file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
ContDiffWithinAt.differentiableWithinAt
|
[] |
[
546,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
544,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
MultilinearMap.curryFinFinset_apply
|
[] |
[
1543,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1539,
1
] |
Mathlib/Analysis/InnerProductSpace/Adjoint.lean
|
IsSelfAdjoint.adjoint_conj
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1357503\ninst✝⁸ : IsROrC 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace 𝕜 F\ninst✝² : InnerProductSpace 𝕜 G\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] E\nhT : ↑adjoint T = T\nS : F →L[𝕜] E\n⊢ ↑adjoint (comp (↑adjoint S) (comp T S)) = comp (↑adjoint S) (comp T S)",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1357503\ninst✝⁸ : IsROrC 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace 𝕜 F\ninst✝² : InnerProductSpace 𝕜 G\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] E\nhT : IsSelfAdjoint T\nS : F →L[𝕜] E\n⊢ IsSelfAdjoint (comp (↑adjoint S) (comp T S))",
"tactic": "rw [isSelfAdjoint_iff'] at hT⊢"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1357503\ninst✝⁸ : IsROrC 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace 𝕜 F\ninst✝² : InnerProductSpace 𝕜 G\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] E\nhT : ↑adjoint T = T\nS : F →L[𝕜] E\n⊢ comp (comp (↑adjoint S) T) S = comp (↑adjoint S) (comp T S)",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1357503\ninst✝⁸ : IsROrC 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace 𝕜 F\ninst✝² : InnerProductSpace 𝕜 G\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] E\nhT : ↑adjoint T = T\nS : F →L[𝕜] E\n⊢ ↑adjoint (comp (↑adjoint S) (comp T S)) = comp (↑adjoint S) (comp T S)",
"tactic": "simp only [hT, adjoint_comp, adjoint_adjoint]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1357503\ninst✝⁸ : IsROrC 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace 𝕜 F\ninst✝² : InnerProductSpace 𝕜 G\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] E\nhT : ↑adjoint T = T\nS : F →L[𝕜] E\n⊢ comp (comp (↑adjoint S) T) S = comp (↑adjoint S) (comp T S)",
"tactic": "exact ContinuousLinearMap.comp_assoc _ _ _"
}
] |
[
305,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Algebra/BigOperators/Fin.lean
|
Fin.prod_ofFn
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.2004\nβ : Type u_1\ninst✝ : CommMonoid β\nn : ℕ\nf : Fin n → β\n⊢ List.prod (List.ofFn f) = ∏ i : Fin n, f i",
"tactic": "rw [List.ofFn_eq_map, prod_univ_def]"
}
] |
[
57,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Std/Data/List/Lemmas.lean
|
List.filter_eq_self
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\np : α✝ → Bool\nl : List α✝\n⊢ filter p l = l ↔ ∀ (a : α✝), a ∈ l → p a = true",
"tactic": "induction l with simp\n| cons a l ih =>\n cases h : p a <;> simp [*]\n intro h; exact Nat.lt_irrefl _ (h ▸ length_filter_le p l)"
},
{
"state_after": "case cons.false\nα✝ : Type u_1\np : α✝ → Bool\na : α✝\nl : List α✝\nih : filter p l = l ↔ ∀ (a : α✝), a ∈ l → p a = true\nh : p a = false\n⊢ ¬filter p l = a :: l",
"state_before": "case cons\nα✝ : Type u_1\np : α✝ → Bool\na : α✝\nl : List α✝\nih : filter p l = l ↔ ∀ (a : α✝), a ∈ l → p a = true\n⊢ filter p (a :: l) = a :: l ↔ p a = true ∧ ∀ (a : α✝), a ∈ l → p a = true",
"tactic": "cases h : p a <;> simp [*]"
},
{
"state_after": "case cons.false\nα✝ : Type u_1\np : α✝ → Bool\na : α✝\nl : List α✝\nih : filter p l = l ↔ ∀ (a : α✝), a ∈ l → p a = true\nh✝ : p a = false\nh : filter p l = a :: l\n⊢ False",
"state_before": "case cons.false\nα✝ : Type u_1\np : α✝ → Bool\na : α✝\nl : List α✝\nih : filter p l = l ↔ ∀ (a : α✝), a ∈ l → p a = true\nh : p a = false\n⊢ ¬filter p l = a :: l",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case cons.false\nα✝ : Type u_1\np : α✝ → Bool\na : α✝\nl : List α✝\nih : filter p l = l ↔ ∀ (a : α✝), a ∈ l → p a = true\nh✝ : p a = false\nh : filter p l = a :: l\n⊢ False",
"tactic": "exact Nat.lt_irrefl _ (h ▸ length_filter_le p l)"
}
] |
[
1246,
62
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1242,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
lowerSemicontinuousOn_tsum
|
[] |
[
653,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
651,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.curryFinFinset_apply_const
|
[
{
"state_after": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nk l : ℕ\ns : Finset (Fin n)\nhk : card s = k\nhl : card (sᶜ) = l\nf : ContinuousMultilinearMap 𝕜 (fun i => G) G'\nx y : G\n⊢ ↑(↑(LinearIsometryEquiv.symm (curryFinFinset 𝕜 G G' hk hl)) (↑(curryFinFinset 𝕜 G G' hk hl) f))\n (piecewise s (fun x_1 => x) fun x => y) =\n ↑f (piecewise s (fun x_1 => x) fun x => y)",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nk l : ℕ\ns : Finset (Fin n)\nhk : card s = k\nhl : card (sᶜ) = l\nf : ContinuousMultilinearMap 𝕜 (fun i => G) G'\nx y : G\n⊢ (↑(↑(↑(curryFinFinset 𝕜 G G' hk hl) f) fun x_1 => x) fun x => y) = ↑f (piecewise s (fun x_1 => x) fun x => y)",
"tactic": "refine' (curryFinFinset_symm_apply_piecewise_const hk hl _ _ _).symm.trans _"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nk l : ℕ\ns : Finset (Fin n)\nhk : card s = k\nhl : card (sᶜ) = l\nf : ContinuousMultilinearMap 𝕜 (fun i => G) G'\nx y : G\n⊢ ↑(↑(LinearIsometryEquiv.symm (curryFinFinset 𝕜 G G' hk hl)) (↑(curryFinFinset 𝕜 G G' hk hl) f))\n (piecewise s (fun x_1 => x) fun x => y) =\n ↑f (piecewise s (fun x_1 => x) fun x => y)",
"tactic": "rw [LinearIsometryEquiv.symm_apply_apply]"
}
] |
[
1897,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1892,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioc_subset_Ioc_union_Ioc
|
[] |
[
1591,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1590,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.edges_dropUntil_subset
|
[] |
[
1164,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1162,
1
] |
Mathlib/Algebra/Order/Sub/Defs.lean
|
le_add_tsub
|
[] |
[
100,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean
|
MulChar.inv_apply_eq_inv'
|
[] |
[
345,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/Control/Fold.lean
|
Traversable.foldlm_toList
|
[
{
"state_after": "no goals",
"state_before": "α β γ : Type u\nt : Type u → Type u\ninst✝³ : Traversable t\ninst✝² : IsLawfulTraversable t\nm : Type u → Type u\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nf : α → β → m α\nx : α\nxs : t β\n⊢ foldlm f x xs = unop (↑(foldlM.ofFreeMonoid f) (↑FreeMonoid.ofList (toList xs))) x",
"tactic": "simp only [foldlm, toList_spec, foldMap_hom_free (foldlM.ofFreeMonoid f),\n foldlm.ofFreeMonoid_comp_of, foldlM.get, FreeMonoid.ofList_toList]"
},
{
"state_after": "no goals",
"state_before": "α β γ : Type u\nt : Type u → Type u\ninst✝³ : Traversable t\ninst✝² : IsLawfulTraversable t\nm : Type u → Type u\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nf : α → β → m α\nx : α\nxs : t β\n⊢ unop (↑(foldlM.ofFreeMonoid f) (↑FreeMonoid.ofList (toList xs))) x = List.foldlM f x (toList xs)",
"tactic": "simp [foldlM.ofFreeMonoid, unop_op, flip]"
}
] |
[
413,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.bijOn_iUnion_of_directed
|
[] |
[
1637,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1635,
1
] |
Mathlib/Topology/Order/Basic.lean
|
dense_of_exists_between
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : Nontrivial α\ns : Set α\nh : ∀ ⦃a b : α⦄, a < b → ∃ c, c ∈ s ∧ a < c ∧ c < b\nU : Set α\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\n⊢ Set.Nonempty (U ∩ s)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : Nontrivial α\ns : Set α\nh : ∀ ⦃a b : α⦄, a < b → ∃ c, c ∈ s ∧ a < c ∧ c < b\n⊢ Dense s",
"tactic": "refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_"
},
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : Nontrivial α\ns : Set α\nh : ∀ ⦃a b : α⦄, a < b → ∃ c, c ∈ s ∧ a < c ∧ c < b\nU : Set α\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\na b : α\nhab : a < b\nH : Ioo a b ⊆ U\n⊢ Set.Nonempty (U ∩ s)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : Nontrivial α\ns : Set α\nh : ∀ ⦃a b : α⦄, a < b → ∃ c, c ∈ s ∧ a < c ∧ c < b\nU : Set α\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\n⊢ Set.Nonempty (U ∩ s)",
"tactic": "obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : Nontrivial α\ns : Set α\nh : ∀ ⦃a b : α⦄, a < b → ∃ c, c ∈ s ∧ a < c ∧ c < b\nU : Set α\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\na b : α\nhab : a < b\nH : Ioo a b ⊆ U\nx : α\nxs : x ∈ s\nhx : a < x ∧ x < b\n⊢ Set.Nonempty (U ∩ s)",
"state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : Nontrivial α\ns : Set α\nh : ∀ ⦃a b : α⦄, a < b → ∃ c, c ∈ s ∧ a < c ∧ c < b\nU : Set α\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\na b : α\nhab : a < b\nH : Ioo a b ⊆ U\n⊢ Set.Nonempty (U ∩ s)",
"tactic": "obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : Nontrivial α\ns : Set α\nh : ∀ ⦃a b : α⦄, a < b → ∃ c, c ∈ s ∧ a < c ∧ c < b\nU : Set α\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\na b : α\nhab : a < b\nH : Ioo a b ⊆ U\nx : α\nxs : x ∈ s\nhx : a < x ∧ x < b\n⊢ Set.Nonempty (U ∩ s)",
"tactic": "exact ⟨x, ⟨H hx, xs⟩⟩"
}
] |
[
1297,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1292,
1
] |
Mathlib/Data/Rat/Floor.lean
|
Rat.le_floor
|
[
{
"state_after": "α : Type ?u.897\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nz n : ℤ\nd : ℕ\nh : d ≠ 0\nc : Nat.coprime (natAbs n) d\n⊢ z ≤ n / ↑d ↔ ↑z ≤ mk' n d",
"state_before": "α : Type ?u.897\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nz n : ℤ\nd : ℕ\nh : d ≠ 0\nc : Nat.coprime (natAbs n) d\n⊢ z ≤ Rat.floor (mk' n d) ↔ ↑z ≤ mk' n d",
"tactic": "simp [Rat.floor_def']"
},
{
"state_after": "α : Type ?u.897\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nz n : ℤ\nd : ℕ\nh : d ≠ 0\nc : Nat.coprime (natAbs n) d\n⊢ z ≤ n / ↑d ↔ ↑z ≤ n /. ↑d",
"state_before": "α : Type ?u.897\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nz n : ℤ\nd : ℕ\nh : d ≠ 0\nc : Nat.coprime (natAbs n) d\n⊢ z ≤ n / ↑d ↔ ↑z ≤ mk' n d",
"tactic": "rw [num_den']"
},
{
"state_after": "α : Type ?u.897\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nz n : ℤ\nd : ℕ\nh : d ≠ 0\nc : Nat.coprime (natAbs n) d\nh' : ↑0 < ↑d\n⊢ z ≤ n / ↑d ↔ ↑z ≤ n /. ↑d",
"state_before": "α : Type ?u.897\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nz n : ℤ\nd : ℕ\nh : d ≠ 0\nc : Nat.coprime (natAbs n) d\n⊢ z ≤ n / ↑d ↔ ↑z ≤ n /. ↑d",
"tactic": "have h' := Int.ofNat_lt.2 (Nat.pos_of_ne_zero h)"
},
{
"state_after": "α : Type ?u.897\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nz n : ℤ\nd : ℕ\nh : d ≠ 0\nc : Nat.coprime (natAbs n) d\nh' : ↑0 < ↑d\n⊢ z ≤ n / ↑d ↔ z * ↑d ≤ n",
"state_before": "α : Type ?u.897\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nz n : ℤ\nd : ℕ\nh : d ≠ 0\nc : Nat.coprime (natAbs n) d\nh' : ↑0 < ↑d\n⊢ z ≤ n / ↑d ↔ ↑z ≤ n /. ↑d",
"tactic": "conv =>\n rhs\n rw [coe_int_eq_divInt, Rat.le_def zero_lt_one h', mul_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.897\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nz n : ℤ\nd : ℕ\nh : d ≠ 0\nc : Nat.coprime (natAbs n) d\nh' : ↑0 < ↑d\n⊢ z ≤ n / ↑d ↔ z * ↑d ≤ n",
"tactic": "exact Int.le_ediv_iff_mul_le h'"
}
] |
[
50,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
11
] |
Mathlib/Topology/Algebra/Star.lean
|
ContinuousOn.star
|
[] |
[
70,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.monotone_spanningSets
|
[] |
[
3468,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3467,
1
] |
Mathlib/Algebra/Support.lean
|
Function.mulSupport_prod_mk'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type ?u.16880\nA : Type ?u.16883\nB : Type ?u.16886\nM : Type u_1\nN : Type u_2\nP : Type ?u.16895\nR : Type ?u.16898\nS : Type ?u.16901\nG : Type ?u.16904\nM₀ : Type ?u.16907\nG₀ : Type ?u.16910\nι : Sort ?u.16913\ninst✝² : One M\ninst✝¹ : One N\ninst✝ : One P\nf : α → M × N\n⊢ mulSupport f = (mulSupport fun x => (f x).fst) ∪ mulSupport fun x => (f x).snd",
"tactic": "simp only [← mulSupport_prod_mk, Prod.mk.eta]"
}
] |
[
238,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
TFAE_exists_lt_isLittleO_pow
|
[
{
"state_after": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "have A : Ico 0 R ⊆ Ioo (-R) R :=\n fun x hx ↦ ⟨(neg_lt_zero.2 (hx.1.trans_lt hx.2)).trans_le hx.1, hx.2⟩"
},
{
"state_after": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "have B : Ioo 0 R ⊆ Ioo (-R) R := Subset.trans Ioo_subset_Ico_self A"
},
{
"state_after": "case tfae_1_to_3\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\n⊢ (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "tfae_have 1 → 3"
},
{
"state_after": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "case tfae_1_to_3\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\n⊢ (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "exact fun ⟨a, ha, H⟩ ↦ ⟨a, ha, H.isBigO⟩"
},
{
"state_after": "case tfae_2_to_1\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "tfae_have 2 → 1"
},
{
"state_after": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "case tfae_2_to_1\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "exact fun ⟨a, ha, H⟩ ↦ ⟨a, B ha, H⟩"
},
{
"state_after": "case tfae_3_to_2\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\n⊢ (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "tfae_have 3 → 2"
},
{
"state_after": "case tfae_2_to_4\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "tfae_have 2 → 4"
},
{
"state_after": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "case tfae_2_to_4\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "exact fun ⟨a, ha, H⟩ ↦ ⟨a, ha, H.isBigO⟩"
},
{
"state_after": "case tfae_4_to_3\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "tfae_have 4 → 3"
},
{
"state_after": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "case tfae_4_to_3\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "exact fun ⟨a, ha, H⟩ ↦ ⟨a, B ha, H⟩"
},
{
"state_after": "case tfae_4_to_6\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "tfae_have 4 → 6"
},
{
"state_after": "case tfae_6_to_5\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "tfae_have 6 → 5"
},
{
"state_after": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "case tfae_6_to_5\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "exact fun ⟨a, ha, C, H₀, H⟩ ↦ ⟨a, ha.2, C, Or.inl H₀, H⟩"
},
{
"state_after": "case tfae_5_to_3\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\n⊢ (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "tfae_have 5 → 3"
},
{
"state_after": "case tfae_2_to_8\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "tfae_have 2 → 8"
},
{
"state_after": "case tfae_8_to_7\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "tfae_have 8 → 7"
},
{
"state_after": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "case tfae_8_to_7\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "exact fun ⟨a, ha, H⟩ ↦ ⟨a, ha.2, H⟩"
},
{
"state_after": "case tfae_7_to_3\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n⊢ (∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_7_to_3 :\n (∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "tfae_have 7 → 3"
},
{
"state_after": "case tfae_6_to_7\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_7_to_3 :\n (∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_7_to_3 :\n (∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_6_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_7_to_3 :\n (∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "tfae_have 6 → 7"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_7_to_3 :\n (∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_6_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n⊢ TFAE\n [∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x,\n ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x, ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x,\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n,\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n,\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n]",
"tactic": "tfae_finish"
},
{
"state_after": "case tfae_3_to_2.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\na : ℝ\nha : a ∈ Set.Ioo (-R) R\nH : f =O[atTop] fun x => a ^ x\n⊢ ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x",
"state_before": "case tfae_3_to_2\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\n⊢ (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x",
"tactic": "rintro ⟨a, ha, H⟩"
},
{
"state_after": "case tfae_3_to_2.intro.intro.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\na : ℝ\nha : a ∈ Set.Ioo (-R) R\nH : f =O[atTop] fun x => a ^ x\nb : ℝ\nhab : abs a < b\nhbR : b < R\n⊢ ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x",
"state_before": "case tfae_3_to_2.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\na : ℝ\nha : a ∈ Set.Ioo (-R) R\nH : f =O[atTop] fun x => a ^ x\n⊢ ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x",
"tactic": "rcases exists_between (abs_lt.2 ha) with ⟨b, hab, hbR⟩"
},
{
"state_after": "no goals",
"state_before": "case tfae_3_to_2.intro.intro.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\na : ℝ\nha : a ∈ Set.Ioo (-R) R\nH : f =O[atTop] fun x => a ^ x\nb : ℝ\nhab : abs a < b\nhbR : b < R\n⊢ ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x",
"tactic": "exact ⟨b, ⟨(abs_nonneg a).trans_lt hab, hbR⟩,\n H.trans_isLittleO (isLittleO_pow_pow_of_abs_lt_left (hab.trans_le (le_abs_self b)))⟩"
},
{
"state_after": "case tfae_4_to_6.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\na : ℝ\nha : a ∈ Set.Ioo 0 R\nH : f =O[atTop] fun x => a ^ x\n⊢ ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n",
"state_before": "case tfae_4_to_6\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n",
"tactic": "rintro ⟨a, ha, H⟩"
},
{
"state_after": "case tfae_4_to_6.intro.intro.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\na : ℝ\nha : a ∈ Set.Ioo 0 R\nH : f =O[atTop] fun x => a ^ x\nC : ℝ\nhC₀ : C > 0\nhC : ∀ ⦃x : ℕ⦄, a ^ x ≠ 0 → ‖f x‖ ≤ C * ‖a ^ x‖\n⊢ ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n",
"state_before": "case tfae_4_to_6.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\na : ℝ\nha : a ∈ Set.Ioo 0 R\nH : f =O[atTop] fun x => a ^ x\n⊢ ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n",
"tactic": "rcases bound_of_isBigO_nat_atTop H with ⟨C, hC₀, hC⟩"
},
{
"state_after": "case tfae_4_to_6.intro.intro.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\na : ℝ\nha : a ∈ Set.Ioo 0 R\nH : f =O[atTop] fun x => a ^ x\nC : ℝ\nhC₀ : C > 0\nhC : ∀ ⦃x : ℕ⦄, a ^ x ≠ 0 → ‖f x‖ ≤ C * ‖a ^ x‖\nn : ℕ\n⊢ abs (f n) ≤ C * a ^ n",
"state_before": "case tfae_4_to_6.intro.intro.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\na : ℝ\nha : a ∈ Set.Ioo 0 R\nH : f =O[atTop] fun x => a ^ x\nC : ℝ\nhC₀ : C > 0\nhC : ∀ ⦃x : ℕ⦄, a ^ x ≠ 0 → ‖f x‖ ≤ C * ‖a ^ x‖\n⊢ ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n",
"tactic": "refine' ⟨a, ha, C, hC₀, fun n ↦ _⟩"
},
{
"state_after": "no goals",
"state_before": "case tfae_4_to_6.intro.intro.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\na : ℝ\nha : a ∈ Set.Ioo 0 R\nH : f =O[atTop] fun x => a ^ x\nC : ℝ\nhC₀ : C > 0\nhC : ∀ ⦃x : ℕ⦄, a ^ x ≠ 0 → ‖f x‖ ≤ C * ‖a ^ x‖\nn : ℕ\n⊢ abs (f n) ≤ C * a ^ n",
"tactic": "simpa only [Real.norm_eq_abs, abs_pow, abs_of_nonneg ha.1.le] using hC (pow_ne_zero n ha.1.ne')"
},
{
"state_after": "case tfae_5_to_3.intro.intro.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\na : ℝ\nha : a < R\nC : ℝ\nh₀ : 0 < C ∨ 0 < R\nH : ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\n⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x",
"state_before": "case tfae_5_to_3\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\n⊢ (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x",
"tactic": "rintro ⟨a, ha, C, h₀, H⟩"
},
{
"state_after": "case tfae_5_to_3.intro.intro.intro.intro.inl\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\na : ℝ\nha : a < R\nh₀ : 0 < 0 ∨ 0 < R\nH : ∀ (n : ℕ), abs (f n) ≤ 0 * a ^ n\n⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n\ncase tfae_5_to_3.intro.intro.intro.intro.inr.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\na : ℝ\nha : a < R\nC : ℝ\nh₀ : 0 < C ∨ 0 < R\nH : ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\nhC₀ : 0 < C\nha₀ : 0 ≤ a\n⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x",
"state_before": "case tfae_5_to_3.intro.intro.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\na : ℝ\nha : a < R\nC : ℝ\nh₀ : 0 < C ∨ 0 < R\nH : ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\n⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x",
"tactic": "rcases sign_cases_of_C_mul_pow_nonneg fun n ↦ (abs_nonneg _).trans (H n) with (rfl | ⟨hC₀, ha₀⟩)"
},
{
"state_after": "no goals",
"state_before": "case tfae_5_to_3.intro.intro.intro.intro.inr.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\na : ℝ\nha : a < R\nC : ℝ\nh₀ : 0 < C ∨ 0 < R\nH : ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\nhC₀ : 0 < C\nha₀ : 0 ≤ a\n⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x",
"tactic": "exact ⟨a, A ⟨ha₀, ha⟩,\n isBigO_of_le' _ fun n ↦ (H n).trans <| mul_le_mul_of_nonneg_left (le_abs_self _) hC₀.le⟩"
},
{
"state_after": "case tfae_5_to_3.intro.intro.intro.intro.inl\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\na : ℝ\nha : a < R\nh₀ : 0 < 0 ∨ 0 < R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n\nH : ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ 0 * a ^ n\n⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x",
"state_before": "case tfae_5_to_3.intro.intro.intro.intro.inl\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\na : ℝ\nha : a < R\nh₀ : 0 < 0 ∨ 0 < R\nH : ∀ (n : ℕ), abs (f n) ≤ 0 * a ^ n\n⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x",
"tactic": "obtain rfl : f = 0 := by\n ext n\n simpa using H n"
},
{
"state_after": "case tfae_5_to_3.intro.intro.intro.intro.inl\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\na : ℝ\nha : a < R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n\nH : ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ 0 * a ^ n\nh₀ : 0 < R\n⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x",
"state_before": "case tfae_5_to_3.intro.intro.intro.intro.inl\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\na : ℝ\nha : a < R\nh₀ : 0 < 0 ∨ 0 < R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n\nH : ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ 0 * a ^ n\n⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x",
"tactic": "simp only [lt_irrefl, false_or_iff] at h₀"
},
{
"state_after": "no goals",
"state_before": "case tfae_5_to_3.intro.intro.intro.intro.inl\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\na : ℝ\nha : a < R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ 0 =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ C * a ^ n\nH : ∀ (n : ℕ), abs (OfNat.ofNat 0 n) ≤ 0 * a ^ n\nh₀ : 0 < R\n⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ 0 =O[atTop] fun x => a ^ x",
"tactic": "exact ⟨0, ⟨neg_lt_zero.2 h₀, h₀⟩, isBigO_zero _ _⟩"
},
{
"state_after": "case h\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\na : ℝ\nha : a < R\nh₀ : 0 < 0 ∨ 0 < R\nH : ∀ (n : ℕ), abs (f n) ≤ 0 * a ^ n\nn : ℕ\n⊢ f n = OfNat.ofNat 0 n",
"state_before": "α : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\na : ℝ\nha : a < R\nh₀ : 0 < 0 ∨ 0 < R\nH : ∀ (n : ℕ), abs (f n) ≤ 0 * a ^ n\n⊢ f = 0",
"tactic": "ext n"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\na : ℝ\nha : a < R\nh₀ : 0 < 0 ∨ 0 < R\nH : ∀ (n : ℕ), abs (f n) ≤ 0 * a ^ n\nn : ℕ\n⊢ f n = OfNat.ofNat 0 n",
"tactic": "simpa using H n"
},
{
"state_after": "case tfae_2_to_8.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\na : ℝ\nha : a ∈ Set.Ioo 0 R\nH : f =o[atTop] fun x => a ^ x\n⊢ ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n",
"state_before": "case tfae_2_to_8\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n",
"tactic": "rintro ⟨a, ha, H⟩"
},
{
"state_after": "case tfae_2_to_8.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\na : ℝ\nha : a ∈ Set.Ioo 0 R\nH : f =o[atTop] fun x => a ^ x\nn : ℕ\nhn : ‖f n‖ ≤ 1 * ‖a ^ n‖\n⊢ abs (f n) ≤ a ^ n",
"state_before": "case tfae_2_to_8.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\na : ℝ\nha : a ∈ Set.Ioo 0 R\nH : f =o[atTop] fun x => a ^ x\n⊢ ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n",
"tactic": "refine' ⟨a, ha, (H.def zero_lt_one).mono fun n hn ↦ _⟩"
},
{
"state_after": "no goals",
"state_before": "case tfae_2_to_8.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\na : ℝ\nha : a ∈ Set.Ioo 0 R\nH : f =o[atTop] fun x => a ^ x\nn : ℕ\nhn : ‖f n‖ ≤ 1 * ‖a ^ n‖\n⊢ abs (f n) ≤ a ^ n",
"tactic": "rwa [Real.norm_eq_abs, Real.norm_eq_abs, one_mul, abs_pow, abs_of_pos ha.1] at hn"
},
{
"state_after": "case tfae_7_to_3.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\na : ℝ\nha : a < R\nH : ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x",
"state_before": "case tfae_7_to_3\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n⊢ (∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x",
"tactic": "rintro ⟨a, ha, H⟩"
},
{
"state_after": "case tfae_7_to_3.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\na : ℝ\nha : a < R\nH : ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\nthis : 0 ≤ a\n⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x",
"state_before": "case tfae_7_to_3.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\na : ℝ\nha : a < R\nH : ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\n⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x",
"tactic": "have : 0 ≤ a := nonneg_of_eventually_pow_nonneg (H.mono fun n ↦ (abs_nonneg _).trans)"
},
{
"state_after": "case tfae_7_to_3.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\na : ℝ\nha : a < R\nH : ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\nthis : 0 ≤ a\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖f x‖ ≤ 1 * ‖a ^ x‖",
"state_before": "case tfae_7_to_3.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\na : ℝ\nha : a < R\nH : ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\nthis : 0 ≤ a\n⊢ ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x",
"tactic": "refine' ⟨a, A ⟨this, ha⟩, IsBigO.of_bound 1 _⟩"
},
{
"state_after": "no goals",
"state_before": "case tfae_7_to_3.intro.intro\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\na : ℝ\nha : a < R\nH : ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\nthis : 0 ≤ a\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖f x‖ ≤ 1 * ‖a ^ x‖",
"tactic": "simpa only [Real.norm_eq_abs, one_mul, abs_pow, abs_of_nonneg this]"
},
{
"state_after": "no goals",
"state_before": "case tfae_6_to_7\nα : Type ?u.73187\nβ : Type ?u.73190\nι : Type ?u.73193\nf : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 :\n (∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_1 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =o[atTop] fun x => a ^ x\ntfae_3_to_2 : (∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x\ntfae_2_to_4 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_3 : (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_4_to_6 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =O[atTop] fun x => a ^ x) →\n ∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_6_to_5 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n\ntfae_5_to_3 :\n (∃ a, a < R ∧ ∃ C x, ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\ntfae_2_to_8 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ f =o[atTop] fun x => a ^ x) → ∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_8_to_7 :\n (∃ a, a ∈ Set.Ioo 0 R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n\ntfae_7_to_3 :\n (∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n) → ∃ a, a ∈ Set.Ioo (-R) R ∧ f =O[atTop] fun x => a ^ x\n⊢ (∃ a, a ∈ Set.Ioo 0 R ∧ ∃ C, C > 0 ∧ ∀ (n : ℕ), abs (f n) ≤ C * a ^ n) →\n ∃ a, a < R ∧ ∀ᶠ (n : ℕ) in atTop, abs (f n) ≤ a ^ n",
"tactic": "exact fun h ↦ tfae_8_to_7 <| tfae_2_to_8 <| tfae_3_to_2 <| tfae_5_to_3 <| tfae_6_to_5 h"
}
] |
[
187,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/SetTheory/Cardinal/Divisibility.lean
|
Cardinal.isUnit_iff
|
[
{
"state_after": "a b : Cardinal\nn m : ℕ\nh : IsUnit a\n⊢ a = 1",
"state_before": "a b : Cardinal\nn m : ℕ\n⊢ IsUnit a ↔ a = 1",
"tactic": "refine'\n ⟨fun h => _, by\n rintro rfl\n exact isUnit_one⟩"
},
{
"state_after": "case inl\nb : Cardinal\nn m : ℕ\nh : IsUnit 0\n⊢ 0 = 1\n\ncase inr\na b : Cardinal\nn m : ℕ\nh : IsUnit a\nha : a ≠ 0\n⊢ a = 1",
"state_before": "a b : Cardinal\nn m : ℕ\nh : IsUnit a\n⊢ a = 1",
"tactic": "rcases eq_or_ne a 0 with (rfl | ha)"
},
{
"state_after": "case inr\na b : Cardinal\nn m : ℕ\nh : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\n⊢ a = 1",
"state_before": "case inr\na b : Cardinal\nn m : ℕ\nh : IsUnit a\nha : a ≠ 0\n⊢ a = 1",
"tactic": "rw [isUnit_iff_forall_dvd] at h"
},
{
"state_after": "case inr.intro\na b : Cardinal\nn m : ℕ\nh : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\nt : Cardinal\nht : 1 = a * t\n⊢ a = 1",
"state_before": "case inr\na b : Cardinal\nn m : ℕ\nh : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\n⊢ a = 1",
"tactic": "cases' h 1 with t ht"
},
{
"state_after": "case inr.intro\na b : Cardinal\nn m : ℕ\nh : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\nt : Cardinal\nht : a = 1 ∧ t = 1\n⊢ a = 1\n\ncase inr.intro.ha\na b : Cardinal\nn m : ℕ\nh : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\nt : Cardinal\nht : a * t = 1\n⊢ 1 ≤ a\n\ncase inr.intro.hb\na b : Cardinal\nn m : ℕ\nh : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\nt : Cardinal\nht : a * t = 1\n⊢ 1 ≤ t",
"state_before": "case inr.intro\na b : Cardinal\nn m : ℕ\nh : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\nt : Cardinal\nht : 1 = a * t\n⊢ a = 1",
"tactic": "rw [eq_comm, mul_eq_one_iff'] at ht"
},
{
"state_after": "b : Cardinal\nn m : ℕ\n⊢ IsUnit 1",
"state_before": "a b : Cardinal\nn m : ℕ\n⊢ a = 1 → IsUnit a",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "b : Cardinal\nn m : ℕ\n⊢ IsUnit 1",
"tactic": "exact isUnit_one"
},
{
"state_after": "no goals",
"state_before": "case inl\nb : Cardinal\nn m : ℕ\nh : IsUnit 0\n⊢ 0 = 1",
"tactic": "exact (not_isUnit_zero h).elim"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\na b : Cardinal\nn m : ℕ\nh : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\nt : Cardinal\nht : a = 1 ∧ t = 1\n⊢ a = 1",
"tactic": "exact ht.1"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.ha\na b : Cardinal\nn m : ℕ\nh : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\nt : Cardinal\nht : a * t = 1\n⊢ 1 ≤ a",
"tactic": "exact one_le_iff_ne_zero.mpr ha"
},
{
"state_after": "case inr.intro.hb\na b : Cardinal\nn m : ℕ\nh : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\nt : Cardinal\nht : a * t = 1\n⊢ t ≠ 0",
"state_before": "case inr.intro.hb\na b : Cardinal\nn m : ℕ\nh : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\nt : Cardinal\nht : a * t = 1\n⊢ 1 ≤ t",
"tactic": "apply one_le_iff_ne_zero.mpr"
},
{
"state_after": "case inr.intro.hb\na b : Cardinal\nn m : ℕ\nh✝ : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\nt : Cardinal\nht : a * t = 1\nh : t = 0\n⊢ False",
"state_before": "case inr.intro.hb\na b : Cardinal\nn m : ℕ\nh : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\nt : Cardinal\nht : a * t = 1\n⊢ t ≠ 0",
"tactic": "intro h"
},
{
"state_after": "case inr.intro.hb\na b : Cardinal\nn m : ℕ\nh✝ : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\nt : Cardinal\nht : 0 = 1\nh : t = 0\n⊢ False",
"state_before": "case inr.intro.hb\na b : Cardinal\nn m : ℕ\nh✝ : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\nt : Cardinal\nht : a * t = 1\nh : t = 0\n⊢ False",
"tactic": "rw [h, mul_zero] at ht"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.hb\na b : Cardinal\nn m : ℕ\nh✝ : ∀ (y : Cardinal), a ∣ y\nha : a ≠ 0\nt : Cardinal\nht : 0 = 1\nh : t = 0\n⊢ False",
"tactic": "exact zero_ne_one ht"
}
] |
[
61,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/Topology/MetricSpace/Antilipschitz.lean
|
AntilipschitzWith.mul_le_edist
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.4707\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf : α → β\nhf : AntilipschitzWith K f\nx y : α\n⊢ edist x y / ↑K ≤ edist (f x) (f y)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.4707\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf : α → β\nhf : AntilipschitzWith K f\nx y : α\n⊢ (↑K)⁻¹ * edist x y ≤ edist (f x) (f y)",
"tactic": "rw [mul_comm, ← div_eq_mul_inv]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.4707\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf : α → β\nhf : AntilipschitzWith K f\nx y : α\n⊢ edist x y / ↑K ≤ edist (f x) (f y)",
"tactic": "exact ENNReal.div_le_of_le_mul' (hf x y)"
}
] |
[
115,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.mk_zpow
|
[] |
[
783,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
781,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.smul_mem
|
[] |
[
121,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/RingTheory/WittVector/IsPoly.lean
|
WittVector.mulIsPoly₂
|
[
{
"state_after": "p : ℕ\nR S : Type u\nσ : Type ?u.696391\nidx : Type ?u.696394\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Fact (Nat.Prime p)\nR✝ : Type u_1\ninst✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\n⊢ (x✝ * y✝).coeff = fun n => peval (wittMul p n) ![x✝.coeff, y✝.coeff]",
"state_before": "p : ℕ\nR S : Type u\nσ : Type ?u.696391\nidx : Type ?u.696394\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\n⊢ ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x y : 𝕎 R), (x * y).coeff = fun n => peval (wittMul p n) ![x.coeff, y.coeff]",
"tactic": "intros"
},
{
"state_after": "case h\np : ℕ\nR S : Type u\nσ : Type ?u.696391\nidx : Type ?u.696394\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Fact (Nat.Prime p)\nR✝ : Type u_1\ninst✝ : CommRing R✝\nx✝¹ y✝ : 𝕎 R✝\nx✝ : ℕ\n⊢ coeff (x✝¹ * y✝) x✝ = peval (wittMul p x✝) ![x✝¹.coeff, y✝.coeff]",
"state_before": "p : ℕ\nR S : Type u\nσ : Type ?u.696391\nidx : Type ?u.696394\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Fact (Nat.Prime p)\nR✝ : Type u_1\ninst✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\n⊢ (x✝ * y✝).coeff = fun n => peval (wittMul p n) ![x✝.coeff, y✝.coeff]",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\np : ℕ\nR S : Type u\nσ : Type ?u.696391\nidx : Type ?u.696394\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Fact (Nat.Prime p)\nR✝ : Type u_1\ninst✝ : CommRing R✝\nx✝¹ y✝ : 𝕎 R✝\nx✝ : ℕ\n⊢ coeff (x✝¹ * y✝) x✝ = peval (wittMul p x✝) ![x✝¹.coeff, y✝.coeff]",
"tactic": "exact mul_coeff _ _ _"
}
] |
[
550,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
547,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.center_le_centralizer
|
[] |
[
1407,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1406,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isBigOWith_principal
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.289553\nE : Type u_2\nF : Type u_3\nG : Type ?u.289562\nE' : Type ?u.289565\nF' : Type ?u.289568\nG' : Type ?u.289571\nE'' : Type ?u.289574\nF'' : Type ?u.289577\nG'' : Type ?u.289580\nR : Type ?u.289583\nR' : Type ?u.289586\n𝕜 : Type ?u.289589\n𝕜' : Type ?u.289592\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\ns : Set α\n⊢ (∀ᶠ (x : α) in 𝓟 s, ‖f x‖ ≤ c * ‖g x‖) ↔ ∀ (x : α), x ∈ s → ‖f x‖ ≤ c * ‖g x‖",
"state_before": "α : Type u_1\nβ : Type ?u.289553\nE : Type u_2\nF : Type u_3\nG : Type ?u.289562\nE' : Type ?u.289565\nF' : Type ?u.289568\nG' : Type ?u.289571\nE'' : Type ?u.289574\nF'' : Type ?u.289577\nG'' : Type ?u.289580\nR : Type ?u.289583\nR' : Type ?u.289586\n𝕜 : Type ?u.289589\n𝕜' : Type ?u.289592\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\ns : Set α\n⊢ IsBigOWith c (𝓟 s) f g ↔ ∀ (x : α), x ∈ s → ‖f x‖ ≤ c * ‖g x‖",
"tactic": "rw [IsBigOWith_def]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.289553\nE : Type u_2\nF : Type u_3\nG : Type ?u.289562\nE' : Type ?u.289565\nF' : Type ?u.289568\nG' : Type ?u.289571\nE'' : Type ?u.289574\nF'' : Type ?u.289577\nG'' : Type ?u.289580\nR : Type ?u.289583\nR' : Type ?u.289586\n𝕜 : Type ?u.289589\n𝕜' : Type ?u.289592\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\ns : Set α\n⊢ (∀ᶠ (x : α) in 𝓟 s, ‖f x‖ ≤ c * ‖g x‖) ↔ ∀ (x : α), x ∈ s → ‖f x‖ ≤ c * ‖g x‖",
"tactic": "rfl"
}
] |
[
1296,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1295,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiffWithinAt.fderivWithin_apply
|
[] |
[
1028,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1022,
1
] |
Mathlib/Logic/Basic.lean
|
BEx.imp_left
|
[] |
[
1059,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1058,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
LinearMap.rTensor_mul
|
[] |
[
1089,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1088,
1
] |
Mathlib/Algebra/GroupWithZero/Units/Basic.lean
|
div_ne_zero_iff
|
[] |
[
294,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/CategoryTheory/Limits/Constructions/LimitsOfProductsAndEqualizers.lean
|
CategoryTheory.Limits.hasFiniteColimits_of_hasInitial_and_pushouts
|
[] |
[
474,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
468,
1
] |
Mathlib/Data/List/Destutter.lean
|
List.destutter_eq_self_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na b : α\n⊢ destutter R [] = [] ↔ Chain' R []",
"tactic": "simp"
}
] |
[
166,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.modEq_of_xn_modEq
|
[
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j % (4 * n)\n⊢ 0 < 4",
"tactic": "decide"
},
{
"state_after": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\n⊢ j % (4 * n) = j' % (4 * n)",
"state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\n⊢ j ≡ j' [MOD 4 * n]",
"tactic": "delta ModEq"
},
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\n⊢ j % (4 * n) = j' % (4 * n)",
"tactic": "rw [Nat.mod_eq_of_lt jl]"
},
{
"state_after": "a : ℕ\na1 : 1 < a\ni j✝ n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j✝ ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j✝ % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j✝ ≡ j' [MOD 4 * n]\nj q : ℕ\n⊢ xn a1 (j + 4 * n * q) ≡ xn a1 j [MOD xn a1 n]",
"state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j ≡ j' [MOD 4 * n]\n⊢ ∀ (j_1 q : ℕ), xn a1 (j_1 + 4 * n * q) ≡ xn a1 j_1 [MOD xn a1 n]",
"tactic": "intro j q"
},
{
"state_after": "case zero\na : ℕ\na1 : 1 < a\ni j✝ n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j✝ ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j✝ % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j✝ ≡ j' [MOD 4 * n]\nj : ℕ\n⊢ xn a1 (j + 4 * n * zero) ≡ xn a1 j [MOD xn a1 n]\n\ncase succ\na : ℕ\na1 : 1 < a\ni j✝ n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j✝ ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j✝ % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j✝ ≡ j' [MOD 4 * n]\nj q : ℕ\nIH : xn a1 (j + 4 * n * q) ≡ xn a1 j [MOD xn a1 n]\n⊢ xn a1 (j + 4 * n * succ q) ≡ xn a1 j [MOD xn a1 n]",
"state_before": "a : ℕ\na1 : 1 < a\ni j✝ n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j✝ ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j✝ % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j✝ ≡ j' [MOD 4 * n]\nj q : ℕ\n⊢ xn a1 (j + 4 * n * q) ≡ xn a1 j [MOD xn a1 n]",
"tactic": "induction' q with q IH"
},
{
"state_after": "case succ\na : ℕ\na1 : 1 < a\ni j✝ n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j✝ ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j✝ % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j✝ ≡ j' [MOD 4 * n]\nj q : ℕ\nIH : xn a1 (j + 4 * n * q) ≡ xn a1 j [MOD xn a1 n]\n⊢ xn a1 (j + 4 * n * succ q) ≡ xn a1 j [MOD xn a1 n]",
"state_before": "case zero\na : ℕ\na1 : 1 < a\ni j✝ n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j✝ ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j✝ % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j✝ ≡ j' [MOD 4 * n]\nj : ℕ\n⊢ xn a1 (j + 4 * n * zero) ≡ xn a1 j [MOD xn a1 n]\n\ncase succ\na : ℕ\na1 : 1 < a\ni j✝ n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j✝ ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j✝ % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j✝ ≡ j' [MOD 4 * n]\nj q : ℕ\nIH : xn a1 (j + 4 * n * q) ≡ xn a1 j [MOD xn a1 n]\n⊢ xn a1 (j + 4 * n * succ q) ≡ xn a1 j [MOD xn a1 n]",
"tactic": ". simp [ModEq.refl]"
},
{
"state_after": "case succ\na : ℕ\na1 : 1 < a\ni j✝ n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j✝ ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j✝ % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j✝ ≡ j' [MOD 4 * n]\nj q : ℕ\nIH : xn a1 (j + 4 * n * q) ≡ xn a1 j [MOD xn a1 n]\n⊢ xn a1 (4 * n + (j + 4 * n * q)) ≡ xn a1 j [MOD xn a1 n]",
"state_before": "case succ\na : ℕ\na1 : 1 < a\ni j✝ n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j✝ ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j✝ % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j✝ ≡ j' [MOD 4 * n]\nj q : ℕ\nIH : xn a1 (j + 4 * n * q) ≡ xn a1 j [MOD xn a1 n]\n⊢ xn a1 (j + 4 * n * succ q) ≡ xn a1 j [MOD xn a1 n]",
"tactic": "rw [Nat.mul_succ, ← add_assoc, add_comm]"
},
{
"state_after": "no goals",
"state_before": "case succ\na : ℕ\na1 : 1 < a\ni j✝ n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j✝ ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j✝ % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j✝ ≡ j' [MOD 4 * n]\nj q : ℕ\nIH : xn a1 (j + 4 * n * q) ≡ xn a1 j [MOD xn a1 n]\n⊢ xn a1 (4 * n + (j + 4 * n * q)) ≡ xn a1 j [MOD xn a1 n]",
"tactic": "exact (xn_modEq_x4n_add _ _ _).trans IH"
},
{
"state_after": "no goals",
"state_before": "case zero\na : ℕ\na1 : 1 < a\ni j✝ n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j✝ ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j✝ % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j✝ ≡ j' [MOD 4 * n]\nj : ℕ\n⊢ xn a1 (j + 4 * n * zero) ≡ xn a1 j [MOD xn a1 n]",
"tactic": "simp [ModEq.refl]"
},
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j ≡ j' [MOD 4 * n]\nthis : ∀ (j_1 q : ℕ), xn a1 (j_1 + 4 * n * q) ≡ xn a1 j_1 [MOD xn a1 n]\nji : j' = i\n⊢ j ≡ i [MOD 4 * n]",
"tactic": "rwa [← ji]"
},
{
"state_after": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j ≡ j' [MOD 4 * n]\nthis : ∀ (j_1 q : ℕ), xn a1 (j_1 + 4 * n * q) ≡ xn a1 j_1 [MOD xn a1 n]\nji : j' + i = 4 * n\n⊢ 4 * n ≡ 0 [MOD 4 * n]",
"state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j ≡ j' [MOD 4 * n]\nthis : ∀ (j_1 q : ℕ), xn a1 (j_1 + 4 * n * q) ≡ xn a1 j_1 [MOD xn a1 n]\nji : j' + i = 4 * n\n⊢ j' + i ≡ 0 [MOD 4 * n]",
"tactic": "rw [ji]"
},
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j ≡ j' [MOD 4 * n]\nthis : ∀ (j_1 q : ℕ), xn a1 (j_1 + 4 * n * q) ≡ xn a1 j_1 [MOD xn a1 n]\nji : j' + i = 4 * n\n⊢ 4 * n ≡ 0 [MOD 4 * n]",
"tactic": "exact dvd_rfl.modEq_zero_nat"
},
{
"state_after": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j ≡ j' [MOD 4 * n]\nthis : ∀ (j_1 q : ℕ), xn a1 (j_1 + 4 * n * q) ≡ xn a1 j_1 [MOD xn a1 n]\n⊢ xn a1 (j % (4 * n) + 4 * n * (j / (4 * n))) ≡ xn a1 j' [MOD xn a1 n]",
"state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j ≡ j' [MOD 4 * n]\nthis : ∀ (j_1 q : ℕ), xn a1 (j_1 + 4 * n * q) ≡ xn a1 j_1 [MOD xn a1 n]\n⊢ xn a1 j ≡ xn a1 j' [MOD xn a1 n]",
"tactic": "rw [← Nat.mod_add_div j (4 * n)]"
},
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nipos : 0 < i\nhin : i ≤ n\nh : xn a1 j ≡ xn a1 i [MOD xn a1 n]\nj' : ℕ := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j ≡ j' [MOD 4 * n]\nthis : ∀ (j_1 q : ℕ), xn a1 (j_1 + 4 * n * q) ≡ xn a1 j_1 [MOD xn a1 n]\n⊢ xn a1 (j % (4 * n) + 4 * n * (j / (4 * n))) ≡ xn a1 j' [MOD xn a1 n]",
"tactic": "exact this j' _"
}
] |
[
823,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
803,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
WithSeminorms.isVonNBounded_iff_finset_seminorm_bounded
|
[
{
"state_after": "𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\n⊢ (∀ (i : Set E), i ∈ SeminormFamily.basisSets p → Absorbs 𝕜 (id i) s) ↔\n ∀ (I : Finset ι), ∃ r, r > 0 ∧ ∀ (x : E), x ∈ s → ↑(Finset.sup I p) x < r",
"state_before": "𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\n⊢ Bornology.IsVonNBounded 𝕜 s ↔ ∀ (I : Finset ι), ∃ r, r > 0 ∧ ∀ (x : E), x ∈ s → ↑(Finset.sup I p) x < r",
"tactic": "rw [hp.hasBasis.isVonNBounded_basis_iff]"
},
{
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{
"state_after": "case mp.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nhr : 0 < r\nh : ∀ (a : 𝕜), r ≤ ‖a‖ → s ⊆ a • ball (Finset.sup I p) 0 1\n⊢ ∃ r, r > 0 ∧ ∀ (x : E), x ∈ s → ↑(Finset.sup I p) x < r",
"state_before": "case mp\n𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nI : Finset ι\nh : Absorbs 𝕜 (ball (Finset.sup I p) 0 1) s\n⊢ ∃ r, r > 0 ∧ ∀ (x : E), x ∈ s → ↑(Finset.sup I p) x < r",
"tactic": "rcases h with ⟨r, hr, h⟩"
},
{
"state_after": "case mp.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nhr : 0 < r\nh : ∀ (a : 𝕜), r ≤ ‖a‖ → s ⊆ a • ball (Finset.sup I p) 0 1\na : 𝕜\nha : r < ‖a‖\n⊢ ∃ r, r > 0 ∧ ∀ (x : E), x ∈ s → ↑(Finset.sup I p) x < r",
"state_before": "case mp.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nhr : 0 < r\nh : ∀ (a : 𝕜), r ≤ ‖a‖ → s ⊆ a • ball (Finset.sup I p) 0 1\n⊢ ∃ r, r > 0 ∧ ∀ (x : E), x ∈ s → ↑(Finset.sup I p) x < r",
"tactic": "cases' NormedField.exists_lt_norm 𝕜 r with a ha"
},
{
"state_after": "case mp.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nhr : 0 < r\na : 𝕜\nha : r < ‖a‖\nh : s ⊆ a • ball (Finset.sup I p) 0 1\n⊢ ∃ r, r > 0 ∧ ∀ (x : E), x ∈ s → ↑(Finset.sup I p) x < r",
"state_before": "case mp.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nhr : 0 < r\nh : ∀ (a : 𝕜), r ≤ ‖a‖ → s ⊆ a • ball (Finset.sup I p) 0 1\na : 𝕜\nha : r < ‖a‖\n⊢ ∃ r, r > 0 ∧ ∀ (x : E), x ∈ s → ↑(Finset.sup I p) x < r",
"tactic": "specialize h a (le_of_lt ha)"
},
{
"state_after": "case mp.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nhr : 0 < r\na : 𝕜\nha : r < ‖a‖\nh : s ⊆ ball (Finset.sup I p) 0 ‖a‖\n⊢ ∃ r, r > 0 ∧ ∀ (x : E), x ∈ s → ↑(Finset.sup I p) x < r",
"state_before": "case mp.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nhr : 0 < r\na : 𝕜\nha : r < ‖a‖\nh : s ⊆ a • ball (Finset.sup I p) 0 1\n⊢ ∃ r, r > 0 ∧ ∀ (x : E), x ∈ s → ↑(Finset.sup I p) x < r",
"tactic": "rw [Seminorm.smul_ball_zero (norm_pos_iff.1 <| hr.trans ha), mul_one] at h"
},
{
"state_after": "case mp.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nhr : 0 < r\na : 𝕜\nha : r < ‖a‖\nh : s ⊆ ball (Finset.sup I p) 0 ‖a‖\n⊢ ∀ (x : E), x ∈ s → ↑(Finset.sup I p) x < ‖a‖",
"state_before": "case mp.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nhr : 0 < r\na : 𝕜\nha : r < ‖a‖\nh : s ⊆ ball (Finset.sup I p) 0 ‖a‖\n⊢ ∃ r, r > 0 ∧ ∀ (x : E), x ∈ s → ↑(Finset.sup I p) x < r",
"tactic": "refine' ⟨‖a‖, lt_trans hr ha, _⟩"
},
{
"state_after": "case mp.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nhr : 0 < r\na : 𝕜\nha : r < ‖a‖\nh : s ⊆ ball (Finset.sup I p) 0 ‖a‖\nx : E\nhx : x ∈ s\n⊢ ↑(Finset.sup I p) x < ‖a‖",
"state_before": "case mp.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nhr : 0 < r\na : 𝕜\nha : r < ‖a‖\nh : s ⊆ ball (Finset.sup I p) 0 ‖a‖\n⊢ ∀ (x : E), x ∈ s → ↑(Finset.sup I p) x < ‖a‖",
"tactic": "intro x hx"
},
{
"state_after": "case mp.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nhr : 0 < r\na : 𝕜\nha : r < ‖a‖\nx : E\nhx : x ∈ s\nh : x ∈ ball (Finset.sup I p) 0 ‖a‖\n⊢ ↑(Finset.sup I p) x < ‖a‖",
"state_before": "case mp.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nhr : 0 < r\na : 𝕜\nha : r < ‖a‖\nh : s ⊆ ball (Finset.sup I p) 0 ‖a‖\nx : E\nhx : x ∈ s\n⊢ ↑(Finset.sup I p) x < ‖a‖",
"tactic": "specialize h hx"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.464932\n𝕝 : Type ?u.464935\n𝕝₂ : Type ?u.464938\nE : Type u_1\nF : Type ?u.464944\nG : Type ?u.464947\nι : Type u_3\nι' : Type ?u.464953\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nhr : 0 < r\na : 𝕜\nha : r < ‖a‖\nx : E\nhx : x ∈ s\nh : x ∈ ball (Finset.sup I p) 0 ‖a‖\n⊢ ↑(Finset.sup I p) x < ‖a‖",
"tactic": "exact (Finset.sup I p).mem_ball_zero.mp h"
}
] |
[
542,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
521,
1
] |
Mathlib/Topology/ContinuousFunction/Basic.lean
|
ContinuousMap.congr_arg
|
[] |
[
159,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
11
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean
|
AlgebraicGeometry.StructureSheaf.res_const'
|
[] |
[
353,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
1
] |
Mathlib/Data/Rat/Defs.lean
|
Rat.add_zero
|
[
{
"state_after": "no goals",
"state_before": "a b c : ℚ\nn : ℤ\nd : ℕ\nh : d ≠ 0\n⊢ n /. ↑d + 0 = n /. ↑d",
"tactic": "rw [← zero_divInt d, add_def'', zero_mul, add_zero, divInt_mul_right] <;> simp [h]"
}
] |
[
203,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
11
] |
Mathlib/Control/Fold.lean
|
Traversable.foldl.unop_ofFreeMonoid
|
[] |
[
275,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.lift.range_le
|
[
{
"state_after": "case intro.mk\nα : Type u\nL✝ L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\ns : Subgroup β\nH : Set.range f ⊆ ↑s\nw✝ : FreeGroup α\nL : List (α × Bool)\n⊢ ↑(↑lift f) (Quot.mk Red.Step L) ∈ s",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\ns : Subgroup β\nH : Set.range f ⊆ ↑s\n⊢ MonoidHom.range (↑lift f) ≤ s",
"tactic": "rintro _ ⟨⟨L⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.mk\nα : Type u\nL✝ L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\ns : Subgroup β\nH : Set.range f ⊆ ↑s\nw✝ : FreeGroup α\nL : List (α × Bool)\n⊢ ↑(↑lift f) (Quot.mk Red.Step L) ∈ s",
"tactic": "exact\n List.recOn L s.one_mem fun ⟨x, b⟩ tl ih =>\n Bool.recOn b (by simp at ih⊢; exact s.mul_mem (s.inv_mem <| H ⟨x, rfl⟩) ih)\n (by simp at ih⊢; exact s.mul_mem (H ⟨x, rfl⟩) ih)"
},
{
"state_after": "α : Type u\nL✝ L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx✝¹ y : FreeGroup α\ns : Subgroup β\nH : Set.range f ⊆ ↑s\nw✝ : FreeGroup α\nL : List (α × Bool)\nx✝ : α × Bool\ntl : List (α × Bool)\nih : List.prod (List.map (fun x => bif x.snd then f x.fst else (f x.fst)⁻¹) tl) ∈ s\nx : α\nb : Bool\n⊢ (f x)⁻¹ * List.prod (List.map (fun x => bif x.snd then f x.fst else (f x.fst)⁻¹) tl) ∈ s",
"state_before": "α : Type u\nL✝ L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx✝¹ y : FreeGroup α\ns : Subgroup β\nH : Set.range f ⊆ ↑s\nw✝ : FreeGroup α\nL : List (α × Bool)\nx✝ : α × Bool\ntl : List (α × Bool)\nih : ↑(↑lift f) (Quot.mk Red.Step tl) ∈ s\nx : α\nb : Bool\n⊢ ↑(↑lift f) (Quot.mk Red.Step ((x, false) :: tl)) ∈ s",
"tactic": "simp at ih⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nL✝ L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx✝¹ y : FreeGroup α\ns : Subgroup β\nH : Set.range f ⊆ ↑s\nw✝ : FreeGroup α\nL : List (α × Bool)\nx✝ : α × Bool\ntl : List (α × Bool)\nih : List.prod (List.map (fun x => bif x.snd then f x.fst else (f x.fst)⁻¹) tl) ∈ s\nx : α\nb : Bool\n⊢ (f x)⁻¹ * List.prod (List.map (fun x => bif x.snd then f x.fst else (f x.fst)⁻¹) tl) ∈ s",
"tactic": "exact s.mul_mem (s.inv_mem <| H ⟨x, rfl⟩) ih"
},
{
"state_after": "α : Type u\nL✝ L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx✝¹ y : FreeGroup α\ns : Subgroup β\nH : Set.range f ⊆ ↑s\nw✝ : FreeGroup α\nL : List (α × Bool)\nx✝ : α × Bool\ntl : List (α × Bool)\nih : List.prod (List.map (fun x => bif x.snd then f x.fst else (f x.fst)⁻¹) tl) ∈ s\nx : α\nb : Bool\n⊢ f x * List.prod (List.map (fun x => bif x.snd then f x.fst else (f x.fst)⁻¹) tl) ∈ s",
"state_before": "α : Type u\nL✝ L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx✝¹ y : FreeGroup α\ns : Subgroup β\nH : Set.range f ⊆ ↑s\nw✝ : FreeGroup α\nL : List (α × Bool)\nx✝ : α × Bool\ntl : List (α × Bool)\nih : ↑(↑lift f) (Quot.mk Red.Step tl) ∈ s\nx : α\nb : Bool\n⊢ ↑(↑lift f) (Quot.mk Red.Step ((x, true) :: tl)) ∈ s",
"tactic": "simp at ih⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nL✝ L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx✝¹ y : FreeGroup α\ns : Subgroup β\nH : Set.range f ⊆ ↑s\nw✝ : FreeGroup α\nL : List (α × Bool)\nx✝ : α × Bool\ntl : List (α × Bool)\nih : List.prod (List.map (fun x => bif x.snd then f x.fst else (f x.fst)⁻¹) tl) ∈ s\nx : α\nb : Bool\n⊢ f x * List.prod (List.map (fun x => bif x.snd then f x.fst else (f x.fst)⁻¹) tl) ∈ s",
"tactic": "exact s.mul_mem (H ⟨x, rfl⟩) ih"
}
] |
[
769,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
764,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.sInter_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nf g : Filter α\ns✝ t : Set α\ns : Set (Set α)\nhfin : Set.Finite s\n⊢ ⋂₀ s ∈ f ↔ ∀ (U : Set α), U ∈ s → U ∈ f",
"tactic": "rw [sInter_eq_biInter, biInter_mem hfin]"
}
] |
[
203,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/ModelTheory/Substructures.lean
|
FirstOrder.Language.Embedding.codRestrict_apply
|
[] |
[
932,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
930,
1
] |
Mathlib/Topology/Order/Lattice.lean
|
Filter.Tendsto.inf_right_nhds'
|
[] |
[
124,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.image₂_insert_left
|
[
{
"state_after": "α : Type u_1\nα' : Type ?u.36962\nβ : Type u_3\nβ' : Type ?u.36968\nγ : Type u_2\nγ' : Type ?u.36974\nδ : Type ?u.36977\nδ' : Type ?u.36980\nε : Type ?u.36983\nε' : Type ?u.36986\nζ : Type ?u.36989\nζ' : Type ?u.36992\nν : Type ?u.36995\ninst✝⁸ : DecidableEq α'\ninst✝⁷ : DecidableEq β'\ninst✝⁶ : DecidableEq γ\ninst✝⁵ : DecidableEq γ'\ninst✝⁴ : DecidableEq δ\ninst✝³ : DecidableEq δ'\ninst✝² : DecidableEq ε\ninst✝¹ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝ : DecidableEq α\n⊢ image2 f (insert a ↑s) ↑t = (fun b => f a b) '' ↑t ∪ image2 f ↑s ↑t",
"state_before": "α : Type u_1\nα' : Type ?u.36962\nβ : Type u_3\nβ' : Type ?u.36968\nγ : Type u_2\nγ' : Type ?u.36974\nδ : Type ?u.36977\nδ' : Type ?u.36980\nε : Type ?u.36983\nε' : Type ?u.36986\nζ : Type ?u.36989\nζ' : Type ?u.36992\nν : Type ?u.36995\ninst✝⁸ : DecidableEq α'\ninst✝⁷ : DecidableEq β'\ninst✝⁶ : DecidableEq γ\ninst✝⁵ : DecidableEq γ'\ninst✝⁴ : DecidableEq δ\ninst✝³ : DecidableEq δ'\ninst✝² : DecidableEq ε\ninst✝¹ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝ : DecidableEq α\n⊢ ↑(image₂ f (insert a s) t) = ↑(image (fun b => f a b) t ∪ image₂ f s t)",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.36962\nβ : Type u_3\nβ' : Type ?u.36968\nγ : Type u_2\nγ' : Type ?u.36974\nδ : Type ?u.36977\nδ' : Type ?u.36980\nε : Type ?u.36983\nε' : Type ?u.36986\nζ : Type ?u.36989\nζ' : Type ?u.36992\nν : Type ?u.36995\ninst✝⁸ : DecidableEq α'\ninst✝⁷ : DecidableEq β'\ninst✝⁶ : DecidableEq γ\ninst✝⁵ : DecidableEq γ'\ninst✝⁴ : DecidableEq δ\ninst✝³ : DecidableEq δ'\ninst✝² : DecidableEq ε\ninst✝¹ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝ : DecidableEq α\n⊢ image2 f (insert a ↑s) ↑t = (fun b => f a b) '' ↑t ∪ image2 f ↑s ↑t",
"tactic": "exact image2_insert_left"
}
] |
[
187,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Data/List/Basic.lean
|
List.dropWhile_eq_nil_iff
|
[
{
"state_after": "case nil\nι : Type ?u.387507\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\n⊢ dropWhile p [] = [] ↔ ∀ (x : α), x ∈ [] → p x = true\n\ncase cons\nι : Type ?u.387507\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\nx : α\nxs : List α\nIH : dropWhile p xs = [] ↔ ∀ (x : α), x ∈ xs → p x = true\n⊢ dropWhile p (x :: xs) = [] ↔ ∀ (x_1 : α), x_1 ∈ x :: xs → p x_1 = true",
"state_before": "ι : Type ?u.387507\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\n⊢ dropWhile p l = [] ↔ ∀ (x : α), x ∈ l → p x = true",
"tactic": "induction' l with x xs IH"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.387507\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\n⊢ dropWhile p [] = [] ↔ ∀ (x : α), x ∈ [] → p x = true",
"tactic": "simp [dropWhile]"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type ?u.387507\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nl : List α\nx : α\nxs : List α\nIH : dropWhile p xs = [] ↔ ∀ (x : α), x ∈ xs → p x = true\n⊢ dropWhile p (x :: xs) = [] ↔ ∀ (x_1 : α), x_1 ∈ x :: xs → p x_1 = true",
"tactic": "by_cases hp : p x <;> simp [hp, dropWhile, IH]"
}
] |
[
3632,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3629,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.preimage_embedding
|
[] |
[
896,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
895,
1
] |
Mathlib/NumberTheory/Padics/PadicVal.lean
|
padicValInt.eq_zero_of_not_dvd
|
[
{
"state_after": "p : ℕ\nz : ℤ\nh : ¬↑p ∣ z\n⊢ (if h : p ≠ 1 ∧ 0 < Int.natAbs z then\n Part.get (multiplicity p (Int.natAbs z)) (_ : multiplicity.Finite p (Int.natAbs z))\n else 0) =\n 0",
"state_before": "p : ℕ\nz : ℤ\nh : ¬↑p ∣ z\n⊢ padicValInt p z = 0",
"tactic": "rw [padicValInt, padicValNat]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nz : ℤ\nh : ¬↑p ∣ z\n⊢ (if h : p ≠ 1 ∧ 0 < Int.natAbs z then\n Part.get (multiplicity p (Int.natAbs z)) (_ : multiplicity.Finite p (Int.natAbs z))\n else 0) =\n 0",
"tactic": "split_ifs <;> simp [multiplicity.Int.natAbs, multiplicity_eq_zero.2 h]"
}
] |
[
143,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/RingTheory/GradedAlgebra/Basic.lean
|
DirectSum.coe_decompose_mul_of_left_mem
|
[
{
"state_after": "case intro\nι : Type u_1\nR : Type ?u.376414\nA : Type u_2\nσ : Type u_3\ninst✝⁹ : Semiring A\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CanonicallyOrderedAddMonoid ι\ninst✝⁶ : SetLike σ A\ninst✝⁵ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝⁴ : GradedRing 𝒜\nb : A\nn✝ i : ι\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nn : ι\ninst✝ : Decidable (i ≤ n)\na : { x // x ∈ 𝒜 i }\n⊢ ↑(↑(↑(decompose 𝒜) (↑a * b)) n) = if i ≤ n then ↑a * ↑(↑(↑(decompose 𝒜) b) (n - i)) else 0",
"state_before": "ι : Type u_1\nR : Type ?u.376414\nA : Type u_2\nσ : Type u_3\ninst✝⁹ : Semiring A\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CanonicallyOrderedAddMonoid ι\ninst✝⁶ : SetLike σ A\ninst✝⁵ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝⁴ : GradedRing 𝒜\na b : A\nn✝ i : ι\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nn : ι\ninst✝ : Decidable (i ≤ n)\na_mem : a ∈ 𝒜 i\n⊢ ↑(↑(↑(decompose 𝒜) (a * b)) n) = if i ≤ n then a * ↑(↑(↑(decompose 𝒜) b) (n - i)) else 0",
"tactic": "lift a to 𝒜 i using a_mem"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type u_1\nR : Type ?u.376414\nA : Type u_2\nσ : Type u_3\ninst✝⁹ : Semiring A\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CanonicallyOrderedAddMonoid ι\ninst✝⁶ : SetLike σ A\ninst✝⁵ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝⁴ : GradedRing 𝒜\nb : A\nn✝ i : ι\ninst✝³ : Sub ι\ninst✝² : OrderedSub ι\ninst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nn : ι\ninst✝ : Decidable (i ≤ n)\na : { x // x ∈ 𝒜 i }\n⊢ ↑(↑(↑(decompose 𝒜) (↑a * b)) n) = if i ≤ n then ↑a * ↑(↑(↑(decompose 𝒜) b) (n - i)) else 0",
"tactic": "rw [decompose_mul, decompose_coe, coe_of_mul_apply]"
}
] |
[
330,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/FieldTheory/Normal.lean
|
AlgHom.normal_bijective
|
[
{
"state_after": "F : Type u_1\nK : Type u_3\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : Normal F E\nϕ : E →ₐ[F] K\nx : K\nthis : Algebra E K := RingHom.toAlgebra ↑ϕ\n⊢ ∃ a, ↑ϕ a = x",
"state_before": "F : Type u_1\nK : Type u_3\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : Normal F E\nϕ : E →ₐ[F] K\nx : K\n⊢ ∃ a, ↑ϕ a = x",
"tactic": "letI : Algebra E K := ϕ.toRingHom.toAlgebra"
},
{
"state_after": "case intro\nF : Type u_1\nK : Type u_3\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : Normal F E\nϕ : E →ₐ[F] K\nx : K\nthis : Algebra E K := RingHom.toAlgebra ↑ϕ\nh1 : IsIntegral F (↑(algebraMap K E) x)\nh2 : Splits (algebraMap F E) (minpoly F (↑(algebraMap K E) x))\n⊢ ∃ a, ↑ϕ a = x",
"state_before": "F : Type u_1\nK : Type u_3\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : Normal F E\nϕ : E →ₐ[F] K\nx : K\nthis : Algebra E K := RingHom.toAlgebra ↑ϕ\n⊢ ∃ a, ↑ϕ a = x",
"tactic": "obtain ⟨h1, h2⟩ := h.out (algebraMap K E x)"
},
{
"state_after": "case intro.intro\nF : Type u_1\nK : Type u_3\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : Normal F E\nϕ : E →ₐ[F] K\nx : K\nthis : Algebra E K := RingHom.toAlgebra ↑ϕ\nh1 : IsIntegral F (↑(algebraMap K E) x)\nh2 : Splits (algebraMap F E) (minpoly F (↑(algebraMap K E) x))\ny : E\nhy : ↑(algebraMap E K) y = x\n⊢ ∃ a, ↑ϕ a = x",
"state_before": "case intro\nF : Type u_1\nK : Type u_3\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : Normal F E\nϕ : E →ₐ[F] K\nx : K\nthis : Algebra E K := RingHom.toAlgebra ↑ϕ\nh1 : IsIntegral F (↑(algebraMap K E) x)\nh2 : Splits (algebraMap F E) (minpoly F (↑(algebraMap K E) x))\n⊢ ∃ a, ↑ϕ a = x",
"tactic": "cases'\n minpoly.mem_range_of_degree_eq_one E x\n (h2.def.resolve_left (minpoly.ne_zero h1)\n (minpoly.irreducible\n (isIntegral_of_isScalarTower\n ((isIntegral_algebraMap_iff (algebraMap K E).injective).mp h1)))\n (minpoly.dvd E x\n ((algebraMap K E).injective\n (by\n rw [RingHom.map_zero, aeval_map_algebraMap, ← aeval_algebraMap_apply]\n exact minpoly.aeval F (algebraMap K E x))))) with\n y hy"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nF : Type u_1\nK : Type u_3\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : Normal F E\nϕ : E →ₐ[F] K\nx : K\nthis : Algebra E K := RingHom.toAlgebra ↑ϕ\nh1 : IsIntegral F (↑(algebraMap K E) x)\nh2 : Splits (algebraMap F E) (minpoly F (↑(algebraMap K E) x))\ny : E\nhy : ↑(algebraMap E K) y = x\n⊢ ∃ a, ↑ϕ a = x",
"tactic": "exact ⟨y, hy⟩"
},
{
"state_after": "F : Type u_1\nK : Type u_3\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : Normal F E\nϕ : E →ₐ[F] K\nx : K\nthis : Algebra E K := RingHom.toAlgebra ↑ϕ\nh1 : IsIntegral F (↑(algebraMap K E) x)\nh2 : Splits (algebraMap F E) (minpoly F (↑(algebraMap K E) x))\n⊢ ↑(aeval (↑(algebraMap K E) x)) (minpoly F (↑(algebraMap K E) x)) = 0",
"state_before": "F : Type u_1\nK : Type u_3\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : Normal F E\nϕ : E →ₐ[F] K\nx : K\nthis : Algebra E K := RingHom.toAlgebra ↑ϕ\nh1 : IsIntegral F (↑(algebraMap K E) x)\nh2 : Splits (algebraMap F E) (minpoly F (↑(algebraMap K E) x))\n⊢ ↑(algebraMap K E) (↑(aeval x) (map (algebraMap F E) (minpoly F (↑(algebraMap K E) x)))) = ↑(algebraMap K E) 0",
"tactic": "rw [RingHom.map_zero, aeval_map_algebraMap, ← aeval_algebraMap_apply]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\nK : Type u_3\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : Normal F E\nϕ : E →ₐ[F] K\nx : K\nthis : Algebra E K := RingHom.toAlgebra ↑ϕ\nh1 : IsIntegral F (↑(algebraMap K E) x)\nh2 : Splits (algebraMap F E) (minpoly F (↑(algebraMap K E) x))\n⊢ ↑(aeval (↑(algebraMap K E) x)) (minpoly F (↑(algebraMap K E) x)) = 0",
"tactic": "exact minpoly.aeval F (algebraMap K E x)"
}
] |
[
131,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Data/Nat/Interval.lean
|
Nat.Ico_succ_right
|
[
{
"state_after": "case a\na b c x : ℕ\n⊢ x ∈ Ico a (succ b) ↔ x ∈ Icc a b",
"state_before": "a b c : ℕ\n⊢ Ico a (succ b) = Icc a b",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case a\na b c x : ℕ\n⊢ x ∈ Ico a (succ b) ↔ x ∈ Icc a b",
"tactic": "rw [mem_Ico, mem_Icc, lt_succ_iff]"
}
] |
[
167,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.nonempty_of_ncard_ne_zero
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.7426\ns t : Set α\na b x y : α\nf : α → β\nhs : ncard s ≠ 0\n⊢ s ≠ ∅",
"state_before": "α : Type u_1\nβ : Type ?u.7426\ns t : Set α\na b x y : α\nf : α → β\nhs : ncard s ≠ 0\n⊢ Set.Nonempty s",
"tactic": "rw [nonempty_iff_ne_empty]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.7426\nt : Set α\na b x y : α\nf : α → β\nhs : ncard ∅ ≠ 0\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.7426\ns t : Set α\na b x y : α\nf : α → β\nhs : ncard s ≠ 0\n⊢ s ≠ ∅",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7426\nt : Set α\na b x y : α\nf : α → β\nhs : ncard ∅ ≠ 0\n⊢ False",
"tactic": "simp at hs"
}
] |
[
129,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.lt_find_iff
|
[
{
"state_after": "P : ℕ → Prop\ninst✝ : DecidablePred P\nn : ℕ\n⊢ ↑n < find P → ∀ (m : ℕ), m ≤ n → ¬P m",
"state_before": "P : ℕ → Prop\ninst✝ : DecidablePred P\nn : ℕ\n⊢ ↑n < find P ↔ ∀ (m : ℕ), m ≤ n → ¬P m",
"tactic": "refine' ⟨_, lt_find P n⟩"
},
{
"state_after": "P : ℕ → Prop\ninst✝ : DecidablePred P\nn : ℕ\nh : ↑n < find P\nm : ℕ\nhm : m ≤ n\n⊢ ¬P m",
"state_before": "P : ℕ → Prop\ninst✝ : DecidablePred P\nn : ℕ\n⊢ ↑n < find P → ∀ (m : ℕ), m ≤ n → ¬P m",
"tactic": "intro h m hm"
},
{
"state_after": "case pos\nP : ℕ → Prop\ninst✝ : DecidablePred P\nn : ℕ\nh : ↑n < find P\nm : ℕ\nhm : m ≤ n\nH : (find P).Dom\n⊢ ¬P m\n\ncase neg\nP : ℕ → Prop\ninst✝ : DecidablePred P\nn : ℕ\nh : ↑n < find P\nm : ℕ\nhm : m ≤ n\nH : ¬(find P).Dom\n⊢ ¬P m",
"state_before": "P : ℕ → Prop\ninst✝ : DecidablePred P\nn : ℕ\nh : ↑n < find P\nm : ℕ\nhm : m ≤ n\n⊢ ¬P m",
"tactic": "by_cases H : (find P).Dom"
},
{
"state_after": "case pos\nP : ℕ → Prop\ninst✝ : DecidablePred P\nn : ℕ\nh : ↑n < find P\nm : ℕ\nhm : m ≤ n\nH : (find P).Dom\n⊢ m < Nat.find H",
"state_before": "case pos\nP : ℕ → Prop\ninst✝ : DecidablePred P\nn : ℕ\nh : ↑n < find P\nm : ℕ\nhm : m ≤ n\nH : (find P).Dom\n⊢ ¬P m",
"tactic": "apply Nat.find_min H"
},
{
"state_after": "case pos\nP : ℕ → Prop\ninst✝ : DecidablePred P\nn : ℕ\nh : ∀ (h : (find P).Dom), n < Part.get (find P) h\nm : ℕ\nhm : m ≤ n\nH : (find P).Dom\n⊢ m < Nat.find H",
"state_before": "case pos\nP : ℕ → Prop\ninst✝ : DecidablePred P\nn : ℕ\nh : ↑n < find P\nm : ℕ\nhm : m ≤ n\nH : (find P).Dom\n⊢ m < Nat.find H",
"tactic": "rw [coe_lt_iff] at h"
},
{
"state_after": "case pos\nP : ℕ → Prop\ninst✝ : DecidablePred P\nn m : ℕ\nhm : m ≤ n\nH : (find P).Dom\nh : n < Part.get (find P) H\n⊢ m < Nat.find H",
"state_before": "case pos\nP : ℕ → Prop\ninst✝ : DecidablePred P\nn : ℕ\nh : ∀ (h : (find P).Dom), n < Part.get (find P) h\nm : ℕ\nhm : m ≤ n\nH : (find P).Dom\n⊢ m < Nat.find H",
"tactic": "specialize h H"
},
{
"state_after": "no goals",
"state_before": "case pos\nP : ℕ → Prop\ninst✝ : DecidablePred P\nn m : ℕ\nhm : m ≤ n\nH : (find P).Dom\nh : n < Part.get (find P) H\n⊢ m < Nat.find H",
"tactic": "exact lt_of_le_of_lt hm h"
},
{
"state_after": "no goals",
"state_before": "case neg\nP : ℕ → Prop\ninst✝ : DecidablePred P\nn : ℕ\nh : ↑n < find P\nm : ℕ\nhm : m ≤ n\nH : ¬(find P).Dom\n⊢ ¬P m",
"tactic": "exact not_exists.mp H m"
}
] |
[
809,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
801,
1
] |
Mathlib/RingTheory/WittVector/StructurePolynomial.lean
|
bind₁_rename_expand_wittPolynomial
|
[
{
"state_after": "case a\np : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\n⊢ ↑(map (Int.castRingHom ℚ)) (↑(bind₁ fun b => ↑(rename fun i => (b, i)) (↑(expand p) (W_ ℤ n))) Φ) =\n ↑(map (Int.castRingHom ℚ)) (↑(bind₁ fun i => ↑(expand p) (wittStructureInt p Φ i)) (W_ ℤ n))",
"state_before": "p : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\n⊢ ↑(bind₁ fun b => ↑(rename fun i => (b, i)) (↑(expand p) (W_ ℤ n))) Φ =\n ↑(bind₁ fun i => ↑(expand p) (wittStructureInt p Φ i)) (W_ ℤ n)",
"tactic": "apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective"
},
{
"state_after": "case a\np : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\n⊢ ↑(bind₁ fun i => ↑(expand p) (↑(rename fun i_1 => (i, i_1)) (W_ ℚ n))) (↑(map (Int.castRingHom ℚ)) Φ) =\n ↑(bind₁ fun i => ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)",
"state_before": "case a\np : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\n⊢ ↑(map (Int.castRingHom ℚ)) (↑(bind₁ fun b => ↑(rename fun i => (b, i)) (↑(expand p) (W_ ℤ n))) Φ) =\n ↑(map (Int.castRingHom ℚ)) (↑(bind₁ fun i => ↑(expand p) (wittStructureInt p Φ i)) (W_ ℤ n))",
"tactic": "simp only [map_bind₁, map_rename, map_expand, rename_expand, map_wittPolynomial]"
},
{
"state_after": "case a\np : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\nkey :\n ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ n)) (↑(map (Int.castRingHom ℚ)) Φ) =\n ↑(bind₁ (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ))) (W_ ℚ n)\n⊢ ↑(bind₁ fun i => ↑(expand p) (↑(rename fun i_1 => (i, i_1)) (W_ ℚ n))) (↑(map (Int.castRingHom ℚ)) Φ) =\n ↑(bind₁ fun i => ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)",
"state_before": "case a\np : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\n⊢ ↑(bind₁ fun i => ↑(expand p) (↑(rename fun i_1 => (i, i_1)) (W_ ℚ n))) (↑(map (Int.castRingHom ℚ)) Φ) =\n ↑(bind₁ fun i => ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)",
"tactic": "have key := (wittStructureRat_prop p (map (Int.castRingHom ℚ) Φ) n).symm"
},
{
"state_after": "case a\np : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\nkey :\n ↑(expand p) (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ n)) (↑(map (Int.castRingHom ℚ)) Φ)) =\n ↑(expand p) (↑(bind₁ (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ))) (W_ ℚ n))\n⊢ ↑(bind₁ fun i => ↑(expand p) (↑(rename fun i_1 => (i, i_1)) (W_ ℚ n))) (↑(map (Int.castRingHom ℚ)) Φ) =\n ↑(bind₁ fun i => ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)",
"state_before": "case a\np : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\nkey :\n ↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ n)) (↑(map (Int.castRingHom ℚ)) Φ) =\n ↑(bind₁ (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ))) (W_ ℚ n)\n⊢ ↑(bind₁ fun i => ↑(expand p) (↑(rename fun i_1 => (i, i_1)) (W_ ℚ n))) (↑(map (Int.castRingHom ℚ)) Φ) =\n ↑(bind₁ fun i => ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)",
"tactic": "apply_fun expand p at key"
},
{
"state_after": "case a\np : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\nkey :\n ↑(bind₁ fun i => ↑(expand p) (↑(rename (Prod.mk i)) (W_ ℚ n))) (↑(map (Int.castRingHom ℚ)) Φ) =\n ↑(bind₁ fun i => ↑(expand p) (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n)\n⊢ ↑(bind₁ fun i => ↑(expand p) (↑(rename fun i_1 => (i, i_1)) (W_ ℚ n))) (↑(map (Int.castRingHom ℚ)) Φ) =\n ↑(bind₁ fun i => ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)",
"state_before": "case a\np : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\nkey :\n ↑(expand p) (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (W_ ℚ n)) (↑(map (Int.castRingHom ℚ)) Φ)) =\n ↑(expand p) (↑(bind₁ (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ))) (W_ ℚ n))\n⊢ ↑(bind₁ fun i => ↑(expand p) (↑(rename fun i_1 => (i, i_1)) (W_ ℚ n))) (↑(map (Int.castRingHom ℚ)) Φ) =\n ↑(bind₁ fun i => ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)",
"tactic": "simp only [expand_bind₁] at key"
},
{
"state_after": "case a\np : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\nkey :\n ↑(bind₁ fun i => ↑(expand p) (↑(rename (Prod.mk i)) (W_ ℚ n))) (↑(map (Int.castRingHom ℚ)) Φ) =\n ↑(bind₁ fun i => ↑(expand p) (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n)\n⊢ ↑(bind₁ fun i => ↑(expand p) (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n) =\n ↑(bind₁ fun i => ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)",
"state_before": "case a\np : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\nkey :\n ↑(bind₁ fun i => ↑(expand p) (↑(rename (Prod.mk i)) (W_ ℚ n))) (↑(map (Int.castRingHom ℚ)) Φ) =\n ↑(bind₁ fun i => ↑(expand p) (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n)\n⊢ ↑(bind₁ fun i => ↑(expand p) (↑(rename fun i_1 => (i, i_1)) (W_ ℚ n))) (↑(map (Int.castRingHom ℚ)) Φ) =\n ↑(bind₁ fun i => ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)",
"tactic": "rw [key]"
},
{
"state_after": "case a\np : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\n⊢ ↑(bind₁ fun i => ↑(expand p) (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n) =\n ↑(bind₁ fun i => ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)",
"state_before": "case a\np : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\nkey :\n ↑(bind₁ fun i => ↑(expand p) (↑(rename (Prod.mk i)) (W_ ℚ n))) (↑(map (Int.castRingHom ℚ)) Φ) =\n ↑(bind₁ fun i => ↑(expand p) (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n)\n⊢ ↑(bind₁ fun i => ↑(expand p) (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n) =\n ↑(bind₁ fun i => ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)",
"tactic": "clear key"
},
{
"state_after": "p : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\n⊢ ∀ (i : ℕ),\n i ∈ vars (W_ ℚ n) →\n i ∈ vars (W_ ℚ n) →\n ↑(expand p) (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) i) =\n ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))",
"state_before": "case a\np : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\n⊢ ↑(bind₁ fun i => ↑(expand p) (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) i)) (W_ ℚ n) =\n ↑(bind₁ fun i => ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))) (W_ ℚ n)",
"tactic": "apply eval₂Hom_congr' rfl _ rfl"
},
{
"state_after": "p : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\ni : ℕ\nhi : i ∈ vars (W_ ℚ n)\n⊢ ↑(expand p) (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) i) =\n ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))",
"state_before": "p : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\n⊢ ∀ (i : ℕ),\n i ∈ vars (W_ ℚ n) →\n i ∈ vars (W_ ℚ n) →\n ↑(expand p) (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) i) =\n ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))",
"tactic": "rintro i hi -"
},
{
"state_after": "p : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\ni : ℕ\nhi : i < n + 1\n⊢ ↑(expand p) (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) i) =\n ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))",
"state_before": "p : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\ni : ℕ\nhi : i ∈ vars (W_ ℚ n)\n⊢ ↑(expand p) (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) i) =\n ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))",
"tactic": "rw [wittPolynomial_vars, Finset.mem_range] at hi"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type ?u.755142\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH :\n ∀ (m : ℕ),\n m < n + 1 →\n ↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) m\ni : ℕ\nhi : i < n + 1\n⊢ ↑(expand p) (wittStructureRat p (↑(map (Int.castRingHom ℚ)) Φ) i) =\n ↑(expand p) (↑(map (Int.castRingHom ℚ)) (wittStructureInt p Φ i))",
"tactic": "simp only [IH i hi]"
}
] |
[
229,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
Module.eval_ker
|
[
{
"state_after": "no goals",
"state_before": "K : Type u₁\nV : Type u₂\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\n⊢ ker (eval K V) = ⊥",
"tactic": "classical exact (Basis.ofVectorSpace K V).eval_ker"
},
{
"state_after": "no goals",
"state_before": "K : Type u₁\nV : Type u₂\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\n⊢ ker (eval K V) = ⊥",
"tactic": "exact (Basis.ofVectorSpace K V).eval_ker"
}
] |
[
545,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
544,
1
] |
Mathlib/Geometry/Euclidean/Inversion.lean
|
EuclideanGeometry.inversion_of_mem_sphere
|
[] |
[
63,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Algebra/BigOperators/Associated.lean
|
Prime.dvd_finset_prod_iff
|
[] |
[
183,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/Topology/Constructions.lean
|
tendsto_pi_nhds
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.206424\nδ : Type ?u.206427\nε : Type ?u.206430\nζ : Type ?u.206433\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.206444\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf✝ : α → (i : ι) → π i\nf : β → (i : ι) → π i\ng : (i : ι) → π i\nu : Filter β\n⊢ Tendsto f u (𝓝 g) ↔ ∀ (x : ι), Tendsto (fun i => f i x) u (𝓝 (g x))",
"tactic": "rw [nhds_pi, Filter.tendsto_pi]"
}
] |
[
1223,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1221,
1
] |
Mathlib/LinearAlgebra/AffineSpace/MidpointZero.lean
|
pi_midpoint_apply
|
[] |
[
58,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.insertNth_last'
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nβ : Type v\nx : β\np : Fin n → β\n⊢ insertNth (last n) x p = snoc p x",
"tactic": "simp [insertNth_last]"
}
] |
[
750,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
749,
1
] |
Mathlib/Combinatorics/Young/SemistandardTableau.lean
|
Ssyt.copy_eq
|
[] |
[
111,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.Relations.realize_irreflexive
|
[] |
[
1005,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1004,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.div_mem
|
[] |
[
590,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
589,
11
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.algebraMap_mk'
|
[
{
"state_after": "R : Type u_1\ninst✝¹² : CommRing R\nM : Submonoid R\nS : Type u_4\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nP : Type ?u.3264412\ninst✝⁹ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_2\ninst✝⁸ : CommRing Rₘ\ninst✝⁷ : CommRing Sₘ\ninst✝⁶ : Algebra R Rₘ\ninst✝⁵ : IsLocalization M Rₘ\ninst✝⁴ : Algebra S Sₘ\ni : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\ninst✝³ : Algebra Rₘ Sₘ\ninst✝² : Algebra R Sₘ\ninst✝¹ : IsScalarTower R Rₘ Sₘ\ninst✝ : IsScalarTower R S Sₘ\nx : R\ny : { x // x ∈ M }\n⊢ ↑(algebraMap Rₘ Sₘ) (mk' Rₘ (↑y * x) y) = ↑(algebraMap Rₘ Sₘ) (↑(algebraMap R Rₘ) x)",
"state_before": "R : Type u_1\ninst✝¹² : CommRing R\nM : Submonoid R\nS : Type u_4\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nP : Type ?u.3264412\ninst✝⁹ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_2\ninst✝⁸ : CommRing Rₘ\ninst✝⁷ : CommRing Sₘ\ninst✝⁶ : Algebra R Rₘ\ninst✝⁵ : IsLocalization M Rₘ\ninst✝⁴ : Algebra S Sₘ\ni : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\ninst✝³ : Algebra Rₘ Sₘ\ninst✝² : Algebra R Sₘ\ninst✝¹ : IsScalarTower R Rₘ Sₘ\ninst✝ : IsScalarTower R S Sₘ\nx : R\ny : { x // x ∈ M }\n⊢ ↑(algebraMap Rₘ Sₘ) (mk' Rₘ x y) =\n mk' Sₘ (↑(algebraMap R S) x)\n { val := ↑(algebraMap R S) ↑y, property := (_ : ↑(algebraMap R S) ↑y ∈ Algebra.algebraMapSubmonoid S M) }",
"tactic": "rw [IsLocalization.eq_mk'_iff_mul_eq, Subtype.coe_mk, ← IsScalarTower.algebraMap_apply, ←\n IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply R Rₘ Sₘ,\n IsScalarTower.algebraMap_apply R Rₘ Sₘ, ← _root_.map_mul, mul_comm,\n IsLocalization.mul_mk'_eq_mk'_of_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹² : CommRing R\nM : Submonoid R\nS : Type u_4\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nP : Type ?u.3264412\ninst✝⁹ : CommRing P\nRₘ : Type u_3\nSₘ : Type u_2\ninst✝⁸ : CommRing Rₘ\ninst✝⁷ : CommRing Sₘ\ninst✝⁶ : Algebra R Rₘ\ninst✝⁵ : IsLocalization M Rₘ\ninst✝⁴ : Algebra S Sₘ\ni : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ\ninst✝³ : Algebra Rₘ Sₘ\ninst✝² : Algebra R Sₘ\ninst✝¹ : IsScalarTower R Rₘ Sₘ\ninst✝ : IsScalarTower R S Sₘ\nx : R\ny : { x // x ∈ M }\n⊢ ↑(algebraMap Rₘ Sₘ) (mk' Rₘ (↑y * x) y) = ↑(algebraMap Rₘ Sₘ) (↑(algebraMap R Rₘ) x)",
"tactic": "exact congr_arg (algebraMap Rₘ Sₘ) (IsLocalization.mk'_mul_cancel_left x y)"
}
] |
[
1341,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1333,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.lcongr_apply_apply
|
[] |
[
907,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
905,
1
] |
Mathlib/Data/ENat/Basic.lean
|
ENat.toNat_top
|
[] |
[
114,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.Integrable.mono
|
[] |
[
466,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
464,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.filterMap_filterMap
|
[] |
[
2144,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2142,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Ico_mem_nhdsWithin_Ioi'
|
[] |
[
422,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
convexOn_iff_pairwise_pos
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.162921\nα : Type ?u.162924\nβ : Type u_3\nι : Type ?u.162930\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : OrderedAddCommMonoid α\ninst✝² : OrderedAddCommMonoid β\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf : E → β\n⊢ (Convex 𝕜 s ∧\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y) ↔\n Convex 𝕜 s ∧\n Set.Pairwise s fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.162921\nα : Type ?u.162924\nβ : Type u_3\nι : Type ?u.162930\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : OrderedAddCommMonoid α\ninst✝² : OrderedAddCommMonoid β\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf : E → β\n⊢ ConvexOn 𝕜 s f ↔\n Convex 𝕜 s ∧\n Set.Pairwise s fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y",
"tactic": "rw [convexOn_iff_forall_pos]"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.162921\nα : Type ?u.162924\nβ : Type u_3\nι : Type ?u.162930\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : OrderedAddCommMonoid α\ninst✝² : OrderedAddCommMonoid β\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf : E → β\nh : Set.Pairwise s fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ f (a • x + b • y) ≤ a • f x + b • f y",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.162921\nα : Type ?u.162924\nβ : Type u_3\nι : Type ?u.162930\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : OrderedAddCommMonoid α\ninst✝² : OrderedAddCommMonoid β\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf : E → β\n⊢ (Convex 𝕜 s ∧\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y) ↔\n Convex 𝕜 s ∧\n Set.Pairwise s fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y",
"tactic": "refine'\n and_congr_right'\n ⟨fun h x hx y hy _ a b ha hb hab => h hx hy ha hb hab, fun h x hx y hy a b ha hb hab => _⟩"
},
{
"state_after": "case inl\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.162921\nα : Type ?u.162924\nβ : Type u_3\nι : Type ?u.162930\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : OrderedAddCommMonoid α\ninst✝² : OrderedAddCommMonoid β\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf : E → β\nh : Set.Pairwise s fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y\nx : E\nhx : x ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhy : x ∈ s\n⊢ f (a • x + b • x) ≤ a • f x + b • f x\n\ncase inr\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.162921\nα : Type ?u.162924\nβ : Type u_3\nι : Type ?u.162930\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : OrderedAddCommMonoid α\ninst✝² : OrderedAddCommMonoid β\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf : E → β\nh : Set.Pairwise s fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhxy : x ≠ y\n⊢ f (a • x + b • y) ≤ a • f x + b • f y",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.162921\nα : Type ?u.162924\nβ : Type u_3\nι : Type ?u.162930\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : OrderedAddCommMonoid α\ninst✝² : OrderedAddCommMonoid β\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf : E → β\nh : Set.Pairwise s fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ f (a • x + b • y) ≤ a • f x + b • f y",
"tactic": "obtain rfl | hxy := eq_or_ne x y"
},
{
"state_after": "no goals",
"state_before": "case inr\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.162921\nα : Type ?u.162924\nβ : Type u_3\nι : Type ?u.162930\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : OrderedAddCommMonoid α\ninst✝² : OrderedAddCommMonoid β\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf : E → β\nh : Set.Pairwise s fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhxy : x ≠ y\n⊢ f (a • x + b • y) ≤ a • f x + b • f y",
"tactic": "exact h hx hy hxy ha hb hab"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.162921\nα : Type ?u.162924\nβ : Type u_3\nι : Type ?u.162930\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : OrderedAddCommMonoid α\ninst✝² : OrderedAddCommMonoid β\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf : E → β\nh : Set.Pairwise s fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y\nx : E\nhx : x ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhy : x ∈ s\n⊢ f (a • x + b • x) ≤ a • f x + b • f x",
"tactic": "rw [Convex.combo_self hab, Convex.combo_self hab]"
}
] |
[
348,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/Data/Polynomial/Div.lean
|
Polynomial.divByMonic_eq_of_not_monic
|
[] |
[
207,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Order/SymmDiff.lean
|
toDual_bihimp
|
[] |
[
230,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/Topology/MetricSpace/Contracting.lean
|
ContractingWith.apriori_dist_iterate_fixedPoint_le
|
[] |
[
337,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
SupBotHom.coe_id
|
[] |
[
777,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
776,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.image_smul
|
[] |
[
314,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/Order/Disjoint.lean
|
Disjoint.eq_bot_of_ge
|
[] |
[
99,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
Equiv.Perm.VectorsProdEqOne.zero_eq
|
[] |
[
384,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/Tactic/Sat/FromLRAT.lean
|
Sat.Fmla.subsumes_left
|
[] |
[
91,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
UniformSpace.mem_closure_iff_symm_ball
|
[
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.62335\ninst✝ : UniformSpace α\ns : Set α\nx : α\n⊢ x ∈ closure s ↔ ∀ {V : Set (α × α)}, V ∈ 𝓤 α → SymmetricRel V → Set.Nonempty (s ∩ ball x V)",
"tactic": "simp [mem_closure_iff_nhds_basis (hasBasis_nhds x), Set.Nonempty]"
}
] |
[
770,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
768,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.continuous_pow
|
[
{
"state_after": "f g : CircleDeg1Lift\nhf : Continuous ↑f\nn : ℕ\n⊢ Continuous (↑f^[n])",
"state_before": "f g : CircleDeg1Lift\nhf : Continuous ↑f\nn : ℕ\n⊢ Continuous ↑(f ^ n)",
"tactic": "rw [coe_pow]"
},
{
"state_after": "no goals",
"state_before": "f g : CircleDeg1Lift\nhf : Continuous ↑f\nn : ℕ\n⊢ Continuous (↑f^[n])",
"tactic": "exact hf.iterate n"
}
] |
[
953,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
951,
1
] |
Mathlib/Topology/Bases.lean
|
Set.Countable.isSeparable
|
[] |
[
413,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
412,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean
|
Ultrafilter.inf_neBot_iff
|
[] |
[
114,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
RingEquiv.restrictRootsOfUnity_coe_apply
|
[] |
[
175,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean
|
Nat.factorial_pos
|
[] |
[
71,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Topology/MetricSpace/Antilipschitz.lean
|
AntilipschitzWith.ediam_preimage_le
|
[] |
[
120,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Std/Logic.lean
|
and_and_and_comm
|
[
{
"state_after": "no goals",
"state_before": "a b c d : Prop\n⊢ (a ∧ b) ∧ c ∧ d ↔ (a ∧ c) ∧ b ∧ d",
"tactic": "rw [← and_assoc, @and_right_comm a, and_assoc]"
}
] |
[
190,
49
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
189,
1
] |
Mathlib/Order/SuccPred/Limit.lean
|
Order.IsPredLimit.isMax_of_noMin
|
[] |
[
320,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
11
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.Monic.degree_le_zero_iff_eq_one
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\n⊢ degree p ≤ 0 ↔ p = 1",
"tactic": "rw [← hp.natDegree_eq_zero_iff_eq_one, natDegree_eq_zero_iff_degree_le_zero]"
}
] |
[
169,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.mem_span
|
[
{
"state_after": "ι : Type u_3\nι' : Type ?u.461943\nR : Type u_2\nR₂ : Type ?u.461949\nK : Type ?u.461952\nM : Type u_1\nM' : Type ?u.461958\nM'' : Type ?u.461961\nV : Type u\nV' : Type ?u.461966\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ x : M\n⊢ ∑ a in (↑b.repr x).support, ↑(↑b.repr x) a • ↑b a ∈ span R (range ↑b)",
"state_before": "ι : Type u_3\nι' : Type ?u.461943\nR : Type u_2\nR₂ : Type ?u.461949\nK : Type ?u.461952\nM : Type u_1\nM' : Type ?u.461958\nM'' : Type ?u.461961\nV : Type u\nV' : Type ?u.461966\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ x : M\n⊢ x ∈ span R (range ↑b)",
"tactic": "rw [← b.total_repr x, Finsupp.total_apply, Finsupp.sum]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_3\nι' : Type ?u.461943\nR : Type u_2\nR₂ : Type ?u.461949\nK : Type ?u.461952\nM : Type u_1\nM' : Type ?u.461958\nM'' : Type ?u.461961\nV : Type u\nV' : Type ?u.461966\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ x : M\n⊢ ∑ a in (↑b.repr x).support, ↑(↑b.repr x) a • ↑b a ∈ span R (range ↑b)",
"tactic": "exact Submodule.sum_mem _ fun i _ => Submodule.smul_mem _ _ (Submodule.subset_span ⟨i, rfl⟩)"
}
] |
[
571,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
569,
11
] |
Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean
|
CategoryTheory.Limits.hasTerminalChangeDiagram
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝ : Category C\nX : C\nF₁ : Discrete PEmpty ⥤ C\nF₂ : Discrete PEmpty ⥤ C\nh : HasLimit F₁\n⊢ (X : Discrete PEmpty) → ((Functor.const (Discrete PEmpty)).obj (limit F₁)).obj X ⟶ F₂.obj X",
"tactic": "aesop_cat"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝ : Category C\nX : C\nF₁ : Discrete PEmpty ⥤ C\nF₂ : Discrete PEmpty ⥤ C\nh : HasLimit F₁\n⊢ ∀ ⦃X Y : Discrete PEmpty⦄ (f : X ⟶ Y),\n ((Functor.const (Discrete PEmpty)).obj (limit F₁)).map f ≫\n id (Discrete.casesOn Y fun as => False.elim (_ : False)) =\n id (Discrete.casesOn X fun as => False.elim (_ : False)) ≫ F₂.map f",
"tactic": "aesop_cat"
}
] |
[
252,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
exists_pos_lt_mul
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.112188\nα : Type u_1\nβ : Type ?u.112194\ninst✝ : LinearOrderedSemifield α\na✝ b✝ c✝ d e : α\nm n : ℤ\na : α\nh : 0 < a\nb c : α\nhc₀ : 0 < c\nhc : b * c < a\n⊢ b < c⁻¹ * a",
"tactic": "rwa [← div_eq_inv_mul, lt_div_iff hc₀]"
}
] |
[
576,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
574,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.coequalizer.cofork_ι_app_one
|
[] |
[
956,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
955,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_sub_const_Ico
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => x - a) ⁻¹' Ico b c = Ico (b + a) (c + a)",
"tactic": "simp [sub_eq_add_neg]"
}
] |
[
202,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.nat_le_card
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.236438\nβ : Type ?u.236441\nγ : Type ?u.236444\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nn : ℕ\n⊢ ↑n ≤ card o ↔ ↑n ≤ o",
"tactic": "rw [← Cardinal.ord_le, Cardinal.ord_nat]"
}
] |
[
1561,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1560,
1
] |
Mathlib/MeasureTheory/Measure/Content.lean
|
MeasureTheory.Content.is_mul_left_invariant_outerMeasure
|
[
{
"state_after": "no goals",
"state_before": "G : Type w\ninst✝³ : TopologicalSpace G\nμ : Content G\ninst✝² : T2Space G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\nh :\n ∀ (g : G) {K : Compacts G},\n (fun s => ↑(toFun μ s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n (fun s => ↑(toFun μ s)) K\ng : G\nA : Set G\n⊢ ↑(Content.outerMeasure μ) ((fun x => g * x) ⁻¹' A) = ↑(Content.outerMeasure μ) A",
"tactic": "convert μ.outerMeasure_preimage (Homeomorph.mulLeft g) (fun K => h g) A"
}
] |
[
325,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.rangeFactorization_eq
|
[] |
[
1049,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1048,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.vecMul_sub
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.913128\nm : Type u_1\nn : Type u_2\no : Type ?u.913137\nm' : o → Type ?u.913142\nn' : o → Type ?u.913147\nR : Type ?u.913150\nS : Type ?u.913153\nα : Type v\nβ : Type w\nγ : Type ?u.913160\ninst✝¹ : NonUnitalNonAssocRing α\ninst✝ : Fintype m\nA B : Matrix m n α\nx : m → α\n⊢ vecMul x (A - B) = vecMul x A - vecMul x B",
"tactic": "simp [sub_eq_add_neg, vecMul_add, vecMul_neg]"
}
] |
[
1921,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1920,
1
] |
Mathlib/Algebra/Star/SelfAdjoint.lean
|
IsSelfAdjoint.pow
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type ?u.29976\ninst✝¹ : Monoid R\ninst✝ : StarSemigroup R\nx : R\nhx : IsSelfAdjoint x\nn : ℕ\n⊢ IsSelfAdjoint (x ^ n)",
"tactic": "simp only [isSelfAdjoint_iff, star_pow, hx.star_eq]"
}
] |
[
186,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Topology/Bornology/Hom.lean
|
LocallyBoundedMap.id_comp
|
[] |
[
191,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/Data/Sum/Order.lean
|
Sum.Lex.le_def
|
[] |
[
340,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.span_singleton_mul_right_inj
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type ?u.240143\ninst✝¹ : CommSemiring R\nI J K L : Ideal R\ninst✝ : IsDomain R\nx : R\nhx : x ≠ 0\n⊢ span {x} * I = span {x} * J ↔ I = J",
"tactic": "simp only [le_antisymm_iff, span_singleton_mul_right_mono hx]"
}
] |
[
581,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
579,
1
] |
Mathlib/Data/Analysis/Filter.lean
|
CFilter.coe_mk
|
[] |
[
63,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
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