ⇔ (A ∨ B) ∨ C
   
   [DISJ_COMM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀A B. A ∨ B ⇔ B ∨ A
   
   [DISJ_EQ_IMP]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀A B. A ∨ B ⇔ ¬A ⇒ B
   
   [DISJ_IMP_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q R. P ∨ Q ⇒ R ⇔ (P ⇒ R) ∧ (Q ⇒ R)
   
   [DISJ_SYM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀A B. A ∨ B ⇔ B ∨ A
   
   [EQ_CLAUSES]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀t. ((T ⇔ t) ⇔ t) ∧ ((t ⇔ T) ⇔ t) ∧ ((F ⇔ t) ⇔ ¬t) ∧ ((t ⇔ F) ⇔ ¬t)
   
   [EQ_EXPAND]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀t1 t2. (t1 ⇔ t2) ⇔ t1 ∧ t2 ∨ ¬t1 ∧ ¬t2
   
   [EQ_EXT]  Theorem
      
      [oracles: ] [axioms: ETA_AX] [] ⊢ ∀f g. (∀x. f x = g x) ⇒ f = g
   
   [EQ_IMP_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀t1 t2. (t1 ⇔ t2) ⇔ (t1 ⇒ t2) ∧ (t2 ⇒ t1)
   
   [EQ_REFL]  Theorem
      
      ⊢ ∀x. x = x
   
   [EQ_SYM]  Theorem
      
      ⊢ ∀x y. x = y ⇒ y = x
   
   [EQ_SYM_EQ]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀x y. x = y ⇔ y = x
   
   [EQ_TRANS]  Theorem
      
      ⊢ ∀x y z. x = y ∧ y = z ⇒ x = z
   
   [ETA_THM]  Theorem
      
      [oracles: ] [axioms: ETA_AX] [] ⊢ ∀M. (λx. M x) = M
   
   [EXCLUDED_MIDDLE]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀t. t ∨ ¬t
   
   [EXISTS_OR_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∃x. P x ∨ Q x) ⇔ (∃x. P x) ∨ ∃x. Q x
   
   [EXISTS_REFL]  Theorem
      
      ⊢ ∀a. ∃x. x = a
   
   [EXISTS_SIMP]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀t. (∃x. t) ⇔ t
   
   [EXISTS_THM]  Theorem
      
      [oracles: ] [axioms: ETA_AX] [] ⊢ $? f ⇔ ∃x. f x
   
   [EXISTS_UNIQUE_ALT']  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ (∃!x. P x) ⇔ ∃x. ∀y. P y ⇔ y = x
   
   [EXISTS_UNIQUE_REFL]  Theorem
      
      ⊢ ∀a. ∃!x. x = a
   
   [EXISTS_UNIQUE_THM]  Theorem
      
      ⊢ (∃!x. P x) ⇔ (∃x. P x) ∧ ∀x y. P x ∧ P y ⇒ x = y
   
   [EXISTS_itself]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ (∃x. P x) ⇔ P (:α)
   
   [FALSITY]  Theorem
      
      ⊢ ∀t. F ⇒ t
   
   [FORALL_AND_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∀x. P x ∧ Q x) ⇔ (∀x. P x) ∧ ∀x. Q x
   
   [FORALL_BOOL]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ (∀b. P b) ⇔ P T ∧ P F
   
   [FORALL_SIMP]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀t. (∀x. t) ⇔ t
   
   [FORALL_THM]  Theorem
      
      [oracles: ] [axioms: ETA_AX] [] ⊢ $! f ⇔ ∀x. f x
   
   [FORALL_itself]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ (∀x. P x) ⇔ P (:α)
   
   [FUN_EQ_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX, ETA_AX] []
      ⊢ ∀f g. f = g ⇔ ∀x. f x = g x
   
   [F_IMP]  Theorem
      
      ⊢ ∀t. ¬t ⇒ t ⇒ F
   
   [IMP_ANTISYM_AX]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀t1 t2. (t1 ⇒ t2) ⇒ (t2 ⇒ t1) ⇒ (t1 ⇔ t2)
   
   [IMP_CLAUSES]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀t. (T ⇒ t ⇔ t) ∧ (t ⇒ T ⇔ T) ∧ (F ⇒ t ⇔ T) ∧ (t ⇒ t ⇔ T) ∧
            (t ⇒ F ⇔ ¬t)
   
   [IMP_CONG]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀x x' y y'. (x ⇔ x') ∧ (x' ⇒ (y ⇔ y')) ⇒ (x ⇒ y ⇔ x' ⇒ y')
   
   [IMP_CONJ_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q R. P ⇒ Q ∧ R ⇔ (P ⇒ Q) ∧ (P ⇒ R)
   
   [IMP_DISJ_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀A B. A ⇒ B ⇔ ¬A ∨ B
   
   [IMP_F]  Theorem
      
      ⊢ ∀t. (t ⇒ F) ⇒ ¬t
   
   [IMP_F_EQ_F]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀t. t ⇒ F ⇔ (t ⇔ F)
   
   [ITSELF_UNIQUE]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀i. i = (:α)
   
   [JRH_INDUCT_UTIL]  Theorem
      
      [oracles: ] [axioms: ETA_AX] [] ⊢ ∀P t. (∀x. x = t ⇒ P x) ⇒ $? P
   
   [LCOMM_THM]  Theorem
      
      ⊢ ∀f. (∀x y z. f x (f y z) = f (f x y) z) ⇒
            (∀x y. f x y = f y x) ⇒
            ∀x y z. f x (f y z) = f y (f x z)
   
   [LEFT_AND_CONG]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P P' Q Q'. (P ⇔ P') ∧ (P' ⇒ (Q ⇔ Q')) ⇒ (P ∧ Q ⇔ P' ∧ Q')
   
   [LEFT_AND_FORALL_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∀x. P x) ∧ Q ⇔ ∀x. P x ∧ Q
   
   [LEFT_AND_OVER_OR]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀A B C. A ∧ (B ∨ C) ⇔ A ∧ B ∨ A ∧ C
   
   [LEFT_EXISTS_AND_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∃x. P x ∧ Q) ⇔ (∃x. P x) ∧ Q
   
   [LEFT_EXISTS_IMP_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∃x. P x ⇒ Q) ⇔ (∀x. P x) ⇒ Q
   
   [LEFT_FORALL_IMP_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∀x. P x ⇒ Q) ⇔ (∃x. P x) ⇒ Q
   
   [LEFT_FORALL_OR_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀Q P. (∀x. P x ∨ Q) ⇔ (∀x. P x) ∨ Q
   
   [LEFT_OR_CONG]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P P' Q Q'. (P ⇔ P') ∧ (¬P' ⇒ (Q ⇔ Q')) ⇒ (P ∨ Q ⇔ P' ∨ Q')
   
   [LEFT_OR_EXISTS_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∃x. P x) ∨ Q ⇔ ∃x. P x ∨ Q
   
   [LEFT_OR_OVER_AND]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀A B C. A ∨ B ∧ C ⇔ (A ∨ B) ∧ (A ∨ C)
   
   [LET_CONG]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀f g M N. M = N ∧ (∀x. x = N ⇒ f x = g x) ⇒ LET f M = LET g N
   
   [LET_RAND]  Theorem
      
      ⊢ P (let x = M in N x) ⇔ (let x = M in P (N x))
   
   [LET_RATOR]  Theorem
      
      ⊢ (let x = M in N x) b = (let x = M in N x b)
   
   [LET_THM]  Theorem
      
      ⊢ ∀f x. LET f x = f x
   
   [MONO_ALL]  Theorem
      
      ⊢ (∀x. P x ⇒ Q x) ⇒ (∀x. P x) ⇒ ∀x. Q x
   
   [MONO_AND]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ (x ⇒ y) ∧ (z ⇒ w) ⇒ x ∧ z ⇒ y ∧ w
   
   [MONO_COND]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX, SELECT_AX] []
      ⊢ (x ⇒ y) ⇒ (z ⇒ w) ⇒ (if b then x else z) ⇒ if b then y else w
   
   [MONO_EXISTS]  Theorem
      
      ⊢ (∀x. P x ⇒ Q x) ⇒ (∃x. P x) ⇒ ∃x. Q x
   
   [MONO_IMP]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ (y ⇒ x) ∧ (z ⇒ w) ⇒ (x ⇒ z) ⇒ y ⇒ w
   
   [MONO_NOT]  Theorem
      
      ⊢ (y ⇒ x) ⇒ ¬x ⇒ ¬y
   
   [MONO_NOT_EQ]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ y ⇒ x ⇔ ¬x ⇒ ¬y
   
   [MONO_OR]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ (x ⇒ y) ∧ (z ⇒ w) ⇒ x ∨ z ⇒ y ∨ w
   
   [NOT_AND]  Theorem
      
      ⊢ ¬(t ∧ ¬t)
   
   [NOT_CLAUSES]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)
   
   [NOT_EXISTS_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀P. ¬(∃x. P x) ⇔ ∀x. ¬P x
   
   [NOT_F]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀t. ¬t ⇒ (t ⇔ F)
   
   [NOT_FORALL_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀P. ¬(∀x. P x) ⇔ ∃x. ¬P x
   
   [NOT_IMP]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀A B. ¬(A ⇒ B) ⇔ A ∧ ¬B
   
   [ONE_ONE_THM]  Theorem
      
      ⊢ ∀f. ONE_ONE f ⇔ ∀x1 x2. f x1 = f x2 ⇒ x1 = x2
   
   [ONTO_THM]  Theorem
      
      ⊢ ∀f. ONTO f ⇔ ∀y. ∃x. y = f x
   
   [OR_CLAUSES]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀t. (T ∨ t ⇔ T) ∧ (t ∨ T ⇔ T) ∧ (F ∨ t ⇔ t) ∧ (t ∨ F ⇔ t) ∧
            (t ∨ t ⇔ t)
   
   [OR_CONG]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P P' Q Q'. (¬Q ⇒ (P ⇔ P')) ∧ (¬P' ⇒ (Q ⇔ Q')) ⇒ (P ∨ Q ⇔ P' ∨ Q')
   
   [OR_ELIM_THM]  Theorem
      
      ⊢ ∀t t1 t2. t1 ∨ t2 ⇒ (t1 ⇒ t) ⇒ (t2 ⇒ t) ⇒ t
   
   [OR_IMP_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀A B. (A ⇔ B ∨ A) ⇔ B ⇒ A
   
   [OR_INTRO_THM1]  Theorem
      
      ⊢ ∀t1 t2. t1 ⇒ t1 ∨ t2
   
   [OR_INTRO_THM2]  Theorem
      
      ⊢ ∀t1 t2. t2 ⇒ t1 ∨ t2
   
   [PEIRCE]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ((P ⇒ Q) ⇒ P) ⇒ P
   
   [PULL_EXISTS]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q.
          ((∃x. P x) ⇒ Q ⇔ ∀x. P x ⇒ Q) ∧ ((∃x. P x) ∧ Q ⇔ ∃x. P x ∧ Q) ∧
          (Q ∧ (∃x. P x) ⇔ ∃x. Q ∧ P x)
   
   [PULL_FORALL]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q.
          (Q ⇒ (∀x. P x) ⇔ ∀x. Q ⇒ P x) ∧ ((∀x. P x) ∧ Q ⇔ ∀x. P x ∧ Q) ∧
          (Q ∧ (∀x. P x) ⇔ ∀x. Q ∧ P x)
   
   [REFL_CLAUSE]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀x. x = x ⇔ T
   
   [RES_EXISTS_CONG]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ P = Q ⇒
        (∀x. x ∈ Q ⇒ (f x ⇔ g x)) ⇒
        (RES_EXISTS P f ⇔ RES_EXISTS Q g)
   
   [RES_EXISTS_FALSE]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ (∃x::P. F) ⇔ F
   
   [RES_EXISTS_THM]  Theorem
      
      ⊢ ∀P f. RES_EXISTS P f ⇔ ∃x. x ∈ P ∧ f x
   
   [RES_EXISTS_UNIQUE_THM]  Theorem
      
      ⊢ ∀P f.
          RES_EXISTS_UNIQUE P f ⇔ (∃x::P. f x) ∧ ∀x y::P. f x ∧ f y ⇒ x = y
   
   [RES_FORALL_CONG]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ P = Q ⇒
        (∀x. x ∈ Q ⇒ (f x ⇔ g x)) ⇒
        (RES_FORALL P f ⇔ RES_FORALL Q g)
   
   [RES_FORALL_THM]  Theorem
      
      ⊢ ∀P f. RES_FORALL P f ⇔ ∀x. x ∈ P ⇒ f x
   
   [RES_FORALL_TRUE]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ (∀x::P. T) ⇔ T
   
   [RES_SELECT_THM]  Theorem
      
      ⊢ ∀P f. RES_SELECT P f = @x. x ∈ P ∧ f x
   
   [RIGHT_AND_FORALL_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. P ∧ (∀x. Q x) ⇔ ∀x. P ∧ Q x
   
   [RIGHT_AND_OVER_OR]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀A B C. (B ∨ C) ∧ A ⇔ B ∧ A ∨ C ∧ A
   
   [RIGHT_EXISTS_AND_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∃x. P ∧ Q x) ⇔ P ∧ ∃x. Q x
   
   [RIGHT_EXISTS_IMP_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∃x. P ⇒ Q x) ⇔ P ⇒ ∃x. Q x
   
   [RIGHT_FORALL_IMP_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∀x. P ⇒ Q x) ⇔ P ⇒ ∀x. Q x
   
   [RIGHT_FORALL_OR_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∀x. P ∨ Q x) ⇔ P ∨ ∀x. Q x
   
   [RIGHT_OR_EXISTS_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. P ∨ (∃x. Q x) ⇔ ∃x. P ∨ Q x
   
   [RIGHT_OR_OVER_AND]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀A B C. B ∧ C ∨ A ⇔ (B ∨ A) ∧ (C ∨ A)
   
   [SELECT_ELIM_THM]  Theorem
      
      [oracles: ] [axioms: SELECT_AX] []
      ⊢ ∀P Q. (∃x. P x) ∧ (∀x. P x ⇒ Q x) ⇒ Q ($@ P)
   
   [SELECT_REFL]  Theorem
      
      [oracles: ] [axioms: SELECT_AX] [] ⊢ ∀x. (@y. y = x) = x
   
   [SELECT_REFL_2]  Theorem
      
      [oracles: ] [axioms: SELECT_AX] [] ⊢ ∀x. (@y. x = y) = x
   
   [SELECT_THM]  Theorem
      
      ⊢ ∀P. P (@x. P x) ⇔ ∃x. P x
   
   [SELECT_UNIQUE]  Theorem
      
      [oracles: ] [axioms: ETA_AX, BOOL_CASES_AX, SELECT_AX] []
      ⊢ ∀P x. (∀y. P y ⇔ y = x) ⇒ $@ P = x
   
   [SKOLEM_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX, SELECT_AX] []
      ⊢ ∀P. (∀x. ∃y. P x y) ⇔ ∃f. ∀x. P x (f x)
   
   [SWAP_EXISTS_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P. (∃x y. P x y) ⇔ ∃y x. P x y
   
   [SWAP_FORALL_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P. (∀x y. P x y) ⇔ ∀y x. P x y
   
   [TRUTH]  Theorem
      
      ⊢ T
   
   [TYPE_DEFINITION_THM]  Theorem
      
      ⊢ ∀P rep.
          TYPE_DEFINITION P rep ⇔
          (∀x' x''. rep x' = rep x'' ⇒ x' = x'') ∧
          ∀x. P x ⇔ ∃x'. x = rep x'
   
   [UEXISTS_OR_THM]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P Q. (∃!x. P x ∨ Q x) ⇒ (∃!x. P x) ∨ ∃!x. Q x
   
   [UEXISTS_SIMP]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ (∃!x. t) ⇔ t ∧ ∀x y. x = y
   
   [UNWIND_FORALL_THM1]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀f v. (∀x. v = x ⇒ f x) ⇔ f v
   
   [UNWIND_FORALL_THM2]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀f v. (∀x. x = v ⇒ f x) ⇔ f v
   
   [UNWIND_THM1]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P a. (∃x. a = x ∧ P x) ⇔ P a
   
   [UNWIND_THM2]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀P a. (∃x. x = a ∧ P x) ⇔ P a
   
   [boolAxiom]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX, SELECT_AX] []
      ⊢ ∀t1 t2. ∃fn. fn T = t1 ∧ fn F = t2
   
   [bool_INDUCT]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀P. P T ∧ P F ⇒ ∀b. P b
   
   [bool_case_CONG]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX, SELECT_AX] []
      ⊢ ∀P Q x x' y y'.
          (P ⇔ Q) ∧ (Q ⇒ x = x') ∧ (¬Q ⇒ y = y') ⇒
          (if P then x else y) = if Q then x' else y'
   
   [bool_case_ID]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX, SELECT_AX] []
      ⊢ ∀b t. (if b then t else t) = t
   
   [bool_case_thm]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX, SELECT_AX] []
      ⊢ (∀t1 t2. (if T then t1 else t2) = t1) ∧
        ∀t1 t2. (if F then t1 else t2) = t2
   
   [itself_Axiom]  Theorem
      
      ⊢ ∀e. ∃f. f (:α) = e
   
   [itself_induction]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] [] ⊢ ∀P. P (:α) ⇒ ∀i. P i
   
   [literal_case_CONG]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX] []
      ⊢ ∀f g M N.
          M = N ∧ (∀x. x = N ⇒ f x = g x) ⇒
          literal_case f M = literal_case g N
   
   [literal_case_RAND]  Theorem
      
      ⊢ P (literal_case (λx. N x) M) = literal_case (λx. P (N x)) M
   
   [literal_case_RATOR]  Theorem
      
      ⊢ literal_case (λx. N x) M b = literal_case (λx. N x b) M
   
   [literal_case_THM]  Theorem
      
      ⊢ ∀f x. literal_case f x = f x
   
   [literal_case_id]  Theorem
      
      [oracles: ] [axioms: BOOL_CASES_AX, SELECT_AX] []
      ⊢ literal_case (λx. if x = a then t else u) a = t
   
   
*)
end

HOL 4, Kananaskis-14