Supporting results for pointwise provable equality.

Logarithmic function #

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_exists_unique_pos {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {y : V} (hy : 0 < y) :

∃! x : V, x < y ∧ ∃ y' ≤ y, Exponential x y' ∧ y < 2 * y'

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_exists_unique {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (y : V) :

∃! x : V, (y = 0 → x = 0) ∧ (0 < y → x < y ∧ ∃ y' ≤ y, Exponential x y' ∧ y < 2 * y')

source

noncomputable def PointwiseComposition.FFL.FirstOrder.Arithmetic.log {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a : V) :

The log construction in FFL.FirstOrder.Arithmetic.

Equations

PointwiseComposition.FFL.FirstOrder.Arithmetic.log a = PointwiseComposition.Classical.chooseUnique ⋯

Instances For

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_zero {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

log 0 = 0

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_pos {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {y : V} (pos : 0 < y) :

∃ y' ≤ y, Exponential (log y) y' ∧ y < 2 * y'

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_lt_self_of_pos {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {y : V} (pos : 0 < y) :

log y < y

source

@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_le_self {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a : V) :

log a ≤ a

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_graph {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {x y : V} :

x = log y ↔ (y = 0 → x = 0) ∧ (0 < y → x < y ∧ ∃ y' ≤ y, Exponential x y' ∧ y < 2 * y')

source

def PointwiseComposition.FFL.FirstOrder.Arithmetic.logDef :

Σ₀.Semisentence 2

The log Def construction in FFL.FirstOrder.Arithmetic.

Equations

One or more equations did not get rendered due to their size.

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instance PointwiseComposition.FFL.FirstOrder.Arithmetic.log_defined {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

Σ₀ -Function₁ log via logDef

source

instance PointwiseComposition.FFL.FirstOrder.Arithmetic.log_definable {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

Σ₀ -Function₁ log

source

instance PointwiseComposition.FFL.FirstOrder.Arithmetic.instBounded₁Log {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

Bounded₁ log

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_eq_of_pos {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {x y : V} (pos : 0 < y) {y' : V} (H : Exponential x y') (hy' : y' ≤ y) (hy : y < 2 * y') :

log y = x

source

@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_one {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

log 1 = 0

source

@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_two {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

log 2 = 1

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_two_mul_of_pos {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {y : V} (pos : 0 < y) :

log (2 * y) = log y + 1

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_two_mul_add_one_of_pos {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {y : V} (pos : 0 < y) :

log (2 * y + 1) = log y + 1

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.Exponential.log_eq_of_exp {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {x y : V} (H : Exponential x y) :

log y = x

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.exponential_of_pow2 {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {p : V} (pp : Pow2 p) :

Exponential (log p) p

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_mul_pow2_add_of_lt {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a p b : V} (pos : 0 < a) (pp : Pow2 p) (hb : b < p) :

log (a * p + b) = log a + log p

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_mul_pow2 {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a p : V} (pos : 0 < a) (pp : Pow2 p) :

log (a * p) = log a + log p

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_monotone {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a b : V} (h : a ≤ b) :

log a ≤ log b

source

noncomputable def PointwiseComposition.FFL.FirstOrder.Arithmetic.binaryLength {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a : V) :

The binary Length construction in FFL.FirstOrder.Arithmetic.

Equations

PointwiseComposition.FFL.FirstOrder.Arithmetic.binaryLength a = if 0 < a then PointwiseComposition.FFL.FirstOrder.Arithmetic.log a + 1 else 0

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@[instance_reducible]

noncomputable def PointwiseComposition.FFL.FirstOrder.Arithmetic.instLength {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

Length V

The inst Length construction in FFL.FirstOrder.Arithmetic.

Equations

PointwiseComposition.FFL.FirstOrder.Arithmetic.instLength = { length := PointwiseComposition.FFL.FirstOrder.Arithmetic.binaryLength }

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_eq_binaryLength {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a : V) :

‖a‖ = if 0 < a then log a + 1 else 0

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_zero {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

‖0‖ = 0

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_of_pos {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a : V} (pos : 0 < a) :

‖a‖ = log a + 1

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_le {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a : V) :

‖a‖ ≤ a

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_graph {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {i a : V} :

i = ‖a‖ ↔ (0 < a → ∃ k ≤ a, k = log a ∧ i = k + 1) ∧ (a = 0 → i = 0)

source

def PointwiseComposition.FFL.FirstOrder.Arithmetic.lengthDef :

Σ₀.Semisentence 2

The length Def construction in FFL.FirstOrder.Arithmetic.

Equations

One or more equations did not get rendered due to their size.

Instances For

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instance PointwiseComposition.FFL.FirstOrder.Arithmetic.length_defined {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

Σ₀ -Function₁ Length.length via lengthDef

source

instance PointwiseComposition.FFL.FirstOrder.Arithmetic.length_definable {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

Σ₀ -Function₁ fun (x : V) => ‖x‖

source

instance PointwiseComposition.FFL.FirstOrder.Arithmetic.instBounded₁Length {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

Bounded₁ fun (x : V) => ‖x‖

source

@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_one {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

‖1‖ = 1

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.Exponential.length_eq {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {x y : V} (H : Exponential x y) :

‖y‖ = x + 1

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_two_mul_of_pos {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a : V} (pos : 0 < a) :

‖2 * a‖ = ‖a‖ + 1

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_two_mul_add_one {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a : V) :

‖2 * a + 1‖ = ‖a‖ + 1

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_mul_pow2_add_of_lt {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a p b : V} (pos : 0 < a) (pp : Pow2 p) (hb : b < p) :

‖a * p + b‖ = ‖a‖ + log p

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_mul_pow2 {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a p : V} (pos : 0 < a) (pp : Pow2 p) :

‖a * p‖ = ‖a‖ + log p

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_monotone {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a b : V} (h : a ≤ b) :

‖a‖ ≤ ‖b‖

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.pos_of_lt_length {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a b : V} (h : a < ‖b‖) :

0 < b

source

@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_pos_iff {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a : V} :

0 < ‖a‖ ↔ 0 < a

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_eq_zero_iff {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a : V} :

‖a‖ = 0 ↔ a = 0

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.le_log_of_lt_length {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a b : V} (h : a < ‖b‖) :

a ≤ log b

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.exponential_log_le_self {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a b : V} (pos : 0 < a) (h : Exponential (log a) b) :

b ≤ a

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.lt_exponential_log_self {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a b : V} (h : Exponential (log a) b) :

a < 2 * b

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.lt_exp_len_self {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a b : V} (h : Exponential ‖a‖ b) :

a < b

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.le_iff_le_log_of_exp {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {x y a : V} (H : Exponential x y) (pos : 0 < a) :

y ≤ a ↔ x ≤ log a

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.le_iff_lt_length_of_exp {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {x y a : V} (H : Exponential x y) :

y ≤ a ↔ x < ‖a‖

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.Exponential.lt_iff_log_lt {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {x y a : V} (H : Exponential x y) (pos : 0 < a) :

a < y ↔ log a < x

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.Exponential.lt_iff_len_le {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {x y a : V} (H : Exponential x y) :

a < y ↔ ‖a‖ ≤ x

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.Exponential.le_of_lt_length {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {x y a : V} (H : Exponential x y) :

x < ‖a‖ → y ≤ a

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.Exponential.le_log {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {x y : V} (H : Exponential x y) :

x ≤ log y

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.Exponential.lt_length {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {x y : V} (H : Exponential x y) :

x < ‖y‖

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.lt_exponential_length {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a b : V} (h : Exponential ‖a‖ b) :

a < b

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.sq_len_le_three_mul {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a : V) :

‖a‖ ^ 2 ≤ 3 * a

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.brange_exists_unique {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a x : V) :

x < ‖a‖ → ∃! y : V, Exponential x y

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_exists_unique {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a x : V) :

∃! y : V, (x < ‖a‖ → Exponential x y) ∧ (‖a‖ ≤ x → y = 0)

source

noncomputable def PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a x : V) :

bexp a x = exp x if x < ‖a‖; = 0 o.w.

Equations

PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp a x = PointwiseComposition.Classical.chooseUnique ⋯

Instances For

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.exp_bexp_of_lt {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a x : V} (h : x < ‖a‖) :

Exponential x (bexp a x)

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_eq_zero_of_le {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a x : V} (h : ‖a‖ ≤ x) :

bexp a x = 0

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_zero {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (x : V) :

bexp 0 x = 0

source

@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.exp_bexp_of_lt_iff {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a x : V} :

Exponential x (bexp a x) ↔ x < ‖a‖

source

@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_le_self {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a x : V) :

bexp a x ≤ a

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_graph {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {y a x : V} :

y = bexp a x ↔ ∃ l ≤ a, l = ‖a‖ ∧ (x < l → Exponential x y) ∧ (l ≤ x → y = 0)

source

def PointwiseComposition.FFL.FirstOrder.Arithmetic.bexpDef :

Σ₀.Semisentence 3

The bexp Def construction in FFL.FirstOrder.Arithmetic.

Equations

One or more equations did not get rendered due to their size.

Instances For

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instance PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_defined {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

Σ₀ -Function₂ bexp via bexpDef

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instance PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_definable {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

Σ₀ -Function₂ bexp

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instance PointwiseComposition.FFL.FirstOrder.Arithmetic.instBounded₂Bexp {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

Bounded₂ bexp

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_monotone_iff {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a i j : V} (hi : i < ‖a‖) (hj : j < ‖a‖) :

bexp a i < bexp a j ↔ i < j

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_monotone_le_iff {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a i j : V} (hi : i < ‖a‖) (hj : j < ‖a‖) :

bexp a i ≤ bexp a j ↔ i ≤ j

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_eq_of_lt_length {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {i a a' : V} (ha : i < ‖a‖) (ha' : i < ‖a'‖) :

bexp a i = bexp a' i

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_pow2 {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a x : V} (h : x < ‖a‖) :

Pow2 (bexp a x)

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.lt_bexp {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a x : V} (h : x < ‖a‖) :

x < bexp a x

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_pos {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a x : V} (h : x < ‖a‖) :

0 < bexp a x

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.lt_bexp_len {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a x : V} (h : ‖x‖ < ‖a‖) :

x < bexp a ‖x‖

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_eq_of_exp {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {y a x : V} (h : x < ‖a‖) (H : Exponential x y) :

bexp a x = y

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_bexp {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a x : V} (h : x < ‖a‖) :

log (bexp a x) = x

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.len_bexp {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a x : V} (h : x < ‖a‖) :

‖bexp a x‖ = x + 1

source

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_zero_zero {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

bexp 0 0 = 0

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_pos_zero {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a : V} (h : 0 < a) :

bexp a 0 = 1

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_monotone {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a₁ x₁ a₂ x₂ : V} (h₁ : x₁ < ‖a₁‖) (h₂ : x₂ < ‖a₂‖) :

bexp a₁ x₁ < bexp a₂ x₂ ↔ x₁ < x₂

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_monotone_le {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a₁ x₁ a₂ x₂ : V} (h₁ : x₁ < ‖a₁‖) (h₂ : x₂ < ‖a₂‖) :

bexp a₁ x₁ ≤ bexp a₂ x₂ ↔ x₁ ≤ x₂

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_add {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {x₁ x₂ a : V} (h : x₁ + x₂ < ‖a‖) :

bexp a (x₁ + x₂) = bexp a x₁ * bexp a x₂

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_two_mul {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a a' x : V} (hx : 2 * x < ‖a‖) (hx' : x < ‖a'‖) :

bexp a (2 * x) = bexp a' x ^ 2

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_two_mul_succ {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a i : V} :

bexp (2 * a) (i + 1) = 2 * bexp a i

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp_two_mul_add_one_succ {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a i : V} :

bexp (2 * a + 1) (i + 1) = 2 * bexp a i

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noncomputable def PointwiseComposition.FFL.FirstOrder.Arithmetic.fbit {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a i : V) :

The fbit construction in FFL.FirstOrder.Arithmetic.

Equations

PointwiseComposition.FFL.FirstOrder.Arithmetic.fbit a i = a / PointwiseComposition.FFL.FirstOrder.Arithmetic.bexp a i % 2

Instances For

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.fbit_lt_two {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a i : V) :

fbit a i < 2

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.fbit_le_one {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a i : V) :

fbit a i ≤ 1

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.fbit_eq_one_iff {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a i : V} :

fbit a i = 1 ↔ LenBit (bexp a i) a

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.fbit_eq_zero_iff {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a i : V} :

fbit a i = 0 ↔ ¬LenBit (bexp a i) a

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.fbit_eq_zero_of_le {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] {a i : V} (hi : ‖a‖ ≤ i) :

fbit a i = 0

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def PointwiseComposition.FFL.FirstOrder.Arithmetic.fbitDef :

Σ₀.Semisentence 3

The fbit Def construction in FFL.FirstOrder.Arithmetic.

Equations

One or more equations did not get rendered due to their size.

Instances For

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instance PointwiseComposition.FFL.FirstOrder.Arithmetic.fbit_defined {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

Σ₀ -Function₂ fbit via fbitDef

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instance PointwiseComposition.FFL.FirstOrder.Arithmetic.fbit_definable {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

Σ₀ -Function₂ fbit

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instance PointwiseComposition.FFL.FirstOrder.Arithmetic.instBounded₂Fbit {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] :

Bounded₂ fbit

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.fbit_zero {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (i : V) :

fbit 0 i = 0

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.fbit_mul_two_mul {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a i : V) :

fbit (2 * a) (i + 1) = fbit a i

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.fbit_mul_two_add_one_mul {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a i : V) :

fbit (2 * a + 1) (i + 1) = fbit a i

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.fbit_two_mul_zero_eq_zero {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a : V) :

fbit (2 * a) 0 = 0

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.fbit_two_mul_add_one_zero_eq_one {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₀] (a : V) :

fbit (2 * a + 1) 0 = 1

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_exponential {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₁] (a : V) :

log (Exp.exp a) = a

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.exp_log_le_self {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₁] {a : V} (pos : 0 < a) :

Exp.exp (log a) ≤ a

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.lt_two_mul_exponential_log {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₁] {a : V} (pos : 0 < a) :

a < 2 * Exp.exp (log a)

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@[simp]

theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_exponential {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₁] (a : V) :

‖Exp.exp a‖ = a + 1

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.exp_add {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₁] (a b : V) :

Exp.exp (a + b) = Exp.exp a * Exp.exp b

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_mul_exp_add_of_lt {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₁] {a b : V} (pos : 0 < a) (i : V) (hb : b < Exp.exp i) :

log (a * Exp.exp i + b) = log a + i

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.log_mul_exp {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₁] {a : V} (pos : 0 < a) (i : V) :

log (a * Exp.exp i) = log a + i

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_mul_exp_add_of_lt {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₁] {a b : V} (pos : 0 < a) (i : V) (hb : b < Exp.exp i) :

‖a * Exp.exp i + b‖ = ‖a‖ + i

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.length_mul_exp {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₁] {a : V} (pos : 0 < a) (i : V) :

‖a * Exp.exp i‖ = ‖a‖ + i

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theorem PointwiseComposition.FFL.FirstOrder.Arithmetic.exp_le_iff_le_log {V : Type u_1} [ORingStructure V] [V↓[ℒₒᵣ ] ⊧* 𝐈Σ₁] {i a : V} (pos : 0 < a) :

Exp.exp i ≤ a ↔ i ≤ log a

Documentation

LeanPool.PointwiseComposition.Foundation.FirstOrder.Arithmetic.Exponential.Log

Logarithmic function #