Well-formed bases #

Main definitions #

WellFormedBasis basis: a predicate meaning that all functions from basis tend to atTop, and basis is sorted such that if g goes after f in basis, then log f =o[atTop] log g.

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@[reducible, inline]

abbrev Tactic.ComputeAsymptotics.Basis :

Type

List of functions used to construct monomials in multiseries.

Equations

Tactic.ComputeAsymptotics.Basis = List (ℝ → ℝ)

Instances For

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def Tactic.ComputeAsymptotics.WellFormedBasis (basis : Basis) :

Prop

WellFormedBasis basis means that all functions from basis tend to atTop, and basis is sorted such that if g goes after f in basis, then log f =o[atTop] log g.

We use two types Basis and WellFormedBasis instead of a single bundled one because it it lets us to use the List API for Basis.

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

Instances For

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.nil :

WellFormedBasis []

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.single (f : ℝ → ℝ) (hf : Filter.Tendsto f Filter.atTop Filter.atTop) :

WellFormedBasis [f]

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.of_sublist {basis basis' : Basis} (h : List.Sublist basis basis') (h_basis : WellFormedBasis basis') :

WellFormedBasis basis

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.tail {basis_hd : ℝ → ℝ} {basis_tl : Basis} (h : WellFormedBasis (basis_hd :: basis_tl)) :

WellFormedBasis basis_tl

The tail of a well-formed basis is well-formed.

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.of_append_right {left right : Basis} (h : WellFormedBasis (left ++ right)) :

WellFormedBasis right

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.compare_left_aux {basis : Basis} {f : ℝ → ℝ} (h : WellFormedBasis basis) (h_comp : ∀ (g : ℝ → ℝ), List.getLast? basis = some g → (Real.log ∘ f) =o[Filter.atTop ] (Real.log ∘ g)) (g : ℝ → ℝ) :

g ∈ basis → (Real.log ∘ f) =o[Filter.atTop ] (Real.log ∘ g)

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.compare_right_aux {basis : Basis} {f : ℝ → ℝ} (h : WellFormedBasis basis) (h_comp : ∀ (g : ℝ → ℝ), List.head? basis = some g → (Real.log ∘ g) =o[Filter.atTop ] (Real.log ∘ f)) (g : ℝ → ℝ) :

g ∈ basis → (Real.log ∘ g) =o[Filter.atTop ] (Real.log ∘ f)

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.append {left right : Basis} (h_left : WellFormedBasis left) (h_right : WellFormedBasis right) (h : ∀ f ∈ left, ∀ g ∈ right, (Real.log ∘ g) =o[Filter.atTop ] (Real.log ∘ f)) :

WellFormedBasis (left ++ right)

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.cons {basis : Basis} {f : ℝ → ℝ} (h_basis : WellFormedBasis basis) (hf_tendsto : Filter.Tendsto f Filter.atTop Filter.atTop) (hf : ∀ g ∈ basis, (Real.log ∘ g) =o[Filter.atTop ] (Real.log ∘ f)) :

WellFormedBasis (f :: basis)

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.insert {left right : Basis} {f : ℝ → ℝ} (h : WellFormedBasis (left ++ right)) (hf_tendsto : Filter.Tendsto f Filter.atTop Filter.atTop) (hf_comp_left : ∀ (g : ℝ → ℝ), List.getLast? left = some g → (Real.log ∘ f) =o[Filter.atTop ] (Real.log ∘ g)) (hf_comp_right : ∀ (g : ℝ → ℝ), List.head? right = some g → (Real.log ∘ g) =o[Filter.atTop ] (Real.log ∘ f)) :

WellFormedBasis (left ++ f :: right)

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.push {basis : Basis} {f : ℝ → ℝ} (h : WellFormedBasis basis) (hf_tendsto : Filter.Tendsto f Filter.atTop Filter.atTop) (hf_comp : ∀ (g : ℝ → ℝ), List.getLast? basis = some g → (Real.log ∘ f) =o[Filter.atTop ] (Real.log ∘ g)) :

WellFormedBasis (basis ++ [f])

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.tendsto_atTop {basis : Basis} (h : WellFormedBasis basis) {f : ℝ → ℝ} (hf : f ∈ basis) :

Filter.Tendsto f Filter.atTop Filter.atTop

All functions from a well-formed basis tend to atTop.

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.eventually_pos {basis : Basis} (h : WellFormedBasis basis) :

∀ᶠ (x : ℝ) in Filter.atTop , ∀ f ∈ basis, 0 < f x

Eventually all functions from a well-formed basis are positive.

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.head_eventually_pos {basis_hd : ℝ → ℝ} {basis_tl : Basis} (h : WellFormedBasis (basis_hd :: basis_tl)) :

∀ᶠ (x : ℝ) in Filter.atTop , 0 < basis_hd x

The first function in a well-formed basis is eventually positive.

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.tail_isLittleO_head {hd : ℝ → ℝ} {tl : Basis} (h : WellFormedBasis (hd :: tl)) {f : ℝ → ℝ} (hf : f ∈ tl) :

(Real.log ∘ f) =o[Filter.atTop ] (Real.log ∘ hd)

All functions in the tail of a well-formed basis are little-o of the basis' head.

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theorem Tactic.ComputeAsymptotics.WellFormedBasis.push_log_last {basis_hd : ℝ → ℝ} {basis_tl : Basis} (h_basis : WellFormedBasis (basis_hd :: basis_tl)) :

WellFormedBasis (basis_hd :: basis_tl ++ [Real.log ∘ (basis_hd :: basis_tl).getLast ⋯])

Basis extensions #

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inductive Tactic.ComputeAsymptotics.BasisExtension :

Basis → Type

The type of extensions of a given basis, defined as an inductive type. Given a basis : Basis and ex : BasisExtension basis of it, one can use getBasis to produce a basis basis' for which basis <+ basis'. Moreover, all such bases for which basis is a sublist can be obtained in this manner. In this sense BasisExtension is a Type-valued analogue of List.Sublist.