mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.SetTheory.Ordinal.Notation

Goal state before:
pp: case oadd.inl.none.inl.some.zero
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b = 0
e✝ : b.fundamentalSequence = Sum.inl none
a' : ONote
e : a.fundamentalSequence = Sum.inl (some a')
e' : m.natPred = 0
⊢ match
    match Sum.inl (some a'), 0 with
    | Sum.inl none, 0 => Sum.inl (some zero)
    | Sum.inl none, m.succ => Sum.inl (some (zero.oadd m.succPNat zero))
    | Sum.inl (some a'), 0 => Sum.inr fun i ↦ a'.oadd i.succPNat zero
    | Sum.inl (some a'), m.succ => Sum.inr fun i ↦ a.oadd m.succPNat (a'.oadd i.succPNat zero)
    | Sum.inr f, 0 => Sum.inr fun i ↦ (f i).oadd 1 zero
    | Sum.inr f, m.succ => Sum.inr fun i ↦ a.oadd m.succPNat ((f i).oadd 1 zero) with
  | Sum.inl none => a.oadd m b = 0
  | Sum.inl (some a_1) => (a.oadd m b).repr = succ a_1.repr ∧ ((a.oadd m b).NF → a_1.NF)
  | Sum.inr f =>
    IsSuccLimit (a.oadd m b).repr ∧
      (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a.oadd m b ∧ ((a.oadd m b).NF → (f i).NF)) ∧
        ∀ a_1 < (a.oadd m b).repr, ∃ i, a_1 < (f i).repr

Tactic:
simp only

Goal state after:
case oadd.inl.none.inl.some.zero
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b = 0
e✝ : b.fundamentalSequence = Sum.inl none
a' : ONote
e : a.fundamentalSequence = Sum.inl (some a')
e' : m.natPred = 0
⊢ IsSuccLimit (a.oadd m b).repr ∧
    (∀ (i : ℕ),
        a'.oadd i.succPNat zero < a'.oadd (i + 1).succPNat zero ∧
          a'.oadd i.succPNat zero < a.oadd m b ∧ ((a.oadd m b).NF → (a'.oadd i.succPNat zero).NF)) ∧
      ∀ a_1 < (a.oadd m b).repr, ∃ i, a_1 < (a'.oadd i.succPNat zero).repr
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: β : Type u_1
γ : Type u_2
basis_hd : ℝ → ℝ
basis_tl : Basis
F : β → Option (ℝ × MultiseriesExpansion basis_tl × γ × β)
op : γ → Multiseries basis_hd basis_tl → Multiseries basis_hd basis_tl
inst✝ : FriendlyOperationClass op
b : β
exp : ℝ
coef : MultiseriesExpansion basis_tl
c : γ
next : β
h : F b = some (exp, coef, c, next)
⊢ gcorec F op b = cons exp coef (op c (gcorec F op next))

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
⊢ nil.destruct = none

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: m : UnitMonomial
left right : Basis
⊢ (List.replicate (List.length left) 0 ++ m).toFun (left ++ right) = m.toFun right

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
exp : ℝ
coef : MultiseriesExpansion basis_tl
tl : Multiseries basis_hd basis_tl
⊢ (cons exp coef tl).destruct = some (exp, coef, tl)

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Ordinal.Notation

Goal state before:
pp: case oadd.inl.none.inl.some.succ
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b = 0
e✝ : b.fundamentalSequence = Sum.inl none
a' : ONote
e : a.fundamentalSequence = Sum.inl (some a')
m' : ℕ
e' : m.natPred = m' + 1
⊢ match
    match Sum.inl (some a'), m' + 1 with
    | Sum.inl none, 0 => Sum.inl (some zero)
    | Sum.inl none, m.succ => Sum.inl (some (zero.oadd m.succPNat zero))
    | Sum.inl (some a'), 0 => Sum.inr fun i ↦ a'.oadd i.succPNat zero
    | Sum.inl (some a'), m.succ => Sum.inr fun i ↦ a.oadd m.succPNat (a'.oadd i.succPNat zero)
    | Sum.inr f, 0 => Sum.inr fun i ↦ (f i).oadd 1 zero
    | Sum.inr f, m.succ => Sum.inr fun i ↦ a.oadd m.succPNat ((f i).oadd 1 zero) with
  | Sum.inl none => a.oadd m b = 0
  | Sum.inl (some a_1) => (a.oadd m b).repr = succ a_1.repr ∧ ((a.oadd m b).NF → a_1.NF)
  | Sum.inr f =>
    IsSuccLimit (a.oadd m b).repr ∧
      (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a.oadd m b ∧ ((a.oadd m b).NF → (f i).NF)) ∧
        ∀ a_1 < (a.oadd m b).repr, ∃ i, a_1 < (f i).repr

Tactic:
simp only

Goal state after:
case oadd.inl.none.inl.some.succ
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b = 0
e✝ : b.fundamentalSequence = Sum.inl none
a' : ONote
e : a.fundamentalSequence = Sum.inl (some a')
m' : ℕ
e' : m.natPred = m' + 1
⊢ IsSuccLimit (a.oadd m b).repr ∧
    (∀ (i : ℕ),
        a.oadd m'.succPNat (a'.oadd i.succPNat zero) < a.oadd m'.succPNat (a'.oadd (i + 1).succPNat zero) ∧
          a.oadd m'.succPNat (a'.oadd i.succPNat zero) < a.oadd m b ∧
            ((a.oadd m b).NF → (a.oadd m'.succPNat (a'.oadd i.succPNat zero)).NF)) ∧
      ∀ a_1 < (a.oadd m b).repr, ∃ i, a_1 < (a.oadd m'.succPNat (a'.oadd i.succPNat zero)).repr
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
ms : Multiseries basis_hd basis_tl
h : ms.destruct = none
⊢ ms = nil

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Ordinal.Notation

Goal state before:
pp: case oadd.inl.none.inr.zero
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b = 0
e✝ : b.fundamentalSequence = Sum.inl none
f : ℕ → ONote
e : a.fundamentalSequence = Sum.inr f
e' : m.natPred = 0
⊢ match
    match Sum.inr f, 0 with
    | Sum.inl none, 0 => Sum.inl (some zero)
    | Sum.inl none, m.succ => Sum.inl (some (zero.oadd m.succPNat zero))
    | Sum.inl (some a'), 0 => Sum.inr fun i ↦ a'.oadd i.succPNat zero
    | Sum.inl (some a'), m.succ => Sum.inr fun i ↦ a.oadd m.succPNat (a'.oadd i.succPNat zero)
    | Sum.inr f, 0 => Sum.inr fun i ↦ (f i).oadd 1 zero
    | Sum.inr f, m.succ => Sum.inr fun i ↦ a.oadd m.succPNat ((f i).oadd 1 zero) with
  | Sum.inl none => a.oadd m b = 0
  | Sum.inl (some a_1) => (a.oadd m b).repr = succ a_1.repr ∧ ((a.oadd m b).NF → a_1.NF)
  | Sum.inr f =>
    IsSuccLimit (a.oadd m b).repr ∧
      (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a.oadd m b ∧ ((a.oadd m b).NF → (f i).NF)) ∧
        ∀ a_1 < (a.oadd m b).repr, ∃ i, a_1 < (f i).repr

Tactic:
simp only

Goal state after:
case oadd.inl.none.inr.zero
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b = 0
e✝ : b.fundamentalSequence = Sum.inl none
f : ℕ → ONote
e : a.fundamentalSequence = Sum.inr f
e' : m.natPred = 0
⊢ IsSuccLimit (a.oadd m b).repr ∧
    (∀ (i : ℕ),
        (f i).oadd 1 zero < (f (i + 1)).oadd 1 zero ∧
          (f i).oadd 1 zero < a.oadd m b ∧ ((a.oadd m b).NF → ((f i).oadd 1 zero).NF)) ∧
      ∀ a_1 < (a.oadd m b).repr, ∃ i, a_1 < ((f i).oadd 1 zero).repr
Module: Mathlib.SetTheory.Ordinal.Notation

Goal state before:
pp: case oadd.inl.none.inr.succ
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b = 0
e✝ : b.fundamentalSequence = Sum.inl none
f : ℕ → ONote
e : a.fundamentalSequence = Sum.inr f
m' : ℕ
e' : m.natPred = m' + 1
⊢ match
    match Sum.inr f, m' + 1 with
    | Sum.inl none, 0 => Sum.inl (some zero)
    | Sum.inl none, m.succ => Sum.inl (some (zero.oadd m.succPNat zero))
    | Sum.inl (some a'), 0 => Sum.inr fun i ↦ a'.oadd i.succPNat zero
    | Sum.inl (some a'), m.succ => Sum.inr fun i ↦ a.oadd m.succPNat (a'.oadd i.succPNat zero)
    | Sum.inr f, 0 => Sum.inr fun i ↦ (f i).oadd 1 zero
    | Sum.inr f, m.succ => Sum.inr fun i ↦ a.oadd m.succPNat ((f i).oadd 1 zero) with
  | Sum.inl none => a.oadd m b = 0
  | Sum.inl (some a_1) => (a.oadd m b).repr = succ a_1.repr ∧ ((a.oadd m b).NF → a_1.NF)
  | Sum.inr f =>
    IsSuccLimit (a.oadd m b).repr ∧
      (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a.oadd m b ∧ ((a.oadd m b).NF → (f i).NF)) ∧
        ∀ a_1 < (a.oadd m b).repr, ∃ i, a_1 < (f i).repr

Tactic:
simp only

Goal state after:
case oadd.inl.none.inr.succ
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b = 0
e✝ : b.fundamentalSequence = Sum.inl none
f : ℕ → ONote
e : a.fundamentalSequence = Sum.inr f
m' : ℕ
e' : m.natPred = m' + 1
⊢ IsSuccLimit (a.oadd m b).repr ∧
    (∀ (i : ℕ),
        a.oadd m'.succPNat ((f i).oadd 1 zero) < a.oadd m'.succPNat ((f (i + 1)).oadd 1 zero) ∧
          a.oadd m'.succPNat ((f i).oadd 1 zero) < a.oadd m b ∧
            ((a.oadd m b).NF → (a.oadd m'.succPNat ((f i).oadd 1 zero)).NF)) ∧
      ∀ a_1 < (a.oadd m b).repr, ∃ i, a_1 < (a.oadd m'.succPNat ((f i).oadd 1 zero)).repr
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
ms : Multiseries basis_hd basis_tl
exp : ℝ
coef : MultiseriesExpansion basis_tl
tl : Multiseries basis_hd basis_tl
h : ms.destruct = some (exp, coef, tl)
⊢ ms = cons exp coef tl

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
⊢ nil.head = none

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
exp : ℝ
coef : MultiseriesExpansion basis_tl
tl : Multiseries basis_hd basis_tl
⊢ (cons exp coef tl).head = some (exp, coef)

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
⊢ nil.tail = nil

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
exp : ℝ
coef : MultiseriesExpansion basis_tl
tl : Multiseries basis_hd basis_tl
⊢ (cons exp coef tl).tail = tl

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: x e : ℝ
es : List ℝ
ih :
  ∀ {basis : Basis},
    WellFormedBasis basis →
      (∀ f ∈ basis, 0 < f x) → 0 < toFun es basis x ∧ (log ∘ toFun es basis) x = toLogFun es basis x
b : ℝ → ℝ
bs : List (ℝ → ℝ)
h_basis : WellFormedBasis (b :: bs)
hx : ∀ f ∈ b :: bs, 0 < f x
x1✝ : ℝ → ℝ
x2✝ : x1✝ ∈ bs
⊢ x1✝ ∈ b :: bs

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
basis_hd' : ℝ → ℝ
basis_tl' : Basis
f : ℝ → ℝ
g : MultiseriesExpansion basis_tl → MultiseriesExpansion basis_tl'
⊢ map f g nil = nil

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
basis_hd' : ℝ → ℝ
basis_tl' : Basis
f : ℝ → ℝ
g : MultiseriesExpansion basis_tl → MultiseriesExpansion basis_tl'
exp : ℝ
coef : MultiseriesExpansion basis_tl
tl : Multiseries basis_hd basis_tl
⊢ map f g (cons exp coef tl) = cons (f exp) (g coef) (map f g tl)

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: b₁ b₂ b₃ : ℝ → ℝ
bs₁ bs₂ bs₃ : Basis
f₁ : ℝ → ℝ
g₁ : MultiseriesExpansion bs₁ → MultiseriesExpansion bs₂
f₂ : ℝ → ℝ
g₂ : MultiseriesExpansion bs₂ → MultiseriesExpansion bs₃
ms : Multiseries b₁ bs₁
⊢ map (f₂ ∘ f₁) (g₂ ∘ g₁) ms = map f₂ g₂ (map f₁ g₁ ms)

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: x e : ℝ
es : List ℝ
ih :
  ∀ {basis : Basis},
    WellFormedBasis basis →
      (∀ f ∈ basis, 0 < f x) → 0 < toFun es basis x ∧ (log ∘ toFun es basis) x = toLogFun es basis x
b : ℝ → ℝ
bs : List (ℝ → ℝ)
h_basis : WellFormedBasis (b :: bs)
hx : ∀ f ∈ b :: bs, 0 < f x
hpos : 0 < toFun es bs x
heq : (log ∘ toFun es bs) x = toLogFun es bs x
⊢ b ∈ b :: bs

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
R : ℝ × MultiseriesExpansion basis_tl → ℝ × MultiseriesExpansion basis_tl → Prop
⊢ Seq.Pairwise R nil

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
R : ℝ × MultiseriesExpansion basis_tl → ℝ × MultiseriesExpansion basis_tl → Prop
exp : ℝ
coef : MultiseriesExpansion basis_tl
⊢ Seq.Pairwise R (cons exp coef nil)

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : List (ℝ → ℝ)
motive : MultiseriesExpansion (basis_hd :: basis_tl) → Sort u_1
nil : (f : ℝ → ℝ) → motive (mk Multiseries.nil f)
cons :
  (exp : ℝ) →
    (coef : MultiseriesExpansion basis_tl) →
      (tl : Multiseries basis_hd basis_tl) → (f : ℝ → ℝ) → motive (mk (Multiseries.cons exp coef tl) f)
ms : MultiseriesExpansion (basis_hd :: basis_tl)
⊢ motive ms

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
s t : Multiseries basis_hd basis_tl
f g : ℝ → ℝ
h : mk s f = mk t g
⊢ s = t ∧ f = g

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
s t : Multiseries basis_hd basis_tl
f g : ℝ → ℝ
h : s = t ∧ f = g
⊢ mk s f = mk t g

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : List (ℝ → ℝ)
ms : MultiseriesExpansion (basis_hd :: basis_tl)
s : Multiseries basis_hd basis_tl
f : ℝ → ℝ
⊢ ms = mk s f ↔ ms.seq = s ∧ ms.toFun = f

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: case cons
x e : ℝ
es : List ℝ
ih :
  ∀ {basis : Basis},
    WellFormedBasis basis →
      (∀ f ∈ basis, 0 < f x) → 0 < toFun es basis x ∧ (log ∘ toFun es basis) x = toLogFun es basis x
basis : Basis
h_basis : WellFormedBasis basis
hx : ∀ f ∈ basis, 0 < f x
⊢ 0 < toFun (e :: es) basis x ∧ (log ∘ toFun (e :: es) basis) x = toLogFun (e :: es) basis x

Tactic:
cases basis with
    | nil => simp
    | cons b bs =>
      simp only [toFun_cons, Pi.mul_apply, Pi.pow_apply, Function.comp_apply, toLogFun_cons,
        Pi.add_apply, Pi.smul_apply, smul_eq_mul]
      obtain ⟨hpos, heq⟩ := ih h_basis.tail (hx · <| by simp [·])
      refine ⟨mul_pos (Real.rpow_pos_of_pos (hx b (by simp)) _) hpos, ?_⟩
      rw [Real.log_mul (Real.rpow_pos_of_pos (hx b (by simp)) _).ne' hpos.ne',
            Real.log_rpow (hx b (by simp)), ← heq]
      rfl

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : List (ℝ → ℝ)
ms : MultiseriesExpansion (basis_hd :: basis_tl)
s : Multiseries basis_hd basis_tl
f : ℝ → ℝ
⊢ mk s f = ms ↔ ms.seq = s ∧ ms.toFun = f

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: m : UnitMonomial
basis : Basis
h_basis : WellFormedBasis basis
⊢ log ∘ m.toFun basis =ᶠ[atTop] m.toLogFun basis

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : List (ℝ → ℝ)
ms₁ ms₂ : MultiseriesExpansion (basis_hd :: basis_tl)
h : ms₁ = ms₂
⊢ ms₁.seq = ms₂.seq ∧ ms₁.toFun = ms₂.toFun

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : List (ℝ → ℝ)
ms₁ ms₂ : MultiseriesExpansion (basis_hd :: basis_tl)
h : ms₁.seq = ms₂.seq ∧ ms₁.toFun = ms₂.toFun
⊢ ms₁ = ms₂

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
s : Multiseries basis_hd basis_tl
⊢ s.leadingExp = ⊥ ↔ s = nil

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis : Basis
exp1 exp2 : ℝ
coef1 coef2 : MultiseriesExpansion basis
⊢ (exp1, coef1) < (exp2, coef2) ↔ exp1 < exp2

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: exps_hd : ℝ
basis_hd : ℝ → ℝ
h_nonzero : exps_hd ≠ 0
e : ℝ
es : List ℝ
ih :
  ∀ {basis_tl : Basis},
    (∀ b ∈ basis_tl, (log ∘ b) =o[atTop] (log ∘ basis_hd)) → toLogFun es basis_tl =o[atTop] (log ∘ basis_hd)
b : ℝ → ℝ
bs : List (ℝ → ℝ)
hlo : ∀ b_1 ∈ b :: bs, (log ∘ b_1) =o[atTop] (log ∘ basis_hd)
⊢ b ∈ b :: bs

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: exps_hd : ℝ
basis_hd : ℝ → ℝ
h_nonzero : exps_hd ≠ 0
e : ℝ
es : List ℝ
ih :
  ∀ {basis_tl : Basis},
    (∀ b ∈ basis_tl, (log ∘ b) =o[atTop] (log ∘ basis_hd)) → toLogFun es basis_tl =o[atTop] (log ∘ basis_hd)
b : ℝ → ℝ
bs : List (ℝ → ℝ)
hlo : ∀ b_1 ∈ b :: bs, (log ∘ b_1) =o[atTop] (log ∘ basis_hd)
⊢ ∀ b ∈ bs, (log ∘ b) =o[atTop] (log ∘ basis_hd)

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
ms : MultiseriesExpansion (basis_hd :: basis_tl)
h : ms.Sorted
⊢ ms.seq.Sorted

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: exps_hd : ℝ
exps_tl : UnitMonomial
basis_hd : ℝ → ℝ
basis_tl : Basis
h_basis : WellFormedBasis (basis_hd :: basis_tl)
h_nonzero : exps_hd ≠ 0
⊢ toLogFun (exps_hd :: exps_tl) (basis_hd :: basis_tl) ~[atTop] exps_hd • log ∘ basis_hd

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
ms : MultiseriesExpansion (basis_hd :: basis_tl)
h : ms.seq.Sorted
⊢ ms.Sorted

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: exps_hd : ℝ
exps_tl : UnitMonomial
basis_hd : ℝ → ℝ
basis_tl : Basis
h_basis : WellFormedBasis (basis_hd :: basis_tl)
h_nonzero : 0 < exps_hd
h_equiv : log ∘ toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl) ~[atTop] exps_hd • log ∘ basis_hd
h_log : Tendsto (log ∘ toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl)) atTop atTop
⊢ Tendsto (toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl)) atTop atTop

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
⊢ Multiseries.nil.Sorted

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: exps_hd : ℝ
exps_tl : UnitMonomial
basis_hd : ℝ → ℝ
basis_tl : Basis
h_basis : WellFormedBasis (basis_hd :: basis_tl)
h_nonzero : 0 < exps_hd
h_equiv : log ∘ toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl) ~[atTop] exps_hd • log ∘ basis_hd
⊢ basis_hd ∈ basis_hd :: basis_tl

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
exp : ℝ
coef : MultiseriesExpansion basis_tl
h_coef : coef.Sorted
⊢ (cons exp coef Multiseries.nil).Sorted

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: exps_hd : ℝ
exps_tl : UnitMonomial
basis_hd : ℝ → ℝ
basis_tl : Basis
h_basis : WellFormedBasis (basis_hd :: basis_tl)
h_nonzero : 0 < exps_hd
⊢ Tendsto (toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl)) atTop atTop

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
exp : ℝ
coef : MultiseriesExpansion basis_tl
h_coef : coef.Sorted
exp✝ : ℝ
coef✝ : MultiseriesExpansion basis_tl
tl✝ : Multiseries basis_hd basis_tl
h_comp : (Multiseries.cons exp✝ coef✝ tl✝).leadingExp < ↑exp
h_tl_coef : ∀ x ∈ (mk (Multiseries.cons exp✝ coef✝ tl✝) 0).seq, x.2.Sorted
h_tl_tl : Seq.Pairwise (fun x1 x2 ↦ x1 > x2) (mk (Multiseries.cons exp✝ coef✝ tl✝) 0).seq
⊢ (exp, coef) > (exp✝, coef✝)

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: exps_hd : ℝ
exps_tl : UnitMonomial
basis_hd : ℝ → ℝ
basis_tl : Basis
h_basis : WellFormedBasis (basis_hd :: basis_tl)
h_nonzero : exps_hd < 0
h_equiv : log ∘ toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl) ~[atTop] exps_hd • log ∘ basis_hd
h_log : Tendsto (log ∘ toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl)) atTop atBot
⊢ Tendsto (toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl)) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: exps_hd : ℝ
exps_tl : UnitMonomial
basis_hd : ℝ → ℝ
basis_tl : Basis
h_basis : WellFormedBasis (basis_hd :: basis_tl)
h_nonzero : exps_hd < 0
h_equiv : log ∘ toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl) ~[atTop] exps_hd • log ∘ basis_hd
⊢ basis_hd ∈ basis_hd :: basis_tl

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
exp : ℝ
coef : MultiseriesExpansion basis_tl
tl : Multiseries basis_hd basis_tl
h_coef : coef.Sorted
h_comp : tl.leadingExp < ↑exp
h_tl : tl.Sorted
⊢ (Multiseries.cons exp coef tl).Sorted

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: exps_hd : ℝ
exps_tl : UnitMonomial
basis_hd : ℝ → ℝ
basis_tl : Basis
h_basis : WellFormedBasis (basis_hd :: basis_tl)
h_nonzero : exps_hd < 0
⊢ Tendsto (toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl)) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs

Goal state before:
pp: basis_hd : ℝ → ℝ
basis_tl : Basis
exp : ℝ
coef : MultiseriesExpansion basis_tl
tl : Multiseries basis_hd basis_tl
h_coef : ∀ x ∈ (mk (Multiseries.cons exp coef tl) 0).seq, x.2.Sorted
h_Pairwise : Seq.Pairwise (fun x1 x2 ↦ x1 > x2) (mk (Multiseries.cons exp coef tl) 0).seq
⊢ (exp, coef) ∈ (mk (Multiseries.cons exp coef tl) 0).seq

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Corecursion

Goal state before:
pp: α : Type u_1
op : Stream'.Seq α → Stream'.Seq α
motive : (Stream'.Seq α → Stream'.Seq α) → Prop
h_base : motive op
h_step :
  ∀ (op : Stream'.Seq α → Stream'.Seq α),
    motive op →
      ∃ T,
        ∀ (s : Stream'.Seq α),
          (op s).destruct =
            Option.map
              (fun x ↦
                match x with
                | (hd, opf, op') => (hd, ↑opf (↑op' s.tail)))
              (T s.head)
⊢ FriendlyOperation op

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: exps_hd : ℝ
exps_tl : List ℝ
basis_hd : ℝ → ℝ
basis_tl : List (ℝ → ℝ)
h_basis : WellFormedBasis (basis_hd :: basis_tl)
h_length : (exps_hd :: exps_tl).length = (basis_hd :: basis_tl).length
h : exps_hd = 0 ∧ FirstNonzeroIsPos exps_tl
⊢ toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl) = toFun exps_tl basis_tl

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: exps_hd : ℝ
exps_tl : List ℝ
basis_hd : ℝ → ℝ
basis_tl : List (ℝ → ℝ)
h_basis : WellFormedBasis (basis_hd :: basis_tl)
h_length : (exps_hd :: exps_tl).length = (basis_hd :: basis_tl).length
h : exps_hd = 0 ∧ FirstNonzeroIsPos exps_tl
h_eq : toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl) = toFun exps_tl basis_tl
⊢ exps_tl.length = basis_tl.length

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: m : UnitMonomial
basis : Basis
h_basis : WellFormedBasis basis
h_length : List.length m = List.length basis
h_firstIsPos : m.FirstNonzeroIsPos
⊢ Tendsto (m.toFun basis) atTop atTop

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: exps_hd : ℝ
exps_tl : List ℝ
basis_hd : ℝ → ℝ
basis_tl : List (ℝ → ℝ)
h_basis : WellFormedBasis (basis_hd :: basis_tl)
h_length : (exps_hd :: exps_tl).length = (basis_hd :: basis_tl).length
h : exps_hd = 0 ∧ FirstNonzeroIsNeg exps_tl
⊢ toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl) = toFun exps_tl basis_tl

Tactic:
by

Goal state after:
Module: Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic

Goal state before:
pp: exps_hd : ℝ
exps_tl : List ℝ
basis_hd : ℝ → ℝ
basis_tl : List (ℝ → ℝ)
h_basis : WellFormedBasis (basis_hd :: basis_tl)
h_length : (exps_hd :: exps_tl).length = (basis_hd :: basis_tl).length
h : exps_hd = 0 ∧ FirstNonzeroIsNeg exps_tl
h_eq : toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl) = toFun exps_tl basis_tl
⊢ exps_tl.length = basis_tl.length

Tactic:
by

Goal state after: