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/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.dpow_apply | [765, 1] | [767, 39] | rw [dpow_def, dpow_apply' hI hIf ha] | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
⊢ (dividedPowers hI hf hIf).dpow n (f a) = f (hI.dpow n a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
⊢ (dividedPowers hI hf hIf).dpow n (f a) = f (hI.dpow n ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.isDPMorphism | [770, 1] | [773, 45] | constructor | A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
⊢ hI.isDPMorphism (dividedPowers hI hf hIf) f | case left
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
⊢ Ideal.map f I ≤ Ideal.map f I
case right
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ :... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
⊢ hI.isDPMorphism (dividedPowers hI hf hIf) f
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.isDPMorphism | [770, 1] | [773, 45] | exact le_refl (Ideal.map f I) | case left
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
⊢ Ideal.map f I ≤ Ideal.map f I
case right
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ :... | case right
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
⊢ ∀ (n : ℕ), ∀ a ∈ I, (dividedPowers hI hf hIf).dpow n (f a) = f (hI.dpow n a) | Please generate a tactic in lean4 to solve the state.
STATE:
case left
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
⊢ Ideal.map f I ≤ Ideal.map f I
case right
A : Type u_1
inst✝¹ : Com... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.isDPMorphism | [770, 1] | [773, 45] | intro n a ha | case right
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
⊢ ∀ (n : ℕ), ∀ a ∈ I, (dividedPowers hI hf hIf).dpow n (f a) = f (hI.dpow n a) | case right
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
⊢ (dividedPowers hI hf hIf).dpow n (f a) = f (hI.dpow n a) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
⊢ ∀ (n : ℕ), ∀ a ∈ I, (dividedPowers hI hf hIf).dpow n (f a) = f (hI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.isDPMorphism | [770, 1] | [773, 45] | rw [dpow_apply hI hf hIf ha] | case right
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
⊢ (dividedPowers hI hf hIf).dpow n (f a) = f (hI.dpow n a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_2
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
n : ℕ
a : A
ha : a ∈ I
⊢ (dividedPowers hI hf hIf).dpow n (f a) = f ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.unique | [776, 1] | [780, 46] | obtain ⟨a, ha, rfl⟩ := (Ideal.mem_map_iff_of_surjective f hf).mp hx | A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
hquot : DividedPowers (Ideal.map f I)
hm : hI.isDPMorphism hquot f
n : ℕ
x : B
hx : x ∈ Ideal.map f I
⊢ hquot.dpow n x = (dividedPowers hI h... | case intro.intro
A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
hquot : DividedPowers (Ideal.map f I)
hm : hI.isDPMorphism hquot f
n : ℕ
a : A
ha : a ∈ I
hx : f a ∈ Ideal.map f I
⊢ hquot.... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
hquot : DividedPowers (Ideal.map f I)
hm : hI.isDPMorphism hquot f
n : ℕ
x : B
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.OfSurjective.unique | [776, 1] | [780, 46] | rw [hm.2 n a ha, dpow_apply hI hf hIf ha] | case intro.intro
A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
hquot : DividedPowers (Ideal.map f I)
hm : hI.isDPMorphism hquot f
n : ℕ
a : A
ha : a ∈ I
hx : f a ∈ Ideal.map f I
⊢ hquot.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
A : Type u_2
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
B : Type u_1
inst✝ : CommRing B
f : A →+* B
hf : Function.Surjective ⇑f
hIf : hI.isSubDPIdeal (RingHom.ker f ⊓ I)
hquot : DividedPowers (Ideal.map f I)
hm : hI.isDPMorphism hqu... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/SubDPIdeal.lean | DividedPowers.Quotient.isSubDPIdeal_aux | [797, 9] | [798, 33] | simpa [Ideal.mk_ker] using hIJ | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
⊢ hI.isSubDPIdeal (RingHom.ker (Ideal.Quotient.mk J) ⊓ I) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hIJ : hI.isSubDPIdeal (J ⊓ I)
⊢ hI.isSubDPIdeal (RingHom.ker (Ideal.Quotient.mk J) ⊓ I)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/CombinatoricsLemmas.lean | mchoose_zero | [19, 1] | [20, 53] | rw [mchoose, Finset.range_zero, Finset.prod_empty] | n : ℕ
⊢ mchoose 0 n = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ mchoose 0 n = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/CombinatoricsLemmas.lean | mchoose_zero' | [23, 1] | [24, 85] | simp only [mchoose, MulZeroClass.mul_zero, Nat.choose_self, Finset.prod_const_one] | m : ℕ
⊢ mchoose m 0 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
⊢ mchoose m 0 = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/CombinatoricsLemmas.lean | mchoose_succ | [27, 1] | [29, 56] | simp only [mchoose, Finset.prod_range_succ, mul_comm] | m n : ℕ
⊢ mchoose (m + 1) n = (m * n + n - 1).choose (n - 1) * mchoose m n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
⊢ mchoose (m + 1) n = (m * n + n - 1).choose (n - 1) * mchoose m n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/CombinatoricsLemmas.lean | mchoose_one | [32, 1] | [34, 31] | simp only [mchoose, Finset.range_one, Finset.prod_singleton, zero_mul,
zero_add, Nat.choose_self] | n : ℕ
⊢ mchoose 1 n = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ mchoose 1 n = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/CombinatoricsLemmas.lean | mchoose_one' | [36, 1] | [38, 70] | simp only [mchoose, mul_one, add_tsub_cancel_right, ge_iff_le, le_refl,
tsub_eq_zero_of_le, Nat.choose_zero_right, Finset.prod_const_one] | m : ℕ
⊢ mchoose m 1 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
⊢ mchoose m 1 = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/CombinatoricsLemmas.lean | mchoose_lemma | [40, 1] | [59, 12] | rw [← zero_lt_iff] at hn | m n : ℕ
hn : n ≠ 0
⊢ m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial | m n : ℕ
hn : 0 < n
⊢ m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
hn : n ≠ 0
⊢ m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/CombinatoricsLemmas.lean | mchoose_lemma | [40, 1] | [59, 12] | induction' m with m ih | m n : ℕ
hn : 0 < n
⊢ m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial | case zero
n : ℕ
hn : 0 < n
⊢ Nat.factorial 0 * n.factorial ^ 0 * mchoose 0 n = (0 * n).factorial
case add_one
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
⊢ (m + 1).factorial * n.factorial ^ (m + 1) * mchoose (m + 1) n = ((m + 1) * n).factorial | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
hn : 0 < n
⊢ m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/CombinatoricsLemmas.lean | mchoose_lemma | [40, 1] | [59, 12] | rw [mchoose_zero, mul_one, MulZeroClass.zero_mul, Nat.factorial_zero, pow_zero, mul_one] | case zero
n : ℕ
hn : 0 < n
⊢ Nat.factorial 0 * n.factorial ^ 0 * mchoose 0 n = (0 * n).factorial | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
n : ℕ
hn : 0 < n
⊢ Nat.factorial 0 * n.factorial ^ 0 * mchoose 0 n = (0 * n).factorial
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/CombinatoricsLemmas.lean | mchoose_lemma | [40, 1] | [59, 12] | have hmn : (m + 1) * (m * n + n - 1).choose (n - 1) = (m * n + n).choose n :=
by
rw [←
mul_left_inj' (Nat.mul_ne_zero (Nat.factorial_ne_zero (m * n)) (Nat.factorial_ne_zero n)), ←
mul_assoc, ← mul_assoc, Nat.add_choose_mul_factorial_mul_factorial, ←
Nat.mul_factorial_pred hn, mul_comm n _, ← mul_assoc, ... | case add_one
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
⊢ (m + 1).factorial * n.factorial ^ (m + 1) * mchoose (m + 1) n = ((m + 1) * n).factorial | case add_one
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
hmn : (m + 1) * (m * n + n - 1).choose (n - 1) = (m * n + n).choose n
⊢ (m + 1).factorial * n.factorial ^ (m + 1) * mchoose (m + 1) n = ((m + 1) * n).factorial | Please generate a tactic in lean4 to solve the state.
STATE:
case add_one
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
⊢ (m + 1).factorial * n.factorial ^ (m + 1) * mchoose (m + 1) n = ((m + 1) * n).factorial
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/CombinatoricsLemmas.lean | mchoose_lemma | [40, 1] | [59, 12] | rw [mchoose_succ, Nat.factorial_succ, pow_succ, ← mul_assoc] | case add_one
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
hmn : (m + 1) * (m * n + n - 1).choose (n - 1) = (m * n + n).choose n
⊢ (m + 1).factorial * n.factorial ^ (m + 1) * mchoose (m + 1) n = ((m + 1) * n).factorial | case add_one
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
hmn : (m + 1) * (m * n + n - 1).choose (n - 1) = (m * n + n).choose n
⊢ (m + 1) * m.factorial * (n.factorial ^ m * n.factorial) * (m * n + n - 1).choose (n - 1) * mchoose m n =
((m + 1) * n).factorial | Please generate a tactic in lean4 to solve the state.
STATE:
case add_one
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
hmn : (m + 1) * (m * n + n - 1).choose (n - 1) = (m * n + n).choose n
⊢ (m + 1).factorial * n.factorial ^ (m + 1) * mchoose (m + 1) n = ((m + 1) * n).fact... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/CombinatoricsLemmas.lean | mchoose_lemma | [40, 1] | [59, 12] | conv_rhs => rw [Nat.succ_mul] | case add_one
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
hmn : (m + 1) * (m * n + n - 1).choose (n - 1) = (m * n + n).choose n
⊢ (m + 1) * m.factorial * (n.factorial ^ m * n.factorial) * (m * n + n - 1).choose (n - 1) * mchoose m n =
((m + 1) * n).factorial | case add_one
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
hmn : (m + 1) * (m * n + n - 1).choose (n - 1) = (m * n + n).choose n
⊢ (m + 1) * m.factorial * (n.factorial ^ m * n.factorial) * (m * n + n - 1).choose (n - 1) * mchoose m n =
(m * n + n).factorial | Please generate a tactic in lean4 to solve the state.
STATE:
case add_one
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
hmn : (m + 1) * (m * n + n - 1).choose (n - 1) = (m * n + n).choose n
⊢ (m + 1) * m.factorial * (n.factorial ^ m * n.factorial) * (m * n + n - 1).choose (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/CombinatoricsLemmas.lean | mchoose_lemma | [40, 1] | [59, 12] | rw [← Nat.add_choose_mul_factorial_mul_factorial, ← ih, ← hmn] | case add_one
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
hmn : (m + 1) * (m * n + n - 1).choose (n - 1) = (m * n + n).choose n
⊢ (m + 1) * m.factorial * (n.factorial ^ m * n.factorial) * (m * n + n - 1).choose (n - 1) * mchoose m n =
(m * n + n).factorial | case add_one
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
hmn : (m + 1) * (m * n + n - 1).choose (n - 1) = (m * n + n).choose n
⊢ (m + 1) * m.factorial * (n.factorial ^ m * n.factorial) * (m * n + n - 1).choose (n - 1) * mchoose m n =
(m + 1) * (m * n + n - 1).choose (... | Please generate a tactic in lean4 to solve the state.
STATE:
case add_one
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
hmn : (m + 1) * (m * n + n - 1).choose (n - 1) = (m * n + n).choose n
⊢ (m + 1) * m.factorial * (n.factorial ^ m * n.factorial) * (m * n + n - 1).choose (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/CombinatoricsLemmas.lean | mchoose_lemma | [40, 1] | [59, 12] | ring_nf | case add_one
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
hmn : (m + 1) * (m * n + n - 1).choose (n - 1) = (m * n + n).choose n
⊢ (m + 1) * m.factorial * (n.factorial ^ m * n.factorial) * (m * n + n - 1).choose (n - 1) * mchoose m n =
(m + 1) * (m * n + n - 1).choose (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case add_one
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
hmn : (m + 1) * (m * n + n - 1).choose (n - 1) = (m * n + n).choose n
⊢ (m + 1) * m.factorial * (n.factorial ^ m * n.factorial) * (m * n + n - 1).choose (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/CombinatoricsLemmas.lean | mchoose_lemma | [40, 1] | [59, 12] | rw [←
mul_left_inj' (Nat.mul_ne_zero (Nat.factorial_ne_zero (m * n)) (Nat.factorial_ne_zero n)), ←
mul_assoc, ← mul_assoc, Nat.add_choose_mul_factorial_mul_factorial, ←
Nat.mul_factorial_pred hn, mul_comm n _, ← mul_assoc, Nat.add_sub_assoc hn (m * n),
mul_comm, mul_assoc ((m + 1) * (m * n + (n - 1)).choose (n ... | n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
⊢ (m + 1) * (m * n + n - 1).choose (n - 1) = (m * n + n).choose n | n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
⊢ n * ((m + 1) * (m * n + n - Nat.succ 0).factorial) = (m * n + n) * (m * n + n - 1).factorial | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
⊢ (m + 1) * (m * n + n - 1).choose (n - 1) = (m * n + n).choose n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/CombinatoricsLemmas.lean | mchoose_lemma | [40, 1] | [59, 12] | ring | n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
⊢ n * ((m + 1) * (m * n + n - Nat.succ 0).factorial) = (m * n + n) * (m * n + n - 1).factorial | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
hn : 0 < n
m : ℕ
ih : m.factorial * n.factorial ^ m * mchoose m n = (m * n).factorial
⊢ n * ((m + 1) * (m * n + n - Nat.succ 0).factorial) = (m * n + n) * (m * n + n - 1).factorial
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.mk_apply_of_mem | [16, 1] | [19, 18] | dsimp only [Finset.coe_sort_coe, mk, AddMonoidHom.coe_mk, ZeroHom.coe_mk, DFinsupp.mk_apply] | ι : Type v
inst✝¹ : DecidableEq ι
β : ι → Type w
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ↑↑s) → β ↑i
n : ι
hn : n ∈ s
⊢ ((mk β s) f) n = f ⟨n, hn⟩ | ι : Type v
inst✝¹ : DecidableEq ι
β : ι → Type w
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ↑↑s) → β ↑i
n : ι
hn : n ∈ s
⊢ (if H : n ∈ s then f ⟨n, H⟩ else 0) = f ⟨n, hn⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝¹ : DecidableEq ι
β : ι → Type w
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ↑↑s) → β ↑i
n : ι
hn : n ∈ s
⊢ ((mk β s) f) n = f ⟨n, hn⟩
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.mk_apply_of_mem | [16, 1] | [19, 18] | rw [dif_pos hn] | ι : Type v
inst✝¹ : DecidableEq ι
β : ι → Type w
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ↑↑s) → β ↑i
n : ι
hn : n ∈ s
⊢ (if H : n ∈ s then f ⟨n, H⟩ else 0) = f ⟨n, hn⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝¹ : DecidableEq ι
β : ι → Type w
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ↑↑s) → β ↑i
n : ι
hn : n ∈ s
⊢ (if H : n ∈ s then f ⟨n, H⟩ else 0) = f ⟨n, hn⟩
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.mk_apply_of_not_mem | [22, 1] | [25, 18] | dsimp only [Finset.coe_sort_coe, mk, AddMonoidHom.coe_mk, ZeroHom.coe_mk, DFinsupp.mk_apply] | ι : Type v
inst✝¹ : DecidableEq ι
β : ι → Type w
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ↑↑s) → β ↑i
n : ι
hn : n ∉ s
⊢ ((mk β s) f) n = 0 | ι : Type v
inst✝¹ : DecidableEq ι
β : ι → Type w
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ↑↑s) → β ↑i
n : ι
hn : n ∉ s
⊢ (if H : n ∈ s then f ⟨n, H⟩ else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝¹ : DecidableEq ι
β : ι → Type w
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ↑↑s) → β ↑i
n : ι
hn : n ∉ s
⊢ ((mk β s) f) n = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.mk_apply_of_not_mem | [22, 1] | [25, 18] | rw [dif_neg hn] | ι : Type v
inst✝¹ : DecidableEq ι
β : ι → Type w
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ↑↑s) → β ↑i
n : ι
hn : n ∉ s
⊢ (if H : n ∈ s then f ⟨n, H⟩ else 0) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝¹ : DecidableEq ι
β : ι → Type w
inst✝ : (i : ι) → AddCommMonoid (β i)
s : Finset ι
f : (i : ↑↑s) → β ↑i
n : ι
hn : n ∉ s
⊢ (if H : n ∈ s then f ⟨n, H⟩ else 0) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.coeAddMonoidHom_eq_dfinsupp_sum | [34, 1] | [39, 37] | simp only [DirectSum.coeAddMonoidHom, DirectSum.toAddMonoid,
DFinsupp.liftAddHom, AddEquiv.coe_mk, Equiv.coe_fn_mk] | ι : Type v
inst✝⁴ : DecidableEq ι
M✝ : Type w
inst✝³ : DecidableEq M✝
inst✝² : AddCommMonoid M✝
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
⊢ (DirectSum.coeAddMonoidHom A) x = DFinsupp.sum x fun i x => ↑x | ι : Type v
inst✝⁴ : DecidableEq ι
M✝ : Type w
inst✝³ : DecidableEq M✝
inst✝² : AddCommMonoid M✝
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
⊢ (DFinsupp.sumAddHom fun i => AddSubmonoidClass.subtype (A i)) x = DFinsupp.sum x fun i x => ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝⁴ : DecidableEq ι
M✝ : Type w
inst✝³ : DecidableEq M✝
inst✝² : AddCommMonoid M✝
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
⊢ (DirectSum.coeAddMonoidHom A) x = DFinsupp.sum x fun i x =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.coeAddMonoidHom_eq_dfinsupp_sum | [34, 1] | [39, 37] | exact DFinsupp.sumAddHom_apply _ x | ι : Type v
inst✝⁴ : DecidableEq ι
M✝ : Type w
inst✝³ : DecidableEq M✝
inst✝² : AddCommMonoid M✝
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
⊢ (DFinsupp.sumAddHom fun i => AddSubmonoidClass.subtype (A i)) x = DFinsupp.sum x fun i x => ↑x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝⁴ : DecidableEq ι
M✝ : Type w
inst✝³ : DecidableEq M✝
inst✝² : AddCommMonoid M✝
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
⊢ (DFinsupp.sumAddHom fun i => AddSubmonoidClass.subtype (A ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.coeLinearMap_eq_dfinsupp_sum | [42, 1] | [48, 65] | simp only [coeLinearMap, toModule, DFinsupp.lsum, LinearEquiv.coe_mk, LinearMap.coe_mk,
AddHom.coe_mk] | ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u
inst✝¹ : Semiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
⊢ (coeLinearMap A) x = DFinsupp.sum x fun i x => ↑x | ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u
inst✝¹ : Semiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
⊢ (DFinsupp.sumAddHom fun i => (A i).subtype.toAddMonoidHom) x = DFinsupp.sum x fun i x => ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u
inst✝¹ : Semiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
⊢ (coeLinearMap A) x = DFinsupp.sum x fun i x => ↑x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.coeLinearMap_eq_dfinsupp_sum | [42, 1] | [48, 65] | rw [DFinsupp.sumAddHom_apply] | ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u
inst✝¹ : Semiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
⊢ (DFinsupp.sumAddHom fun i => (A i).subtype.toAddMonoidHom) x = DFinsupp.sum x fun i x => ↑x | ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u
inst✝¹ : Semiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
⊢ (DFinsupp.sum x fun x => ⇑(A x).subtype.toAddMonoidHom) = DFinsupp.sum x fun i x => ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u
inst✝¹ : Semiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
⊢ (DFinsupp.sumAddHom fun i => (A i).subtype.toAddMonoidHom) x = DFinsup... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.coeLinearMap_eq_dfinsupp_sum | [42, 1] | [48, 65] | simp only [LinearMap.toAddMonoidHom_coe, Submodule.coeSubtype] | ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u
inst✝¹ : Semiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
⊢ (DFinsupp.sum x fun x => ⇑(A x).subtype.toAddMonoidHom) = DFinsupp.sum x fun i x => ↑x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u
inst✝¹ : Semiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
⊢ (DFinsupp.sum x fun x => ⇑(A x).subtype.toAddMonoidHom) = DFinsupp.sum... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.support_subset | [51, 1] | [57, 34] | intro m | ι : Type v
inst✝² : DecidableEq ι
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
⊢ (Function.support fun i => ↑(x i)) ⊆ ↑(DFinsupp.support x) | ι : Type v
inst✝² : DecidableEq ι
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
m : ι
⊢ (m ∈ Function.support fun i => ↑(x i)) → m ∈ ↑(DFinsupp.support x) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝² : DecidableEq ι
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
⊢ (Function.support fun i => ↑(x i)) ⊆ ↑(DFinsupp.support x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.support_subset | [51, 1] | [57, 34] | rw [Function.mem_support, Finset.mem_coe, DFinsupp.mem_support_toFun, not_imp_not] | ι : Type v
inst✝² : DecidableEq ι
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
m : ι
⊢ (m ∈ Function.support fun i => ↑(x i)) → m ∈ ↑(DFinsupp.support x) | ι : Type v
inst✝² : DecidableEq ι
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
m : ι
⊢ x m = 0 → ↑(x m) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝² : DecidableEq ι
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
m : ι
⊢ (m ∈ Function.support fun i => ↑(x i)) → m ∈ ↑(DFinsupp.support x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.support_subset | [51, 1] | [57, 34] | intro hm' | ι : Type v
inst✝² : DecidableEq ι
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
m : ι
⊢ x m = 0 → ↑(x m) = 0 | ι : Type v
inst✝² : DecidableEq ι
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
m : ι
hm' : x m = 0
⊢ ↑(x m) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝² : DecidableEq ι
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
m : ι
⊢ x m = 0 → ↑(x m) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.support_subset | [51, 1] | [57, 34] | rw [hm', AddSubmonoid.coe_zero] | ι : Type v
inst✝² : DecidableEq ι
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
m : ι
hm' : x m = 0
⊢ ↑(x m) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝² : DecidableEq ι
M : Type w
inst✝¹ : DecidableEq M
inst✝ : AddCommMonoid M
A : ι → AddSubmonoid M
x : ⨁ (i : ι), ↥(A i)
m : ι
hm' : x m = 0
⊢ ↑(x m) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.support_subset_submodule | [60, 1] | [67, 38] | intro m | ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
⊢ (Function.support fun i => ↑(x i)) ⊆ ↑(DFinsupp.support x) | ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
m : ι
⊢ (m ∈ Function.support fun i => ↑(x i)) → m ∈ ↑(DFinsupp.support x) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
⊢ (Function.support fun i => ↑(x i)) ⊆ ↑(DFinsupp.support x)
TACTI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.support_subset_submodule | [60, 1] | [67, 38] | rw [Function.mem_support, Finset.mem_coe, DFinsupp.mem_support_toFun, not_imp_not] | ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
m : ι
⊢ (m ∈ Function.support fun i => ↑(x i)) → m ∈ ↑(DFinsupp.support x) | ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
m : ι
⊢ x m = 0 → ↑(x m) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
m : ι
⊢ (m ∈ Function.support fun i => ↑(x i)) → m ∈ ↑(DFinsupp.su... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.support_subset_submodule | [60, 1] | [67, 38] | intro hm' | ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
m : ι
⊢ x m = 0 → ↑(x m) = 0 | ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
m : ι
hm' : x m = 0
⊢ ↑(x m) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
m : ι
⊢ x m = 0 → ↑(x m) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | DirectSum.support_subset_submodule | [60, 1] | [67, 38] | simp only [hm', Submodule.coe_zero] | ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
m : ι
hm' : x m = 0
⊢ ↑(x m) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type v
inst✝⁴ : DecidableEq ι
M : Type w
inst✝³ : DecidableEq M
inst✝² : AddCommMonoid M
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Module R M
A : ι → Submodule R M
x : ⨁ (i : ι), ↥(A i)
m : ι
hm' : x m = 0
⊢ ↑(x m) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | LinearMap.map_finsum | [81, 1] | [87, 37] | rw [← LinearMap.toAddMonoidHom_coe] | α : Type u_1
R : Type u_2
S : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : Semiring R
inst✝⁴ : Semiring S
σ : R →+* S
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module S N
f : α → M
g : M →ₛₗ[σ] N
hf : (Function.support f).Finite
⊢ g (∑ᶠ (i : α), f i) = ∑ᶠ (i : α), g (f i) | α : Type u_1
R : Type u_2
S : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : Semiring R
inst✝⁴ : Semiring S
σ : R →+* S
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module S N
f : α → M
g : M →ₛₗ[σ] N
hf : (Function.support f).Finite
⊢ g.toAddMonoidHom (∑ᶠ (i : α), f i) = ∑ᶠ (i : α), g.toA... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
R : Type u_2
S : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : Semiring R
inst✝⁴ : Semiring S
σ : R →+* S
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module S N
f : α → M
g : M →ₛₗ[σ] N
hf : (Function.support f).Fi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/GradedAlgebra.lean | LinearMap.map_finsum | [81, 1] | [87, 37] | exact AddMonoidHom.map_finsum _ hf | α : Type u_1
R : Type u_2
S : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : Semiring R
inst✝⁴ : Semiring S
σ : R →+* S
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module S N
f : α → M
g : M →ₛₗ[σ] N
hf : (Function.support f).Finite
⊢ g.toAddMonoidHom (∑ᶠ (i : α), f i) = ∑ᶠ (i : α), g.toA... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
R : Type u_2
S : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : Semiring R
inst✝⁴ : Semiring S
σ : R →+* S
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module S N
f : α → M
g : M →ₛₗ[σ] N
hf : (Function.support f).Fi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/LinearEquiv.lean | TensorProduct.congr'_coe | [78, 1] | [80, 6] | rfl | R : Type u
M : Type v
N : Type w
inst✝¹⁴ : CommSemiring R
inst✝¹³ : AddCommMonoid M
inst✝¹² : Module R M
S : Type u_1
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : Module S M
inst✝⁶ : IsScalarTower R S M
inst✝⁵ : Module S N
inst✝⁴ : IsScalarTower R S N
P : Type u_2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
M : Type v
N : Type w
inst✝¹⁴ : CommSemiring R
inst✝¹³ : AddCommMonoid M
inst✝¹² : Module R M
S : Type u_1
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : Module S M
inst✝⁶ : IsScalarTower R S M
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.mem_basis | [49, 1] | [52, 6] | simp only [basis, Submodule.mem_mk, AddSubmonoid.mem_mk, Set.mem_setOf_eq] | σ : Type u_1
α : Type u_2
inst✝ : CommRing α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ f ∈ basis σ α d ↔ ∀ e ≤ d, (coeff α e) f = 0 | σ : Type u_1
α : Type u_2
inst✝ : CommRing α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ f ∈ { carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ↔ ∀ e ≤ d, (coeff α e) f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝ : CommRing α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ f ∈ basis σ α d ↔ ∀ e ≤ d, (coeff α e) f = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.mem_basis | [49, 1] | [52, 6] | rfl | σ : Type u_1
α : Type u_2
inst✝ : CommRing α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ f ∈ { carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ↔ ∀ e ≤ d, (coeff α e) f = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝ : CommRing α
f : MvPowerSeries σ α
d : σ →₀ ℕ
⊢ f ∈ { carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ↔ ∀ e ≤ d, (coeff α e) f = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | refine' ⟨_, basis_le _ _⟩ | σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
⊢ basis σ α d ≤ basis σ α e ↔ e ≤ d | σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
⊢ basis σ α d ≤ basis σ α e → e ≤ d | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
⊢ basis σ α d ≤ basis σ α e ↔ e ≤ d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | simp only [basis, Submodule.mk_le_mk, AddSubmonoid.mk_le_mk, setOf_subset_setOf] | σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
⊢ basis σ α d ≤ basis σ α e → e ≤ d | σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
⊢ { carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ } →
e ≤ d | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
⊢ basis σ α d ≤ basis σ α e → e ≤ d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | intro h | σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
⊢ { carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ } →
e ≤ d | σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ e ≤ d | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
⊢ { carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ } →
e ≤ d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | rw [← inf_eq_right] | σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ e ≤ d | σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ d ⊓ e = e | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ e ≤ d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | apply le_antisymm | σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ d ⊓ e = e | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ d ⊓ e ≤ e
case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e ... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ d ⊓ e = e
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | . exact inf_le_right | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ d ⊓ e ≤ e
case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e ... | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ e ≤ d ⊓ e | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ d ⊓ e ≤ e
case a
σ : Type u... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | . by_contra h'
simp only [AddSubsemigroup.mk_le_mk, setOf_subset_setOf] at h
specialize h (monomial α e 1) _
. intro e' he'
apply coeff_monomial_ne
intro hee'
rw [hee'] at he'
apply h'
exact le_inf_iff.mpr ⟨he', le_rfl⟩
apply one_ne_zero' α
convert h e le_rfl
rw [coeff_monomial_same] | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ e ≤ d ⊓ e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ e ≤ d ⊓ e
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | exact inf_le_right | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ d ⊓ e ≤ e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ d ⊓ e ≤ e
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | by_contra h' | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ e ≤ d ⊓ e | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
h' : ¬e ≤ d ⊓ e
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
⊢ e ≤ d ⊓ e
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | simp only [AddSubsemigroup.mk_le_mk, setOf_subset_setOf] at h | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
h' : ¬e ≤ d ⊓ e
⊢ False | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h :
{ carrier := {f | ∀ e ≤ d, (coeff α e) f = 0}, add_mem' := ⋯ } ≤
{ carrier := {f | ∀ e_1 ≤ e, (coeff α e_1) f = 0}, add_mem' := ⋯ }
h' : ¬e ≤ d ⊓ e
⊢ False
TACTIC... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | specialize h (monomial α e 1) _ | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
⊢ False | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
⊢ ∀ e_1 ≤ d, (coeff α e_1) ((monomial α e) 1) = 0
case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | . intro e' he'
apply coeff_monomial_ne
intro hee'
rw [hee'] at he'
apply h'
exact le_inf_iff.mpr ⟨he', le_rfl⟩ | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
⊢ ∀ e_1 ≤ d, (coeff α e_1) ((monomial α e) 1) = 0
case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α... | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ e_1 ≤ e, (coeff α e_1) ((monomial α e) 1) = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
⊢ ∀ e_1 ≤ d, (coeff α e_1) ((monomial α e) 1) = 0
case a
σ : T... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | apply one_ne_zero' α | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ e_1 ≤ e, (coeff α e_1) ((monomial α e) 1) = 0
⊢ False | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ e_1 ≤ e, (coeff α e_1) ((monomial α e) 1) = 0
⊢ 1 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ e_1 ≤ e, (coeff α e_1) ((monomial α e) 1) = 0
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | convert h e le_rfl | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ e_1 ≤ e, (coeff α e_1) ((monomial α e) 1) = 0
⊢ 1 = 0 | case h.e'_2
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ e_1 ≤ e, (coeff α e_1) ((monomial α e) 1) = 0
⊢ 1 = (coeff α e) ((monomial α e) 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ e_1 ≤ e, (coeff α e_1) ((monomial α e) 1) = 0
⊢ 1 = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | rw [coeff_monomial_same] | case h.e'_2
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ e_1 ≤ e, (coeff α e_1) ((monomial α e) 1) = 0
⊢ 1 = (coeff α e) ((monomial α e) 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ e_1 ≤ e, (coeff α e_1) ((monomial α e) 1) = 0
⊢ 1 = (coeff α e) ((monomial α e) 1)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | intro e' he' | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
⊢ ∀ e_1 ≤ d, (coeff α e_1) ((monomial α e) 1) = 0 | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
e' : σ →₀ ℕ
he' : e' ≤ d
⊢ (coeff α e') ((monomial α e) 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
⊢ ∀ e_1 ≤ d, (coeff α e_1) ((monomial α e) 1) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | apply coeff_monomial_ne | case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
e' : σ →₀ ℕ
he' : e' ≤ d
⊢ (coeff α e') ((monomial α e) 1) = 0 | case a.h
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
e' : σ →₀ ℕ
he' : e' ≤ d
⊢ e' ≠ e | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
e' : σ →₀ ℕ
he' : e' ≤ d
⊢ (coeff α e') ((monomial α e) 1) = 0
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | intro hee' | case a.h
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
e' : σ →₀ ℕ
he' : e' ≤ d
⊢ e' ≠ e | case a.h
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
e' : σ →₀ ℕ
he' : e' ≤ d
hee' : e' = e
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
e' : σ →₀ ℕ
he' : e' ≤ d
⊢ e' ≠ e
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | rw [hee'] at he' | case a.h
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
e' : σ →₀ ℕ
he' : e' ≤ d
hee' : e' = e
⊢ False | case a.h
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
e' : σ →₀ ℕ
he' : e ≤ d
hee' : e' = e
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
e' : σ →₀ ℕ
he' : e' ≤ d
hee' : e' = e
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | apply h' | case a.h
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
e' : σ →₀ ℕ
he' : e ≤ d
hee' : e' = e
⊢ False | case a.h
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
e' : σ →₀ ℕ
he' : e ≤ d
hee' : e' = e
⊢ e ≤ d ⊓ e | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
e' : σ →₀ ℕ
he' : e ≤ d
hee' : e' = e
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_le_iff | [61, 1] | [80, 29] | exact le_inf_iff.mpr ⟨he', le_rfl⟩ | case a.h
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
e' : σ →₀ ℕ
he' : e ≤ d
hee' : e' = e
⊢ e ≤ d ⊓ e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
σ : Type u_1
α : Type u_2
inst✝¹ : CommRing α
inst✝ : Nontrivial α
d e : σ →₀ ℕ
h' : ¬e ≤ d ⊓ e
h : ∀ (a : MvPowerSeries σ α), (∀ e ≤ d, (coeff α e) a = 0) → ∀ e_1 ≤ e, (coeff α e_1) a = 0
e' : σ →₀ ℕ
he' : e ≤ d
hee' : e' = e
⊢ e ≤ d ⊓ e
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.idealIsBasis | [89, 1] | [90, 95] | use d ⊔ e | σ : Type u_1
α : Type u_2
inst✝ : CommRing α
d e : σ →₀ ℕ
⊢ ∃ k, basis σ α k ≤ basis σ α d ⊓ basis σ α e | case h
σ : Type u_1
α : Type u_2
inst✝ : CommRing α
d e : σ →₀ ℕ
⊢ basis σ α (d ⊔ e) ≤ basis σ α d ⊓ basis σ α e | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝ : CommRing α
d e : σ →₀ ℕ
⊢ ∃ k, basis σ α k ≤ basis σ α d ⊓ basis σ α e
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.idealIsBasis | [89, 1] | [90, 95] | apply Antitone.map_sup_le (basis_antitone σ α) | case h
σ : Type u_1
α : Type u_2
inst✝ : CommRing α
d e : σ →₀ ℕ
⊢ basis σ α (d ⊔ e) ≤ basis σ α d ⊓ basis σ α e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝ : CommRing α
d e : σ →₀ ℕ
⊢ basis σ α (d ⊔ e) ≤ basis σ α d ⊓ basis σ α e
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_mem_nhds_zero | [111, 1] | [124, 45] | rw [nhds_pi, Filter.mem_pi] | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
⊢ ↑(basis σ α d) ∈ nhds 0 | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
⊢ ∃ I, I.Finite ∧ ∃ t, (∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)) ∧ I.pi t ⊆ ↑(basis σ α d) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
⊢ ↑(basis σ α d) ∈ nhds 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_mem_nhds_zero | [111, 1] | [124, 45] | use Finset.Iic d, Finset.finite_toSet _, (fun e => if e ≤ d then {0} else univ) | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
⊢ ∃ I, I.Finite ∧ ∃ t, (∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)) ∧ I.pi t ⊆ ↑(basis σ α d) | case h
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
⊢ (∀ (i : σ →₀ ℕ), (if i ≤ d then {0} else univ) ∈ nhds (0 i)) ∧
((↑(Finset.Iic d)).pi fun e => if e ≤ d then {0} else univ) ⊆ ↑(basis σ α d) | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
⊢ ∃ I, I.Finite ∧ ∃ t, (∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)) ∧ I.pi t ⊆ ↑(basis σ α d)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_mem_nhds_zero | [111, 1] | [124, 45] | constructor | case h
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
⊢ (∀ (i : σ →₀ ℕ), (if i ≤ d then {0} else univ) ∈ nhds (0 i)) ∧
((↑(Finset.Iic d)).pi fun e => if e ≤ d then {0} else univ) ⊆ ↑(basis σ α d) | case h.left
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
⊢ ∀ (i : σ →₀ ℕ), (if i ≤ d then {0} else univ) ∈ nhds (0 i)
case h.right
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
⊢ ((↑(Fin... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
⊢ (∀ (i : σ →₀ ℕ), (if i ≤ d then {0} else univ) ∈ nhds (0 i)) ∧
((↑(Finset.Iic d)).pi fun e => if e ≤ d then {0} else univ) ⊆ ↑(basis σ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_mem_nhds_zero | [111, 1] | [124, 45] | intro e | case h.left
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
⊢ ∀ (i : σ →₀ ℕ), (if i ≤ d then {0} else univ) ∈ nhds (0 i) | case h.left
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
⊢ (if e ≤ d then {0} else univ) ∈ nhds (0 e) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
⊢ ∀ (i : σ →₀ ℕ), (if i ≤ d then {0} else univ) ∈ nhds (0 i)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_mem_nhds_zero | [111, 1] | [124, 45] | split_ifs with h | case h.left
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
⊢ (if e ≤ d then {0} else univ) ∈ nhds (0 e) | case pos
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
h : e ≤ d
⊢ {0} ∈ nhds (0 e)
case neg
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
h : ¬e ≤ d
⊢ univ ∈ nhds (0 e) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
⊢ (if e ≤ d then {0} else univ) ∈ nhds (0 e)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_mem_nhds_zero | [111, 1] | [124, 45] | . simp only [nhds_discrete, Filter.mem_pure, mem_singleton_iff]
rfl | case pos
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
h : e ≤ d
⊢ {0} ∈ nhds (0 e)
case neg
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
h : ¬e ≤ d
⊢ univ ∈ nhds (0 e) | case neg
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
h : ¬e ≤ d
⊢ univ ∈ nhds (0 e) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
h : e ≤ d
⊢ {0} ∈ nhds (0 e)
case neg
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopolog... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_mem_nhds_zero | [111, 1] | [124, 45] | . simp only [Filter.univ_mem] | case neg
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
h : ¬e ≤ d
⊢ univ ∈ nhds (0 e) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
h : ¬e ≤ d
⊢ univ ∈ nhds (0 e)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_mem_nhds_zero | [111, 1] | [124, 45] | simp only [nhds_discrete, Filter.mem_pure, mem_singleton_iff] | case pos
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
h : e ≤ d
⊢ {0} ∈ nhds (0 e) | case pos
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
h : e ≤ d
⊢ OfNat.ofNat 0 e = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
h : e ≤ d
⊢ {0} ∈ nhds (0 e)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_mem_nhds_zero | [111, 1] | [124, 45] | rfl | case pos
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
h : e ≤ d
⊢ OfNat.ofNat 0 e = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
h : e ≤ d
⊢ OfNat.ofNat 0 e = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_mem_nhds_zero | [111, 1] | [124, 45] | simp only [Filter.univ_mem] | case neg
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
h : ¬e ≤ d
⊢ univ ∈ nhds (0 e) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d e : σ →₀ ℕ
h : ¬e ≤ d
⊢ univ ∈ nhds (0 e)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_mem_nhds_zero | [111, 1] | [124, 45] | intro f | case h.right
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
⊢ ((↑(Finset.Iic d)).pi fun e => if e ≤ d then {0} else univ) ⊆ ↑(basis σ α d) | case h.right
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
f : (σ →₀ ℕ) → α
⊢ (f ∈ (↑(Finset.Iic d)).pi fun e => if e ≤ d then {0} else univ) → f ∈ ↑(basis σ α d) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
⊢ ((↑(Finset.Iic d)).pi fun e => if e ≤ d then {0} else univ) ⊆ ↑(basis σ α d)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_mem_nhds_zero | [111, 1] | [124, 45] | simp only [Finset.coe_Iic, mem_pi, mem_Iic, mem_ite_univ_right, mem_singleton_iff, mem_coe] | case h.right
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
f : (σ →₀ ℕ) → α
⊢ (f ∈ (↑(Finset.Iic d)).pi fun e => if e ≤ d then {0} else univ) → f ∈ ↑(basis σ α d) | case h.right
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
f : (σ →₀ ℕ) → α
⊢ (∀ i ≤ d, i ≤ d → f i = 0) → f ∈ basis σ α d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
f : (σ →₀ ℕ) → α
⊢ (f ∈ (↑(Finset.Iic d)).pi fun e => if e ≤ d then {0} else univ) → f ∈ ↑(basis σ α d)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.basis_mem_nhds_zero | [111, 1] | [124, 45] | exact forall_imp (fun e h he => h he he) | case h.right
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
f : (σ →₀ ℕ) → α
⊢ (∀ i ≤ d, i ≤ d → f i = 0) → f ∈ basis σ α d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
d : σ →₀ ℕ
f : (σ →₀ ℕ) → α
⊢ (∀ i ≤ d, i ≤ d → f i = 0) → f ∈ basis σ α d
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | let τ := MvPowerSeries.WithPiTopology.topologicalSpace σ α | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
⊢ topologicalSpace σ α = ⋯.topology | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ topologicalSpace σ α = ⋯.topology | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
⊢ topologicalSpace σ α = ⋯.topology
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | let τ' := (toRingSubgroupsBasis σ α).topology | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ topologicalSpace σ α = ⋯.topology | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
⊢ topologicalSpace σ α = ⋯.topology | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
⊢ topologicalSpace σ α = ⋯.topology
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | erw [TopologicalAddGroup.ext_iff_nhds_zero τ τ'] | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
⊢ topologicalSpace σ α = ⋯.topology | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
⊢ nhds 0 = nhds 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
⊢ topologicalSpace σ α = ⋯.topology
T... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | ext s | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
⊢ nhds 0 = nhds 0 | case a
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ s ∈ nhds 0 ↔ s ∈ nhds 0 | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
⊢ nhds 0 = nhds 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | rw [(RingSubgroupsBasis.hasBasis_nhds (toRingSubgroupsBasis σ α) 0).mem_iff] | case a
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ s ∈ nhds 0 ↔ s ∈ nhds 0 | case a
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ s ∈ nhds 0 ↔ ∃ i, True ∧ {b | b - 0 ∈ Submodule.toAddSubgroup... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | simp only [sub_zero, Submodule.mem_toAddSubgroup, exists_true_left, true_and] | case a
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ s ∈ nhds 0 ↔ ∃ i, True ∧ {b | b - 0 ∈ Submodule.toAddSubgroup... | case a
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ s ∈ nhds 0 ↔ ∃ i, {b | b ∈ basis σ α i} ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | refine' ⟨_, fun ⟨d, hd⟩ => (@nhds _ τ 0).sets_of_superset (basis_mem_nhds_zero σ α d) hd⟩ | case a
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ s ∈ nhds 0 ↔ ∃ i, {b | b ∈ basis σ α i} ⊆ s | case a
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ s ∈ nhds 0 → ∃ i, {b | b ∈ basis σ α i} ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | rw [nhds_pi, Filter.mem_pi] | case a
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ s ∈ nhds 0 → ∃ i, {b | b ∈ basis σ α i} ⊆ s | case a
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ (∃ I, I.Finite ∧ ∃ t, (∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)) ∧ I.... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | rintro ⟨D, hD, t, ht, ht'⟩ | case a
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ (∃ I, I.Finite ∧ ∃ t, (∀ (i : σ →₀ ℕ), t i ∈ nhds (0 i)) ∧ I.... | case a.intro.intro.intro.intro
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
⊢ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | use Finset.sup hD.toFinset id | case a.intro.intro.intro.intro
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →... | case h
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i :... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro.intro.intro
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Se... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | apply subset_trans _ ht' | case h
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i :... | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | intro f hf e he | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ... | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | rw [← coeff_eq_apply f e, hf e] | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ... | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | exact mem_of_mem_nhds (ht e) | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ... | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | . have he' : e ∈ (Finite.toFinset hD) := by
simp only [id_eq, Finite.mem_toFinset]
exact he
apply Finset.le_sup he' | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | have he' : e ∈ (Finite.toFinset hD) := by
simp only [id_eq, Finite.mem_toFinset]
exact he | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ... | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | apply Finset.le_sup he' | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | simp only [id_eq, Finite.mem_toFinset] | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ... | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ... | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/LinearTopology.lean | MvPowerSeries.topology_eq_ideals_basis_topology | [129, 1] | [149, 28] | exact he | σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (σ →₀ ℕ)
hD : D.Finite
t : (σ →₀ ℕ) → Set α
ht : ∀ (i : σ →₀ ℕ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
σ : Type u_1
α : Type u_2
inst✝² : CommRing α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
τ : TopologicalSpace (MvPowerSeries σ α) := topologicalSpace σ α
τ' : TopologicalSpace (MvPowerSeries σ α) := ⋯.topology
s : Set (MvPowerSeries σ α)
D : Set (... |
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