Structure hrealTheory

signature hrealTheory =
sig
  type thm = Thm.thm
  
  (*  Definitions  *)
    val cut_of_hrat : thm
    val hrat_lt : thm
    val hreal_1 : thm
    val hreal_TY_DEF : thm
    val hreal_add : thm
    val hreal_inv : thm
    val hreal_lt : thm
    val hreal_mul : thm
    val hreal_sub : thm
    val hreal_sup : thm
    val hreal_tybij : thm
    val isacut : thm
  
  (*  Theorems  *)
    val CUT_BOUNDED : thm
    val CUT_DOWN : thm
    val CUT_ISACUT : thm
    val CUT_NEARTOP_ADD : thm
    val CUT_NEARTOP_MUL : thm
    val CUT_NONEMPTY : thm
    val CUT_STRADDLE : thm
    val CUT_UBOUND : thm
    val CUT_UP : thm
    val EQUAL_CUTS : thm
    val HRAT_DOWN : thm
    val HRAT_DOWN2 : thm
    val HRAT_EQ_LADD : thm
    val HRAT_EQ_LMUL : thm
    val HRAT_GT_L1 : thm
    val HRAT_GT_LMUL1 : thm
    val HRAT_INV_MUL : thm
    val HRAT_LT_ADD2 : thm
    val HRAT_LT_ADDL : thm
    val HRAT_LT_ADDR : thm
    val HRAT_LT_ANTISYM : thm
    val HRAT_LT_GT : thm
    val HRAT_LT_L1 : thm
    val HRAT_LT_LADD : thm
    val HRAT_LT_LMUL : thm
    val HRAT_LT_LMUL1 : thm
    val HRAT_LT_MUL2 : thm
    val HRAT_LT_NE : thm
    val HRAT_LT_R1 : thm
    val HRAT_LT_RADD : thm
    val HRAT_LT_REFL : thm
    val HRAT_LT_RMUL : thm
    val HRAT_LT_RMUL1 : thm
    val HRAT_LT_TOTAL : thm
    val HRAT_LT_TRANS : thm
    val HRAT_MEAN : thm
    val HRAT_MUL_RID : thm
    val HRAT_MUL_RINV : thm
    val HRAT_RDISTRIB : thm
    val HRAT_UP : thm
    val HREAL_ADD_ASSOC : thm
    val HREAL_ADD_ISACUT : thm
    val HREAL_ADD_SYM : thm
    val HREAL_ADD_TOTAL : thm
    val HREAL_INV_ISACUT : thm
    val HREAL_LDISTRIB : thm
    val HREAL_LT : thm
    val HREAL_LT_LEMMA : thm
    val HREAL_LT_TOTAL : thm
    val HREAL_MUL_ASSOC : thm
    val HREAL_MUL_ISACUT : thm
    val HREAL_MUL_LID : thm
    val HREAL_MUL_LINV : thm
    val HREAL_MUL_SYM : thm
    val HREAL_NOZERO : thm
    val HREAL_SUB_ADD : thm
    val HREAL_SUB_ISACUT : thm
    val HREAL_SUP : thm
    val HREAL_SUP_ISACUT : thm
    val ISACUT_HRAT : thm
  
  val hreal_grammars : type_grammar.grammar * term_grammar.grammar
(*
   [hrat] Parent theory of "hreal"
   
   [cut_of_hrat]  Definition
      
      ⊢ ∀x. cut_of_hrat x = (λy. y hrat_lt x)
   
   [hrat_lt]  Definition
      
      ⊢ ∀x y. x hrat_lt y ⇔ ∃d. y = x hrat_add d
   
   [hreal_1]  Definition
      
      ⊢ hreal_1 = hreal (cut_of_hrat hrat_1)
   
   [hreal_TY_DEF]  Definition
      
      ⊢ ∃rep. TYPE_DEFINITION isacut rep
   
   [hreal_add]  Definition
      
      ⊢ ∀X Y.
            X hreal_add Y =
            hreal (λw. ∃x y. (w = x hrat_add y) ∧ cut X x ∧ cut Y y)
   
   [hreal_inv]  Definition
      
      ⊢ ∀X.
            hreal_inv X =
            hreal
              (λw.
                   ∃d.
                       d hrat_lt hrat_1 ∧
                       ∀x. cut X x ⇒ w hrat_mul x hrat_lt d)
   
   [hreal_lt]  Definition
      
      ⊢ ∀X Y. X hreal_lt Y ⇔ X ≠ Y ∧ ∀x. cut X x ⇒ cut Y x
   
   [hreal_mul]  Definition
      
      ⊢ ∀X Y.
            X hreal_mul Y =
            hreal (λw. ∃x y. (w = x hrat_mul y) ∧ cut X x ∧ cut Y y)
   
   [hreal_sub]  Definition
      
      ⊢ ∀Y X.
            Y hreal_sub X = hreal (λw. ∃x. ¬cut X x ∧ cut Y (x hrat_add w))
   
   [hreal_sup]  Definition
      
      ⊢ ∀P. hreal_sup P = hreal (λw. ∃X. P X ∧ cut X w)
   
   [hreal_tybij]  Definition
      
      ⊢ (∀a. hreal (cut a) = a) ∧ ∀r. isacut r ⇔ (cut (hreal r) = r)
   
   [isacut]  Definition
      
      ⊢ ∀C.
            isacut C ⇔
            (∃x. C x) ∧ (∃x. ¬C x) ∧ (∀x y. C x ∧ y hrat_lt x ⇒ C y) ∧
            ∀x. C x ⇒ ∃y. C y ∧ x hrat_lt y
   
   [CUT_BOUNDED]  Theorem
      
      ⊢ ∀X. ∃x. ¬cut X x
   
   [CUT_DOWN]  Theorem
      
      ⊢ ∀X x y. cut X x ∧ y hrat_lt x ⇒ cut X y
   
   [CUT_ISACUT]  Theorem
      
      ⊢ ∀X. isacut (cut X)
   
   [CUT_NEARTOP_ADD]  Theorem
      
      ⊢ ∀X e. ∃x. cut X x ∧ ¬cut X (x hrat_add e)
   
   [CUT_NEARTOP_MUL]  Theorem
      
      ⊢ ∀X u. hrat_1 hrat_lt u ⇒ ∃x. cut X x ∧ ¬cut X (u hrat_mul x)
   
   [CUT_NONEMPTY]  Theorem
      
      ⊢ ∀X. ∃x. cut X x
   
   [CUT_STRADDLE]  Theorem
      
      ⊢ ∀X x y. cut X x ∧ ¬cut X y ⇒ x hrat_lt y
   
   [CUT_UBOUND]  Theorem
      
      ⊢ ∀X x y. ¬cut X x ∧ x hrat_lt y ⇒ ¬cut X y
   
   [CUT_UP]  Theorem
      
      ⊢ ∀X x. cut X x ⇒ ∃y. cut X y ∧ x hrat_lt y
   
   [EQUAL_CUTS]  Theorem
      
      ⊢ ∀X Y. (cut X = cut Y) ⇒ (X = Y)
   
   [HRAT_DOWN]  Theorem
      
      ⊢ ∀x. ∃y. y hrat_lt x
   
   [HRAT_DOWN2]  Theorem
      
      ⊢ ∀x y. ∃z. z hrat_lt x ∧ z hrat_lt y
   
   [HRAT_EQ_LADD]  Theorem
      
      ⊢ ∀x y z. (x hrat_add y = x hrat_add z) ⇔ (y = z)
   
   [HRAT_EQ_LMUL]  Theorem
      
      ⊢ ∀x y z. (x hrat_mul y = x hrat_mul z) ⇔ (y = z)
   
   [HRAT_GT_L1]  Theorem
      
      ⊢ ∀x y. hrat_1 hrat_lt hrat_inv x hrat_mul y ⇔ x hrat_lt y
   
   [HRAT_GT_LMUL1]  Theorem
      
      ⊢ ∀x y. y hrat_lt x hrat_mul y ⇔ hrat_1 hrat_lt x
   
   [HRAT_INV_MUL]  Theorem
      
      ⊢ ∀x y. hrat_inv (x hrat_mul y) = hrat_inv x hrat_mul hrat_inv y
   
   [HRAT_LT_ADD2]  Theorem
      
      ⊢ ∀u v x y.
            u hrat_lt x ∧ v hrat_lt y ⇒ u hrat_add v hrat_lt x hrat_add y
   
   [HRAT_LT_ADDL]  Theorem
      
      ⊢ ∀x y. x hrat_lt x hrat_add y
   
   [HRAT_LT_ADDR]  Theorem
      
      ⊢ ∀x y. y hrat_lt x hrat_add y
   
   [HRAT_LT_ANTISYM]  Theorem
      
      ⊢ ∀x y. ¬(x hrat_lt y ∧ y hrat_lt x)
   
   [HRAT_LT_GT]  Theorem
      
      ⊢ ∀x y. x hrat_lt y ⇒ ¬(y hrat_lt x)
   
   [HRAT_LT_L1]  Theorem
      
      ⊢ ∀x y. hrat_inv x hrat_mul y hrat_lt hrat_1 ⇔ y hrat_lt x
   
   [HRAT_LT_LADD]  Theorem
      
      ⊢ ∀x y z. z hrat_add x hrat_lt z hrat_add y ⇔ x hrat_lt y
   
   [HRAT_LT_LMUL]  Theorem
      
      ⊢ ∀x y z. z hrat_mul x hrat_lt z hrat_mul y ⇔ x hrat_lt y
   
   [HRAT_LT_LMUL1]  Theorem
      
      ⊢ ∀x y. x hrat_mul y hrat_lt y ⇔ x hrat_lt hrat_1
   
   [HRAT_LT_MUL2]  Theorem
      
      ⊢ ∀u v x y.
            u hrat_lt x ∧ v hrat_lt y ⇒ u hrat_mul v hrat_lt x hrat_mul y
   
   [HRAT_LT_NE]  Theorem
      
      ⊢ ∀x y. x hrat_lt y ⇒ x ≠ y
   
   [HRAT_LT_R1]  Theorem
      
      ⊢ ∀x y. x hrat_mul hrat_inv y hrat_lt hrat_1 ⇔ x hrat_lt y
   
   [HRAT_LT_RADD]  Theorem
      
      ⊢ ∀x y z. x hrat_add z hrat_lt y hrat_add z ⇔ x hrat_lt y
   
   [HRAT_LT_REFL]  Theorem
      
      ⊢ ∀x. ¬(x hrat_lt x)
   
   [HRAT_LT_RMUL]  Theorem
      
      ⊢ ∀x y z. x hrat_mul z hrat_lt y hrat_mul z ⇔ x hrat_lt y
   
   [HRAT_LT_RMUL1]  Theorem
      
      ⊢ ∀x y. x hrat_mul y hrat_lt x ⇔ y hrat_lt hrat_1
   
   [HRAT_LT_TOTAL]  Theorem
      
      ⊢ ∀x y. (x = y) ∨ x hrat_lt y ∨ y hrat_lt x
   
   [HRAT_LT_TRANS]  Theorem
      
      ⊢ ∀x y z. x hrat_lt y ∧ y hrat_lt z ⇒ x hrat_lt z
   
   [HRAT_MEAN]  Theorem
      
      ⊢ ∀x y. x hrat_lt y ⇒ ∃z. x hrat_lt z ∧ z hrat_lt y
   
   [HRAT_MUL_RID]  Theorem
      
      ⊢ ∀x. x hrat_mul hrat_1 = x
   
   [HRAT_MUL_RINV]  Theorem
      
      ⊢ ∀x. x hrat_mul hrat_inv x = hrat_1
   
   [HRAT_RDISTRIB]  Theorem
      
      ⊢ ∀x y z.
            (x hrat_add y) hrat_mul z = x hrat_mul z hrat_add y hrat_mul z
   
   [HRAT_UP]  Theorem
      
      ⊢ ∀x. ∃y. x hrat_lt y
   
   [HREAL_ADD_ASSOC]  Theorem
      
      ⊢ ∀X Y Z. X hreal_add (Y hreal_add Z) = X hreal_add Y hreal_add Z
   
   [HREAL_ADD_ISACUT]  Theorem
      
      ⊢ ∀X Y. isacut (λw. ∃x y. (w = x hrat_add y) ∧ cut X x ∧ cut Y y)
   
   [HREAL_ADD_SYM]  Theorem
      
      ⊢ ∀X Y. X hreal_add Y = Y hreal_add X
   
   [HREAL_ADD_TOTAL]  Theorem
      
      ⊢ ∀X Y. (X = Y) ∨ (∃D. Y = X hreal_add D) ∨ ∃D. X = Y hreal_add D
   
   [HREAL_INV_ISACUT]  Theorem
      
      ⊢ ∀X.
            isacut
              (λw.
                   ∃d.
                       d hrat_lt hrat_1 ∧
                       ∀x. cut X x ⇒ w hrat_mul x hrat_lt d)
   
   [HREAL_LDISTRIB]  Theorem
      
      ⊢ ∀X Y Z.
            X hreal_mul (Y hreal_add Z) =
            X hreal_mul Y hreal_add X hreal_mul Z
   
   [HREAL_LT]  Theorem
      
      ⊢ ∀X Y. X hreal_lt Y ⇔ ∃D. Y = X hreal_add D
   
   [HREAL_LT_LEMMA]  Theorem
      
      ⊢ ∀X Y. X hreal_lt Y ⇒ ∃x. ¬cut X x ∧ cut Y x
   
   [HREAL_LT_TOTAL]  Theorem
      
      ⊢ ∀X Y. (X = Y) ∨ X hreal_lt Y ∨ Y hreal_lt X
   
   [HREAL_MUL_ASSOC]  Theorem
      
      ⊢ ∀X Y Z. X hreal_mul (Y hreal_mul Z) = X hreal_mul Y hreal_mul Z
   
   [HREAL_MUL_ISACUT]  Theorem
      
      ⊢ ∀X Y. isacut (λw. ∃x y. (w = x hrat_mul y) ∧ cut X x ∧ cut Y y)
   
   [HREAL_MUL_LID]  Theorem
      
      ⊢ ∀X. hreal_1 hreal_mul X = X
   
   [HREAL_MUL_LINV]  Theorem
      
      ⊢ ∀X. hreal_inv X hreal_mul X = hreal_1
   
   [HREAL_MUL_SYM]  Theorem
      
      ⊢ ∀X Y. X hreal_mul Y = Y hreal_mul X
   
   [HREAL_NOZERO]  Theorem
      
      ⊢ ∀X Y. X hreal_add Y ≠ X
   
   [HREAL_SUB_ADD]  Theorem
      
      ⊢ ∀X Y. X hreal_lt Y ⇒ (Y hreal_sub X hreal_add X = Y)
   
   [HREAL_SUB_ISACUT]  Theorem
      
      ⊢ ∀X Y.
            X hreal_lt Y ⇒ isacut (λw. ∃x. ¬cut X x ∧ cut Y (x hrat_add w))
   
   [HREAL_SUP]  Theorem
      
      ⊢ ∀P.
            (∃X. P X) ∧ (∃Y. ∀X. P X ⇒ X hreal_lt Y) ⇒
            ∀Y. (∃X. P X ∧ Y hreal_lt X) ⇔ Y hreal_lt hreal_sup P
   
   [HREAL_SUP_ISACUT]  Theorem
      
      ⊢ ∀P.
            (∃X. P X) ∧ (∃Y. ∀X. P X ⇒ X hreal_lt Y) ⇒
            isacut (λw. ∃X. P X ∧ cut X w)
   
   [ISACUT_HRAT]  Theorem
      
      ⊢ ∀h. isacut (cut_of_hrat h)
   
   
*)
end

HOL 4, Kananaskis-13