Theory "util_prob"

Signature

Definitions

prod_sets_def

⊢ ∀a b. prod_sets a b = {s × t | s ∈ a ∧ t ∈ b}

powr_def

⊢ ∀x a. x powr a = exp (a * ln x)

pair_def

⊢ ∀X Y. pair X Y = (λ(x,y). x ∈ X ∧ y ∈ Y)

minimal_def

⊢ ∀p. minimal p = @n. p n ∧ ∀m. m < n ⇒ ¬p m

logr_def

⊢ ∀a x. logr a x = ln x / ln a

lg_def

⊢ ∀x. lg x = logr 2 x

from_def

⊢ ∀n. from n = {m | n ≤ m}

disjoint_def

⊢ ∀A. disjoint A ⇔ ∀a b. a ∈ A ∧ b ∈ A ∧ a ≠ b ⇒ DISJOINT a b

Theorems

UNION_FROM_COUNT

⊢ ∀n. from n ∪ count n = 𝕌(:num)

UNION_COUNT_FROM

⊢ ∀n. count n ∪ from n = 𝕌(:num)

REAL_X_LE_SUP

⊢ ∀P x. (∃r. P r) ∧ (∃z. ∀r. P r ⇒ r ≤ z) ∧ (∃r. P r ∧ x ≤ r) ⇒ x ≤ sup P

REAL_SUP_LE_X

⊢ ∀P x. (∃r. P r) ∧ (∀r. P r ⇒ r ≤ x) ⇒ sup P ≤ x

REAL_NEG_NZ

⊢ ∀x. x < 0 ⇒ x ≠ 0

REAL_MUL_IDEMPOT

⊢ ∀r. (r * r = r) ⇔ (r = 0) ∨ (r = 1)

REAL_LT_RMUL_NEG_0_NEG

⊢ ∀x y. x * y < 0 ∧ y < 0 ⇒ 0 < x

REAL_LT_RMUL_NEG_0

⊢ ∀x y. x * y < 0 ∧ 0 < y ⇒ x < 0

REAL_LT_RMUL_0_NEG

⊢ ∀x y. 0 < x * y ∧ y < 0 ⇒ x < 0

REAL_LT_RDIV_EQ_NEG

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

REAL_LT_LMUL_NEG_0_NEG

⊢ ∀x y. x * y < 0 ∧ x < 0 ⇒ 0 < y

REAL_LT_LMUL_NEG_0

⊢ ∀x y. x * y < 0 ∧ 0 < x ⇒ y < 0

REAL_LT_LMUL_0_NEG

⊢ ∀x y. 0 < x * y ∧ x < 0 ⇒ y < 0

REAL_LT_LE_MUL

⊢ ∀x y. 0 < x ∧ 0 ≤ y ⇒ 0 ≤ x * y

REAL_LE_RDIV_EQ_NEG

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

REAL_LE_LT_MUL

⊢ ∀x y. 0 ≤ x ∧ 0 < y ⇒ 0 ≤ x * y

PREIMAGE_REAL_COMPL4

⊢ ∀c. COMPL {x | x < c} = {x | c ≤ x}

PREIMAGE_REAL_COMPL3

⊢ ∀c. COMPL {x | x ≤ c} = {x | c < x}

PREIMAGE_REAL_COMPL2

⊢ ∀c. COMPL {x | c ≤ x} = {x | x < c}

PREIMAGE_REAL_COMPL1

⊢ ∀c. COMPL {x | c < x} = {x | x ≤ c}

POW_POS_EVEN

⊢ ∀x. x < 0 ⇒ (0 < x pow n ⇔ EVEN n)

POW_NEG_ODD

⊢ ∀x. x < 0 ⇒ (x pow n < 0 ⇔ ODD n)

POW_HALF_SMALL

⊢ ∀e. 0 < e ⇒ ∃n. (1 / 2) pow n < e

POW_HALF_POS

⊢ ∀n. 0 < (1 / 2) pow n

POW_HALF_MONO

⊢ ∀m n. m ≤ n ⇒ (1 / 2) pow n ≤ (1 / 2) pow m

PAIRED_BETA_THM

⊢ ∀f z. UNCURRY f z = f (FST z) (SND z)

PAIR_UNIV

⊢ pair 𝕌(:α) 𝕌(:β) = 𝕌(:α # β)

NUM_2D_BIJ_NZ_INV

⊢ ∃f. BIJ f 𝕌(:num) (𝕌(:num) × (𝕌(:num) DIFF {0}))

NUM_2D_BIJ_NZ_ALT_INV

⊢ ∃f. BIJ f (𝕌(:num) DIFF {0}) (𝕌(:num) × 𝕌(:num))

NUM_2D_BIJ_NZ_ALT2_INV

⊢ ∃f. BIJ f 𝕌(:num) ((𝕌(:num) DIFF {0}) × (𝕌(:num) DIFF {0}))

NUM_2D_BIJ_NZ_ALT2

⊢ ∃f. BIJ f ((𝕌(:num) DIFF {0}) × (𝕌(:num) DIFF {0})) 𝕌(:num)

NUM_2D_BIJ_NZ_ALT

⊢ ∃f. BIJ f (𝕌(:num) × 𝕌(:num)) (𝕌(:num) DIFF {0})

NUM_2D_BIJ_NZ

⊢ ∃f. BIJ f (𝕌(:num) × (𝕌(:num) DIFF {0})) 𝕌(:num)

NUM_2D_BIJ_INV

⊢ ∃f. BIJ f 𝕌(:num) (𝕌(:num) × 𝕌(:num))

NUM_2D_BIJ

⊢ ∃f. BIJ f (𝕌(:num) × 𝕌(:num)) 𝕌(:num)

neg_logr

⊢ ∀b x. 0 < x ⇒ (-logr b x = logr b x⁻¹)

neg_lg

⊢ ∀x. 0 < x ⇒ (-lg x = lg x⁻¹)

MINIMAL_SUC_IMP

⊢ ∀n p. p (SUC n) ∧ ¬p 0 ∧ (n = minimal (p ∘ SUC)) ⇒ (SUC n = minimal p)

MINIMAL_SUC

⊢ ∀n p.
      (SUC n = minimal p) ∧ p (SUC n) ⇔
      ¬p 0 ∧ (n = minimal (p ∘ SUC)) ∧ p (SUC n)

MINIMAL_EXISTS_IMP

⊢ ∀P. (∃n. P n) ⇒ ∃m. P m ∧ ∀n. n < m ⇒ ¬P n

MINIMAL_EXISTS0

⊢ (∃n. P n) ⇔ ∃n. P n ∧ ∀m. m < n ⇒ ¬P m

MINIMAL_EXISTS

⊢ ∀P. (∃n. P n) ⇔ P (minimal P) ∧ ∀n. n < minimal P ⇒ ¬P n

MINIMAL_EQ_IMP

⊢ ∀m p. p m ∧ (∀n. n < m ⇒ ¬p n) ⇒ (m = minimal p)

MINIMAL_EQ

⊢ ∀p m. p m ∧ (m = minimal p) ⇔ p m ∧ ∀n. n < m ⇒ ¬p n

logr_mul

⊢ ∀b x y. 0 < x ∧ 0 < y ⇒ (logr b (x * y) = logr b x + logr b y)

LOGR_MONO_LE_IMP

⊢ ∀x y b. 0 < x ∧ x ≤ y ∧ 1 ≤ b ⇒ logr b x ≤ logr b y

LOGR_MONO_LE

⊢ ∀x y b. 0 < x ∧ 0 < y ∧ 1 < b ⇒ (logr b x ≤ logr b y ⇔ x ≤ y)

logr_inv

⊢ ∀b x. 0 < x ⇒ (logr b x⁻¹ = -logr b x)

logr_div

⊢ ∀b x y. 0 < x ∧ 0 < y ⇒ (logr b (x / y) = logr b x − logr b y)

logr_1

⊢ ∀b. logr b 1 = 0

lg_pow

⊢ ∀n. lg (2 pow n) = &n

lg_nonzero

⊢ ∀x. x ≠ 0 ∧ 0 ≤ x ⇒ (lg x ≠ 0 ⇔ x ≠ 1)

lg_mul

⊢ ∀x y. 0 < x ∧ 0 < y ⇒ (lg (x * y) = lg x + lg y)

lg_inv

⊢ ∀x. 0 < x ⇒ (lg x⁻¹ = -lg x)

lg_2

⊢ lg 2 = 1

lg_1

⊢ lg 1 = 0

LE_INF

⊢ ∀p r. (∃x. x ∈ p) ∧ (∀x. x ∈ p ⇒ r ≤ x) ⇒ r ≤ inf p

INF_LE

⊢ ∀p r. (∃z. ∀x. x ∈ p ⇒ z ≤ x) ∧ (∃x. x ∈ p ∧ x ≤ r) ⇒ inf p ≤ r

INF_GREATER

⊢ ∀p z. (∃x. x ∈ p) ∧ inf p < z ⇒ ∃x. x ∈ p ∧ x < z

INF_DEF_ALT

⊢ ∀p. inf p = -sup (λr. -r ∈ p)

INF_CLOSE

⊢ ∀p e. (∃x. x ∈ p) ∧ 0 < e ⇒ ∃x. x ∈ p ∧ x < inf p + e

IN_PROD_SETS

⊢ ∀s a b. s ∈ prod_sets a b ⇔ ∃t u. (s = t × u) ∧ t ∈ a ∧ u ∈ b

IN_PAIR

⊢ ∀x X Y. x ∈ pair X Y ⇔ FST x ∈ X ∧ SND x ∈ Y

IN_o

⊢ ∀x f s. x ∈ s ∘ f ⇔ f x ∈ s

IN_FROM

⊢ ∀m n. m ∈ from n ⇔ n ≤ m

GBIGUNION_IMAGE

⊢ ∀s p n. {s | ∃n. p s n} = BIGUNION (IMAGE (λn. {s | p s n}) 𝕌(:γ))

FROM_0

⊢ from 0 = 𝕌(:num)

finite_enumeration_of_sets_has_max_non_empty

⊢ ∀f s.
      FINITE s ∧ (∀x. f x ∈ s) ∧ (∀m n. m ≠ n ⇒ DISJOINT (f m) (f n)) ⇒
      ∃N. ∀n. n ≥ N ⇒ (f n = ∅)

EQ_T_IMP

⊢ ∀x. x ⇔ T ⇒ x

disjointI

⊢ ∀A. (∀a b. a ∈ A ⇒ b ∈ A ⇒ a ≠ b ⇒ DISJOINT a b) ⇒ disjoint A

disjointD

⊢ ∀A a b. disjoint A ⇒ a ∈ A ⇒ b ∈ A ⇒ a ≠ b ⇒ DISJOINT a b

disjoint_union

⊢ ∀A B.
      disjoint A ∧ disjoint B ∧ (BIGUNION A ∩ BIGUNION B = ∅) ⇒
      disjoint (A ∪ B)

disjoint_sing

⊢ ∀a. disjoint {a}

DISJOINT_FROM_COUNT

⊢ ∀n. DISJOINT (from n) (count n)

disjoint_empty

⊢ disjoint ∅

DISJOINT_COUNT_FROM

⊢ ∀n. DISJOINT (count n) (from n)