url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalSpace.le_iff_nhds_le | [34, 1] | [48, 58] | intro a | case h
α : Type u_1
τ τ' : TopologicalSpace α
⊢ ∀ (a : α), nhds a ≤ nhds a ↔ ∀ (a_1 : Set α), a ∈ a_1 → a_1 ∈ nhds a → a_1 ∈ nhds a | case h
α : Type u_1
τ τ' : TopologicalSpace α
a : α
⊢ nhds a ≤ nhds a ↔ ∀ (a_1 : Set α), a ∈ a_1 → a_1 ∈ nhds a → a_1 ∈ nhds a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
τ τ' : TopologicalSpace α
⊢ ∀ (a : α), nhds a ≤ nhds a ↔ ∀ (a_1 : Set α), a ∈ a_1 → a_1 ∈ nhds a → a_1 ∈ nhds a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalSpace.le_iff_nhds_le | [34, 1] | [48, 58] | rw [Filter.le_def] | case h
α : Type u_1
τ τ' : TopologicalSpace α
a : α
⊢ nhds a ≤ nhds a ↔ ∀ (a_1 : Set α), a ∈ a_1 → a_1 ∈ nhds a → a_1 ∈ nhds a | case h
α : Type u_1
τ τ' : TopologicalSpace α
a : α
⊢ (∀ x ∈ nhds a, x ∈ nhds a) ↔ ∀ (a_1 : Set α), a ∈ a_1 → a_1 ∈ nhds a → a_1 ∈ nhds a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
τ τ' : TopologicalSpace α
a : α
⊢ nhds a ≤ nhds a ↔ ∀ (a_1 : Set α), a ∈ a_1 → a_1 ∈ nhds a → a_1 ∈ nhds a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalSpace.le_iff_nhds_le | [34, 1] | [48, 58] | apply forall_congr' | case h
α : Type u_1
τ τ' : TopologicalSpace α
a : α
⊢ (∀ x ∈ nhds a, x ∈ nhds a) ↔ ∀ (a_1 : Set α), a ∈ a_1 → a_1 ∈ nhds a → a_1 ∈ nhds a | case h.h
α : Type u_1
τ τ' : TopologicalSpace α
a : α
⊢ ∀ (a_1 : Set α), a_1 ∈ nhds a → a_1 ∈ nhds a ↔ a ∈ a_1 → a_1 ∈ nhds a → a_1 ∈ nhds a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
τ τ' : TopologicalSpace α
a : α
⊢ (∀ x ∈ nhds a, x ∈ nhds a) ↔ ∀ (a_1 : Set α), a ∈ a_1 → a_1 ∈ nhds a → a_1 ∈ nhds a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalSpace.le_iff_nhds_le | [34, 1] | [48, 58] | intro s | case h.h
α : Type u_1
τ τ' : TopologicalSpace α
a : α
⊢ ∀ (a_1 : Set α), a_1 ∈ nhds a → a_1 ∈ nhds a ↔ a ∈ a_1 → a_1 ∈ nhds a → a_1 ∈ nhds a | case h.h
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
⊢ s ∈ nhds a → s ∈ nhds a ↔ a ∈ s → s ∈ nhds a → s ∈ nhds a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
τ τ' : TopologicalSpace α
a : α
⊢ ∀ (a_1 : Set α), a_1 ∈ nhds a → a_1 ∈ nhds a ↔ a ∈ a_1 → a_1 ∈ nhds a → a_1 ∈ nhds a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalSpace.le_iff_nhds_le | [34, 1] | [48, 58] | constructor | case h.h
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
⊢ s ∈ nhds a → s ∈ nhds a ↔ a ∈ s → s ∈ nhds a → s ∈ nhds a | case h.h.mp
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
⊢ (s ∈ nhds a → s ∈ nhds a) → a ∈ s → s ∈ nhds a → s ∈ nhds a
case h.h.mpr
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
⊢ (a ∈ s → s ∈ nhds a → s ∈ nhds a) → s ∈ nhds a → s ∈ nhds a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
⊢ s ∈ nhds a → s ∈ nhds a ↔ a ∈ s → s ∈ nhds a → s ∈ nhds a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalSpace.le_iff_nhds_le | [34, 1] | [48, 58] | exact fun h _ ↦ h | case h.h.mp
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
⊢ (s ∈ nhds a → s ∈ nhds a) → a ∈ s → s ∈ nhds a → s ∈ nhds a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.mp
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
⊢ (s ∈ nhds a → s ∈ nhds a) → a ∈ s → s ∈ nhds a → s ∈ nhds a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalSpace.le_iff_nhds_le | [34, 1] | [48, 58] | intro h | case h.h.mpr
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
⊢ (a ∈ s → s ∈ nhds a → s ∈ nhds a) → s ∈ nhds a → s ∈ nhds a | case h.h.mpr
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
h : a ∈ s → s ∈ nhds a → s ∈ nhds a
⊢ s ∈ nhds a → s ∈ nhds a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.mpr
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
⊢ (a ∈ s → s ∈ nhds a → s ∈ nhds a) → s ∈ nhds a → s ∈ nhds a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalSpace.le_iff_nhds_le | [34, 1] | [48, 58] | by_cases ha : a ∈ s | case h.h.mpr
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
h : a ∈ s → s ∈ nhds a → s ∈ nhds a
⊢ s ∈ nhds a → s ∈ nhds a | case pos
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
h : a ∈ s → s ∈ nhds a → s ∈ nhds a
ha : a ∈ s
⊢ s ∈ nhds a → s ∈ nhds a
case neg
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
h : a ∈ s → s ∈ nhds a → s ∈ nhds a
ha : a ∉ s
⊢ s ∈ nhds a → s ∈ nhds a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.mpr
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
h : a ∈ s → s ∈ nhds a → s ∈ nhds a
⊢ s ∈ nhds a → s ∈ nhds a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalSpace.le_iff_nhds_le | [34, 1] | [48, 58] | exact h ha | case pos
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
h : a ∈ s → s ∈ nhds a → s ∈ nhds a
ha : a ∈ s
⊢ s ∈ nhds a → s ∈ nhds a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
h : a ∈ s → s ∈ nhds a → s ∈ nhds a
ha : a ∈ s
⊢ s ∈ nhds a → s ∈ nhds a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalSpace.le_iff_nhds_le | [34, 1] | [48, 58] | exact fun hs ↦ False.elim (ha (mem_of_mem_nhds hs)) | case neg
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
h : a ∈ s → s ∈ nhds a → s ∈ nhds a
ha : a ∉ s
⊢ s ∈ nhds a → s ∈ nhds a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
τ τ' : TopologicalSpace α
a : α
s : Set α
h : a ∈ s → s ∈ nhds a → s ∈ nhds a
ha : a ∉ s
⊢ s ∈ nhds a → s ∈ nhds a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | mem_nhds_add_iff | [69, 1] | [81, 32] | constructor | α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
⊢ V ∈ nhds (a + b) ↔ Add.add a ⁻¹' V ∈ nhds b | case mp
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
⊢ V ∈ nhds (a + b) → Add.add a ⁻¹' V ∈ nhds b
case mpr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
⊢ Add.add a ⁻¹' V ∈ nhds b → V... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
⊢ V ∈ nhds (a + b) ↔ Add.add a ⁻¹' V ∈ nhds b
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | mem_nhds_add_iff | [69, 1] | [81, 32] | . exact fun hV => ContinuousAt.preimage_mem_nhds (continuous_add_left a).continuousAt hV | case mp
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
⊢ V ∈ nhds (a + b) → Add.add a ⁻¹' V ∈ nhds b
case mpr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
⊢ Add.add a ⁻¹' V ∈ nhds b → V... | case mpr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
⊢ Add.add a ⁻¹' V ∈ nhds b → V ∈ nhds (a + b) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
⊢ V ∈ nhds (a + b) → Add.add a ⁻¹' V ∈ nhds b
case mpr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : Topologic... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | mem_nhds_add_iff | [69, 1] | [81, 32] | exact fun hV => ContinuousAt.preimage_mem_nhds (continuous_add_left a).continuousAt hV | case mp
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
⊢ V ∈ nhds (a + b) → Add.add a ⁻¹' V ∈ nhds b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
⊢ V ∈ nhds (a + b) → Add.add a ⁻¹' V ∈ nhds b
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | mem_nhds_add_iff | [69, 1] | [81, 32] | intro hV | case mpr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
⊢ Add.add a ⁻¹' V ∈ nhds b → V ∈ nhds (a + b) | case mpr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
⊢ V ∈ nhds (a + b) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
⊢ Add.add a ⁻¹' V ∈ nhds b → V ∈ nhds (a + b)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | mem_nhds_add_iff | [69, 1] | [81, 32] | suffices h : V = Add.add (-a) ⁻¹' (Add.add a ⁻¹' V) by
rw [h]
apply ContinuousAt.preimage_mem_nhds (continuous_add_left (-a)).continuousAt
convert hV
apply neg_add_cancel_left | case mpr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
⊢ V ∈ nhds (a + b) | case mpr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
⊢ V = Add.add (-a) ⁻¹' (Add.add a ⁻¹' V) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
⊢ V ∈ nhds (a + b)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | mem_nhds_add_iff | [69, 1] | [81, 32] | rw [Set.preimage_preimage, eq_comm] | case mpr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
⊢ V = Add.add (-a) ⁻¹' (Add.add a ⁻¹' V) | case mpr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
⊢ (fun x => Add.add a (Add.add (-a) x)) ⁻¹' V = V | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
⊢ V = Add.add (-a) ⁻¹' (Add.add a ⁻¹' V)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | mem_nhds_add_iff | [69, 1] | [81, 32] | convert Set.preimage_id' | case mpr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
⊢ (fun x => Add.add a (Add.add (-a) x)) ⁻¹' V = V | case h.e'_2.h.e'_3.h
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
x✝ : α
⊢ Add.add a (Add.add (-a) x✝) = x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
⊢ (fun x => Add.add a (Add.add (-a) x)) ⁻¹' V = V
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | mem_nhds_add_iff | [69, 1] | [81, 32] | apply add_neg_cancel_left a | case h.e'_2.h.e'_3.h
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
x✝ : α
⊢ Add.add a (Add.add (-a) x✝) = x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_3.h
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
x✝ : α
⊢ Add.add a (Add.add (-a) x✝) = x✝
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | mem_nhds_add_iff | [69, 1] | [81, 32] | rw [h] | α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
h : V = Add.add (-a) ⁻¹' (Add.add a ⁻¹' V)
⊢ V ∈ nhds (a + b) | α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
h : V = Add.add (-a) ⁻¹' (Add.add a ⁻¹' V)
⊢ Add.add (-a) ⁻¹' (Add.add a ⁻¹' V) ∈ nhds (a + b) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
h : V = Add.add (-a) ⁻¹' (Add.add a ⁻¹' V)
⊢ V ∈ nhds (a + b)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | mem_nhds_add_iff | [69, 1] | [81, 32] | apply ContinuousAt.preimage_mem_nhds (continuous_add_left (-a)).continuousAt | α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
h : V = Add.add (-a) ⁻¹' (Add.add a ⁻¹' V)
⊢ Add.add (-a) ⁻¹' (Add.add a ⁻¹' V) ∈ nhds (a + b) | α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
h : V = Add.add (-a) ⁻¹' (Add.add a ⁻¹' V)
⊢ Add.add a ⁻¹' V ∈ nhds (-a + (a + b)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
h : V = Add.add (-a) ⁻¹' (Add.add a ⁻¹' V)
⊢ Add.add (-a) ⁻¹' (Add.add a ⁻¹' V) ∈ nhds (a + b)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | mem_nhds_add_iff | [69, 1] | [81, 32] | convert hV | α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
h : V = Add.add (-a) ⁻¹' (Add.add a ⁻¹' V)
⊢ Add.add a ⁻¹' V ∈ nhds (-a + (a + b)) | case h.e'_5.h.e'_3
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
h : V = Add.add (-a) ⁻¹' (Add.add a ⁻¹' V)
⊢ -a + (a + b) = b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
h : V = Add.add (-a) ⁻¹' (Add.add a ⁻¹' V)
⊢ Add.add a ⁻¹' V ∈ nhds (-a + (a + b))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | mem_nhds_add_iff | [69, 1] | [81, 32] | apply neg_add_cancel_left | case h.e'_5.h.e'_3
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
h : V = Add.add (-a) ⁻¹' (Add.add a ⁻¹' V)
⊢ -a + (a + b) = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.e'_3
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalAddGroup α
V : Set α
a b : α
hV : Add.add a ⁻¹' V ∈ nhds b
h : V = Add.add (-a) ⁻¹' (Add.add a ⁻¹' V)
⊢ -a + (a + b) = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalAddGroup.ext_iff_nhds_zero | [84, 1] | [95, 36] | rw [TopologicalSpace.ext_iff_nhds] | α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
⊢ τ = τ' ↔ nhds 0 = nhds 0 | α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
⊢ (∀ (x : α), nhds x = nhds x) ↔ nhds 0 = nhds 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
⊢ τ = τ' ↔ nhds 0 = nhds 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalAddGroup.ext_iff_nhds_zero | [84, 1] | [95, 36] | constructor | α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
⊢ (∀ (x : α), nhds x = nhds x) ↔ nhds 0 = nhds 0 | case mp
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
⊢ (∀ (x : α), nhds x = nhds x) → nhds 0 = nhds 0
case mpr
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalS... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
⊢ (∀ (x : α), nhds x = nhds x) ↔ nhds 0 = nhds 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalAddGroup.ext_iff_nhds_zero | [84, 1] | [95, 36] | intro h | case mp
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
⊢ (∀ (x : α), nhds x = nhds x) → nhds 0 = nhds 0 | case mp
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
h : ∀ (x : α), nhds x = nhds x
⊢ nhds 0 = nhds 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
⊢ (∀ (x : α), nhds x = nhds x) → nhds 0 = nhds 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalAddGroup.ext_iff_nhds_zero | [84, 1] | [95, 36] | exact h 0 | case mp
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
h : ∀ (x : α), nhds x = nhds x
⊢ nhds 0 = nhds 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
h : ∀ (x : α), nhds x = nhds x
⊢ nhds 0 = nhds 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalAddGroup.ext_iff_nhds_zero | [84, 1] | [95, 36] | intro h a | case mpr
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
⊢ nhds 0 = nhds 0 → ∀ (x : α), nhds x = nhds x | case mpr
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
h : nhds 0 = nhds 0
a : α
⊢ nhds a = nhds a | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
⊢ nhds 0 = nhds 0 → ∀ (x : α), nhds x = nhds x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalAddGroup.ext_iff_nhds_zero | [84, 1] | [95, 36] | ext s | case mpr
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
h : nhds 0 = nhds 0
a : α
⊢ nhds a = nhds a | case mpr.a
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
h : nhds 0 = nhds 0
a : α
s : Set α
⊢ s ∈ nhds a ↔ s ∈ nhds a | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
h : nhds 0 = nhds 0
a : α
⊢ nhds a = nhds a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalAddGroup.ext_iff_nhds_zero | [84, 1] | [95, 36] | rw [← add_zero a] | case mpr.a
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
h : nhds 0 = nhds 0
a : α
s : Set α
⊢ s ∈ nhds a ↔ s ∈ nhds a | case mpr.a
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
h : nhds 0 = nhds 0
a : α
s : Set α
⊢ s ∈ nhds (a + 0) ↔ s ∈ nhds (a + 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.a
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
h : nhds 0 = nhds 0
a : α
s : Set α
⊢ s ∈ nhds a ↔ s ∈ nhds a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | TopologicalAddGroup.ext_iff_nhds_zero | [84, 1] | [95, 36] | simp only [mem_nhds_add_iff, h] | case mpr.a
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
h : nhds 0 = nhds 0
a : α
s : Set α
⊢ s ∈ nhds (a + 0) ↔ s ∈ nhds (a + 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.a
α : Type u_1
inst✝² : AddCommGroup α
τ : TopologicalSpace α
inst✝¹ : TopologicalAddGroup α
τ' : TopologicalSpace α
inst✝ : TopologicalAddGroup α
h : nhds 0 = nhds 0
a : α
s : Set α
⊢ s ∈ nhds (a + 0) ↔ s ∈ nhds (a + 0)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofComm | [147, 1] | [153, 60] | rw [mul_comm] | α✝ : Type u_1
inst✝² : Ring α✝
α : Type u_2
inst✝¹ : CommRing α
ι : Type u_3
inst✝ : Nonempty ι
B : ι → Ideal α
inter : ∀ (i j : ι), ∃ k, B k ≤ B i ⊓ B j
i : ι
a r : α
h : a ∈ B i
⊢ a * r ∈ B i | α✝ : Type u_1
inst✝² : Ring α✝
α : Type u_2
inst✝¹ : CommRing α
ι : Type u_3
inst✝ : Nonempty ι
B : ι → Ideal α
inter : ∀ (i j : ι), ∃ k, B k ≤ B i ⊓ B j
i : ι
a r : α
h : a ∈ B i
⊢ r * a ∈ B i | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
inst✝² : Ring α✝
α : Type u_2
inst✝¹ : CommRing α
ι : Type u_3
inst✝ : Nonempty ι
B : ι → Ideal α
inter : ∀ (i j : ι), ∃ k, B k ≤ B i ⊓ B j
i : ι
a r : α
h : a ∈ B i
⊢ a * r ∈ B i
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofComm | [147, 1] | [153, 60] | refine' Ideal.mul_mem_left (B i) r h | α✝ : Type u_1
inst✝² : Ring α✝
α : Type u_2
inst✝¹ : CommRing α
ι : Type u_3
inst✝ : Nonempty ι
B : ι → Ideal α
inter : ∀ (i j : ι), ∃ k, B k ≤ B i ⊓ B j
i : ι
a r : α
h : a ∈ B i
⊢ r * a ∈ B i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
inst✝² : Ring α✝
α : Type u_2
inst✝¹ : CommRing α
ι : Type u_3
inst✝ : Nonempty ι
B : ι → Ideal α
inter : ∀ (i j : ι), ∃ k, B k ≤ B i ⊓ B j
i : ι
a r : α
h : a ∈ B i
⊢ r * a ∈ B i
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.toRingSubgroupsBasis | [155, 1] | [163, 56] | rintro ⟨x, _, _, hy, rfl⟩ | α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
i : ι
u : α
⊢ u ∈ ↑(Submodule.toAddSubgroup (B i)) * ↑(Submodule.toAddSubgroup (B i)) → u ∈ ↑(Submodule.toAddSubgroup (B i)) | case intro.intro.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
i : ι
x : α
left✝ : x ∈ ↑(Submodule.toAddSubgroup (B i))
w✝ : α
hy : w✝ ∈ ↑(Submodule.toAddSubgroup (B i))
⊢ (fun x x_1 => x * x_1) x w✝ ∈ ↑(Submodule.toAddSubgroup (B i)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
i : ι
u : α
⊢ u ∈ ↑(Submodule.toAddSubgroup (B i)) * ↑(Submodule.toAddSubgroup (B i)) → u ∈ ↑(Submodule.toAddSubgroup (B i))
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.toRingSubgroupsBasis | [155, 1] | [163, 56] | exact Ideal.mul_mem_left _ _ hy | case intro.intro.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
i : ι
x : α
left✝ : x ∈ ↑(Submodule.toAddSubgroup (B i))
w✝ : α
hy : w✝ ∈ ↑(Submodule.toAddSubgroup (B i))
⊢ (fun x x_1 => x * x_1) x w✝ ∈ ↑(Submodule.toAddSubgroup (B i)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
i : ι
x : α
left✝ : x ∈ ↑(Submodule.toAddSubgroup (B i))
w✝ : α
hy : w✝ ∈ ↑(Submodule.toAddSubgroup (B i))
⊢ (fun x x_1 => x * x_1) x w✝ ∈ ↑(Submodule.toAddSu... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.mem_nhds_zero_iff | [176, 1] | [187, 27] | simp only [AddGroupFilterBasis.nhds_eq, AddGroupFilterBasis.N_zero,
Filter.IsBasis.mem_filter_iff, FilterBasis.mem_filter_iff] | α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
⊢ s ∈ nhds 0 ↔ ∃ i, ↑(B i) ∈ nhds 0 ∧ ↑(B i) ⊆ s | α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
⊢ (∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s) ↔
∃ i, (∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)) ∧ ↑(B i) ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
⊢ s ∈ nhds 0 ↔ ∃ i, ↑(B i) ∈ nhds 0 ∧ ↑(B i) ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.mem_nhds_zero_iff | [176, 1] | [187, 27] | constructor | α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
⊢ (∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s) ↔
∃ i, (∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)) ∧ ↑(B i) ⊆ s | case mp
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
⊢ (∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s) →
∃ i, (∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)) ∧ ↑(B i) ⊆ s
case mpr
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
⊢ (∃ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
⊢ (∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s) ↔
∃ i, (∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)) ∧ ↑(B i) ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.mem_nhds_zero_iff | [176, 1] | [187, 27] | rintro ⟨t, ⟨i, rfl⟩, hts⟩ | case mp
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
⊢ (∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s) →
∃ i, (∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)) ∧ ↑(B i) ⊆ s | case mp.intro.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
i : ι
hts : ↑((fun i => Submodule.toAddSubgroup (B i)) i) ⊆ s
⊢ ∃ i, (∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)) ∧ ↑(B i) ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
⊢ (∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s) →
∃ i, (∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)) ∧ ↑(B i) ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.mem_nhds_zero_iff | [176, 1] | [187, 27] | simp only [Submodule.coe_toAddSubgroup] at hts | case mp.intro.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
i : ι
hts : ↑((fun i => Submodule.toAddSubgroup (B i)) i) ⊆ s
⊢ ∃ i, (∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)) ∧ ↑(B i) ⊆ s | case mp.intro.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
i : ι
hts : ↑(B i) ⊆ s
⊢ ∃ i, (∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)) ∧ ↑(B i) ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
i : ι
hts : ↑((fun i => Submodule.toAddSubgroup (B i)) i) ⊆ s
⊢ ∃ i, (∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)) ∧ ↑(B i) ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.mem_nhds_zero_iff | [176, 1] | [187, 27] | exact ⟨i, ⟨B i, ⟨i, rfl⟩, subset_of_eq rfl⟩, hts⟩ | case mp.intro.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
i : ι
hts : ↑(B i) ⊆ s
⊢ ∃ i, (∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)) ∧ ↑(B i) ⊆ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
i : ι
hts : ↑(B i) ⊆ s
⊢ ∃ i, (∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)) ∧ ↑(B i) ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.mem_nhds_zero_iff | [176, 1] | [187, 27] | rintro ⟨i, _, his⟩ | case mpr
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
⊢ (∃ i, (∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)) ∧ ↑(B i) ⊆ s) →
∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s | case mpr.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
i : ι
left✝ : ∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)
his : ↑(B i) ⊆ s
⊢ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
⊢ (∃ i, (∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)) ∧ ↑(B i) ⊆ s) →
∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.mem_nhds_zero_iff | [176, 1] | [187, 27] | use B i, ⟨i, rfl⟩, his | case mpr.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
i : ι
left✝ : ∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)
his : ↑(B i) ⊆ s
⊢ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
s : Set α
i : ι
left✝ : ∃ s ∈ AddGroupFilterBasis.toFilterBasis, s ⊆ ↑(B i)
his : ↑(B i) ⊆ s
⊢ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofIdealBasis_topology_eq | [189, 1] | [205, 46] | rw [TopologicalSpace.ext_iff_nhds] | α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
⊢ ⋯.topology = hB.topology | α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
⊢ ∀ (x : α), nhds x = nhds x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
⊢ ⋯.topology = hB.topology
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofIdealBasis_topology_eq | [189, 1] | [205, 46] | intro a | α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
⊢ ∀ (x : α), nhds x = nhds x | α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
⊢ nhds a = nhds a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
⊢ ∀ (x : α), nhds x = nhds x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofIdealBasis_topology_eq | [189, 1] | [205, 46] | simp [AddGroupFilterBasis.nhds_eq] | α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
⊢ nhds a = nhds a | α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
⊢ RingFilterBasis.toAddGroupFilterBasis.N a = RingFilterBasis.toAddGroupFilterBasis.N a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
⊢ nhds a = nhds a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofIdealBasis_topology_eq | [189, 1] | [205, 46] | simp only [AddGroupFilterBasis.N] | α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
⊢ RingFilterBasis.toAddGroupFilterBasis.N a = RingFilterBasis.toAddGroupFilterBasis.N a | α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
⊢ Filter.map (fun y => a + y) AddGroupFilterBasis.toFilterBasis.filter =
Filter.map (fun y => a + y) AddGroupFilterBasis.toFilterBasis.filter | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
⊢ RingFilterBasis.toAddGroupFilterBasis.N a = RingFilterBasis.toAddGroupFilterBasis.N a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofIdealBasis_topology_eq | [189, 1] | [205, 46] | apply congr_arg₂ _ rfl | α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
⊢ Filter.map (fun y => a + y) AddGroupFilterBasis.toFilterBasis.filter =
Filter.map (fun y => a + y) AddGroupFilterBasis.toFilterBasis.filter | α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
⊢ AddGroupFilterBasis.toFilterBasis.filter = AddGroupFilterBasis.toFilterBasis.filter | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
⊢ Filter.map (fun y => a + y) AddGroupFilterBasis.toFilterBasis.filter =
Filter.map (fun y => a + y) AddGroupFilterBasis.toFilterBasis.filter
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofIdealBasis_topology_eq | [189, 1] | [205, 46] | ext s | α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
⊢ AddGroupFilterBasis.toFilterBasis.filter = AddGroupFilterBasis.toFilterBasis.filter | case a
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
⊢ s ∈ AddGroupFilterBasis.toFilterBasis.filter ↔ s ∈ AddGroupFilterBasis.toFilterBasis.filter | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
⊢ AddGroupFilterBasis.toFilterBasis.filter = AddGroupFilterBasis.toFilterBasis.filter
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofIdealBasis_topology_eq | [189, 1] | [205, 46] | simp only [FilterBasis.mem_filter_iff] | case a
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
⊢ s ∈ AddGroupFilterBasis.toFilterBasis.filter ↔ s ∈ AddGroupFilterBasis.toFilterBasis.filter | case a
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
⊢ (∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s) ↔ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
⊢ s ∈ AddGroupFilterBasis.toFilterBasis.filter ↔ s ∈ AddGroupFilterBasis.toFilterBasis.filter
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofIdealBasis_topology_eq | [189, 1] | [205, 46] | constructor | case a
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
⊢ (∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s) ↔ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s | case a.mp
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
⊢ (∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s) → ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s
case a.mpr
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
⊢ (∃ s_1 ∈ Add... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
⊢ (∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s) ↔ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofIdealBasis_topology_eq | [189, 1] | [205, 46] | rintro ⟨u, ⟨⟨v, ⟨i, rfl⟩⟩, rfl⟩, hus⟩ | case a.mp
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
⊢ (∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s) → ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s | case a.mp.intro.intro.intro.mk.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
i : ι
hus : ↑((fun i => Submodule.toAddSubgroup ((fun x => ↑x) i)) ⟨B i, ⋯⟩) ⊆ s
⊢ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
⊢ (∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s) → ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofIdealBasis_topology_eq | [189, 1] | [205, 46] | simp only [Submodule.coe_toAddSubgroup] at hus | case a.mp.intro.intro.intro.mk.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
i : ι
hus : ↑((fun i => Submodule.toAddSubgroup ((fun x => ↑x) i)) ⟨B i, ⋯⟩) ⊆ s
⊢ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s | case a.mp.intro.intro.intro.mk.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
i : ι
hus : ↑(B i) ⊆ s
⊢ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp.intro.intro.intro.mk.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
i : ι
hus : ↑((fun i => Submodule.toAddSubgroup ((fun x => ↑x) i)) ⟨B i, ⋯⟩) ⊆ s
⊢ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofIdealBasis_topology_eq | [189, 1] | [205, 46] | exact ⟨B i, ⟨i, rfl⟩, hus⟩ | case a.mp.intro.intro.intro.mk.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
i : ι
hus : ↑(B i) ⊆ s
⊢ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp.intro.intro.intro.mk.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
i : ι
hus : ↑(B i) ⊆ s
⊢ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofIdealBasis_topology_eq | [189, 1] | [205, 46] | rintro ⟨u, ⟨i, rfl⟩, hus⟩ | case a.mpr
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
⊢ (∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s) → ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s | case a.mpr.intro.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
i : ι
hus : ↑((fun i => Submodule.toAddSubgroup (B i)) i) ⊆ s
⊢ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
⊢ (∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s) → ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofIdealBasis_topology_eq | [189, 1] | [205, 46] | simp only [Submodule.coe_toAddSubgroup] at hus | case a.mpr.intro.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
i : ι
hus : ↑((fun i => Submodule.toAddSubgroup (B i)) i) ⊆ s
⊢ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s | case a.mpr.intro.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
i : ι
hus : ↑(B i) ⊆ s
⊢ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
i : ι
hus : ↑((fun i => Submodule.toAddSubgroup (B i)) i) ⊆ s
⊢ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | Ideal.IsBasis.ofIdealBasis_topology_eq | [189, 1] | [205, 46] | refine ⟨B i, ⟨⟨B i, ⟨i, rfl⟩⟩, rfl⟩, hus⟩ | case a.mpr.intro.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
i : ι
hus : ↑(B i) ⊆ s
⊢ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.intro.intro.intro
α : Type u_1
inst✝ : Ring α
ι : Type u_2
B : ι → Ideal α
hB : IsBasis B
a : α
s : Set α
i : ι
hus : ↑(B i) ⊆ s
⊢ ∃ s_1 ∈ AddGroupFilterBasis.toFilterBasis, s_1 ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | IdealBasis.mem_nhds_zero_iff | [265, 1] | [275, 38] | rw [Ideal.IsBasis.mem_nhds_zero_iff] | α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
⊢ s ∈ nhds 0 ↔ ∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ≤ s | α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
⊢ (∃ i, ↑↑i ∈ nhds 0 ∧ ↑↑i ⊆ s) ↔ ∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ≤ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
⊢ s ∈ nhds 0 ↔ ∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ≤ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | IdealBasis.mem_nhds_zero_iff | [265, 1] | [275, 38] | simp only [Subtype.exists, exists_and_left, exists_prop, Set.le_eq_subset] | α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
⊢ (∃ i, ↑↑i ∈ nhds 0 ∧ ↑↑i ⊆ s) ↔ ∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ≤ s | α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
⊢ (∃ a, ↑a ∈ nhds 0 ∧ a ∈ B.sets ∧ ↑a ⊆ s) ↔ ∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
⊢ (∃ i, ↑↑i ∈ nhds 0 ∧ ↑↑i ⊆ s) ↔ ∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ≤ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | IdealBasis.mem_nhds_zero_iff | [265, 1] | [275, 38] | constructor | α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
⊢ (∃ a, ↑a ∈ nhds 0 ∧ a ∈ B.sets ∧ ↑a ⊆ s) ↔ ∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ⊆ s | case mp
α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
⊢ (∃ a, ↑a ∈ nhds 0 ∧ a ∈ B.sets ∧ ↑a ⊆ s) → ∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ⊆ s
case mpr
α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
⊢ (∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ⊆ s) → ∃ a, ↑a ∈ nhds 0 ∧ a ∈ B.sets ∧ ↑a ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
⊢ (∃ a, ↑a ∈ nhds 0 ∧ a ∈ B.sets ∧ ↑a ⊆ s) ↔ ∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | IdealBasis.mem_nhds_zero_iff | [265, 1] | [275, 38] | rintro ⟨a, mem_nhds, mem_sets, subset_s⟩ | case mp
α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
⊢ (∃ a, ↑a ∈ nhds 0 ∧ a ∈ B.sets ∧ ↑a ⊆ s) → ∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ⊆ s | case mp.intro.intro.intro
α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
a : Ideal α
mem_nhds : ↑a ∈ nhds 0
mem_sets : a ∈ B.sets
subset_s : ↑a ⊆ s
⊢ ∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
⊢ (∃ a, ↑a ∈ nhds 0 ∧ a ∈ B.sets ∧ ↑a ⊆ s) → ∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | IdealBasis.mem_nhds_zero_iff | [265, 1] | [275, 38] | exact ⟨a, mem_sets, mem_nhds, subset_s⟩ | case mp.intro.intro.intro
α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
a : Ideal α
mem_nhds : ↑a ∈ nhds 0
mem_sets : a ∈ B.sets
subset_s : ↑a ⊆ s
⊢ ∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ⊆ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro
α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
a : Ideal α
mem_nhds : ↑a ∈ nhds 0
mem_sets : a ∈ B.sets
subset_s : ↑a ⊆ s
⊢ ∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | IdealBasis.mem_nhds_zero_iff | [265, 1] | [275, 38] | rintro ⟨i, hi, mem_nhds, subset_s⟩ | case mpr
α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
⊢ (∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ⊆ s) → ∃ a, ↑a ∈ nhds 0 ∧ a ∈ B.sets ∧ ↑a ⊆ s | case mpr.intro.intro.intro
α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
i : Ideal α
hi : i ∈ B.sets
mem_nhds : ↑i ∈ nhds 0
subset_s : ↑i ⊆ s
⊢ ∃ a, ↑a ∈ nhds 0 ∧ a ∈ B.sets ∧ ↑a ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
⊢ (∃ i ∈ B.sets, ↑i ∈ nhds 0 ∧ ↑i ⊆ s) → ∃ a, ↑a ∈ nhds 0 ∧ a ∈ B.sets ∧ ↑a ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | IdealBasis.mem_nhds_zero_iff | [265, 1] | [275, 38] | exact ⟨i, mem_nhds, hi, subset_s⟩ | case mpr.intro.intro.intro
α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
i : Ideal α
hi : i ∈ B.sets
mem_nhds : ↑i ∈ nhds 0
subset_s : ↑i ⊆ s
⊢ ∃ a, ↑a ∈ nhds 0 ∧ a ∈ B.sets ∧ ↑a ⊆ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro
α : Type u_1
inst✝ : Ring α
B : IdealBasis α
s : Set α
i : Ideal α
hi : i ∈ B.sets
mem_nhds : ↑i ∈ nhds 0
subset_s : ↑i ⊆ s
⊢ ∃ a, ↑a ∈ nhds 0 ∧ a ∈ B.sets ∧ ↑a ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | LinearTopology.mem_nhds_zero_iff | [323, 1] | [328, 39] | rw [TopologicalSpace.ext_iff_nhds.mp hL.isTopology,
hL.toIdealBasis.mem_nhds_zero_iff] | α : Type u
inst✝¹ : Ring α
inst✝ : TopologicalSpace α
hL : LinearTopology α
s : Set α
⊢ s ∈ nhds 0 ↔ ∃ i ∈ toIdealBasis.sets, ↑i ∈ nhds 0 ∧ ↑i ≤ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝¹ : Ring α
inst✝ : TopologicalSpace α
hL : LinearTopology α
s : Set α
⊢ s ∈ nhds 0 ↔ ∃ i ∈ toIdealBasis.sets, ↑i ∈ nhds 0 ∧ ↑i ≤ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | LinearTopology.tendsto_zero_mul | [330, 1] | [346, 34] | intro v hv | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
⊢ Filter.Tendsto (a * b) f (nhds 0) | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
hv : v ∈ nhds 0
⊢ v ∈ Filter.map (a * b) f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
⊢ Filter.Tendsto (a * b) f (nhds 0)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | LinearTopology.tendsto_zero_mul | [330, 1] | [346, 34] | rw [LinearTopology.mem_nhds_zero_iff] at hv | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
hv : v ∈ nhds 0
⊢ v ∈ Filter.map (a * b) f | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
hv : ∃ i ∈ toIdealBasis.sets, ↑i ∈ nhds 0 ∧ ↑i ≤ v
⊢ v ∈ Filter.map (a * b) f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
hv : v ∈ nhds 0
⊢ v ∈ Filter.map (a * b) f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | LinearTopology.tendsto_zero_mul | [330, 1] | [346, 34] | obtain ⟨I, _, I_mem, I_le⟩ := hv | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
hv : ∃ i ∈ toIdealBasis.sets, ↑i ∈ nhds 0 ∧ ↑i ≤ v
⊢ v ∈ Filter.map (a * b) f | case intro.intro.intro
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ≤ v
⊢ v ∈ Filter.map (a * b) f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
hv : ∃ i ∈ toIdealBasis.sets, ↑i ∈ nhds 0 ∧ ↑i ≤ v
⊢ v ∈ Filter.map (a * b) f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | LinearTopology.tendsto_zero_mul | [330, 1] | [346, 34] | simp only [Set.le_eq_subset] at I_le | case intro.intro.intro
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ≤ v
⊢ v ∈ Filter.map (a * b) f | case intro.intro.intro
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
⊢ v ∈ Filter.map (a * b) f | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ≤ v... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | LinearTopology.tendsto_zero_mul | [330, 1] | [346, 34] | apply Filter.sets_of_superset _ _ I_le | case intro.intro.intro
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
⊢ v ∈ Filter.map (a * b) f | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
⊢ ↑I ∈ (Filter.map (a * b) f).sets | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | LinearTopology.tendsto_zero_mul | [330, 1] | [346, 34] | simp only [Filter.mem_sets, Filter.mem_map] | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
⊢ ↑I ∈ (Filter.map (a * b) f).sets | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
⊢ (a * b) ⁻¹' ↑I ∈ f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
⊢ ↑I ∈ (Filter.map (a ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | LinearTopology.tendsto_zero_mul | [330, 1] | [346, 34] | rw [Filter.tendsto_def] at hb | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
⊢ (a * b) ⁻¹' ↑I ∈ f | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : ∀ s ∈ nhds 0, b ⁻¹' s ∈ f
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
⊢ (a * b) ⁻¹' ↑I ∈ f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : Filter.Tendsto b f (nhds 0)
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
⊢ (a * b) ⁻¹' ↑I ∈ f
T... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | LinearTopology.tendsto_zero_mul | [330, 1] | [346, 34] | specialize hb _ I_mem | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : ∀ s ∈ nhds 0, b ⁻¹' s ∈ f
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
⊢ (a * b) ⁻¹' ↑I ∈ f | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
hb : b ⁻¹' ↑I ∈ f
⊢ (a * b) ⁻¹' ↑I ∈ f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
hb : ∀ s ∈ nhds 0, b ⁻¹' s ∈ f
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
⊢ (a * b) ⁻¹' ↑I ∈ f
TAC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | LinearTopology.tendsto_zero_mul | [330, 1] | [346, 34] | apply Filter.sets_of_superset _ hb | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
hb : b ⁻¹' ↑I ∈ f
⊢ (a * b) ⁻¹' ↑I ∈ f | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
hb : b ⁻¹' ↑I ∈ f
⊢ b ⁻¹' ↑I ⊆ (a * b) ⁻¹' ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
hb : b ⁻¹' ↑I ∈ f
⊢ (a * b) ⁻¹' ↑I ∈ f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | LinearTopology.tendsto_zero_mul | [330, 1] | [346, 34] | intro x | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
hb : b ⁻¹' ↑I ∈ f
⊢ b ⁻¹' ↑I ⊆ (a * b) ⁻¹' ↑I | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
hb : b ⁻¹' ↑I ∈ f
x : ι
⊢ x ∈ b ⁻¹' ↑I → x ∈ (a * b) ⁻¹' ↑I | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
hb : b ⁻¹' ↑I ∈ f
⊢ b ⁻¹' ↑I ⊆ (a * b) ⁻¹' ↑I
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | LinearTopology.tendsto_zero_mul | [330, 1] | [346, 34] | simp only [Set.mem_preimage, Pi.mul_apply, SetLike.mem_coe] | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
hb : b ⁻¹' ↑I ∈ f
x : ι
⊢ x ∈ b ⁻¹' ↑I → x ∈ (a * b) ⁻¹' ↑I | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
hb : b ⁻¹' ↑I ∈ f
x : ι
⊢ b x ∈ I → a x * b x ∈ I | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
hb : b ⁻¹' ↑I ∈ f
x : ι
⊢ x ∈ b ⁻¹' ↑I → x ∈ (a * b) ⁻¹... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | LinearTopology.tendsto_zero_mul | [330, 1] | [346, 34] | intro hx | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
hb : b ⁻¹' ↑I ∈ f
x : ι
⊢ b x ∈ I → a x * b x ∈ I | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
hb : b ⁻¹' ↑I ∈ f
x : ι
hx : b x ∈ I
⊢ a x * b x ∈ I | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
hb : b ⁻¹' ↑I ∈ f
x : ι
⊢ b x ∈ I → a x * b x ∈ I
TACTI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/Topology/LinearTopology.lean | LinearTopology.tendsto_zero_mul | [330, 1] | [346, 34] | apply Ideal.mul_mem_left _ _ hx | α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
hb : b ⁻¹' ↑I ∈ f
x : ι
hx : b x ∈ I
⊢ a x * b x ∈ I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
inst✝² : Ring α
inst✝¹ : TopologicalSpace α
inst✝ : LinearTopology α
ι : Type u_1
f : Filter ι
a b : ι → α
v : Set α
I : Ideal α
left✝ : I ∈ toIdealBasis.sets
I_mem : ↑I ∈ nhds 0
I_le : ↑I ⊆ v
hb : b ⁻¹' ↑I ∈ f
x : ι
hx : b x ∈ I
⊢ a x * b x ∈ I
TA... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_pow | [25, 1] | [30, 37] | suffices f ^ n = (Finset.range n).prod fun _ => f by
rw [this, coeff_prod] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
n : ℕ
d : σ →₀ ℕ
⊢ (coeff α d) (f ^ n) = ∑ l ∈ (range n).piAntidiagonal d, ∏ i ∈ range n, (coeff α (l i)) f | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
n : ℕ
d : σ →₀ ℕ
⊢ f ^ n = ∏ x ∈ range n, f | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
n : ℕ
d : σ →₀ ℕ
⊢ (coeff α d) (f ^ n) = ∑ l ∈ (range n).piAntidiagonal d, ∏ i ∈ ran... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_pow | [25, 1] | [30, 37] | rw [Finset.prod_const, card_range] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
n : ℕ
d : σ →₀ ℕ
⊢ f ^ n = ∏ x ∈ range n, f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
n : ℕ
d : σ →₀ ℕ
⊢ f ^ n = ∏ x ∈ range n, f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_pow | [25, 1] | [30, 37] | rw [this, coeff_prod] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
n : ℕ
d : σ →₀ ℕ
this : f ^ n = ∏ x ∈ range n, f
⊢ (coeff α d) (f ^ n) = ∑ l ∈ (range n).piAntidiagonal d, ∏ i ∈ range n, (coeff α (l i)) f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
n : ℕ
d : σ →₀ ℕ
this : f ^ n = ∏ x ∈ range n, f
⊢ (coeff α d) (f ^ n) = ∑ l ∈ (rang... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | rw [coeff_pow] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
⊢ (coeff α d) (f ^ n) = 0 | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
⊢ ∑ l ∈ (range n).piAntidiagonal d, ∏ i ∈ range n, (coeff α (l i)... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
⊢ (c... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | apply sum_eq_zero | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
⊢ ∑ l ∈ (range n).piAntidiagonal d, ∏ i ∈ range n, (coeff α (l i)... | case h
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
⊢ ∀ x ∈ (range n).piAntidiagonal d, ∏ i ∈ range n, (coeff ... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
⊢ ∑ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | intro k hk | case h
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
⊢ ∀ x ∈ (range n).piAntidiagonal d, ∏ i ∈ range n, (coeff ... | case h
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k ∈ (range n).piAntidiagonal d
⊢ ∏ i ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | rw [mem_piAntidiagonal'] at hk | case h
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k ∈ (range n).piAntidiagonal d
⊢ ∏ i ... | case h
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | set s := (range n).filter fun i => k i = 0 with hs_def | case h
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑... | case h
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | apply filter_subset | case h
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | rw [← prod_sdiff hs] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | refine' mul_eq_zero_of_right _ _ | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | have hs' : ∀ i ∈ s, coeff α (k i) f = constantCoeff σ α f := by
intro i hi
simp only [hs_def, mem_filter] at hi
rw [hi.2, coeff_zero_eq_constantCoeff] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | rw [prod_congr rfl hs', prod_const] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | suffices m ≤ s.card by
obtain ⟨m', hm'⟩ := Nat.exists_eq_add_of_le this
rw [hm', pow_add, hf, MulZeroClass.zero_mul] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | rw [← Nat.add_le_add_iff_right, add_comm s.card, Finset.card_sdiff_add_card_eq_card hs] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | simp only [card_range] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | apply le_trans _ hn | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | simp only [add_comm m, Nat.add_le_add_iff_right] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | rw [← hk.2, map_sum, ← sum_sdiff hs] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | have hs'' : ∀ i ∈ s, degree (k i) = 0 := by
intro i hi
simp only [hs_def, mem_filter] at hi
rw [hi.2, map_zero] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | rw [sum_eq_zero hs'', add_zero] | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | convert Finset.card_nsmul_le_sum (range n \ s) _ 1 _ | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | case h.e'_3
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | simp only [Algebra.id.smul_eq_mul, mul_one] | case h.e'_3
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).... | case convert_2
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degre... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Basic.lean | MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent | [33, 1] | [70, 22] | intro i hi | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k.support ⊆ range n ∧ (range n).sum ⇑k = d
s... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
inst✝³ : DecidableEq σ
ι : Type u_2
inst✝² : DecidableEq (ι → σ →₀ ℕ)
α : Type u_3
inst✝¹ : CommSemiring α
inst✝ : DecidableEq (ℕ → σ →₀ ℕ)
f : MvPowerSeries σ α
m : ℕ
hf : (constantCoeff σ α) f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ... |
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