- l2n_def
-
|- (∀b. l2n b [] = 0) ∧ ∀b h t. l2n b (h::t) = h MOD b + b * l2n b t
- n2l_tupled_primitive_def
-
|- n2l_tupled =
WFREC (@R. WF R ∧ ∀b n. ¬(n < b ∨ b < 2) ⇒ R (b,n DIV b) (b,n))
(λn2l_tupled a.
case a of
(b,n) =>
I
(if n < b ∨ b < 2 then [n MOD b]
else n MOD b::n2l_tupled (b,n DIV b)))
- n2l_curried_def
-
|- ∀x x1. n2l x x1 = n2l_tupled (x,x1)
- num_from_bin_list_def
-
|- num_from_bin_list = l2n 2
- num_from_oct_list_def
-
|- num_from_oct_list = l2n 8
- num_from_dec_list_def
-
|- num_from_dec_list = l2n 10
- num_from_hex_list_def
-
|- num_from_hex_list = l2n 16
- num_to_bin_list_def
-
|- num_to_bin_list = n2l 2
- num_to_oct_list_def
-
|- num_to_oct_list = n2l 8
- num_to_dec_list_def
-
|- num_to_dec_list = n2l 10
- num_to_hex_list_def
-
|- num_to_hex_list = n2l 16
- BOOLIFY_def
-
|- (∀m a. BOOLIFY 0 m a = a) ∧
∀n m a. BOOLIFY (SUC n) m a = BOOLIFY n (DIV2 m) (ODD m::a)
- l2n2
-
|- numposrep$l2n2 = l2n 2
- n2l_ind
-
|- ∀P. (∀b n. (¬(n < b ∨ b < 2) ⇒ P b (n DIV b)) ⇒ P b n) ⇒ ∀v v1. P v v1
- n2l_def
-
|- ∀n b.
n2l b n = if n < b ∨ b < 2 then [n MOD b] else n MOD b::n2l b (n DIV b)
- LENGTH_n2l
-
|- ∀b n. 1 < b ⇒ (LENGTH (n2l b n) = if n = 0 then 1 else SUC (LOG b n))
- l2n_n2l
-
|- ∀b n. 1 < b ⇒ (l2n b (n2l b n) = n)
- l2n_lt
-
|- ∀l b. 0 < b ⇒ l2n b l < b ** LENGTH l
- LENGTH_l2n
-
|- ∀b l.
1 < b ∧ EVERY ($> b) l ∧ l2n b l ≠ 0 ⇒ SUC (LOG b (l2n b l)) ≤ LENGTH l
- EL_TAKE
-
|- ∀x n l. x < n ∧ n ≤ LENGTH l ⇒ (EL x (TAKE n l) = EL x l)
- l2n_DIGIT
-
|- ∀b l x.
1 < b ∧ EVERY ($> b) l ∧ x < LENGTH l ⇒
((l2n b l DIV b ** x) MOD b = EL x l)
- DIV_0_IMP_LT
-
|- ∀b n. 1 < b ∧ (n DIV b = 0) ⇒ n < b
- EL_n2l
-
|- ∀b x n.
1 < b ∧ x < LENGTH (n2l b n) ⇒ (EL x (n2l b n) = (n DIV b ** x) MOD b)
- n2l_l2n
-
|- ∀b l.
1 < b ∧ EVERY ($> b) l ⇒
(n2l b (l2n b l) =
if l2n b l = 0 then [0] else TAKE (SUC (LOG b (l2n b l))) l)
- l2n_eq_0
-
|- ∀b. 0 < b ⇒ ∀l. (l2n b l = 0) ⇔ EVERY ($= 0 o combin$C $MOD b) l
- l2n_SNOC_0
-
|- ∀b ls. 0 < b ⇒ (l2n b (SNOC 0 ls) = l2n b ls)
- LOG_l2n
-
|- ∀b.
1 < b ⇒
∀l.
l ≠ [] ∧ 0 < LAST l ∧ EVERY ($> b) l ⇒
(LOG b (l2n b l) = PRE (LENGTH l))
- l2n_dropWhile_0
-
|- ∀b ls. 0 < b ⇒ (l2n b (REVERSE (dropWhile ($= 0) (REVERSE ls))) = l2n b ls)
- LOG_l2n_dropWhile
-
|- ∀b l.
1 < b ∧ EXISTS (λy. 0 ≠ y) l ∧ EVERY ($> b) l ⇒
(LOG b (l2n b l) = PRE (LENGTH (dropWhile ($= 0) (REVERSE l))))
- n2l_BOUND
-
|- ∀b n. 0 < b ⇒ EVERY ($> b) (n2l b n)
- l2n_pow2_compute
-
|- (∀p. l2n (2 ** p) [] = 0) ∧
∀p h t. l2n (2 ** p) (h::t) = MOD_2EXP p h + TIMES_2EXP p (l2n (2 ** p) t)
- n2l_pow2_compute
-
|- ∀p n.
0 < p ⇒
(n2l (2 ** p) n =
(let (q,r) = DIVMOD_2EXP p n
in
if q = 0 then [r] else r::n2l (2 ** p) q))
- l2n_2_thms
-
|- (∀t. l2n 2 (0::t) = NUMERAL (numposrep$l2n2 (0::t))) ∧
(∀t. l2n 2 (1::t) = NUMERAL (numposrep$l2n2 (1::t))) ∧
(numposrep$l2n2 [] = ZERO) ∧
(∀t. numposrep$l2n2 (0::t) = numeral$iDUB (numposrep$l2n2 t)) ∧
∀t. numposrep$l2n2 (1::t) = BIT1 (numposrep$l2n2 t)
- BIT_num_from_bin_list
-
|- ∀x l.
EVERY ($> 2) l ∧ x < LENGTH l ⇒
(BIT x (num_from_bin_list l) ⇔ (EL x l = 1))
- EL_num_to_bin_list
-
|- ∀x n.
x < LENGTH (num_to_bin_list n) ⇒ (EL x (num_to_bin_list n) = BITV n x)
- num_bin_list
-
|- num_from_bin_list o num_to_bin_list = I
- num_oct_list
-
|- num_from_oct_list o num_to_oct_list = I
- num_dec_list
-
|- num_from_dec_list o num_to_dec_list = I
- num_hex_list
-
|- num_from_hex_list o num_to_hex_list = I