mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case cons.refine_1
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
s : Finset ι
f : ι → M
j : ι
t : Finset ι
hj : j ∉ t
hs✝ : t.Nonempty
IH : ∃ b ∈ t, ‖∏ i ∈ t, f i‖ ≤ ‖f b‖
h : ‖∏ i ∈ t, f i‖ ≤ ‖f j‖
⊢ ‖f j * ∏ i ∈ t, f i‖ ≤ ‖f j‖

pp: case cons.refine_2
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
s : Finset ι
f : ι → M
j : ι
t : Finset ι
hj : j ∉ t
hs✝ : t.Nonempty
IH : ∃ b ∈ t, ‖∏ i ∈ t, f i‖ ≤ ‖f b‖
h : ‖f j‖ ≤ ‖∏ i ∈ t, f i‖
⊢ ∃ a ∈ t, ‖f j * ∏ i ∈ t, f i‖ ≤ ‖f a‖

Tactic:
·

Goal state after:
case cons.refine_1
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
s : Finset ι
f : ι → M
j : ι
t : Finset ι
hj : j ∉ t
hs✝ : t.Nonempty
IH : ∃ b ∈ t, ‖∏ i ∈ t, f i‖ ≤ ‖f b‖
h : ‖∏ i ∈ t, f i‖ ≤ ‖f j‖
⊢ ‖f j * ∏ i ∈ t, f i‖ ≤ ‖f j‖
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case cons.refine_2
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
s : Finset ι
f : ι → M
j : ι
t : Finset ι
hj : j ∉ t
hs✝ : t.Nonempty
IH : ∃ b ∈ t, ‖∏ i ∈ t, f i‖ ≤ ‖f b‖
h : ‖f j‖ ≤ ‖∏ i ∈ t, f i‖
⊢ ∃ a ∈ t, ‖f j * ∏ i ∈ t, f i‖ ≤ ‖f a‖

Tactic:
·

Goal state after:
case cons.refine_2
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
s : Finset ι
f : ι → M
j : ι
t : Finset ι
hj : j ∉ t
hs✝ : t.Nonempty
IH : ∃ b ∈ t, ‖∏ i ∈ t, f i‖ ≤ ‖f b‖
h : ‖f j‖ ≤ ‖∏ i ∈ t, f i‖
⊢ ∃ a ∈ t, ‖f j * ∏ i ∈ t, f i‖ ≤ ‖f a‖
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case inl
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
⊢ ‖∏ i ∈ ∅, f i‖₊ ≤ ∅.sup fun x ↦ ‖f x‖₊

pp: case inr
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
s : Finset ι
f : ι → M
hs : s.Nonempty
⊢ ‖∏ i ∈ s, f i‖₊ ≤ s.sup fun x ↦ ‖f x‖₊

Tactic:
·

Goal state after:
case inl
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
⊢ ‖∏ i ∈ ∅, f i‖₊ ≤ ∅.sup fun x ↦ ‖f x‖₊
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case inr
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
s : Finset ι
f : ι → M
hs : s.Nonempty
⊢ ‖∏ i ∈ s, f i‖₊ ≤ s.sup fun x ↦ ‖f x‖₊

Tactic:
·

Goal state after:
case inr
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
s : Finset ι
f : ι → M
hs : s.Nonempty
⊢ ‖∏ i ∈ s, f i‖₊ ≤ s.sup fun x ↦ ‖f x‖₊
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case inl
M : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M
inst✝¹ : IsUltrametricDist M
inst✝ : Nonempty ι
f : ι → M
⊢ ∃ i, (∅.Nonempty → i ∈ ∅) ∧ ‖∏ j ∈ ∅, f j‖ ≤ ‖f i‖

pp: case inr
M : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M
inst✝¹ : IsUltrametricDist M
t : Finset ι
inst✝ : Nonempty ι
f : ι → M
ht : t.Nonempty
⊢ ∃ i, (t.Nonempty → i ∈ t) ∧ ‖∏ j ∈ t, f j‖ ≤ ‖f i‖

Tactic:
·

Goal state after:
case inl
M : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M
inst✝¹ : IsUltrametricDist M
inst✝ : Nonempty ι
f : ι → M
⊢ ∃ i, (∅.Nonempty → i ∈ ∅) ∧ ‖∏ j ∈ ∅, f j‖ ≤ ‖f i‖
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case pos
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a : ι
t : Multiset ι
M : ι
hMs : t ≠ 0 → M ∈ t
hM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖
hMa : ‖f M‖ ≤ ‖f a‖
⊢ ∃ i, (a ::ₘ t ≠ 0 → i ∈ a ::ₘ t) ∧ ‖(Multiset.map f (a ::ₘ t)).prod‖ ≤ ‖f i‖

pp: case neg
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a : ι
t : Multiset ι
M : ι
hMs : t ≠ 0 → M ∈ t
hM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖
hMa : ‖f a‖ < ‖f M‖
⊢ ∃ i, (a ::ₘ t ≠ 0 → i ∈ a ::ₘ t) ∧ ‖(Multiset.map f (a ::ₘ t)).prod‖ ≤ ‖f i‖

Tactic:
·

Goal state after:
case pos
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a : ι
t : Multiset ι
M : ι
hMs : t ≠ 0 → M ∈ t
hM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖
hMa : ‖f M‖ ≤ ‖f a‖
⊢ ∃ i, (a ::ₘ t ≠ 0 → i ∈ a ::ₘ t) ∧ ‖(Multiset.map f (a ::ₘ t)).prod‖ ≤ ‖f i‖
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case pos
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a : ι
t : Multiset ι
M : ι
hMs : t ≠ 0 → M ∈ t
hM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖
hMa : ‖f M‖ ≤ ‖f a‖
⊢ ‖(Multiset.map f (a ::ₘ t)).prod‖ ≤ ‖f a‖

Tactic:
·

Goal state after:
case pos
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a : ι
t : Multiset ι
M : ι
hMs : t ≠ 0 → M ∈ t
hM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖
hMa : ‖f M‖ ≤ ‖f a‖
⊢ ‖(Multiset.map f (a ::ₘ t)).prod‖ ≤ ‖f a‖
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case neg
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a : ι
t : Multiset ι
M : ι
hMs : t ≠ 0 → M ∈ t
hM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖
hMa : ‖f a‖ < ‖f M‖
⊢ ∃ i, (a ::ₘ t ≠ 0 → i ∈ a ::ₘ t) ∧ ‖(Multiset.map f (a ::ₘ t)).prod‖ ≤ ‖f i‖

Tactic:
·

Goal state after:
case neg
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a : ι
t : Multiset ι
M : ι
hMs : t ≠ 0 → M ∈ t
hM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖
hMa : ‖f a‖ < ‖f M‖
⊢ ∃ i, (a ::ₘ t ≠ 0 → i ∈ a ::ₘ t) ∧ ‖(Multiset.map f (a ::ₘ t)).prod‖ ≤ ‖f i‖
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case neg.inl
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a M : ι
hMa : ‖f a‖ < ‖f M‖
hMs : 0 ≠ 0 → M ∈ 0
hM : ‖(Multiset.map f 0).prod‖ ≤ ‖f M‖
⊢ ∃ i, (a ::ₘ 0 ≠ 0 → i ∈ a ::ₘ 0) ∧ ‖(Multiset.map f (a ::ₘ 0)).prod‖ ≤ ‖f i‖

pp: case neg.inr
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a : ι
t : Multiset ι
M : ι
hMs : t ≠ 0 → M ∈ t
hM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖
hMa : ‖f a‖ < ‖f M‖
ht : t ≠ 0
⊢ ∃ i, (a ::ₘ t ≠ 0 → i ∈ a ::ₘ t) ∧ ‖(Multiset.map f (a ::ₘ t)).prod‖ ≤ ‖f i‖

Tactic:
·

Goal state after:
case neg.inl
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a M : ι
hMa : ‖f a‖ < ‖f M‖
hMs : 0 ≠ 0 → M ∈ 0
hM : ‖(Multiset.map f 0).prod‖ ≤ ‖f M‖
⊢ ∃ i, (a ::ₘ 0 ≠ 0 → i ∈ a ::ₘ 0) ∧ ‖(Multiset.map f (a ::ₘ 0)).prod‖ ≤ ‖f i‖
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case neg.inr
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a : ι
t : Multiset ι
M : ι
hMs : t ≠ 0 → M ∈ t
hM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖
hMa : ‖f a‖ < ‖f M‖
ht : t ≠ 0
⊢ ∃ i, (a ::ₘ t ≠ 0 → i ∈ a ::ₘ t) ∧ ‖(Multiset.map f (a ::ₘ t)).prod‖ ≤ ‖f i‖

Tactic:
·

Goal state after:
case neg.inr
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a : ι
t : Multiset ι
M : ι
hMs : t ≠ 0 → M ∈ t
hM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖
hMa : ‖f a‖ < ‖f M‖
ht : t ≠ 0
⊢ ∃ i, (a ::ₘ t ≠ 0 → i ∈ a ::ₘ t) ∧ ‖(Multiset.map f (a ::ₘ t)).prod‖ ≤ ‖f i‖
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case neg.inr.refine_1
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a : ι
t : Multiset ι
M : ι
hMs : t ≠ 0 → M ∈ t
hM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖
hMa : ‖f a‖ < ‖f M‖
ht : t ≠ 0
⊢ a ::ₘ t ≠ 0 → M ∈ a ::ₘ t

pp: case neg.inr.refine_2
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a : ι
t : Multiset ι
M : ι
hMs : t ≠ 0 → M ∈ t
hM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖
hMa : ‖f a‖ < ‖f M‖
ht : t ≠ 0
⊢ ‖(Multiset.map f (a ::ₘ t)).prod‖ ≤ ‖f M‖

Tactic:
·

Goal state after:
case neg.inr.refine_1
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a : ι
t : Multiset ι
M : ι
hMs : t ≠ 0 → M ∈ t
hM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖
hMa : ‖f a‖ < ‖f M‖
ht : t ≠ 0
⊢ a ::ₘ t ≠ 0 → M ∈ a ::ₘ t
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case inl
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : IsEmpty ι
⊢ ‖∏' (i : ι), f i‖ ≤ ⨆ i, ‖f i‖

pp: case inr
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : Nonempty ι
⊢ ‖∏' (i : ι), f i‖ ≤ ⨆ i, ‖f i‖

Tactic:
·

Goal state after:
case inl
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : IsEmpty ι
⊢ ‖∏' (i : ι), f i‖ ≤ ⨆ i, ‖f i‖
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case neg
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : Nonempty ι
h : ¬Multipliable f
⊢ ‖∏' (i : ι), f i‖ ≤ ⨆ i, ‖f i‖

pp: case pos
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : Nonempty ι
h : Multipliable f
⊢ ‖∏' (i : ι), f i‖ ≤ ⨆ i, ‖f i‖

Tactic:
·

Goal state after:
case neg
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : Nonempty ι
h : ¬Multipliable f
⊢ ‖∏' (i : ι), f i‖ ≤ ⨆ i, ‖f i‖
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: case refine_1
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
⊢ (∃ ε > 0, Metric.ball 0 ε ⊆ U) → ∃ γ, {x | valuation.restrict x < ↑γ} ⊆ U

pp: case refine_2
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
⊢ (∃ γ, {x | valuation.restrict x < ↑γ} ⊆ U) → ∃ ε > 0, Metric.ball 0 ε ⊆ U

Tactic:
·

Goal state after:
case refine_1
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
⊢ (∃ ε > 0, Metric.ball 0 ε ⊆ U) → ∃ γ, {x | valuation.restrict x < ↑γ} ⊆ U
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case pos
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : Nonempty ι
h : ¬Multipliable f
h_bd : BddAbove (Set.range fun i ↦ ‖f i‖)
⊢ 0 ≤ ⨆ i, ‖f i‖

pp: case neg
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : Nonempty ι
h : ¬Multipliable f
h_bd : ¬BddAbove (Set.range fun i ↦ ‖f i‖)
⊢ 0 ≤ ⨆ i, ‖f i‖

Tactic:
·

Goal state after:
case pos
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : Nonempty ι
h : ¬Multipliable f
h_bd : BddAbove (Set.range fun i ↦ ‖f i‖)
⊢ 0 ≤ ⨆ i, ‖f i‖
Module: Mathlib.RingTheory.MvPowerSeries.LexOrder

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
h : ¬φ = 0
ne : (⇑toLex '' Function.support φ).Nonempty
⊢ Lex (σ →₀ ℕ) → Lex (σ →₀ ℕ) → Prop

pp: σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
h : ¬φ = 0
ne : (⇑toLex '' Function.support φ).Nonempty
⊢ WellFounded ?m.40

Tactic:
·

Goal state after:
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
h : ¬φ = 0
ne : (⇑toLex '' Function.support φ).Nonempty
⊢ Lex (σ →₀ ℕ) → Lex (σ →₀ ℕ) → Prop
Module: Mathlib.RingTheory.MvPowerSeries.LexOrder

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
h : ¬φ = 0
ne : (⇑toLex '' Function.support φ).Nonempty
⊢ WellFounded LT.lt

Tactic:
·

Goal state after:
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
h : ¬φ = 0
ne : (⇑toLex '' Function.support φ).Nonempty
⊢ WellFounded LT.lt
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: case refine_1.inl
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
ε : ℝ
hε : ε > 0
h : Metric.ball 0 ε ⊆ U
H : Subsingleton (ofClass valuation).ValueGroup₀ˣ
⊢ ∃ γ, {x | valuation.restrict x < ↑γ} ⊆ U

pp: case refine_1.inr
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
ε : ℝ
hε : ε > 0
h : Metric.ball 0 ε ⊆ U
H : ∀ {γ : ℝ≥0}, γ ≠ 0 → ∃ x, x ≠ 0 ∧ (RankLeOne.hom' valuation) (valuation.restrict x) < γ
⊢ ∃ γ, {x | valuation.restrict x < ↑γ} ⊆ U

Tactic:
·

Goal state after:
case refine_1.inl
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
ε : ℝ
hε : ε > 0
h : Metric.ball 0 ε ⊆ U
H : Subsingleton (ofClass valuation).ValueGroup₀ˣ
⊢ ∃ γ, {x | valuation.restrict x < ↑γ} ⊆ U
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case neg
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : Nonempty ι
h : ¬Multipliable f
h_bd : ¬BddAbove (Set.range fun i ↦ ‖f i‖)
⊢ 0 ≤ ⨆ i, ‖f i‖

Tactic:
·

Goal state after:
case neg
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : Nonempty ι
h : ¬Multipliable f
h_bd : ¬BddAbove (Set.range fun i ↦ ‖f i‖)
⊢ 0 ≤ ⨆ i, ‖f i‖
Module: Mathlib.RingTheory.MvPowerSeries.LexOrder

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
h : φ = 0
⊢ ⊤ = ⊤ ↔ φ = 0

pp: case neg
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
h : ¬φ = 0
⊢ (have ne := ⋯;
      ↑(⋯.min (⇑toLex '' Function.support φ) ne)) =
      ⊤ ↔
    φ = 0

Tactic:
·

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
h : φ = 0
⊢ ⊤ = ⊤ ↔ φ = 0
Module: Mathlib.RingTheory.MvPowerSeries.LexOrder

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
h : ¬φ = 0
⊢ (have ne := ⋯;
      ↑(⋯.min (⇑toLex '' Function.support φ) ne)) =
      ⊤ ↔
    φ = 0

Tactic:
·

Goal state after:
case neg
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
h : ¬φ = 0
⊢ (have ne := ⋯;
      ↑(⋯.min (⇑toLex '' Function.support φ) ne)) =
      ⊤ ↔
    φ = 0
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case pos.refine_1
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : Nonempty ι
h : Multipliable f
h_bd : BddAbove (Set.range fun i ↦ ‖f i‖)
s : Finset ι
⊢ 0 ≤ ⨆ i, ‖f i‖

pp: case pos.refine_2
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : Nonempty ι
h : Multipliable f
h_bd : BddAbove (Set.range fun i ↦ ‖f i‖)
s : Finset ι
⊢ ∀ i ∈ s, ‖f i‖ ≤ ⨆ i, ‖f i‖

Tactic:
·

Goal state after:
case pos.refine_1
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : Nonempty ι
h : Multipliable f
h_bd : BddAbove (Set.range fun i ↦ ‖f i‖)
s : Finset ι
⊢ 0 ≤ ⨆ i, ‖f i‖
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case pos.refine_2
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : Nonempty ι
h : Multipliable f
h_bd : BddAbove (Set.range fun i ↦ ‖f i‖)
s : Finset ι
⊢ ∀ i ∈ s, ‖f i‖ ≤ ⨆ i, ‖f i‖

Tactic:
·

Goal state after:
case pos.refine_2
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
hι : Nonempty ι
h : Multipliable f
h_bd : BddAbove (Set.range fun i ↦ ‖f i‖)
s : Finset ι
⊢ ∀ i ∈ s, ‖f i‖ ≤ ⨆ i, ‖f i‖
Module: Mathlib.RingTheory.MvPowerSeries.LexOrder

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑(toLex d) < φ.lexOrder
hφ : φ = 0
⊢ (coeff d) φ = 0

pp: case neg
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑(toLex d) < φ.lexOrder
hφ : ¬φ = 0
⊢ (coeff d) φ = 0

Tactic:
·

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑(toLex d) < φ.lexOrder
hφ : φ = 0
⊢ (coeff d) φ = 0
Module: Mathlib.RingTheory.MvPowerSeries.LexOrder

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑(toLex d) < φ.lexOrder
hφ : ¬φ = 0
⊢ (coeff d) φ = 0

Tactic:
·

Goal state after:
case neg
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑(toLex d) < φ.lexOrder
hφ : ¬φ = 0
⊢ (coeff d) φ = 0
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: case pos
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
ε : ℝ
hε : ε > 0
h : Metric.ball 0 ε ⊆ U
H : Subsingleton (ofClass valuation).ValueGroup₀ˣ
x : K
hx : valuation.restrict x < 1
hx0 : x = 0
⊢ x ∈ U

pp: case neg
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
ε : ℝ
hε : ε > 0
h : Metric.ball 0 ε ⊆ U
H : Subsingleton (ofClass valuation).ValueGroup₀ˣ
x : K
hx : valuation.restrict x < 1
hx0 : ¬x = 0
⊢ x ∈ U

Tactic:
·

Goal state after:
case pos
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
ε : ℝ
hε : ε > 0
h : Metric.ball 0 ε ⊆ U
H : Subsingleton (ofClass valuation).ValueGroup₀ˣ
x : K
hx : valuation.restrict x < 1
hx0 : x = 0
⊢ x ∈ U
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case inl
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
C : ℝ
hC : 0 ≤ C
h : ∀ (i : ι), ‖f i‖ ≤ C
h✝ : IsEmpty ι
⊢ ‖∏' (i : ι), f i‖ ≤ C

pp: case inr
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
C : ℝ
hC : 0 ≤ C
h : ∀ (i : ι), ‖f i‖ ≤ C
h✝ : Nonempty ι
⊢ ‖∏' (i : ι), f i‖ ≤ C

Tactic:
·

Goal state after:
case inl
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
C : ℝ
hC : 0 ≤ C
h : ∀ (i : ι), ‖f i‖ ≤ C
h✝ : IsEmpty ι
⊢ ‖∏' (i : ι), f i‖ ≤ C
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: case neg
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
ε : ℝ
hε : ε > 0
h : Metric.ball 0 ε ⊆ U
H : Subsingleton (ofClass valuation).ValueGroup₀ˣ
x : K
hx : valuation.restrict x < 1
hx0 : ¬x = 0
⊢ x ∈ U

Tactic:
·

Goal state after:
case neg
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
ε : ℝ
hε : ε > 0
h : Metric.ball 0 ε ⊆ U
H : Subsingleton (ofClass valuation).ValueGroup₀ˣ
x : K
hx : valuation.restrict x < 1
hx0 : ¬x = 0
⊢ x ∈ U
Module: Mathlib.RingTheory.MvPowerSeries.LexOrder

Goal state before:
pp: case mp
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
w : WithTop (Lex (σ →₀ ℕ))
⊢ w ≤ φ.lexOrder → ∀ (d : σ →₀ ℕ), ↑(toLex d) < w → (coeff d) φ = 0

pp: case mpr
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
w : WithTop (Lex (σ →₀ ℕ))
⊢ (∀ (d : σ →₀ ℕ), ↑(toLex d) < w → (coeff d) φ = 0) → w ≤ φ.lexOrder

Tactic:
·

Goal state after:
case mp
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
w : WithTop (Lex (σ →₀ ℕ))
⊢ w ≤ φ.lexOrder → ∀ (d : σ →₀ ℕ), ↑(toLex d) < w → (coeff d) φ = 0
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case inr
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
C : ℝ
hC : 0 ≤ C
h : ∀ (i : ι), ‖f i‖ ≤ C
h✝ : Nonempty ι
⊢ ‖∏' (i : ι), f i‖ ≤ C

Tactic:
·

Goal state after:
case inr
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
C : ℝ
hC : 0 ≤ C
h : ∀ (i : ι), ‖f i‖ ≤ C
h✝ : Nonempty ι
⊢ ‖∏' (i : ι), f i‖ ≤ C
Module: Mathlib.RingTheory.MvPowerSeries.LexOrder

Goal state before:
pp: case mpr
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
w : WithTop (Lex (σ →₀ ℕ))
⊢ (∀ (d : σ →₀ ℕ), ↑(toLex d) < w → (coeff d) φ = 0) → w ≤ φ.lexOrder

Tactic:
·

Goal state after:
case mpr
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
w : WithTop (Lex (σ →₀ ℕ))
⊢ (∀ (d : σ →₀ ℕ), ↑(toLex d) < w → (coeff d) φ = 0) → w ≤ φ.lexOrder
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: case cons.inl
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
a : ι
ha : a ∉ ∅
IH : ((↑∅).Pairwise fun i j ↦ ‖f i‖₊ ≠ ‖f j‖₊) → ‖∏ i ∈ ∅, f i‖₊ = ∅.sup fun i ↦ ‖f i‖₊
hs : (↑(Finset.cons a ∅ ha)).Pairwise fun i j ↦ ‖f i‖₊ ≠ ‖f j‖₊
⊢ ‖∏ i ∈ Finset.cons a ∅ ha, f i‖₊ = (Finset.cons a ∅ ha).sup fun i ↦ ‖f i‖₊

pp: case cons.inr
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
a : ι
s : Finset ι
ha : a ∉ s
IH : ((↑s).Pairwise fun i j ↦ ‖f i‖₊ ≠ ‖f j‖₊) → ‖∏ i ∈ s, f i‖₊ = s.sup fun i ↦ ‖f i‖₊
hs : (↑(Finset.cons a s ha)).Pairwise fun i j ↦ ‖f i‖₊ ≠ ‖f j‖₊
hs' : s.Nonempty
⊢ ‖∏ i ∈ Finset.cons a s ha, f i‖₊ = (Finset.cons a s ha).sup fun i ↦ ‖f i‖₊

Tactic:
·

Goal state after:
case cons.inl
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
a : ι
ha : a ∉ ∅
IH : ((↑∅).Pairwise fun i j ↦ ‖f i‖₊ ≠ ‖f j‖₊) → ‖∏ i ∈ ∅, f i‖₊ = ∅.sup fun i ↦ ‖f i‖₊
hs : (↑(Finset.cons a ∅ ha)).Pairwise fun i j ↦ ‖f i‖₊ ≠ ‖f j‖₊
⊢ ‖∏ i ∈ Finset.cons a ∅ ha, f i‖₊ = (Finset.cons a ∅ ha).sup fun i ↦ ‖f i‖₊
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: case refine_1.inr
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
ε : ℝ
hε : ε > 0
h : Metric.ball 0 ε ⊆ U
H : ∀ {γ : ℝ≥0}, γ ≠ 0 → ∃ x, x ≠ 0 ∧ (RankLeOne.hom' valuation) (valuation.restrict x) < γ
⊢ ∃ γ, {x | valuation.restrict x < ↑γ} ⊆ U

Tactic:
·

Goal state after:
case refine_1.inr
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
ε : ℝ
hε : ε > 0
h : Metric.ball 0 ε ⊆ U
H : ∀ {γ : ℝ≥0}, γ ≠ 0 → ∃ x, x ≠ 0 ∧ (RankLeOne.hom' valuation) (valuation.restrict x) < γ
⊢ ∃ γ, {x | valuation.restrict x < ↑γ} ⊆ U