Theory "real"

Signature

Definitions

real_of_num

|- (0 = real_0) ∧ ∀n. &SUC n = &n + real_1

real_sub

|- ∀x y. x − y = x + -y

real_lte

|- ∀x y. x ≤ y ⇔ ¬(y < x)

real_gt

|- ∀x y. x > y ⇔ y < x

real_ge

|- ∀x y. x ≥ y ⇔ y ≤ x

real_div

|- ∀x y. x / y = x * inv y

abs

|- ∀x. abs x = if 0 ≤ x then x else -x

pow

|- (∀x. x pow 0 = 1) ∧ ∀x n. x pow SUC n = x * x pow n

sup

|- ∀P. sup P = @s. ∀y. (∃x. P x ∧ y < x) ⇔ y < s

sum_tupled_primitive

|- sum_tupled =
   WFREC (@R. WF R ∧ ∀f m n. R ((n,m),f) ((n,SUC m),f))
     (λsum_tupled a.
        case a of
          ((n,0),f) => I 0
        | ((n,SUC m),f) => I (sum_tupled ((n,m),f) + f (n + m)))

sum_curried

|- ∀x x1. sum x x1 = sum_tupled (x,x1)

pos_def

|- ∀x. pos x = if 0 ≤ x then x else 0

min_def

|- ∀x y. min x y = if x ≤ y then x else y

max_def

|- ∀x y. max x y = if x ≤ y then y else x

inf_def

|- ∀p. inf p = -sup (λr. p (-r))

NUM_FLOOR_def

|- ∀x. flr x = LEAST n. &(n + 1) > x

NUM_CEILING_def

|- ∀x. clg x = LEAST n. x ≤ &n

Theorems

REAL_0

|- real_0 = 0

REAL_1

|- real_1 = 1

REAL_10

|- 1 ≠ 0

REAL_ADD_SYM

|- ∀x y. x + y = y + x

REAL_ADD_COMM

|- ∀x y. x + y = y + x

REAL_ADD_ASSOC

|- ∀x y z. x + (y + z) = x + y + z

REAL_ADD_LID

|- ∀x. 0 + x = x

REAL_ADD_LINV

|- ∀x. -x + x = 0

REAL_LDISTRIB

|- ∀x y z. x * (y + z) = x * y + x * z

REAL_LT_TOTAL

|- ∀x y. (x = y) ∨ x < y ∨ y < x

REAL_LT_REFL

|- ∀x. ¬(x < x)

REAL_LT_TRANS

|- ∀x y z. x < y ∧ y < z ⇒ x < z

REAL_LT_IADD

|- ∀x y z. y < z ⇒ x + y < x + z

REAL_SUP_ALLPOS

|- ∀P.
     (∀x. P x ⇒ 0 < x) ∧ (∃x. P x) ∧ (∃z. ∀x. P x ⇒ x < z) ⇒
     ∃s. ∀y. (∃x. P x ∧ y < x) ⇔ y < s

REAL_MUL_SYM

|- ∀x y. x * y = y * x

REAL_MUL_COMM

|- ∀x y. x * y = y * x

REAL_MUL_ASSOC

|- ∀x y z. x * (y * z) = x * y * z

REAL_MUL_LID

|- ∀x. 1 * x = x

REAL_MUL_LINV

|- ∀x. x ≠ 0 ⇒ (inv x * x = 1)

REAL_LT_MUL

|- ∀x y. 0 < x ∧ 0 < y ⇒ 0 < x * y

REAL_INV_0

|- inv 0 = 0

REAL_ADD_RID

|- ∀x. x + 0 = x

REAL_ADD_RINV

|- ∀x. x + -x = 0

REAL_MUL_RID

|- ∀x. x * 1 = x

REAL_MUL_RINV

|- ∀x. x ≠ 0 ⇒ (x * inv x = 1)

REAL_RDISTRIB

|- ∀x y z. (x + y) * z = x * z + y * z

REAL_EQ_LADD

|- ∀x y z. (x + y = x + z) ⇔ (y = z)

REAL_EQ_RADD

|- ∀x y z. (x + z = y + z) ⇔ (x = y)

REAL_ADD_LID_UNIQ

|- ∀x y. (x + y = y) ⇔ (x = 0)

REAL_ADD_RID_UNIQ

|- ∀x y. (x + y = x) ⇔ (y = 0)

REAL_LNEG_UNIQ

|- ∀x y. (x + y = 0) ⇔ (x = -y)

REAL_RNEG_UNIQ

|- ∀x y. (x + y = 0) ⇔ (y = -x)

REAL_NEG_ADD

|- ∀x y. -(x + y) = -x + -y

REAL_MUL_LZERO

|- ∀x. 0 * x = 0

REAL_MUL_RZERO

|- ∀x. x * 0 = 0

REAL_NEG_LMUL

|- ∀x y. -(x * y) = -x * y

REAL_NEG_RMUL

|- ∀x y. -(x * y) = x * -y

REAL_NEGNEG

|- ∀x. - -x = x

REAL_NEG_MUL2

|- ∀x y. -x * -y = x * y

REAL_ENTIRE

|- ∀x y. (x * y = 0) ⇔ (x = 0) ∨ (y = 0)

REAL_LT_LADD

|- ∀x y z. x + y < x + z ⇔ y < z

REAL_LT_RADD

|- ∀x y z. x + z < y + z ⇔ x < y

REAL_NOT_LT

|- ∀x y. ¬(x < y) ⇔ y ≤ x

REAL_LT_ANTISYM

|- ∀x y. ¬(x < y ∧ y < x)

REAL_LT_GT

|- ∀x y. x < y ⇒ ¬(y < x)

REAL_NOT_LE

|- ∀x y. ¬(x ≤ y) ⇔ y < x

REAL_LE_TOTAL

|- ∀x y. x ≤ y ∨ y ≤ x

REAL_LET_TOTAL

|- ∀x y. x ≤ y ∨ y < x

REAL_LTE_TOTAL

|- ∀x y. x < y ∨ y ≤ x

REAL_LE_REFL

|- ∀x. x ≤ x

REAL_LE_LT

|- ∀x y. x ≤ y ⇔ x < y ∨ (x = y)

REAL_LT_LE

|- ∀x y. x < y ⇔ x ≤ y ∧ x ≠ y

REAL_LT_IMP_LE

|- ∀x y. x < y ⇒ x ≤ y

REAL_LTE_TRANS

|- ∀x y z. x < y ∧ y ≤ z ⇒ x < z

REAL_LET_TRANS

|- ∀x y z. x ≤ y ∧ y < z ⇒ x < z

REAL_LE_TRANS

|- ∀x y z. x ≤ y ∧ y ≤ z ⇒ x ≤ z

REAL_LE_ANTISYM

|- ∀x y. x ≤ y ∧ y ≤ x ⇔ (x = y)

REAL_LET_ANTISYM

|- ∀x y. ¬(x < y ∧ y ≤ x)

REAL_LTE_ANTSYM

|- ∀x y. ¬(x ≤ y ∧ y < x)

REAL_NEG_LT0

|- ∀x. -x < 0 ⇔ 0 < x

REAL_NEG_GT0

|- ∀x. 0 < -x ⇔ x < 0

REAL_NEG_LE0

|- ∀x. -x ≤ 0 ⇔ 0 ≤ x

REAL_NEG_GE0

|- ∀x. 0 ≤ -x ⇔ x ≤ 0

REAL_LT_NEGTOTAL

|- ∀x. (x = 0) ∨ 0 < x ∨ 0 < -x

REAL_LE_NEGTOTAL

|- ∀x. 0 ≤ x ∨ 0 ≤ -x

REAL_LE_MUL

|- ∀x y. 0 ≤ x ∧ 0 ≤ y ⇒ 0 ≤ x * y

REAL_LE_SQUARE

|- ∀x. 0 ≤ x * x

REAL_LE_01

|- 0 ≤ 1

REAL_LT_01

|- 0 < 1

REAL_LE_LADD

|- ∀x y z. x + y ≤ x + z ⇔ y ≤ z

REAL_LE_RADD

|- ∀x y z. x + z ≤ y + z ⇔ x ≤ y

REAL_LT_ADD2

|- ∀w x y z. w < x ∧ y < z ⇒ w + y < x + z

REAL_LE_ADD2

|- ∀w x y z. w ≤ x ∧ y ≤ z ⇒ w + y ≤ x + z

REAL_LE_ADD

|- ∀x y. 0 ≤ x ∧ 0 ≤ y ⇒ 0 ≤ x + y

REAL_LT_ADD

|- ∀x y. 0 < x ∧ 0 < y ⇒ 0 < x + y

REAL_LT_ADDNEG

|- ∀x y z. y < x + -z ⇔ y + z < x

REAL_LT_ADDNEG2

|- ∀x y z. x + -y < z ⇔ x < z + y

REAL_LT_ADD1

|- ∀x y. x ≤ y ⇒ x < y + 1

REAL_SUB_ADD

|- ∀x y. x − y + y = x

REAL_SUB_ADD2

|- ∀x y. y + (x − y) = x

REAL_SUB_REFL

|- ∀x. x − x = 0

REAL_SUB_0

|- ∀x y. (x − y = 0) ⇔ (x = y)

REAL_LE_DOUBLE

|- ∀x. 0 ≤ x + x ⇔ 0 ≤ x

REAL_LE_NEGL

|- ∀x. -x ≤ x ⇔ 0 ≤ x

REAL_LE_NEGR

|- ∀x. x ≤ -x ⇔ x ≤ 0

REAL_NEG_EQ0

|- ∀x. (-x = 0) ⇔ (x = 0)

REAL_NEG_0

|- -0 = 0

REAL_NEG_SUB

|- ∀x y. -(x − y) = y − x

REAL_SUB_LT

|- ∀x y. 0 < x − y ⇔ y < x

REAL_SUB_LE

|- ∀x y. 0 ≤ x − y ⇔ y ≤ x

REAL_ADD_SUB

|- ∀x y. x + y − x = y

REAL_EQ_LMUL

|- ∀x y z. (x * y = x * z) ⇔ (x = 0) ∨ (y = z)

REAL_EQ_RMUL

|- ∀x y z. (x * z = y * z) ⇔ (z = 0) ∨ (x = y)

REAL_SUB_LDISTRIB

|- ∀x y z. x * (y − z) = x * y − x * z

REAL_SUB_RDISTRIB

|- ∀x y z. (x − y) * z = x * z − y * z

REAL_NEG_EQ

|- ∀x y. (-x = y) ⇔ (x = -y)

REAL_NEG_MINUS1

|- ∀x. -x = -1 * x

REAL_INV_NZ

|- ∀x. x ≠ 0 ⇒ inv x ≠ 0

REAL_INVINV

|- ∀x. x ≠ 0 ⇒ (inv (inv x) = x)

REAL_LT_IMP_NE

|- ∀x y. x < y ⇒ x ≠ y

REAL_INV_POS

|- ∀x. 0 < x ⇒ 0 < inv x

REAL_LT_LMUL_0

|- ∀x y. 0 < x ⇒ (0 < x * y ⇔ 0 < y)

REAL_LT_RMUL_0

|- ∀x y. 0 < y ⇒ (0 < x * y ⇔ 0 < x)

REAL_LT_LMUL

|- ∀x y z. 0 < x ⇒ (x * y < x * z ⇔ y < z)

REAL_LT_RMUL

|- ∀x y z. 0 < z ⇒ (x * z < y * z ⇔ x < y)

REAL_LT_RMUL_IMP

|- ∀x y z. x < y ∧ 0 < z ⇒ x * z < y * z

REAL_LT_LMUL_IMP

|- ∀x y z. y < z ∧ 0 < x ⇒ x * y < x * z

REAL_LINV_UNIQ

|- ∀x y. (x * y = 1) ⇒ (x = inv y)

REAL_RINV_UNIQ

|- ∀x y. (x * y = 1) ⇒ (y = inv x)

REAL_INV_INV

|- ∀x. inv (inv x) = x

REAL_INV_EQ_0

|- ∀x. (inv x = 0) ⇔ (x = 0)

REAL_NEG_INV

|- ∀x. x ≠ 0 ⇒ (-inv x = inv (-x))

REAL_INV_1OVER

|- ∀x. inv x = 1 / x

REAL_LT_INV_EQ

|- ∀x. 0 < inv x ⇔ 0 < x

REAL_LE_INV_EQ

|- ∀x. 0 ≤ inv x ⇔ 0 ≤ x

REAL_LE_INV

|- ∀x. 0 ≤ x ⇒ 0 ≤ inv x

REAL_LE_ADDR

|- ∀x y. x ≤ x + y ⇔ 0 ≤ y

REAL_LE_ADDL

|- ∀x y. y ≤ x + y ⇔ 0 ≤ x

REAL_LT_ADDR

|- ∀x y. x < x + y ⇔ 0 < y

REAL_LT_ADDL

|- ∀x y. y < x + y ⇔ 0 < x

REAL

|- ∀n. &SUC n = &n + 1

REAL_POS

|- ∀n. 0 ≤ &n

REAL_LE

|- ∀m n. &m ≤ &n ⇔ m ≤ n

REAL_LT

|- ∀m n. &m < &n ⇔ m < n

REAL_INJ

|- ∀m n. (&m = &n) ⇔ (m = n)

REAL_ADD

|- ∀m n. &m + &n = &(m + n)

REAL_MUL

|- ∀m n. &m * &n = &(m * n)

REAL_INV1

|- inv 1 = 1

REAL_OVER1

|- ∀x. x / 1 = x

REAL_DIV_REFL

|- ∀x. x ≠ 0 ⇒ (x / x = 1)

REAL_DIV_LZERO

|- ∀x. 0 / x = 0

REAL_LT_NZ

|- ∀n. &n ≠ 0 ⇔ 0 < &n

REAL_NZ_IMP_LT

|- ∀n. n ≠ 0 ⇒ 0 < &n

REAL_LT_RDIV_0

|- ∀y z. 0 < z ⇒ (0 < y / z ⇔ 0 < y)

REAL_LT_RDIV

|- ∀x y z. 0 < z ⇒ (x / z < y / z ⇔ x < y)

REAL_LT_FRACTION_0

|- ∀n d. n ≠ 0 ⇒ (0 < d / &n ⇔ 0 < d)

REAL_LT_MULTIPLE

|- ∀n d. 1 < n ⇒ (d < &n * d ⇔ 0 < d)

REAL_LT_FRACTION

|- ∀n d. 1 < n ⇒ (d / &n < d ⇔ 0 < d)

REAL_LT_HALF1

|- ∀d. 0 < d / 2 ⇔ 0 < d

REAL_LT_HALF2

|- ∀d. d / 2 < d ⇔ 0 < d

REAL_DOUBLE

|- ∀x. x + x = 2 * x

REAL_DIV_LMUL

|- ∀x y. y ≠ 0 ⇒ (y * (x / y) = x)

REAL_DIV_RMUL

|- ∀x y. y ≠ 0 ⇒ (x / y * y = x)

REAL_HALF_DOUBLE

|- ∀x. x / 2 + x / 2 = x

REAL_DOWN

|- ∀x. 0 < x ⇒ ∃y. 0 < y ∧ y < x

REAL_DOWN2

|- ∀x y. 0 < x ∧ 0 < y ⇒ ∃z. 0 < z ∧ z < x ∧ z < y

REAL_SUB_SUB

|- ∀x y. x − y − x = -y

REAL_LT_ADD_SUB

|- ∀x y z. x + y < z ⇔ x < z − y

REAL_LT_SUB_RADD

|- ∀x y z. x − y < z ⇔ x < z + y

REAL_LT_SUB_LADD

|- ∀x y z. x < y − z ⇔ x + z < y

REAL_LE_SUB_LADD

|- ∀x y z. x ≤ y − z ⇔ x + z ≤ y

REAL_LE_SUB_RADD

|- ∀x y z. x − y ≤ z ⇔ x ≤ z + y

REAL_LT_NEG

|- ∀x y. -x < -y ⇔ y < x

REAL_LE_NEG

|- ∀x y. -x ≤ -y ⇔ y ≤ x

REAL_ADD2_SUB2

|- ∀a b c d. a + b − (c + d) = a − c + (b − d)

REAL_SUB_LZERO

|- ∀x. 0 − x = -x

REAL_SUB_RZERO

|- ∀x. x − 0 = x

REAL_LET_ADD2

|- ∀w x y z. w ≤ x ∧ y < z ⇒ w + y < x + z

REAL_LTE_ADD2

|- ∀w x y z. w < x ∧ y ≤ z ⇒ w + y < x + z

REAL_LET_ADD

|- ∀x y. 0 ≤ x ∧ 0 < y ⇒ 0 < x + y

REAL_LTE_ADD

|- ∀x y. 0 < x ∧ 0 ≤ y ⇒ 0 < x + y

REAL_LT_MUL2

|- ∀x1 x2 y1 y2. 0 ≤ x1 ∧ 0 ≤ y1 ∧ x1 < x2 ∧ y1 < y2 ⇒ x1 * y1 < x2 * y2

REAL_LT_INV

|- ∀x y. 0 < x ∧ x < y ⇒ inv y < inv x

REAL_SUB_LNEG

|- ∀x y. -x − y = -(x + y)

REAL_SUB_RNEG

|- ∀x y. x − -y = x + y

REAL_SUB_NEG2

|- ∀x y. -x − -y = y − x

REAL_SUB_TRIANGLE

|- ∀a b c. a − b + (b − c) = a − c

REAL_EQ_SUB_LADD

|- ∀x y z. (x = y − z) ⇔ (x + z = y)

REAL_EQ_SUB_RADD

|- ∀x y z. (x − y = z) ⇔ (x = z + y)

REAL_INV_MUL

|- ∀x y. x ≠ 0 ∧ y ≠ 0 ⇒ (inv (x * y) = inv x * inv y)

REAL_LE_LMUL

|- ∀x y z. 0 < x ⇒ (x * y ≤ x * z ⇔ y ≤ z)

REAL_LE_RMUL

|- ∀x y z. 0 < z ⇒ (x * z ≤ y * z ⇔ x ≤ y)

REAL_SUB_INV2

|- ∀x y. x ≠ 0 ∧ y ≠ 0 ⇒ (inv x − inv y = (y − x) / (x * y))

REAL_SUB_SUB2

|- ∀x y. x − (x − y) = y

REAL_ADD_SUB2

|- ∀x y. x − (x + y) = -y

REAL_MEAN

|- ∀x y. x < y ⇒ ∃z. x < z ∧ z < y

REAL_EQ_LMUL2

|- ∀x y z. x ≠ 0 ⇒ ((y = z) ⇔ (x * y = x * z))

REAL_LE_MUL2

|- ∀x1 x2 y1 y2. 0 ≤ x1 ∧ 0 ≤ y1 ∧ x1 ≤ x2 ∧ y1 ≤ y2 ⇒ x1 * y1 ≤ x2 * y2

REAL_LE_LDIV

|- ∀x y z. 0 < x ∧ y ≤ z * x ⇒ y / x ≤ z

REAL_LE_RDIV

|- ∀x y z. 0 < x ∧ y * x ≤ z ⇒ y ≤ z / x

REAL_LT_DIV

|- ∀x y. 0 < x ∧ 0 < y ⇒ 0 < x / y

REAL_LE_DIV

|- ∀x y. 0 ≤ x ∧ 0 ≤ y ⇒ 0 ≤ x / y

REAL_LT_1

|- ∀x y. 0 ≤ x ∧ x < y ⇒ x / y < 1

REAL_LE_LMUL_IMP

|- ∀x y z. 0 ≤ x ∧ y ≤ z ⇒ x * y ≤ x * z

REAL_LE_RMUL_IMP

|- ∀x y z. 0 ≤ x ∧ y ≤ z ⇒ y * x ≤ z * x

REAL_EQ_IMP_LE

|- ∀x y. (x = y) ⇒ x ≤ y

REAL_INV_LT1

|- ∀x. 0 < x ∧ x < 1 ⇒ 1 < inv x

REAL_POS_NZ

|- ∀x. 0 < x ⇒ x ≠ 0

REAL_EQ_RMUL_IMP

|- ∀x y z. z ≠ 0 ∧ (x * z = y * z) ⇒ (x = y)

REAL_EQ_LMUL_IMP

|- ∀x y z. x ≠ 0 ∧ (x * y = x * z) ⇒ (y = z)

REAL_FACT_NZ

|- ∀n. &FACT n ≠ 0

REAL_DIFFSQ

|- ∀x y. (x + y) * (x − y) = x * x − y * y

REAL_POASQ

|- ∀x. 0 < x * x ⇔ x ≠ 0

REAL_SUMSQ

|- ∀x y. (x * x + y * y = 0) ⇔ (x = 0) ∧ (y = 0)

REAL_EQ_NEG

|- ∀x y. (-x = -y) ⇔ (x = y)

REAL_DIV_MUL2

|- ∀x z. x ≠ 0 ∧ z ≠ 0 ⇒ ∀y. y / z = x * y / (x * z)

REAL_MIDDLE1

|- ∀a b. a ≤ b ⇒ a ≤ (a + b) / 2

REAL_MIDDLE2

|- ∀a b. a ≤ b ⇒ (a + b) / 2 ≤ b

ABS_ZERO

|- ∀x. (abs x = 0) ⇔ (x = 0)

ABS_0

|- abs 0 = 0

ABS_1

|- abs 1 = 1

ABS_NEG

|- ∀x. abs (-x) = abs x

ABS_TRIANGLE

|- ∀x y. abs (x + y) ≤ abs x + abs y

ABS_POS

|- ∀x. 0 ≤ abs x

ABS_MUL

|- ∀x y. abs (x * y) = abs x * abs y

ABS_LT_MUL2

|- ∀w x y z. abs w < y ∧ abs x < z ⇒ abs (w * x) < y * z

ABS_SUB

|- ∀x y. abs (x − y) = abs (y − x)

ABS_NZ

|- ∀x. x ≠ 0 ⇔ 0 < abs x

ABS_INV

|- ∀x. x ≠ 0 ⇒ (abs (inv x) = inv (abs x))

ABS_ABS

|- ∀x. abs (abs x) = abs x

ABS_LE

|- ∀x. x ≤ abs x

ABS_REFL

|- ∀x. (abs x = x) ⇔ 0 ≤ x

ABS_N

|- ∀n. abs (&n) = &n

ABS_BETWEEN

|- ∀x y d. 0 < d ∧ x − d < y ∧ y < x + d ⇔ abs (y − x) < d

ABS_BOUND

|- ∀x y d. abs (x − y) < d ⇒ y < x + d

ABS_STILLNZ

|- ∀x y. abs (x − y) < abs y ⇒ x ≠ 0

ABS_CASES

|- ∀x. (x = 0) ∨ 0 < abs x

ABS_BETWEEN1

|- ∀x y z. x < z ∧ abs (y − x) < z − x ⇒ y < z

ABS_SIGN

|- ∀x y. abs (x − y) < y ⇒ 0 < x

ABS_SIGN2

|- ∀x y. abs (x − y) < -y ⇒ x < 0

ABS_DIV

|- ∀y. y ≠ 0 ⇒ ∀x. abs (x / y) = abs x / abs y

ABS_CIRCLE

|- ∀x y h. abs h < abs y − abs x ⇒ abs (x + h) < abs y

REAL_SUB_ABS

|- ∀x y. abs x − abs y ≤ abs (x − y)

ABS_SUB_ABS

|- ∀x y. abs (abs x − abs y) ≤ abs (x − y)

ABS_BETWEEN2

|- ∀x0 x y0 y.
     x0 < y0 ∧ abs (x − x0) < (y0 − x0) / 2 ∧ abs (y − y0) < (y0 − x0) / 2 ⇒
     x < y

ABS_BOUNDS

|- ∀x k. abs x ≤ k ⇔ -k ≤ x ∧ x ≤ k

POW_0

|- ∀n. 0 pow SUC n = 0

POW_NZ

|- ∀c n. c ≠ 0 ⇒ c pow n ≠ 0

POW_INV

|- ∀c. c ≠ 0 ⇒ ∀n. inv (c pow n) = inv c pow n

POW_ABS

|- ∀c n. abs c pow n = abs (c pow n)

POW_PLUS1

|- ∀e. 0 < e ⇒ ∀n. 1 + &n * e ≤ (1 + e) pow n

POW_ADD

|- ∀c m n. c pow (m + n) = c pow m * c pow n

POW_1

|- ∀x. x pow 1 = x

POW_2

|- ∀x. x pow 2 = x * x

POW_ONE

|- ∀n. 1 pow n = 1

POW_POS

|- ∀x. 0 ≤ x ⇒ ∀n. 0 ≤ x pow n

POW_LE

|- ∀n x y. 0 ≤ x ∧ x ≤ y ⇒ x pow n ≤ y pow n

POW_M1

|- ∀n. abs (-1 pow n) = 1

POW_MUL

|- ∀n x y. (x * y) pow n = x pow n * y pow n

REAL_LE_POW2

|- ∀x. 0 ≤ x pow 2

ABS_POW2

|- ∀x. abs (x pow 2) = x pow 2

REAL_POW2_ABS

|- ∀x. abs x pow 2 = x pow 2

REAL_LE1_POW2

|- ∀x. 1 ≤ x ⇒ 1 ≤ x pow 2

REAL_LT1_POW2

|- ∀x. 1 < x ⇒ 1 < x pow 2

POW_POS_LT

|- ∀x n. 0 < x ⇒ 0 < x pow SUC n

POW_2_LE1

|- ∀n. 1 ≤ 2 pow n

POW_2_LT

|- ∀n. &n < 2 pow n

POW_MINUS1

|- ∀n. -1 pow (2 * n) = 1

POW_LT

|- ∀n x y. 0 ≤ x ∧ x < y ⇒ x pow SUC n < y pow SUC n

REAL_POW_LT

|- ∀x n. 0 < x ⇒ 0 < x pow n

POW_EQ

|- ∀n x y. 0 ≤ x ∧ 0 ≤ y ∧ (x pow SUC n = y pow SUC n) ⇒ (x = y)

POW_ZERO

|- ∀n x. (x pow n = 0) ⇒ (x = 0)

POW_ZERO_EQ

|- ∀n x. (x pow SUC n = 0) ⇔ (x = 0)

REAL_POW_LT2

|- ∀n x y. n ≠ 0 ∧ 0 ≤ x ∧ x < y ⇒ x pow n < y pow n

REAL_POW_MONO_LT

|- ∀m n x. 1 < x ∧ m < n ⇒ x pow m < x pow n

REAL_POW_POW

|- ∀x m n. (x pow m) pow n = x pow (m * n)

REAL_SUP_SOMEPOS

|- ∀P.
     (∃x. P x ∧ 0 < x) ∧ (∃z. ∀x. P x ⇒ x < z) ⇒
     ∃s. ∀y. (∃x. P x ∧ y < x) ⇔ y < s

SUP_LEMMA1

|- ∀P s d.
     (∀y. (∃x. (λx. P (x + d)) x ∧ y < x) ⇔ y < s) ⇒
     ∀y. (∃x. P x ∧ y < x) ⇔ y < s + d

SUP_LEMMA2

|- ∀P. (∃x. P x) ⇒ ∃d x. (λx. P (x + d)) x ∧ 0 < x

SUP_LEMMA3

|- ∀d. (∃z. ∀x. P x ⇒ x < z) ⇒ ∃z. ∀x. (λx. P (x + d)) x ⇒ x < z

REAL_SUP_EXISTS

|- ∀P. (∃x. P x) ∧ (∃z. ∀x. P x ⇒ x < z) ⇒ ∃s. ∀y. (∃x. P x ∧ y < x) ⇔ y < s

REAL_SUP

|- ∀P. (∃x. P x) ∧ (∃z. ∀x. P x ⇒ x < z) ⇒ ∀y. (∃x. P x ∧ y < x) ⇔ y < sup P

REAL_SUP_UBOUND

|- ∀P. (∃x. P x) ∧ (∃z. ∀x. P x ⇒ x < z) ⇒ ∀y. P y ⇒ y ≤ sup P

SETOK_LE_LT

|- ∀P. (∃x. P x) ∧ (∃z. ∀x. P x ⇒ x ≤ z) ⇔ (∃x. P x) ∧ ∃z. ∀x. P x ⇒ x < z

REAL_SUP_LE

|- ∀P. (∃x. P x) ∧ (∃z. ∀x. P x ⇒ x ≤ z) ⇒ ∀y. (∃x. P x ∧ y < x) ⇔ y < sup P

REAL_SUP_UBOUND_LE

|- ∀P. (∃x. P x) ∧ (∃z. ∀x. P x ⇒ x ≤ z) ⇒ ∀y. P y ⇒ y ≤ sup P

REAL_ARCH

|- ∀x. 0 < x ⇒ ∀y. ∃n. y < &n * x

REAL_ARCH_LEAST

|- ∀y. 0 < y ⇒ ∀x. 0 ≤ x ⇒ ∃n. &n * y ≤ x ∧ x < &SUC n * y

sum_ind

|- ∀P.
     (∀n f. P (n,0) f) ∧ (∀n m f. P (n,m) f ⇒ P (n,SUC m) f) ⇒
     ∀v v1 v2. P (v,v1) v2

sum

|- (∀n f. sum (n,0) f = 0) ∧ ∀n m f. sum (n,SUC m) f = sum (n,m) f + f (n + m)

sum_compute

|- (∀n f. sum (n,0) f = 0) ∧
   (∀n m f.
      sum (n,NUMERAL (BIT1 m)) f =
      sum (n,NUMERAL (BIT1 m) − 1) f + f (n + (NUMERAL (BIT1 m) − 1))) ∧
   ∀n m f.
     sum (n,NUMERAL (BIT2 m)) f =
     sum (n,NUMERAL (BIT1 m)) f + f (n + NUMERAL (BIT1 m))

SUM_TWO

|- ∀f n p. sum (0,n) f + sum (n,p) f = sum (0,n + p) f

SUM_DIFF

|- ∀f m n. sum (m,n) f = sum (0,m + n) f − sum (0,m) f

ABS_SUM

|- ∀f m n. abs (sum (m,n) f) ≤ sum (m,n) (λn. abs (f n))

SUM_LE

|- ∀f g m n. (∀r. m ≤ r ∧ r < n + m ⇒ f r ≤ g r) ⇒ sum (m,n) f ≤ sum (m,n) g

SUM_EQ

|- ∀f g m n.
     (∀r. m ≤ r ∧ r < n + m ⇒ (f r = g r)) ⇒ (sum (m,n) f = sum (m,n) g)

SUM_POS

|- ∀f. (∀n. 0 ≤ f n) ⇒ ∀m n. 0 ≤ sum (m,n) f

SUM_POS_GEN

|- ∀f m. (∀n. m ≤ n ⇒ 0 ≤ f n) ⇒ ∀n. 0 ≤ sum (m,n) f

SUM_ABS

|- ∀f m n. abs (sum (m,n) (λm. abs (f m))) = sum (m,n) (λm. abs (f m))

SUM_ABS_LE

|- ∀f m n. abs (sum (m,n) f) ≤ sum (m,n) (λn. abs (f n))

SUM_ZERO

|- ∀f N. (∀n. n ≥ N ⇒ (f n = 0)) ⇒ ∀m n. m ≥ N ⇒ (sum (m,n) f = 0)

SUM_ADD

|- ∀f g m n. sum (m,n) (λn. f n + g n) = sum (m,n) f + sum (m,n) g

SUM_CMUL

|- ∀f c m n. sum (m,n) (λn. c * f n) = c * sum (m,n) f

SUM_NEG

|- ∀f n d. sum (n,d) (λn. -f n) = -sum (n,d) f

SUM_SUB

|- ∀f g m n. sum (m,n) (λn. f n − g n) = sum (m,n) f − sum (m,n) g

SUM_SUBST

|- ∀f g m n.
     (∀p. m ≤ p ∧ p < m + n ⇒ (f p = g p)) ⇒ (sum (m,n) f = sum (m,n) g)

SUM_NSUB

|- ∀n f c. sum (0,n) f − &n * c = sum (0,n) (λp. f p − c)

SUM_BOUND

|- ∀f k m n. (∀p. m ≤ p ∧ p < m + n ⇒ f p ≤ k) ⇒ sum (m,n) f ≤ &n * k

SUM_GROUP

|- ∀n k f. sum (0,n) (λm. sum (m * k,k) f) = sum (0,n * k) f

SUM_1

|- ∀f n. sum (n,1) f = f n

SUM_2

|- ∀f n. sum (n,2) f = f n + f (n + 1)

SUM_OFFSET

|- ∀f n k. sum (0,n) (λm. f (m + k)) = sum (0,n + k) f − sum (0,k) f

SUM_REINDEX

|- ∀f m k n. sum (m + k,n) f = sum (m,n) (λr. f (r + k))

SUM_0

|- ∀m n. sum (m,n) (λr. 0) = 0

SUM_PERMUTE_0

|- ∀n p.
     (∀y. y < n ⇒ ∃!x. x < n ∧ (p x = y)) ⇒
     ∀f. sum (0,n) (λn. f (p n)) = sum (0,n) f

SUM_CANCEL

|- ∀f n d. sum (n,d) (λn. f (SUC n) − f n) = f (n + d) − f n

REAL_MUL_RNEG

|- ∀x y. x * -y = -(x * y)

REAL_MUL_LNEG

|- ∀x y. -x * y = -(x * y)

real_lt

|- ∀y x. x < y ⇔ ¬(y ≤ x)

REAL_LE_LADD_IMP

|- ∀x y z. y ≤ z ⇒ x + y ≤ x + z

REAL_LE_LNEG

|- ∀x y. -x ≤ y ⇔ 0 ≤ x + y

REAL_LE_NEG2

|- ∀x y. -x ≤ -y ⇔ y ≤ x

REAL_NEG_NEG

|- ∀x. - -x = x

REAL_LE_RNEG

|- ∀x y. x ≤ -y ⇔ x + y ≤ 0

REAL_POW_INV

|- ∀x n. inv x pow n = inv (x pow n)

REAL_POW_DIV

|- ∀x y n. (x / y) pow n = x pow n / y pow n

REAL_POW_ADD

|- ∀x m n. x pow (m + n) = x pow m * x pow n

REAL_LE_RDIV_EQ

|- ∀x y z. 0 < z ⇒ (x ≤ y / z ⇔ x * z ≤ y)

REAL_LE_LDIV_EQ

|- ∀x y z. 0 < z ⇒ (x / z ≤ y ⇔ x ≤ y * z)

REAL_LT_RDIV_EQ

|- ∀x y z. 0 < z ⇒ (x < y / z ⇔ x * z < y)

REAL_LT_LDIV_EQ

|- ∀x y z. 0 < z ⇒ (x / z < y ⇔ x < y * z)

REAL_EQ_RDIV_EQ

|- ∀x y z. 0 < z ⇒ ((x = y / z) ⇔ (x * z = y))

REAL_EQ_LDIV_EQ

|- ∀x y z. 0 < z ⇒ ((x / z = y) ⇔ (x = y * z))

REAL_OF_NUM_POW

|- ∀x n. &x pow n = &(x ** n)

REAL_ADD_LDISTRIB

|- ∀x y z. x * (y + z) = x * y + x * z

REAL_ADD_RDISTRIB

|- ∀x y z. (x + y) * z = x * z + y * z

REAL_OF_NUM_ADD

|- ∀m n. &m + &n = &(m + n)

REAL_OF_NUM_LE

|- ∀m n. &m ≤ &n ⇔ m ≤ n

REAL_OF_NUM_MUL

|- ∀m n. &m * &n = &(m * n)

REAL_OF_NUM_SUC

|- ∀n. &n + 1 = &SUC n

REAL_OF_NUM_EQ

|- ∀m n. (&m = &n) ⇔ (m = n)

REAL_EQ_MUL_LCANCEL

|- ∀x y z. (x * y = x * z) ⇔ (x = 0) ∨ (y = z)

REAL_ABS_0

|- abs 0 = 0

REAL_ABS_TRIANGLE

|- ∀x y. abs (x + y) ≤ abs x + abs y

REAL_ABS_MUL

|- ∀x y. abs (x * y) = abs x * abs y

REAL_ABS_POS

|- ∀x. 0 ≤ abs x

REAL_LE_EPSILON

|- ∀x y. (∀e. 0 < e ⇒ x ≤ y + e) ⇒ x ≤ y

REAL_BIGNUM

|- ∀r. ∃n. r < &n

REAL_INV_LT_ANTIMONO

|- ∀x y. 0 < x ∧ 0 < y ⇒ (inv x < inv y ⇔ y < x)

REAL_INV_INJ

|- ∀x y. (inv x = inv y) ⇔ (x = y)

REAL_DIV_RMUL_CANCEL

|- ∀c a b. c ≠ 0 ⇒ (a * c / (b * c) = a / b)

REAL_DIV_LMUL_CANCEL

|- ∀c a b. c ≠ 0 ⇒ (c * a / (c * b) = a / b)

REAL_DIV_ADD

|- ∀x y z. y / x + z / x = (y + z) / x

REAL_ADD_RAT

|- ∀a b c d. b ≠ 0 ∧ d ≠ 0 ⇒ (a / b + c / d = (a * d + b * c) / (b * d))

REAL_SUB_RAT

|- ∀a b c d. b ≠ 0 ∧ d ≠ 0 ⇒ (a / b − c / d = (a * d − b * c) / (b * d))

REAL_SUB

|- ∀m n. &m − &n = if m − n = 0 then -&(n − m) else &(m − n)

REAL_POS_POS

|- ∀x. 0 ≤ pos x

REAL_POS_ID

|- ∀x. 0 ≤ x ⇒ (pos x = x)

REAL_POS_INFLATE

|- ∀x. x ≤ pos x

REAL_POS_MONO

|- ∀x y. x ≤ y ⇒ pos x ≤ pos y

REAL_POS_EQ_ZERO

|- ∀x. (pos x = 0) ⇔ x ≤ 0

REAL_POS_LE_ZERO

|- ∀x. pos x ≤ 0 ⇔ x ≤ 0

REAL_MIN_REFL

|- ∀x. min x x = x

REAL_LE_MIN

|- ∀z x y. z ≤ min x y ⇔ z ≤ x ∧ z ≤ y

REAL_MIN_LE

|- ∀z x y. min x y ≤ z ⇔ x ≤ z ∨ y ≤ z

REAL_MIN_LE1

|- ∀x y. min x y ≤ x

REAL_MIN_LE2

|- ∀x y. min x y ≤ y

REAL_MIN_ALT

|- ∀x y. (x ≤ y ⇒ (min x y = x)) ∧ (y ≤ x ⇒ (min x y = y))

REAL_MIN_LE_LIN

|- ∀z x y. 0 ≤ z ∧ z ≤ 1 ⇒ min x y ≤ z * x + (1 − z) * y

REAL_MIN_ADD

|- ∀z x y. min (x + z) (y + z) = min x y + z

REAL_MIN_SUB

|- ∀z x y. min (x − z) (y − z) = min x y − z

REAL_IMP_MIN_LE2

|- ∀x1 x2 y1 y2. x1 ≤ y1 ∧ x2 ≤ y2 ⇒ min x1 x2 ≤ min y1 y2

REAL_MAX_REFL

|- ∀x. max x x = x

REAL_LE_MAX

|- ∀z x y. z ≤ max x y ⇔ z ≤ x ∨ z ≤ y

REAL_LE_MAX1

|- ∀x y. x ≤ max x y

REAL_LE_MAX2

|- ∀x y. y ≤ max x y

REAL_MAX_LE

|- ∀z x y. max x y ≤ z ⇔ x ≤ z ∧ y ≤ z

REAL_MAX_ALT

|- ∀x y. (x ≤ y ⇒ (max x y = y)) ∧ (y ≤ x ⇒ (max x y = x))

REAL_MAX_MIN

|- ∀x y. max x y = -min (-x) (-y)

REAL_MIN_MAX

|- ∀x y. min x y = -max (-x) (-y)

REAL_LIN_LE_MAX

|- ∀z x y. 0 ≤ z ∧ z ≤ 1 ⇒ z * x + (1 − z) * y ≤ max x y

REAL_MAX_ADD

|- ∀z x y. max (x + z) (y + z) = max x y + z

REAL_MAX_SUB

|- ∀z x y. max (x − z) (y − z) = max x y − z

REAL_IMP_MAX_LE2

|- ∀x1 x2 y1 y2. x1 ≤ y1 ∧ x2 ≤ y2 ⇒ max x1 x2 ≤ max y1 y2

REAL_SUP_EXISTS_UNIQUE

|- ∀p. (∃x. p x) ∧ (∃z. ∀x. p x ⇒ x ≤ z) ⇒ ∃!s. ∀y. (∃x. p x ∧ y < x) ⇔ y < s

REAL_SUP_MAX

|- ∀p z. p z ∧ (∀x. p x ⇒ x ≤ z) ⇒ (sup p = z)

REAL_IMP_SUP_LE

|- ∀p x. (∃r. p r) ∧ (∀r. p r ⇒ r ≤ x) ⇒ sup p ≤ x

REAL_IMP_LE_SUP

|- ∀p x. (∃r. p r) ∧ (∃z. ∀r. p r ⇒ r ≤ z) ∧ (∃r. p r ∧ x ≤ r) ⇒ x ≤ sup p

REAL_INF_MIN

|- ∀p z. p z ∧ (∀x. p x ⇒ z ≤ x) ⇒ (inf p = z)

REAL_IMP_LE_INF

|- ∀p r. (∃x. p x) ∧ (∀x. p x ⇒ r ≤ x) ⇒ r ≤ inf p

REAL_IMP_INF_LE

|- ∀p r. (∃z. ∀x. p x ⇒ z ≤ x) ∧ (∃x. p x ∧ x ≤ r) ⇒ inf p ≤ r

REAL_INF_LT

|- ∀p z. (∃x. p x) ∧ inf p < z ⇒ ∃x. p x ∧ x < z

REAL_INF_CLOSE

|- ∀p e. (∃x. p x) ∧ 0 < e ⇒ ∃x. p x ∧ x < inf p + e

SUP_EPSILON

|- ∀p e. 0 < e ∧ (∃x. p x) ∧ (∃z. ∀x. p x ⇒ x ≤ z) ⇒ ∃x. p x ∧ sup p ≤ x + e

REAL_LE_SUP

|- ∀p x.
     (∃y. p y) ∧ (∃y. ∀z. p z ⇒ z ≤ y) ⇒
     (x ≤ sup p ⇔ ∀y. (∀z. p z ⇒ z ≤ y) ⇒ x ≤ y)

REAL_INF_LE

|- ∀p x.
     (∃y. p y) ∧ (∃y. ∀z. p z ⇒ y ≤ z) ⇒
     (inf p ≤ x ⇔ ∀y. (∀z. p z ⇒ y ≤ z) ⇒ y ≤ x)

REAL_SUP_CONST

|- ∀x. sup (λr. r = x) = x

REAL_MUL_SUB2_CANCEL

|- ∀z x y. x * y + (z − x) * y = z * y

REAL_MUL_SUB1_CANCEL

|- ∀z x y. y * x + y * (z − x) = y * z

REAL_NEG_HALF

|- ∀x. x − x / 2 = x / 2

REAL_NEG_THIRD

|- ∀x. x − x / 3 = 2 * x / 3

REAL_DIV_DENOM_CANCEL

|- ∀x y z. x ≠ 0 ⇒ (y / x / (z / x) = y / z)

REAL_DIV_DENOM_CANCEL2

|- ∀y z. y / 2 / (z / 2) = y / z

REAL_DIV_DENOM_CANCEL3

|- ∀y z. y / 3 / (z / 3) = y / z

REAL_DIV_INNER_CANCEL

|- ∀x y z. x ≠ 0 ⇒ (y / x * (x / z) = y / z)

REAL_DIV_INNER_CANCEL2

|- ∀y z. y / 2 * (2 / z) = y / z

REAL_DIV_INNER_CANCEL3

|- ∀y z. y / 3 * (3 / z) = y / z

REAL_DIV_OUTER_CANCEL

|- ∀x y z. x ≠ 0 ⇒ (x / y * (z / x) = z / y)

REAL_DIV_OUTER_CANCEL2

|- ∀y z. 2 / y * (z / 2) = z / y

REAL_DIV_OUTER_CANCEL3

|- ∀y z. 3 / y * (z / 3) = z / y

REAL_DIV_REFL2

|- 2 / 2 = 1

REAL_DIV_REFL3

|- 3 / 3 = 1

REAL_HALF_BETWEEN

|- (0 < 1 / 2 ∧ 1 / 2 < 1) ∧ 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1

REAL_THIRDS_BETWEEN

|- (0 < 1 / 3 ∧ 1 / 3 < 1 ∧ 0 < 2 / 3 ∧ 2 / 3 < 1) ∧ 0 ≤ 1 / 3 ∧ 1 / 3 ≤ 1 ∧
   0 ≤ 2 / 3 ∧ 2 / 3 ≤ 1

REAL_LE_SUB_CANCEL2

|- ∀x y z. x − z ≤ y − z ⇔ x ≤ y

REAL_ADD_SUB_ALT

|- ∀x y. x + y − y = x

INFINITE_REAL_UNIV

|- INFINITE 𝕌(:real)

add_rat

|- x / y + u / v =
   if y = 0 then unint (x / y) + u / v
   else if v = 0 then x / y + unint (u / v)
   else if y = v then (x + u) / v
   else (x * v + u * y) / (y * v)

add_ratl

|- x / y + z = if y = 0 then unint (x / y) + z else (x + z * y) / y

add_ratr

|- x + y / z = if z = 0 then x + unint (y / z) else (x * z + y) / z

add_ints

|- (&n + &m = &(n + m)) ∧ (-&n + &m = if n ≤ m then &(m − n) else -&(n − m)) ∧
   (&n + -&m = if n < m then -&(m − n) else &(n − m)) ∧
   (-&n + -&m = -&(n + m))

mult_rat

|- x / y * (u / v) =
   if y = 0 then unint (x / y) * (u / v)
   else if v = 0 then x / y * unint (u / v)
   else x * u / (y * v)

mult_ratl

|- x / y * z = if y = 0 then unint (x / y) * z else x * z / y

mult_ratr

|- x * (y / z) = if z = 0 then x * unint (y / z) else x * y / z

mult_ints

|- (&a * &b = &(a * b)) ∧ (-&a * &b = -&(a * b)) ∧ (&a * -&b = -&(a * b)) ∧
   (-&a * -&b = &(a * b))

pow_rat

|- (x pow 0 = 1) ∧ (0 pow NUMERAL (BIT1 n) = 0) ∧
   (0 pow NUMERAL (BIT2 n) = 0) ∧
   (&NUMERAL a pow NUMERAL n = &(NUMERAL a ** NUMERAL n)) ∧
   (-&NUMERAL a pow NUMERAL n =
    (if ODD (NUMERAL n) then numeric_negate else (λx. x))
      (&(NUMERAL a ** NUMERAL n))) ∧ ((x / y) pow n = x pow n / y pow n)

neg_rat

|- (-(x / y) = if y = 0 then -unint (x / y) else -x / y) ∧
   (x / -y = if y = 0 then unint (x / y) else -x / y)

eq_rat

|- (x / y = u / v) ⇔
   if y = 0 then unint (x / y) = u / v
   else if v = 0 then x / y = unint (u / v)
   else if y = v then x = u
   else x * v = y * u

eq_ratl

|- (x / y = z) ⇔ if y = 0 then unint (x / y) = z else x = y * z

eq_ratr

|- (z = x / y) ⇔ if y = 0 then z = unint (x / y) else y * z = x

eq_ints

|- ((&n = &m) ⇔ (n = m)) ∧ ((-&n = &m) ⇔ (n = 0) ∧ (m = 0)) ∧
   ((&n = -&m) ⇔ (n = 0) ∧ (m = 0)) ∧ ((-&n = -&m) ⇔ (n = m))

div_ratr

|- x / (y / z) = if (y = 0) ∨ (z = 0) then x / unint (y / z) else x * z / y

div_ratl

|- x / y / z =
   if y = 0 then unint (x / y) / z
   else if z = 0 then unint (x / y / z)
   else x / (y * z)

div_rat

|- x / y / (u / v) =
   if (u = 0) ∨ (v = 0) then x / y / unint (u / v)
   else if y = 0 then unint (x / y) / (u / v)
   else x * v / (y * u)

le_rat

|- x / &n ≤ u / &m ⇔
   if n = 0 then unint (x / 0) ≤ u / &m
   else if m = 0 then x / &n ≤ unint (u / 0)
   else &m * x ≤ &n * u

le_ratl

|- x / &n ≤ u ⇔ if n = 0 then unint (x / 0) ≤ u else x ≤ &n * u

le_ratr

|- x ≤ u / &m ⇔ if m = 0 then x ≤ unint (u / 0) else &m * x ≤ u

le_int

|- (&n ≤ &m ⇔ n ≤ m) ∧ (-&n ≤ &m ⇔ T) ∧ (&n ≤ -&m ⇔ (n = 0) ∧ (m = 0)) ∧
   (-&n ≤ -&m ⇔ m ≤ n)

lt_rat

|- x / &n < u / &m ⇔
   if n = 0 then unint (x / 0) < u / &m
   else if m = 0 then x / &n < unint (u / 0)
   else &m * x < &n * u

lt_ratl

|- x / &n < u ⇔ if n = 0 then unint (x / 0) < u else x < &n * u

lt_ratr

|- x < u / &m ⇔ if m = 0 then x < unint (u / 0) else &m * x < u

lt_int

|- (&n < &m ⇔ n < m) ∧ (-&n < &m ⇔ n ≠ 0 ∨ m ≠ 0) ∧ (&n < -&m ⇔ F) ∧
   (-&n < -&m ⇔ m < n)

NUM_FLOOR_LE

|- 0 ≤ x ⇒ &flr x ≤ x

NUM_FLOOR_LE2

|- 0 ≤ y ⇒ (x ≤ flr y ⇔ &x ≤ y)

NUM_FLOOR_LET

|- flr x ≤ y ⇔ x < &y + 1

NUM_FLOOR_DIV

|- 0 ≤ x ∧ 0 < y ⇒ &flr (x / y) * y ≤ x

NUM_FLOOR_DIV_LOWERBOUND

|- 0 ≤ x ∧ 0 < y ⇒ x < &(flr (x / y) + 1) * y

NUM_FLOOR_EQNS

|- (flr (&n) = n) ∧ (0 < m ⇒ (flr (&n / &m) = n DIV m))

NUM_FLOOR_LOWER_BOUND

|- x < &n ⇔ flr (x + 1) ≤ n

NUM_FLOOR_upper_bound

|- &n ≤ x ⇔ n < flr (x + 1)

LE_NUM_CEILING

|- ∀x. x ≤ &clg x

NUM_CEILING_LE

|- ∀x n. x ≤ &n ⇒ clg x ≤ n