Structure logrootTheory

signature logrootTheory =
sig
  type thm = Thm.thm
  
  (*  Definitions  *)
    val LOG : thm
    val ROOT : thm
    val SQRTd_def : thm
    val iSQRT0_def : thm
    val iSQRT1_def : thm
    val iSQRT2_def : thm
    val iSQRT3_def : thm
  
  (*  Theorems  *)
    val EXP_LE_ISO : thm
    val EXP_LE_LOG_SIMP : thm
    val EXP_LT_ISO : thm
    val EXP_LT_LOG_SIMP : thm
    val EXP_MUL : thm
    val LE_EXP_ISO : thm
    val LE_EXP_LOG_SIMP : thm
    val LOG_1 : thm
    val LOG_ADD : thm
    val LOG_ADD1 : thm
    val LOG_BASE : thm
    val LOG_DIV : thm
    val LOG_EQ_0 : thm
    val LOG_EXP : thm
    val LOG_LE_MONO : thm
    val LOG_MOD : thm
    val LOG_MULT : thm
    val LOG_NUMERAL : thm
    val LOG_POW : thm
    val LOG_ROOT : thm
    val LOG_RWT : thm
    val LOG_UNIQUE : thm
    val LOG_add_digit : thm
    val LOG_exists : thm
    val LT_EXP_ISO : thm
    val LT_EXP_LOG : thm
    val LT_EXP_LOG_SIMP : thm
    val ROOT_COMPUTE : thm
    val ROOT_DIV : thm
    val ROOT_EVAL : thm
    val ROOT_EXP : thm
    val ROOT_LE_MONO : thm
    val ROOT_UNIQUE : thm
    val ROOT_exists : thm
    val numeral_root2 : thm
    val numeral_sqrt : thm
  
  val logroot_grammars : type_grammar.grammar * term_grammar.grammar
(*
   [basicSize] Parent theory of "logroot"
   
   [cv] Parent theory of "logroot"
   
   [while] Parent theory of "logroot"
   
   [LOG]  Definition
      
      ⊢ ∀a n. 1 < a ∧ 0 < n ⇒ a ** LOG a n ≤ n ∧ n < a ** SUC (LOG a n)
   
   [ROOT]  Definition
      
      ⊢ ∀r n. 0 < r ⇒ ROOT r n ** r ≤ n ∧ n < SUC (ROOT r n) ** r
   
   [SQRTd_def]  Definition
      
      ⊢ ∀n. SQRTd n = (ROOT 2 n,n − ROOT 2 n * ROOT 2 n)
   
   [iSQRT0_def]  Definition
      
      ⊢ ∀n. iSQRT0 n =
            (let
               p = SQRTd n;
               d = SND p − FST p
             in
               if d = 0 then (2 * FST p,4 * SND p)
               else (SUC (2 * FST p),4 * d − 1))
   
   [iSQRT1_def]  Definition
      
      ⊢ ∀n. iSQRT1 n =
            (let
               p = SQRTd n;
               d = SUC (SND p) − FST p
             in
               if d = 0 then (2 * FST p,SUC (4 * SND p))
               else (SUC (2 * FST p),4 * (d − 1)))
   
   [iSQRT2_def]  Definition
      
      ⊢ ∀n. iSQRT2 n =
            (let
               p = SQRTd n;
               d = 2 * FST p;
               c = SUC (2 * SND p);
               e = c − d
             in
               if e = 0 then (d,2 * c) else (SUC d,2 * e − 1))
   
   [iSQRT3_def]  Definition
      
      ⊢ ∀n. iSQRT3 n =
            (let
               p = SQRTd n;
               d = 2 * FST p;
               c = SUC (2 * SND p);
               e = SUC c − d
             in
               if e = 0 then (d,SUC (2 * c)) else (SUC d,2 * (e − 1)))
   
   [EXP_LE_ISO]  Theorem
      
      ⊢ ∀a b r. 0 < r ⇒ (a ≤ b ⇔ a ** r ≤ b ** r)
   
   [EXP_LE_LOG_SIMP]  Theorem
      
      ⊢ (NUMERAL b ** e ≤ NUMERAL (BIT1 x) ⇔
         NUMERAL b < 2 ∨ e ≤ LOG (NUMERAL b) (NUMERAL (BIT1 x))) ∧
        (NUMERAL b ** e ≤ NUMERAL (BIT2 x) ⇔
         NUMERAL b < 2 ∨ e ≤ LOG (NUMERAL b) (NUMERAL (BIT2 x)))
   
   [EXP_LT_ISO]  Theorem
      
      ⊢ ∀a b r. 0 < r ⇒ (a < b ⇔ a ** r < b ** r)
   
   [EXP_LT_LOG_SIMP]  Theorem
      
      ⊢ (NUMERAL b ** e < NUMERAL (BIT1 (BIT1 x)) ⇔
         NUMERAL b < 2 ∨ e ≤ LOG (NUMERAL b) (NUMERAL (BIT1 (BIT1 x)) − 1)) ∧
        (NUMERAL b ** e < NUMERAL (BIT1 (BIT2 x)) ⇔
         NUMERAL b < 2 ∨ e ≤ LOG (NUMERAL b) (NUMERAL (BIT1 (BIT2 x)) − 1)) ∧
        (NUMERAL b ** e < NUMERAL (BIT2 x) ⇔
         NUMERAL b < 2 ∨ e ≤ LOG (NUMERAL b) (NUMERAL (BIT2 x) − 1))
   
   [EXP_MUL]  Theorem
      
      ⊢ ∀a b c. (a ** b) ** c = a ** (b * c)
   
   [LE_EXP_ISO]  Theorem
      
      ⊢ ∀e a b. 1 < e ⇒ (a ≤ b ⇔ e ** a ≤ e ** b)
   
   [LE_EXP_LOG_SIMP]  Theorem
      
      ⊢ (NUMERAL (BIT1 (BIT1 x)) ≤ NUMERAL b ** e ⇔
         2 ≤ NUMERAL b ∧ LOG (NUMERAL b) (NUMERAL (BIT1 (BIT1 x)) − 1) < e) ∧
        (NUMERAL (BIT1 (BIT2 x)) ≤ NUMERAL b ** e ⇔
         2 ≤ NUMERAL b ∧ LOG (NUMERAL b) (NUMERAL (BIT1 (BIT2 x)) − 1) < e) ∧
        (NUMERAL (BIT2 x) ≤ NUMERAL b ** e ⇔
         2 ≤ NUMERAL b ∧ LOG (NUMERAL b) (NUMERAL (BIT2 x) − 1) < e)
   
   [LOG_1]  Theorem
      
      ⊢ ∀a. 1 < a ⇒ LOG a 1 = 0
   
   [LOG_ADD]  Theorem
      
      ⊢ ∀a b c. 1 < a ∧ b < a ** c ⇒ LOG a (b + a ** c) = c
   
   [LOG_ADD1]  Theorem
      
      ⊢ ∀n a b.
          0 < n ∧ 1 < a ∧ 0 < b ⇒
          LOG a (a ** SUC n * b) = SUC (LOG a (a ** n * b))
   
   [LOG_BASE]  Theorem
      
      ⊢ ∀a. 1 < a ⇒ LOG a a = 1
   
   [LOG_DIV]  Theorem
      
      ⊢ ∀a x. 1 < a ∧ a ≤ x ⇒ LOG a x = 1 + LOG a (x DIV a)
   
   [LOG_EQ_0]  Theorem
      
      ⊢ ∀a b. 1 < a ∧ 0 < b ⇒ (LOG a b = 0 ⇔ b < a)
   
   [LOG_EXP]  Theorem
      
      ⊢ ∀n a b. 1 < a ∧ 0 < b ⇒ LOG a (a ** n * b) = n + LOG a b
   
   [LOG_LE_MONO]  Theorem
      
      ⊢ ∀a x y. 1 < a ∧ 0 < x ⇒ x ≤ y ⇒ LOG a x ≤ LOG a y
   
   [LOG_MOD]  Theorem
      
      ⊢ ∀n. 0 < n ⇒ n = 2 ** LOG 2 n + n MOD 2 ** LOG 2 n
   
   [LOG_MULT]  Theorem
      
      ⊢ ∀b x. 1 < b ∧ 0 < x ⇒ LOG b (b * x) = SUC (LOG b x)
   
   [LOG_NUMERAL]  Theorem
      
      ⊢ LOG (NUMERAL (BIT1 (BIT1 b))) (NUMERAL (BIT1 n)) =
        (if NUMERAL (BIT1 n) < NUMERAL (BIT1 (BIT1 b)) then 0
         else
           LOG (NUMERAL (BIT1 (BIT1 b)))
             (NUMERAL (BIT1 n) DIV NUMERAL (BIT1 (BIT1 b))) + 1) ∧
        LOG (NUMERAL (BIT1 (BIT1 b))) (NUMERAL (BIT2 n)) =
        (if NUMERAL (BIT2 n) < NUMERAL (BIT1 (BIT1 b)) then 0
         else
           LOG (NUMERAL (BIT1 (BIT1 b)))
             (NUMERAL (BIT2 n) DIV NUMERAL (BIT1 (BIT1 b))) + 1) ∧
        LOG (NUMERAL (BIT1 (BIT2 b))) (NUMERAL (BIT1 n)) =
        (if NUMERAL (BIT1 n) < NUMERAL (BIT1 (BIT2 b)) then 0
         else
           LOG (NUMERAL (BIT1 (BIT2 b)))
             (NUMERAL (BIT1 n) DIV NUMERAL (BIT1 (BIT2 b))) + 1) ∧
        LOG (NUMERAL (BIT1 (BIT2 b))) (NUMERAL (BIT2 n)) =
        (if NUMERAL (BIT2 n) < NUMERAL (BIT1 (BIT2 b)) then 0
         else
           LOG (NUMERAL (BIT1 (BIT2 b)))
             (NUMERAL (BIT2 n) DIV NUMERAL (BIT1 (BIT2 b))) + 1) ∧
        LOG (NUMERAL (BIT2 b)) (NUMERAL (BIT1 n)) =
        (if NUMERAL (BIT1 n) < NUMERAL (BIT2 b) then 0
         else
           LOG (NUMERAL (BIT2 b)) (NUMERAL (BIT1 n) DIV NUMERAL (BIT2 b)) +
           1) ∧
        LOG (NUMERAL (BIT2 b)) (NUMERAL (BIT2 n)) =
        if NUMERAL (BIT2 n) < NUMERAL (BIT2 b) then 0
        else
          LOG (NUMERAL (BIT2 b)) (NUMERAL (BIT2 n) DIV NUMERAL (BIT2 b)) +
          1
   
   [LOG_POW]  Theorem
      
      ⊢ ∀b n. 1 < b ⇒ LOG b (b ** n) = n
   
   [LOG_ROOT]  Theorem
      
      ⊢ ∀a x r. 1 < a ∧ 0 < x ∧ 0 < r ⇒ LOG a (ROOT r x) = LOG a x DIV r
   
   [LOG_RWT]  Theorem
      
      ⊢ ∀m n.
          1 < m ∧ 0 < n ⇒
          LOG m n = if n < m then 0 else SUC (LOG m (n DIV m))
   
   [LOG_UNIQUE]  Theorem
      
      ⊢ ∀a n p. a ** p ≤ n ∧ n < a ** SUC p ⇒ LOG a n = p
   
   [LOG_add_digit]  Theorem
      
      ⊢ ∀b x y. 1 < b ∧ 0 < y ∧ x < b ⇒ LOG b (b * y + x) = SUC (LOG b y)
   
   [LOG_exists]  Theorem
      
      ⊢ ∃f. ∀a n. 1 < a ∧ 0 < n ⇒ a ** f a n ≤ n ∧ n < a ** SUC (f a n)
   
   [LT_EXP_ISO]  Theorem
      
      ⊢ ∀e a b. 1 < e ⇒ (a < b ⇔ e ** a < e ** b)
   
   [LT_EXP_LOG]  Theorem
      
      ⊢ x < b ** e ⇔
        b = 0 ∧ e = 0 ∧ x = 0 ∨ b = 1 ∧ x = 0 ∨
        2 ≤ b ∧ (LOG b x < e ∨ x = 0)
   
   [LT_EXP_LOG_SIMP]  Theorem
      
      ⊢ (NUMERAL (BIT1 x) < NUMERAL b ** e ⇔
         2 ≤ NUMERAL b ∧ LOG (NUMERAL b) (NUMERAL (BIT1 x)) < e) ∧
        (NUMERAL (BIT2 x) < NUMERAL b ** e ⇔
         2 ≤ NUMERAL b ∧ LOG (NUMERAL b) (NUMERAL (BIT2 x)) < e)
   
   [ROOT_COMPUTE]  Theorem
      
      ⊢ ∀r n.
          0 < r ⇒
          ROOT r 0 = 0 ∧
          ROOT r n =
          (let
             x = 2 * ROOT r (n DIV 2 ** r)
           in
             if n < SUC x ** r then x else SUC x)
   
   [ROOT_DIV]  Theorem
      
      ⊢ ∀r x y. 0 < r ∧ 0 < y ⇒ ROOT r x DIV y = ROOT r (x DIV y ** r)
   
   [ROOT_EVAL]  Theorem
      
      ⊢ ∀r n.
          ROOT r n =
          if r = 0 then ROOT 0 n
          else if n = 0 then 0
          else
            (let
               m = 2 * ROOT r (n DIV 2 ** r)
             in
               m + if SUC m ** r ≤ n then 1 else 0)
   
   [ROOT_EXP]  Theorem
      
      ⊢ ∀n r. 0 < r ⇒ ROOT r (n ** r) = n
   
   [ROOT_LE_MONO]  Theorem
      
      ⊢ ∀r x y. 0 < r ⇒ x ≤ y ⇒ ROOT r x ≤ ROOT r y
   
   [ROOT_UNIQUE]  Theorem
      
      ⊢ ∀r n p. p ** r ≤ n ∧ n < SUC p ** r ⇒ ROOT r n = p
   
   [ROOT_exists]  Theorem
      
      ⊢ ∀r n. 0 < r ⇒ ∃rt. rt ** r ≤ n ∧ n < SUC rt ** r
   
   [numeral_root2]  Theorem
      
      ⊢ ROOT 2 (NUMERAL n) = FST (SQRTd n)
   
   [numeral_sqrt]  Theorem
      
      ⊢ SQRTd ZERO = (0,0) ∧ SQRTd (BIT1 ZERO) = (1,0) ∧
        SQRTd (BIT2 ZERO) = (1,1) ∧ SQRTd (BIT1 (BIT1 n)) = iSQRT3 n ∧
        SQRTd (BIT2 (BIT1 n)) = iSQRT0 (SUC n) ∧
        SQRTd (BIT1 (BIT2 n)) = iSQRT1 (SUC n) ∧
        SQRTd (BIT2 (BIT2 n)) = iSQRT2 (SUC n) ∧
        SQRTd (SUC (BIT1 (BIT1 n))) = iSQRT0 (SUC n) ∧
        SQRTd (SUC (BIT2 (BIT1 n))) = iSQRT1 (SUC n) ∧
        SQRTd (SUC (BIT1 (BIT2 n))) = iSQRT2 (SUC n) ∧
        SQRTd (SUC (BIT2 (BIT2 n))) = iSQRT3 (SUC n)
   
   
*)
end

HOL 4, Trindemossen-1