problem_id stringlengths 17 84 | category stringclasses 5
values | query stringlengths 10 8.6k |
|---|---|---|
common.misc.SUB_LIST_TOPSPLIT | generic | `!l n. APPEND (SUB_LIST(0,n) l) (SUB_LIST(n,LENGTH l - n) l) = l` |
common.misc.LENGTH_SUB_LIST | generic | `!l m n. LENGTH(SUB_LIST(m,n) l) = MIN n (LENGTH l - m)` |
common.misc.SUB_LIST_TRIVIAL | generic | `!l m n. LENGTH l <= m ==> SUB_LIST(m,n) l = []` |
common.misc.SUB_LIST_IDEMPOTENT | generic | `!n (l:(A)list). SUB_LIST (0,n) (SUB_LIST (0,n) l) = SUB_LIST (0,n) l` |
common.misc.SUB_LIST_MIN | generic | `!(l:(A)list) (n:num) m. SUB_LIST (0,n) (SUB_LIST (0,m) l) = SUB_LIST (0, MIN n m) l` |
common.misc.SUB_LIST_REFL | generic | `!(l:A list) n. LENGTH l <= n ==> SUB_LIST(0,n) l = l` |
common.misc.SUB_LIST_1 | generic | `!(l:A list) n. SUB_LIST(n,1) l = if n < LENGTH l then [EL n l] else []` |
common.misc.SUB_LIST_APPEND_LEFT | generic | `!(l:A list) m n.
n <= LENGTH l ==> SUB_LIST(0,n) (APPEND l m) = SUB_LIST(0,n) l` |
common.misc.APPEND_EXISTS | generic | `forall (x:(A)list). exists (x1:(A)list) x2. x = APPEND x1 x2` |
common.misc.RELPOW_ITER | generic | `!f n x y:A. RELPOW n (\a b. f a = b) x y <=> ITER n f x = y` |
common.misc.MOD_64_CLAUSES | bit_vector | `(!c. val(word(c MOD 64):int64) = c MOD 64) /\
(!c. val(word(c MOD 256):byte) = c MOD 256) /\
(!c. val(word c:int64) MOD 64 = c MOD 64) /\
(!n. n MOD 256 MOD 64 = n MOD 64) /\
(!n. n MOD 64 MOD 64 = n MOD 64) /\
(!n. n MOD 2 EXP 64 MOD 64 = n MOD 64) /\
(!n. n MOD 64 MOD 2 EXP 64 = n MOD 64)` |
common.misc.WORD_UNMASK_64 | bit_vector | `(if p then word 18446744073709551615 else word 0):int64 =
word_neg(word(bitval p)) /\
(if p then word 0 else word 18446744073709551615):int64 =
word_neg(word(bitval(~p)))` |
common.misc.EQUAL_FROM_CONGRUENT_REAL | generic | `!n x y z.
(&0 <= x /\ x < &2 pow n) /\
(&0 <= y /\ y < &2 pow n) /\
integer z /\ integer((x - y) / &2 pow n + z)
==> x = y` |
common.misc.EQUAL_FROM_CONGRUENT_MOD | generic | `!n x y z.
&y:real < &2 pow n /\
integer z /\ integer((&x - &y) / &2 pow n + z)
==> x MOD (2 EXP n) = y` |
common.misc.EQUAL_FROM_CONGRUENT_MOD_MOD | generic | `!n m r k r'.
r < 2 EXP k /\ m < 2 EXP k /\ ~(m = 0) /\
integer((&r - r') / &2 pow k) /\
&(n MOD m) = r'
==> n MOD m = r` |
common.misc.LIMB_0 | generic | `!w i. limb w 0 i = 0` |
common.misc.LIMB_BOUND | generic | `!w n i. limb w n i < 2 EXP w` |
common.misc.DIGITSUM_WORDS_LIMB_GEN | generic | `!w n k. nsum {i | i < k} (\i. 2 EXP (w * i) * limb w n i) =
n MOD (2 EXP (w * k))` |
common.misc.DIGITSUM_WORKS_LIMB | generic | `!w n k. n < 2 EXP (w * k)
==> nsum {i | i < k} (\i. 2 EXP (w * i) * limb w n i) = n` |
common.misc.LIMB_DIGITSUM | generic | `!w n k d.
(!i. i < k ==> d i < 2 EXP w)
==> limb w (nsum {i | i < k} (\i. 2 EXP (w * i) * d i)) n =
if n < k then d n else 0` |
common.misc.DIGITSUM_LT_STEP | generic | `!B b k n.
0 < B /\ (!i. i < k ==> b i < B)
==> (nsum {i | i < k} (\i. B EXP i * b i) < B EXP n <=>
k <= n \/
b n = 0 /\ nsum {i | i < k} (\i. B EXP i * b i) < B EXP (n + 1))` |
common.misc.VAL_BOUND_64 | bit_vector | `!x:int64. val x < 2 EXP 64` |
common.misc.VAL_BOUND_128 | bit_vector | `!x:int128. val x < 2 EXP 128` |
common.misc.VAL_BOUND_256 | bit_vector | `!x:int256. val x < 2 EXP 256` |
common.misc.LEXICOGRAPHIC_LT | generic | `!B h l h' l':num.
l < B /\ l' < B
==> (B * h + l < B * h' + l' <=> h < h' \/ h = h' /\ l < l')` |
common.misc.LEXICOGRAPHIC_LE | generic | `!B h l h' l':num.
l < B /\ l' < B
==> (B * h + l <= B * h' + l' <=> h < h' \/ h = h' /\ l <= l')` |
common.misc.LEXICOGRAPHIC_EQ | generic | `!B h l h' l':num.
l < B /\ l' < B
==> (B * h + l = B * h' + l' <=> h = h' /\ l = l')` |
common.misc.LEXICOGRAPHIC_LT_64 | generic | `!h l h' l'.
l < 2 EXP 64 /\ l' < 2 EXP 64
==> (2 EXP 64 * h + l < 2 EXP 64 * h' + l' <=>
h < h' \/ h = h' /\ l < l')` |
common.misc.LEXICOGRAPHIC_LE_64 | generic | `!h l h' l'.
l < 2 EXP 64 /\ l' < 2 EXP 64
==> (2 EXP 64 * h + l <= 2 EXP 64 * h' + l' <=>
h < h' \/ h = h' /\ l <= l')` |
common.misc.LEXICOGRAPHIC_EQ_64 | generic | `!h l h' l'.
l < 2 EXP 64 /\ l' < 2 EXP 64
==> (2 EXP 64 * h + l = 2 EXP 64 * h' + l' <=> h = h' /\ l = l')` |
common.misc.LEXICOGRAPHIC_LT_INT64 | bit_vector | `!h (l:int64) h' (l':int64).
2 EXP 64 * h + val l < 2 EXP 64 * h' + val l' <=>
h < h' \/ h = h' /\ val l < val l'` |
common.misc.LEXICOGRAPHIC_LE_INT64 | bit_vector | `!h (l:int64) h' (l':int64).
2 EXP 64 * h + val l <= 2 EXP 64 * h' + val l' <=>
h < h' \/ h = h' /\ val l <= val l'` |
common.misc.LEXICOGRAPHIC_EQ_INT64 | bit_vector | `!h (l:int64) h' (l':int64).
2 EXP 64 * h + val l = 2 EXP 64 * h' + val l' <=> h = h' /\ l = l'` |
common.misc.WORD_SUBWORD_EQUAL_SUBWORD_THEN_ZX | bit_vector | `!(x:N word) (pos:num) (len1:num) (len2:num).
len1 >= len2 /\ len1 <= dimindex(:P) /\ len2 = dimindex(:M)
==> ((word_subword x (pos,len2)):M word) =
word_zx ((word_subword x (pos,len1)):P word)` |
common.misc.WORD_SUBWORD_N_EQUAL | bit_vector | `!(x:N word). (word_subword x (0,dimindex (:N))) = x` |
common.misc.WORD_SUBWORD_EQUAL_WORD_ZX_POS0 | bit_vector | `!(x:N word) (len:num).
len = dimindex(:M) /\ len <= dimindex(:N)
==> ((word_subword x (0,len)):M word) = word_zx x` |
common.misc.WORD_CTZ_INDEX | bit_vector | `!x:N word.
word_ctz x = if x = word 0 then dimindex(:N) else index 2 (val x)` |
common.misc.WORD_CLZ_BITSIZE | bit_vector | `!x:N word. word_clz x = dimindex(:N) - bitsize(val x)` |
common.misc.WORD_INDEX_WRAP | bit_vector | `!i. word(8 * (i + 1) + 18446744073709551608):int64 = word(8 * i)` |
common.misc.MOD_UNIQ_BALANCED | generic | `!n p z q.
q * p <= n + p /\ n < q * p + p /\
q * p + z = bitval(n < q * p) * p + n
==> n MOD p = z` |
common.misc.MOD_UNIQ_BALANCED_REAL | generic | `!n p z q k.
p < 2 EXP k /\ z < 2 EXP k /\
q * p <= n + p /\ n < q * p + p /\
integer((&n - &z - (&q - &(bitval(n < q * p))) * &p) / &2 pow k)
==> n MOD p = z` |
common.misc.INT_REM_UNIQ_BALANCED | generic | `!n p z q.
q * p <= n + p /\ n < q * p + p /\
q * p + z = &(bitval(n < q * p)) * p + n
==> n rem p = z` |
common.misc.INT_REM_UNIQ_BALANCED_MOD | generic | `!n p z q k.
&0 <= p /\ p < &2 pow k /\
&0 <= z /\ z < &2 pow k /\
q * p <= n + p /\ n < q * p + p /\
(q * p + z == &(bitval(n < q * p)) * p + n) (mod (&2 pow k))
==> n rem p = z` |
common.misc.VAL_WORD_SUBWORD_JOIN_64 | bit_vector | `!(h:int64) (l:int64) k.
k <= 64
==> val(word_subword (word_join h l:int128) (k,64) :int64) =
2 EXP (64 - k) * val h MOD 2 EXP k + val l DIV (2 EXP k)` |
common.misc.NOCARRY_WORD_ADC | bit_vector | `!b x y:N word.
val x + val y + bitval b =
val(word_add (word_add x y) (word (bitval b))) <=>
val x + val y + bitval b < 2 EXP dimindex(:N)` |
common.misc.CARRY_WORD_ADC | bit_vector | `!b x y:N word.
~(val x + val y + bitval b =
val(word_add (word_add x y) (word (bitval b)))) <=>
2 EXP dimindex(:N) <= val x + val y + bitval b` |
common.misc.NOCARRY_WORD_ADD | bit_vector | `!x y:N word.
val x + val y = val(word_add x y) <=>
val x + val y < 2 EXP dimindex(:N)` |
common.misc.CARRY_WORD_ADD | bit_vector | `!x y:N word.
~(val x + val y = val(word_add x y)) <=>
2 EXP dimindex(:N) <= val x + val y` |
common.misc.NOCARRY_WORD_SBB | bit_vector | `!b x y:N word.
&(val x) - (&(val y) + &(bitval b)):int =
&(val(word_sub x (word_add y (word(bitval b))))) <=>
if b then val y < val x else val y <= val x` |
common.misc.CARRY_WORD_SBB | bit_vector | `!b x y:N word.
~(&(val x) - (&(val y) + &(bitval b)):int =
&(val(word_sub x (word_add y (word(bitval b)))))) <=>
if b then val x <= val y else val x < val y` |
common.misc.NOCARRY_WORD_SUB | bit_vector | `!x y:N word.
&(val x) - &(val y):int = &(val(word_sub x y)) <=>
val y <= val x` |
common.misc.CARRY_WORD_SUB | bit_vector | `!x y:N word.
~(&(val x) - &(val y):int = &(val(word_sub x y))) <=>
val x < val y` |
common.misc.NOCARRY_THM | bit_vector | `!x:int64. 2 EXP 64 <= val x <=> F` |
common.misc.NOCARRY64_ADC | bit_vector | `!b x y:int64.
(val x + val y + bitval b =
val(word_add (word_add x y) (word (bitval b)))) <=>
val x + val y + bitval b < 2 EXP 64` |
common.misc.CARRY64_ADC | bit_vector | `!b x y:int64.
~(val x + val y + bitval b =
val(word_add (word_add x y) (word (bitval b)))) <=>
2 EXP 64 <= val x + val y + bitval b` |
common.misc.NOCARRY64_ADD | bit_vector | `!x y:int64.
(val x + val y = val(word_add x y)) <=>
val x + val y < 2 EXP 64` |
common.misc.CARRY64_ADD | bit_vector | `!x y:int64.
~(val x + val y = val(word_add x y)) <=>
2 EXP 64 <= val x + val y` |
common.misc.NOCARRY64_ADD_1 | bit_vector | `!x:int64.
(val x + 1 = val(word_add x (word 1))) <=>
val x + 1 < 2 EXP 64` |
common.misc.CARRY64_ADD_1 | bit_vector | `!x:int64.
~(val x + 1 = val(word_add x (word 1))) <=>
2 EXP 64 <= val x + 1` |
common.misc.NOCARRY64_SBB | bit_vector | `!b x y:int64.
(&(val x) - (&(val y) + &(bitval b)):int =
&(val(word_sub x (word_add y (word (bitval b)))))) <=>
val y + bitval b <= val x` |
common.misc.CARRY64_SBB | bit_vector | `!b x y:int64.
~(&(val x) - (&(val y) + &(bitval b)):int =
&(val(word_sub x (word_add y (word (bitval b)))))) <=>
val x < val y + bitval b` |
common.misc.NOCARRY64_SUB | bit_vector | `!x y:int64.
(&(val x) - &(val y):int = &(val(word_sub x y))) <=>
val y <= val x` |
common.misc.CARRY64_SUB | bit_vector | `!x y:int64.
~(&(val x) - &(val y):int = &(val(word_sub x y))) <=>
val x < val y` |
common.misc.ACCUMULATE_ADC | bit_vector | `!b x y:int64.
val(x) + val(y) + val(word(bitval b):int64) =
2 EXP 64 * bitval(2 EXP 64 <= val x + val y + bitval b) +
val(word_add (word_add x y) (word (bitval b)))` |
common.misc.ACCUMULATE_ADC_0 | bit_vector | `!b x:int64.
val(x) + val(word(bitval b):int64) =
2 EXP 64 * bitval(2 EXP 64 <= val x + bitval b) +
val(word_add x (word (bitval b)))` |
common.misc.ACCUMULATE_ADD | bit_vector | `!x y:int64.
val(x) + val(y) =
2 EXP 64 * bitval(2 EXP 64 <= val x + val y) +
val(word_add x y)` |
common.misc.ACCUMULATE_SBB | bit_vector | `!b x y:int64.
2 EXP 64 * bitval(val x < val y + bitval b) + val x =
val(word_sub x (word_add y (word (bitval b)))) + val y + bitval b` |
common.misc.ACCUMULATE_SBB_RZERO | bit_vector | `!b x:int64.
2 EXP 64 * bitval(val x < bitval b) + val x =
val(word_sub x (word (bitval b))) + bitval b` |
common.misc.ACCUMULATE_SBB_LZERO | bit_vector | `!b x:int64.
2 EXP 64 * bitval(0 < val x + bitval b) =
val(word_neg(word_add x (word (bitval b)))) + val x + bitval b` |
common.misc.ACCUMULATE_SUB | bit_vector | `!x y:int64.
2 EXP 64 * bitval(val x < val y) + val x =
val(word_sub x y) + val y` |
common.misc.ACCUMULATE_MUL_GEN | bit_vector | `!(x:N word) (y:N word).
2 EXP dimindex(:N) *
val(word_zx (word_ushr
(word(val x * val y):(N tybit0)word) (dimindex(:N))):N word) +
val(word_zx (word(val x * val y):(N tybit0)word):N word) =
val x * val y` |
common.misc.ACCUMULATE_MUL | bit_vector | `!(x:int64) (y:int64).
2 EXP 64 *
val(word_zx (word_ushr
(word(val x * val y):(128)word) 64):int64) +
val(word_zx (word(val x * val y):(128)word):int64) =
val x * val y` |
common.misc.APPROXIMATE_WORD_USHR | bit_vector | `!(dest:int64) a n.
dest = word_ushr a n
==> ?b e. dest = b /\
&0 <= e /\ e <= &1 - inv(&2 pow n) /\
&(val b) = &(val a) / &2 pow n - e` |
common.misc.APPROXIMATE_WORD_SHL | bit_vector | `!(dest:int64) a n.
dest = word_shl a n
==> &2 pow n * &(val a):real < &2 pow 64
==> ?c. dest = c /\
&(val c):real = &2 pow n * &(val a)` |
common.misc.APPROXIMATE_WORD_ADD | bit_vector | `!(dest:int64) a b.
dest = word_add a b
==> &(val a) + &(val b):real < &2 pow 64
==> ?c. dest = c /\
&(val c):real = &(val a) + &(val b)` |
common.misc.APPROXIMATE_WORD_SUB | bit_vector | `!(dest:int64) a b.
dest = word_sub a b
==> &0 <= &(val a) - &(val b)
==> ?c. dest = c /\
&(val c) = &(val a) - &(val b)` |
common.misc.APPROXIMATE_WORD_MUL | bit_vector | `!(dest:int64) (a:int64) (b:int64).
dest = word(0 + val a * val b)
==> &(val a) * &(val b):real < &2 pow 64
==> ?c. dest = c /\
&(val c):real = &(val a) * &(val b)` |
common.misc.APPROXIMATE_WORD_MADD | bit_vector | `!(dest:int64) (a:int64) (b:int64) (c:int64).
dest = word(val a + val b * val c)
==> &(val a) + &(val b) * &(val c):real < &2 pow 64
==> ?d. dest = d /\
&(val d):real = &(val a) + &(val b) * &(val c)` |
common.misc.APPROXIMATE_WORD_MNEG | bit_vector | `!(dest:int64) (a:int64) (b:int64).
dest = iword(&0 - ival a * ival b)
==> &0 < &(val a) * &(val b):real /\
&(val a) * &(val b):real <= &2 pow 64
==> ?c. dest = c /\
&(val c):real = &2 pow 64 - &(val a) * &(val b)` |
common.misc.APPROXIMATE_WORD_IWORD | bit_vector | `!(dest:int64) x x'.
dest = iword x
==> &0 <= x' /\ x' < &2 pow 64 /\ (x == x') (mod (&2 pow 64))
==> ?c. dest = c /\
&(val c) = real_of_int x'` |
common.misc.FLAG_FROM_CARRY_REAL_LT | bit_vector | `!k x (y:real) p.
&0 <= &2 pow k * &(bitval p) + x - y /\
&2 pow k * &(bitval p) + x - y < &2 pow k
==> (p <=> x < y)` |
common.misc.FLAG_FROM_CARRY_REAL_LE | bit_vector | `!k x (y:real) p.
&0 <= &2 pow k * (&1 - &(bitval p)) + y - x /\
&2 pow k * (&1 - &(bitval p)) + y - x < &2 pow k
==> (p <=> x <= y)` |
common.misc.FLAG_FROM_CARRY_LT | bit_vector | `!k m n p.
&0:real <= &2 pow k * &(bitval p) + &m - &n /\
&2 pow k * &(bitval p) + &m - &n:real < &2 pow k
==> (p <=> m < n)` |
common.misc.FLAG_FROM_CARRY_LE | bit_vector | `!k m n p.
&0:real <= &2 pow k * (&1 - &(bitval p)) + &n - &m /\
&2 pow k * (&1 - &(bitval p)) + &n - &m:real < &2 pow k
==> (p <=> m <= n)` |
common.misc.WORD_FROM_CARRY_REAL_LT | bit_vector | `!k x (y:real) (w:N word).
y - x <= &2 pow k /\
&0 <= &2 pow k * &(val w) + x - y /\
&2 pow k * &(val w) + x - y < &2 pow k
==> w = word(bitval(x < y))` |
common.misc.WORD_FROM_CARRY_REAL_LE | bit_vector | `!k x (y:real) (w:N word).
y - x < &2 pow k /\
&0 <= &2 pow k * (&1 - &(val w)) + y - x /\
&2 pow k * (&1 - &(val w)) + y - x < &2 pow k
==> w = word(bitval(x <= y))` |
common.misc.WORD_FROM_CARRY_LT | bit_vector | `!k x y (w:N word).
y - x:real <= &2 pow k /\
&0:real <= &2 pow k * &(val w) + x - y /\
&2 pow k * &(val w) + x - y:real < &2 pow k
==> w = word(bitval(x < y))` |
common.misc.WORD_FROM_CARRY_LE | bit_vector | `!k x (w:N word).
&y - &x < &2 pow k /\
&0 <= &2 pow k * (&1 - &(val w)) + &y - &x /\
&2 pow k * (&1 - &(val w)) + &y - &x < &2 pow k
==> w = word(bitval(x <= y))` |
common.misc.FLAG_AND_MASK_FROM_CARRY_REAL_LT | bit_vector | `!c (m:int64) k (x:real).
--(&2 pow k) <= x /\ x < &2 pow k /\
&0 <= x - &2 pow k * (&(val m) - &2 pow 64 * &(bitval c)) /\
x - &2 pow k * (&(val m) - &2 pow 64 * &(bitval c)) < &2 pow k
==> m = word_neg(word(bitval(x < &0))) /\
(c <=> x < &0)` |
common.misc.FLAG_AND_MASK_FROM_CARRY_LT | bit_vector | `!c (m:int64) k x y.
--(&2 pow k):real <= &x - &y /\ &x - &y:real < &2 pow k /\
&0:real <= &x - &y - &2 pow k * (&(val m) - &2 pow 64 * &(bitval c)) /\
&x - &y - &2 pow k * (&(val m) - &2 pow 64 * &(bitval c)):real
< &2 pow k
==> m = word_neg(word(bitval(x < y))) /\
(... |
common.misc.MASK_AND_VALUE_FROM_CARRY_REAL_LT | bit_vector | `!(m:int64) l k (x:real).
(--(&2 pow k) <= x /\ x < &2 pow k) /\
(&0 <= l /\ l < &2 pow k) /\
integer(((&2 pow k * &(val m) + l) - x) / &2 pow (k + 64))
==> m = word_neg(word(bitval(x < &0))) /\
l = if x < &0 then x + &2 pow k else x` |
common.misc.MASK_AND_VALUE_FROM_CARRY_LT | bit_vector | `!(m:int64) l k x y.
(--(&2 pow k):real <= &x - &y /\ &x - &y:real < &2 pow k) /\
(&0 <= l /\ l < &2 pow k) /\
integer(((&2 pow k * &(val m) + l) - (&x - &y)) / &2 pow (k + 64))
==> m = word_neg(word(bitval(x < y))) /\
l = if x < y then (&x - &y) + &2 pow k else &x - &y` |
common.misc.WORD32_ADD_SUB_OF_LT | bit_vector | `!pc tgt. pc <= 2 EXP 31 /\ tgt < 2 EXP 31 ==>
word_add (word pc) (word_sx (iword (&tgt - &pc):int32)):int64 = word tgt` |
common.misc.WORD_MULTIPLICATION_LOHI | bit_vector | `!(x:N word) (y:N word).
word_mul x y = word_zx(word(val x * val y):(N tybit0)word) /\
word((val x * val y) DIV 2 EXP dimindex(:N)):N word =
word_zx (word_ushr (word(val x * val y):(N tybit0)word)
(dimindex(:N)))` |
common.misc.WORD_MULTIPLICATION_LOHI_64 | bit_vector | `!(x:int64) (y:int64).
word_mul x y = word_zx(word(val x * val y):int128) /\
word((val x * val y) DIV 2 EXP 64):int64 =
word_zx (word_ushr (word(val x * val y):int128) 64)` |
common.misc.WORD_DEINTERLEAVE_CLAUSES | bit_vector | `(word_deinterleave:num->num->M word->(N word)list) 2 n z =
[word_subdeinterleave 2 n z 0; word_subdeinterleave 2 n z 1] /\
(word_deinterleave:num->num->M word->(N word)list) 3 n z =
[word_subdeinterleave 3 n z 0;
word_subdeinterleave 3 n z 1;
word_subdeinterleave 3 n z 2]` |
common.misc.WORD_OF_BITS_AS_WORD_ALT | bit_vector | `!s. word_of_bits s : N word =
word(nsum (0..dimindex (:N) - 1) (\i. 2 EXP i * bitval(i IN s)))` |
common.misc.WORD_CONDENSE_ALT | bit_vector | `!x y:N word.
word_condense x y =
word(nsum (0..dimindex(:N)-1)
(\i. 2 EXP word_popcount(word_and y (word(2 EXP i - 1))) *
bitval(bit i (word_and x y))))` |
common.misc.NUM_CONDENSE_0 | generic | `(!m. num_condense m 0 = 0) /\
(!n. num_condense 0 n = 0)` |
common.misc.NUM_CONDENSE_CLAUSES | generic | `(!m n. num_condense (2 * n) (2 * m) = num_condense n m) /\
(!m n. num_condense (2 * n + 1) (2 * m) = num_condense n m) /\
(!m n. num_condense (2 * n) (2 * m + 1) = 2 * num_condense n m) /\
(!m n. num_condense (2 * n + 1) (2 * m + 1) = 2 * num_condense n m + 1)` |
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