Split (1)

train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M	input stringlengths 73 2.09M
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_add_of_weightedOrder_lt.aux	[205, 9]	[221, 13]	rw [(coeff _ _).map_add, coeff_of_lt_weightedOrder w g H, add_zero]	case right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ H : ↑((weight w) d) < weightedOrder w g hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 ⊢ (coeff α d) (f + g) ≠ 0	case right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ H : ↑((weight w) d) < weightedOrder w g hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 ⊢ (coeff α d) f ≠ 0	Please generate a tactic in lean4 to solve the state. STATE: case right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ H : ↑((weight w) d) < weightedOrder w g hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 ⊢ (coeff α d) (f + g) ≠ 0 TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_add_of_weightedOrder_lt.aux	[205, 9]	[221, 13]	exact hd'	case right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ H : ↑((weight w) d) < weightedOrder w g hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 ⊢ (coeff α d) f ≠ 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ H : ↑((weight w) d) < weightedOrder w g hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 ⊢ (coeff α d) f ≠ 0 TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_add_of_weightedOrder_lt.aux	[205, 9]	[221, 13]	intro b hb	case intro.intro.intro.right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α H : weightedOrder w f < weightedOrder w g n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 ⊢ ∀ (d : σ →₀ ℕ), (weight w) d < n → (coeff α d) (f + g) = 0	case intro.intro.intro.right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α H : weightedOrder w f < weightedOrder w g n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 b : σ →₀ ℕ hb : (weight w) b < n ⊢ (coeff α b) (f + g) = 0	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α H : weightedOrder w f < weightedOrder w g n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 ⊢ ∀ (d : σ →₀ ℕ...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_add_of_weightedOrder_lt.aux	[205, 9]	[221, 13]	suffices ↑(weight w b) < weightedOrder w f by rw [(coeff _ _).map_add, coeff_of_lt_weightedOrder w f this, coeff_of_lt_weightedOrder w g (lt_trans this H), add_zero]	case intro.intro.intro.right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α H : weightedOrder w f < weightedOrder w g n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 b : σ →₀ ℕ hb : (weight w) b < n ⊢ (coeff α b) (f + g) = 0	case intro.intro.intro.right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α H : weightedOrder w f < weightedOrder w g n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 b : σ →₀ ℕ hb : (weight w) b < n ⊢ ↑((weight w) b) < weightedOrder w f	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α H : weightedOrder w f < weightedOrder w g n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 b : σ →₀ ℕ hb :...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_add_of_weightedOrder_lt.aux	[205, 9]	[221, 13]	rw [← hn, ENat.some_eq_coe, Nat.cast_lt]	case intro.intro.intro.right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α H : weightedOrder w f < weightedOrder w g n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 b : σ →₀ ℕ hb : (weight w) b < n ⊢ ↑((weight w) b) < weightedOrder w f	case intro.intro.intro.right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α H : weightedOrder w f < weightedOrder w g n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 b : σ →₀ ℕ hb : (weight w) b < n ⊢ (weight w) b < n	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α H : weightedOrder w f < weightedOrder w g n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 b : σ →₀ ℕ hb :...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_add_of_weightedOrder_lt.aux	[205, 9]	[221, 13]	exact hb	case intro.intro.intro.right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α H : weightedOrder w f < weightedOrder w g n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 b : σ →₀ ℕ hb : (weight w) b < n ⊢ (weight w) b < n	no goals	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.right σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α H : weightedOrder w f < weightedOrder w g n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 b : σ →₀ ℕ hb :...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_add_of_weightedOrder_lt.aux	[205, 9]	[221, 13]	rw [(coeff _ _).map_add, coeff_of_lt_weightedOrder w f this, coeff_of_lt_weightedOrder w g (lt_trans this H), add_zero]	σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α H : weightedOrder w f < weightedOrder w g n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 b : σ →₀ ℕ hb : (weight w) b < n this : ↑((weight w) b) < weightedOrder w f ⊢ (coeff α b) (f + g) = 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α H : weightedOrder w f < weightedOrder w g n : ℕ hn : ↑n = weightedOrder w f d : σ →₀ ℕ hd : (weight w) d = n hd' : (coeff α d) f ≠ 0 b : σ →₀ ℕ hb : (weight w) b < n this : ↑((w...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_add_of_weightedOrder_eq	[225, 1]	[235, 63]	refine' le_antisymm _ (le_weightedOrder_add w _ _)	σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g ⊢ weightedOrder w (f + g) = weightedOrder w f ⊓ weightedOrder w g	σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g ⊢ weightedOrder w (f + g) ≤ weightedOrder w f ⊓ weightedOrder w g	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g ⊢ weightedOrder w (f + g) = weightedOrder w f ⊓ weightedOrder w g TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_add_of_weightedOrder_eq	[225, 1]	[235, 63]	by_cases H₁ : f.weightedOrder w < g.weightedOrder w	σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g ⊢ weightedOrder w (f + g) ≤ weightedOrder w f ⊓ weightedOrder w g	case pos σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : weightedOrder w f < weightedOrder w g ⊢ weightedOrder w (f + g) ≤ weightedOrder w f ⊓ weightedOrder w g case neg σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPo...	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g ⊢ weightedOrder w (f + g) ≤ weightedOrder w f ⊓ weightedOrder w g TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_add_of_weightedOrder_eq	[225, 1]	[235, 63]	simp only [le_inf_iff, weightedOrder_add_of_weightedOrder_lt.aux w H₁]	case pos σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : weightedOrder w f < weightedOrder w g ⊢ weightedOrder w (f + g) ≤ weightedOrder w f ⊓ weightedOrder w g	case pos σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : weightedOrder w f < weightedOrder w g ⊢ weightedOrder w f ≤ weightedOrder w f ∧ weightedOrder w f ≤ weightedOrder w g	Please generate a tactic in lean4 to solve the state. STATE: case pos σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : weightedOrder w f < weightedOrder w g ⊢ weightedOrder w (f + g) ≤ weightedOrder w f ⊓ weightedOrder w g TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_add_of_weightedOrder_eq	[225, 1]	[235, 63]	exact ⟨le_rfl, le_of_lt H₁⟩	case pos σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : weightedOrder w f < weightedOrder w g ⊢ weightedOrder w f ≤ weightedOrder w f ∧ weightedOrder w f ≤ weightedOrder w g	no goals	Please generate a tactic in lean4 to solve the state. STATE: case pos σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : weightedOrder w f < weightedOrder w g ⊢ weightedOrder w f ≤ weightedOrder w f ∧ weightedOrder w f ≤ weightedOrder w g TAC...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_add_of_weightedOrder_eq	[225, 1]	[235, 63]	by_cases H₂ : g.weightedOrder w < f.weightedOrder w	case neg σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : ¬weightedOrder w f < weightedOrder w g ⊢ weightedOrder w (f + g) ≤ weightedOrder w f ⊓ weightedOrder w g	case pos σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : ¬weightedOrder w f < weightedOrder w g H₂ : weightedOrder w g < weightedOrder w f ⊢ weightedOrder w (f + g) ≤ weightedOrder w f ⊓ weightedOrder w g case neg σ : Type u_2 α : Type u_...	Please generate a tactic in lean4 to solve the state. STATE: case neg σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : ¬weightedOrder w f < weightedOrder w g ⊢ weightedOrder w (f + g) ≤ weightedOrder w f ⊓ weightedOrder w g TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_add_of_weightedOrder_eq	[225, 1]	[235, 63]	simp only [add_comm f g, le_inf_iff, weightedOrder_add_of_weightedOrder_lt.aux w H₂]	case pos σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : ¬weightedOrder w f < weightedOrder w g H₂ : weightedOrder w g < weightedOrder w f ⊢ weightedOrder w (f + g) ≤ weightedOrder w f ⊓ weightedOrder w g	case pos σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : ¬weightedOrder w f < weightedOrder w g H₂ : weightedOrder w g < weightedOrder w f ⊢ weightedOrder w g ≤ weightedOrder w f ∧ weightedOrder w g ≤ weightedOrder w g	Please generate a tactic in lean4 to solve the state. STATE: case pos σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : ¬weightedOrder w f < weightedOrder w g H₂ : weightedOrder w g < weightedOrder w f ⊢ weightedOrder w (f + g) ≤ weightedOrd...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_add_of_weightedOrder_eq	[225, 1]	[235, 63]	exact ⟨le_of_lt H₂, le_rfl⟩	case pos σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : ¬weightedOrder w f < weightedOrder w g H₂ : weightedOrder w g < weightedOrder w f ⊢ weightedOrder w g ≤ weightedOrder w f ∧ weightedOrder w g ≤ weightedOrder w g	no goals	Please generate a tactic in lean4 to solve the state. STATE: case pos σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : ¬weightedOrder w f < weightedOrder w g H₂ : weightedOrder w g < weightedOrder w f ⊢ weightedOrder w g ≤ weightedOrder w f...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_add_of_weightedOrder_eq	[225, 1]	[235, 63]	exact absurd (le_antisymm (not_lt.1 H₂) (not_lt.1 H₁)) h	case neg σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : ¬weightedOrder w f < weightedOrder w g H₂ : ¬weightedOrder w g < weightedOrder w f ⊢ weightedOrder w (f + g) ≤ weightedOrder w f ⊓ weightedOrder w g	no goals	Please generate a tactic in lean4 to solve the state. STATE: case neg σ : Type u_2 α : Type u_1 inst✝ : Semiring α w : σ → ℕ f✝ f g : MvPowerSeries σ α h : weightedOrder w f ≠ weightedOrder w g H₁ : ¬weightedOrder w f < weightedOrder w g H₂ : ¬weightedOrder w g < weightedOrder w f ⊢ weightedOrder w (f + g) ≤ weightedOr...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_mul_ge	[241, 1]	[255, 40]	apply le_weightedOrder	σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α ⊢ weightedOrder w f + weightedOrder w g ≤ weightedOrder w (f * g)	case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α ⊢ ∀ (d : σ →₀ ℕ), ↑((weight w) d) < weightedOrder w f + weightedOrder w g → (coeff α d) (f * g) = 0	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α ⊢ weightedOrder w f + weightedOrder w g ≤ weightedOrder w (f * g) TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_mul_ge	[241, 1]	[255, 40]	intro d hd	case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α ⊢ ∀ (d : σ →₀ ℕ), ↑((weight w) d) < weightedOrder w f + weightedOrder w g → (coeff α d) (f * g) = 0	case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g ⊢ (coeff α d) (f * g) = 0	Please generate a tactic in lean4 to solve the state. STATE: case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α ⊢ ∀ (d : σ →₀ ℕ), ↑((weight w) d) < weightedOrder w f + weightedOrder w g → (coeff α d) (f * g) = 0 TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_mul_ge	[241, 1]	[255, 40]	rw [coeff_mul, Finset.sum_eq_zero]	case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g ⊢ (coeff α d) (f * g) = 0	case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g ⊢ ∀ x ∈ Finset.antidiagonal d, (coeff α x.1) f * (coeff α x.2) g = 0	Please generate a tactic in lean4 to solve the state. STATE: case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g ⊢ (coeff α d) (f * g) = 0 TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_mul_ge	[241, 1]	[255, 40]	rintro ⟨i, j⟩ hij	case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g ⊢ ∀ x ∈ Finset.antidiagonal d, (coeff α x.1) f * (coeff α x.2) g = 0	case h.mk σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d ⊢ (coeff α (i, j).1) f * (coeff α (i, j).2) g = 0	Please generate a tactic in lean4 to solve the state. STATE: case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g ⊢ ∀ x ∈ Finset.antidiagonal d, (coeff α x.1) f * (coeff...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_mul_ge	[241, 1]	[255, 40]	by_cases hi : ↑(weight w i) < f.weightedOrder w	case h.mk σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d ⊢ (coeff α (i, j).1) f * (coeff α (i, j).2) g = 0	case pos σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d hi : ↑((weight w) i) < weightedOrder w f ⊢ (coeff α (i, j).1) f ...	Please generate a tactic in lean4 to solve the state. STATE: case h.mk σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d ⊢ ...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_mul_ge	[241, 1]	[255, 40]	rw [coeff_of_lt_weightedOrder w f hi, MulZeroClass.zero_mul]	case pos σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d hi : ↑((weight w) i) < weightedOrder w f ⊢ (coeff α (i, j).1) f ...	no goals	Please generate a tactic in lean4 to solve the state. STATE: case pos σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d hi ...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_mul_ge	[241, 1]	[255, 40]	by_cases hj : ↑(weight w j) < g.weightedOrder w	case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d hi : ¬↑((weight w) i) < weightedOrder w f ⊢ (coeff α (i, j).1) f...	case pos σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d hi : ¬↑((weight w) i) < weightedOrder w f hj : ↑((weight w) j) <...	Please generate a tactic in lean4 to solve the state. STATE: case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d hi ...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_mul_ge	[241, 1]	[255, 40]	rw [coeff_of_lt_weightedOrder w g hj, MulZeroClass.mul_zero]	case pos σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d hi : ¬↑((weight w) i) < weightedOrder w f hj : ↑((weight w) j) <...	no goals	Please generate a tactic in lean4 to solve the state. STATE: case pos σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d hi ...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_mul_ge	[241, 1]	[255, 40]	rw [not_lt] at hi hj	case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d hi : ¬↑((weight w) i) < weightedOrder w f hj : ¬↑((weight w) j) ...	case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d hi : weightedOrder w f ≤ ↑((weight w) i) hj : weightedOrder w g ...	Please generate a tactic in lean4 to solve the state. STATE: case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d hi ...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_mul_ge	[241, 1]	[255, 40]	simp only [Finset.mem_antidiagonal] at hij	case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d hi : weightedOrder w f ≤ ↑((weight w) i) hj : weightedOrder w g ...	case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hi : weightedOrder w f ≤ ↑((weight w) i) hj : weightedOrder w g ≤ ↑((weight w) j) hij : i + j = d ⊢ (...	Please generate a tactic in lean4 to solve the state. STATE: case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d hi ...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_mul_ge	[241, 1]	[255, 40]	exfalso	case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hi : weightedOrder w f ≤ ↑((weight w) i) hj : weightedOrder w g ≤ ↑((weight w) j) hij : i + j = d ⊢ (...	case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hi : weightedOrder w f ≤ ↑((weight w) i) hj : weightedOrder w g ≤ ↑((weight w) j) hij : i + j = d ⊢ F...	Please generate a tactic in lean4 to solve the state. STATE: case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hi : weightedOrder w f ≤ ↑((weight w) i)...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_mul_ge	[241, 1]	[255, 40]	apply ne_of_lt (lt_of_lt_of_le hd <\| add_le_add hi hj)	case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hi : weightedOrder w f ≤ ↑((weight w) i) hj : weightedOrder w g ≤ ↑((weight w) j) hij : i + j = d ⊢ F...	case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hi : weightedOrder w f ≤ ↑((weight w) i) hj : weightedOrder w g ≤ ↑((weight w) j) hij : i + j = d ⊢ ↑...	Please generate a tactic in lean4 to solve the state. STATE: case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hi : weightedOrder w f ≤ ↑((weight w) i)...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_mul_ge	[241, 1]	[255, 40]	rw [← hij, map_add, Nat.cast_add]	case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hi : weightedOrder w f ≤ ↑((weight w) i) hj : weightedOrder w g ≤ ↑((weight w) j) hij : i + j = d ⊢ ↑...	no goals	Please generate a tactic in lean4 to solve the state. STATE: case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ hd : ↑((weight w) d) < weightedOrder w f + weightedOrder w g i j : σ →₀ ℕ hi : weightedOrder w f ≤ ↑((weight w) i)...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_monomial	[259, 1]	[270, 45]	split_ifs with h	σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) ⊢ weightedOrder w ((monomial α d) a) = if a = 0 then ⊤ else ↑((weight w) d)	case pos σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : a = 0 ⊢ weightedOrder w ((monomial α d) a) = ⊤ case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a =...	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) ⊢ weightedOrder w ((monomial α d) a) = if a = 0 then ⊤ else ↑((weight w) d) TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_monomial	[259, 1]	[270, 45]	rw [h, weightedOrder_eq_top_iff, LinearMap.map_zero]	case pos σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : a = 0 ⊢ weightedOrder w ((monomial α d) a) = ⊤	no goals	Please generate a tactic in lean4 to solve the state. STATE: case pos σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : a = 0 ⊢ weightedOrder w ((monomial α d) a) = ⊤ TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_monomial	[259, 1]	[270, 45]	rw [weightedOrder_eq_nat]	case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ weightedOrder w ((monomial α d) a) = ↑((weight w) d)	case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ (∃ d_1, (weight w) d_1 = (weight w) d ∧ (coeff α d_1) ((monomial α d) a) ≠ 0) ∧ ∀ (d_1 : σ →₀ ℕ), (weight w) d_1 < (weight w) d → (coeff α d_1) ((monomial α d) a) = 0	Please generate a tactic in lean4 to solve the state. STATE: case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ weightedOrder w ((monomial α d) a) = ↑((weight w) d) TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_monomial	[259, 1]	[270, 45]	constructor	case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ (∃ d_1, (weight w) d_1 = (weight w) d ∧ (coeff α d_1) ((monomial α d) a) ≠ 0) ∧ ∀ (d_1 : σ →₀ ℕ), (weight w) d_1 < (weight w) d → (coeff α d_1) ((monomial α d) a) = 0	case neg.left σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ ∃ d_1, (weight w) d_1 = (weight w) d ∧ (coeff α d_1) ((monomial α d) a) ≠ 0 case neg.right σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d ...	Please generate a tactic in lean4 to solve the state. STATE: case neg σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ (∃ d_1, (weight w) d_1 = (weight w) d ∧ (coeff α d_1) ((monomial α d) a) ≠ 0) ∧ ∀ (d_1 : σ →₀ ℕ), (weight w) d_1...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_monomial	[259, 1]	[270, 45]	use d	case neg.left σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ ∃ d_1, (weight w) d_1 = (weight w) d ∧ (coeff α d_1) ((monomial α d) a) ≠ 0	case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ (weight w) d = (weight w) d ∧ (coeff α d) ((monomial α d) a) ≠ 0	Please generate a tactic in lean4 to solve the state. STATE: case neg.left σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ ∃ d_1, (weight w) d_1 = (weight w) d ∧ (coeff α d_1) ((monomial α d) a) ≠ 0 TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_monomial	[259, 1]	[270, 45]	simp only [coeff_monomial_same, eq_self_iff_true, ne_eq, true_and_iff]	case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ (weight w) d = (weight w) d ∧ (coeff α d) ((monomial α d) a) ≠ 0	case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ ¬a = 0	Please generate a tactic in lean4 to solve the state. STATE: case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ (weight w) d = (weight w) d ∧ (coeff α d) ((monomial α d) a) ≠ 0 TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_monomial	[259, 1]	[270, 45]	exact h	case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ ¬a = 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ ¬a = 0 TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_monomial	[259, 1]	[270, 45]	intro b hb	case neg.right σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ ∀ (d_1 : σ →₀ ℕ), (weight w) d_1 < (weight w) d → (coeff α d_1) ((monomial α d) a) = 0	case neg.right σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 b : σ →₀ ℕ hb : (weight w) b < (weight w) d ⊢ (coeff α b) ((monomial α d) a) = 0	Please generate a tactic in lean4 to solve the state. STATE: case neg.right σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 ⊢ ∀ (d_1 : σ →₀ ℕ), (weight w) d_1 < (weight w) d → (coeff α d_1) ((monomial α d) a) = 0 TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_monomial	[259, 1]	[270, 45]	rw [coeff_monomial, if_neg]	case neg.right σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 b : σ →₀ ℕ hb : (weight w) b < (weight w) d ⊢ (coeff α b) ((monomial α d) a) = 0	case neg.right.hnc σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 b : σ →₀ ℕ hb : (weight w) b < (weight w) d ⊢ ¬b = d	Please generate a tactic in lean4 to solve the state. STATE: case neg.right σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 b : σ →₀ ℕ hb : (weight w) b < (weight w) d ⊢ (coeff α b) ((monomial α d) a) = 0 TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_monomial	[259, 1]	[270, 45]	intro h	case neg.right.hnc σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 b : σ →₀ ℕ hb : (weight w) b < (weight w) d ⊢ ¬b = d	case neg.right.hnc σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h✝ : ¬a = 0 b : σ →₀ ℕ hb : (weight w) b < (weight w) d h : b = d ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: case neg.right.hnc σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h : ¬a = 0 b : σ →₀ ℕ hb : (weight w) b < (weight w) d ⊢ ¬b = d TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_monomial	[259, 1]	[270, 45]	simp only [h, lt_self_iff_false] at hb	case neg.right.hnc σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h✝ : ¬a = 0 b : σ →₀ ℕ hb : (weight w) b < (weight w) d h : b = d ⊢ False	no goals	Please generate a tactic in lean4 to solve the state. STATE: case neg.right.hnc σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α inst✝ : Decidable (a = 0) h✝ : ¬a = 0 b : σ →₀ ℕ hb : (weight w) b < (weight w) d h : b = d ⊢ False TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_monomial_of_ne_zero	[274, 1]	[277, 40]	classical rw [weightedOrder_monomial, if_neg h]	σ : Type u_1 α : Type u_2 inst✝ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α h : a ≠ 0 ⊢ weightedOrder w ((monomial α d) a) = ↑((weight w) d)	no goals	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_1 α : Type u_2 inst✝ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α h : a ≠ 0 ⊢ weightedOrder w ((monomial α d) a) = ↑((weight w) d) TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.weightedOrder_monomial_of_ne_zero	[274, 1]	[277, 40]	rw [weightedOrder_monomial, if_neg h]	σ : Type u_1 α : Type u_2 inst✝ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α h : a ≠ 0 ⊢ weightedOrder w ((monomial α d) a) = ↑((weight w) d)	no goals	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_1 α : Type u_2 inst✝ : Semiring α w : σ → ℕ f : MvPowerSeries σ α d : σ →₀ ℕ a : α h : a ≠ 0 ⊢ weightedOrder w ((monomial α d) a) = ↑((weight w) d) TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.coeff_mul_of_lt_weightedOrder	[283, 1]	[292, 88]	rw [coeff_mul]	σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g ⊢ (coeff α d) (f * g) = 0	σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g ⊢ ∑ p ∈ Finset.antidiagonal d, (coeff α p.1) f * (coeff α p.2) g = 0	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g ⊢ (coeff α d) (f * g) = 0 TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.coeff_mul_of_lt_weightedOrder	[283, 1]	[292, 88]	apply Finset.sum_eq_zero	σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g ⊢ ∑ p ∈ Finset.antidiagonal d, (coeff α p.1) f * (coeff α p.2) g = 0	case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g ⊢ ∀ x ∈ Finset.antidiagonal d, (coeff α x.1) f * (coeff α x.2) g = 0	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g ⊢ ∑ p ∈ Finset.antidiagonal d, (coeff α p.1) f * (coeff α p.2) g = 0 TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.coeff_mul_of_lt_weightedOrder	[283, 1]	[292, 88]	rintro ⟨i, j⟩ hij	case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g ⊢ ∀ x ∈ Finset.antidiagonal d, (coeff α x.1) f * (coeff α x.2) g = 0	case h.mk σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d ⊢ (coeff α (i, j).1) f * (coeff α (i, j).2) g = 0	Please generate a tactic in lean4 to solve the state. STATE: case h σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g ⊢ ∀ x ∈ Finset.antidiagonal d, (coeff α x.1) f * (coeff α x.2) g = 0 TACTIC:...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.coeff_mul_of_lt_weightedOrder	[283, 1]	[292, 88]	refine' mul_eq_zero_of_right (coeff α i f) _	case h.mk σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d ⊢ (coeff α (i, j).1) f * (coeff α (i, j).2) g = 0	case h.mk σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d ⊢ (coeff α (i, j).2) g = 0	Please generate a tactic in lean4 to solve the state. STATE: case h.mk σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d ⊢ (coeff α (i, j).1) f ...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.coeff_mul_of_lt_weightedOrder	[283, 1]	[292, 88]	refine' coeff_of_lt_weightedOrder w g (lt_of_le_of_lt _ h)	case h.mk σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d ⊢ (coeff α (i, j).2) g = 0	case h.mk σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d ⊢ ↑((weight w) (i, j).2) ≤ ↑((weight w) d)	Please generate a tactic in lean4 to solve the state. STATE: case h.mk σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d ⊢ (coeff α (i, j).2) g ...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.coeff_mul_of_lt_weightedOrder	[283, 1]	[292, 88]	simp only [Finset.mem_antidiagonal] at hij	case h.mk σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d ⊢ ↑((weight w) (i, j).2) ≤ ↑((weight w) d)	case h.mk σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g i j : σ →₀ ℕ hij : i + j = d ⊢ ↑((weight w) (i, j).2) ≤ ↑((weight w) d)	Please generate a tactic in lean4 to solve the state. STATE: case h.mk σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g i j : σ →₀ ℕ hij : (i, j) ∈ Finset.antidiagonal d ⊢ ↑((weight w) (i, j).2...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.coeff_mul_of_lt_weightedOrder	[283, 1]	[292, 88]	simp only [Nat.cast_le, coe_le_coe, ← hij, map_add, le_add_iff_nonneg_left, zero_le']	case h.mk σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g i j : σ →₀ ℕ hij : i + j = d ⊢ ↑((weight w) (i, j).2) ≤ ↑((weight w) d)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.mk σ : Type u_1 α : Type u_2 inst✝¹ : Semiring α w : σ → ℕ f✝ : MvPowerSeries σ α inst✝ : DecidableEq σ f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g i j : σ →₀ ℕ hij : i + j = d ⊢ ↑((weight w) (i, j).2) ≤ ↑((weight w) d) T...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.coeff_mul_one_sub_of_lt_weightedOrder	[295, 1]	[298, 97]	simp only [coeff_mul_of_lt_weightedOrder w f h, mul_sub, mul_one, _root_.map_sub, sub_zero]	σ : Type u_1 α✝ : Type ?u.58468 inst✝² : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ inst✝¹ : DecidableEq σ α : Type u_2 inst✝ : CommRing α f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g ⊢ (coeff α d) (f * (1 - g)) = (coeff α d) f	no goals	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_1 α✝ : Type ?u.58468 inst✝² : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ inst✝¹ : DecidableEq σ α : Type u_2 inst✝ : CommRing α f g : MvPowerSeries σ α d : σ →₀ ℕ h : ↑((weight w) d) < weightedOrder w g ⊢ (coeff α d) (f * (1 - g)) = (coeff α d) ...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.coeff_mul_prod_one_sub_of_lt_weightedOrder	[302, 1]	[313, 17]	refine' Finset.induction_on s _ _	σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α ⊢ (∀ i ∈ s, ↑((weight w) d) < weightedOrder w (g i)) → (coeff α d) (f * ∏ i ∈ s, (1 - g i)) = (coeff α d) f	case refine'_1 σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α ⊢ (∀ i ∈ ∅, ↑((weight w) d) < weightedOrder w (g i)) → (coeff α d) (f * ∏ i ∈ ∅, (1 - g i)) = (coeff ...	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α ⊢ (∀ i ∈ s, ↑((weight w) d) < weightedOrder w (g i)) → ...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.coeff_mul_prod_one_sub_of_lt_weightedOrder	[302, 1]	[313, 17]	simp only [imp_true_iff, Finset.prod_empty, mul_one, eq_self_iff_true]	case refine'_1 σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α ⊢ (∀ i ∈ ∅, ↑((weight w) d) < weightedOrder w (g i)) → (coeff α d) (f * ∏ i ∈ ∅, (1 - g i)) = (coeff ...	no goals	Please generate a tactic in lean4 to solve the state. STATE: case refine'_1 σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α ⊢ (∀ i ∈ ∅, ↑((weight w) d) < weightedOr...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.coeff_mul_prod_one_sub_of_lt_weightedOrder	[302, 1]	[313, 17]	intro a s ha ih t	case refine'_2 σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α ⊢ ∀ ⦃a : ι⦄ {s : Finset ι}, a ∉ s → ((∀ i ∈ s, ↑((weight w) d) < weightedOrder w (g i)) → (...	case refine'_2 σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s✝ : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α a : ι s : Finset ι ha : a ∉ s ih : (∀ i ∈ s, ↑((weight w) d) < weightedOrder w (g i)) → (coeff α d) ...	Please generate a tactic in lean4 to solve the state. STATE: case refine'_2 σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α ⊢ ∀ ⦃a : ι⦄ {s : Finset ι}, a ∉ s → ...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.coeff_mul_prod_one_sub_of_lt_weightedOrder	[302, 1]	[313, 17]	simp only [Finset.mem_insert, forall_eq_or_imp] at t	case refine'_2 σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s✝ : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α a : ι s : Finset ι ha : a ∉ s ih : (∀ i ∈ s, ↑((weight w) d) < weightedOrder w (g i)) → (coeff α d) ...	case refine'_2 σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s✝ : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α a : ι s : Finset ι ha : a ∉ s ih : (∀ i ∈ s, ↑((weight w) d) < weightedOrder w (g i)) → (coeff α d) ...	Please generate a tactic in lean4 to solve the state. STATE: case refine'_2 σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s✝ : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α a : ι s : Finset ι ha : a ∉ s ih : (∀ i...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.coeff_mul_prod_one_sub_of_lt_weightedOrder	[302, 1]	[313, 17]	rw [Finset.prod_insert ha, ← mul_assoc, mul_right_comm, coeff_mul_one_sub_of_lt_weightedOrder w _ t.1]	case refine'_2 σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s✝ : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α a : ι s : Finset ι ha : a ∉ s ih : (∀ i ∈ s, ↑((weight w) d) < weightedOrder w (g i)) → (coeff α d) ...	case refine'_2 σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s✝ : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α a : ι s : Finset ι ha : a ∉ s ih : (∀ i ∈ s, ↑((weight w) d) < weightedOrder w (g i)) → (coeff α d) ...	Please generate a tactic in lean4 to solve the state. STATE: case refine'_2 σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s✝ : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α a : ι s : Finset ι ha : a ∉ s ih : (∀ i...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.coeff_mul_prod_one_sub_of_lt_weightedOrder	[302, 1]	[313, 17]	exact ih t.2	case refine'_2 σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s✝ : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α a : ι s : Finset ι ha : a ∉ s ih : (∀ i ∈ s, ↑((weight w) d) < weightedOrder w (g i)) → (coeff α d) ...	no goals	Please generate a tactic in lean4 to solve the state. STATE: case refine'_2 σ : Type u_3 α✝ : Type ?u.62270 inst✝¹ : Semiring α✝ w : σ → ℕ f✝ : MvPowerSeries σ α✝ α : Type u_1 ι : Type u_2 inst✝ : CommRing α d : σ →₀ ℕ s✝ : Finset ι f : MvPowerSeries σ α g : ι → MvPowerSeries σ α a : ι s : Finset ι ha : a ∉ s ih : (∀ i...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.degree_apply	[327, 1]	[332, 28]	rw [degree, weight_apply]	σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ ⊢ degree d = d.sum fun x => id	σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ ⊢ (d.sum fun x => Mul.mul 1) = d.sum fun x => id	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ ⊢ degree d = d.sum fun x => id TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.degree_apply	[327, 1]	[332, 28]	apply congr_arg	σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ ⊢ (d.sum fun x => Mul.mul 1) = d.sum fun x => id	case h σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ ⊢ (fun x => Mul.mul 1) = fun x => id	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ ⊢ (d.sum fun x => Mul.mul 1) = d.sum fun x => id TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.degree_apply	[327, 1]	[332, 28]	ext _ n	case h σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ ⊢ (fun x => Mul.mul 1) = fun x => id	case h.h.h σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ x✝ : σ n : ℕ ⊢ Mul.mul 1 n = id n	Please generate a tactic in lean4 to solve the state. STATE: case h σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ ⊢ (fun x => Mul.mul 1) = fun x => id TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.degree_apply	[327, 1]	[332, 28]	have h_eq : Mul.mul 1 n = 1 * n := by rfl	case h.h.h σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ x✝ : σ n : ℕ ⊢ Mul.mul 1 n = id n	case h.h.h σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ x✝ : σ n : ℕ h_eq : Mul.mul 1 n = 1 * n ⊢ Mul.mul 1 n = id n	Please generate a tactic in lean4 to solve the state. STATE: case h.h.h σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ x✝ : σ n : ℕ ⊢ Mul.mul 1 n = id n TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.degree_apply	[327, 1]	[332, 28]	rw [h_eq, id_eq, one_mul]	case h.h.h σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ x✝ : σ n : ℕ h_eq : Mul.mul 1 n = 1 * n ⊢ Mul.mul 1 n = id n	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.h.h σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ x✝ : σ n : ℕ h_eq : Mul.mul 1 n = 1 * n ⊢ Mul.mul 1 n = id n TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.degree_apply	[327, 1]	[332, 28]	rfl	σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ x✝ : σ n : ℕ ⊢ Mul.mul 1 n = 1 * n	no goals	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_1 α : Type ?u.65991 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ x✝ : σ n : ℕ ⊢ Mul.mul 1 n = 1 * n TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.degree_eq_zero_iff	[335, 1]	[338, 76]	simp only [degree, weight, one_mul, AddMonoidHom.coe_mk, Finsupp.sum, Finset.sum_eq_zero_iff, Finsupp.mem_support_iff, _root_.not_imp_self, DFunLike.ext_iff, Finsupp.coe_zero, Pi.zero_apply, ZeroHom.coe_mk, Finset.sum_eq_zero_iff, Finsupp.mem_support_iff, ne_eq]	σ : Type u_1 α : Type ?u.66556 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ ⊢ degree d = 0 ↔ d = 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_1 α : Type ?u.66556 inst✝ : Semiring α f : MvPowerSeries σ α d : σ →₀ ℕ ⊢ degree d = 0 ↔ d = 0 TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.le_degree	[341, 1]	[343, 20]	convert le_weight _ x _ d	σ : Type u_1 α : Type ?u.68078 inst✝ : Semiring α f : MvPowerSeries σ α x : σ d : σ →₀ ℕ ⊢ d x ≤ degree d	case convert_2 σ : Type u_1 α : Type ?u.68078 inst✝ : Semiring α f : MvPowerSeries σ α x : σ d : σ →₀ ℕ ⊢ 1 ≠ 0	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_1 α : Type ?u.68078 inst✝ : Semiring α f : MvPowerSeries σ α x : σ d : σ →₀ ℕ ⊢ d x ≤ degree d TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.le_degree	[341, 1]	[343, 20]	exact NeZero.ne 1	case convert_2 σ : Type u_1 α : Type ?u.68078 inst✝ : Semiring α f : MvPowerSeries σ α x : σ d : σ →₀ ℕ ⊢ 1 ≠ 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case convert_2 σ : Type u_1 α : Type ?u.68078 inst✝ : Semiring α f : MvPowerSeries σ α x : σ d : σ →₀ ℕ ⊢ 1 ≠ 0 TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.finite_of_degree_le	[346, 1]	[350, 74]	refine' finite_of_weight_le (Function.const σ 1) _ n	σ : Type u_1 α : Type ?u.69196 inst✝¹ : Semiring α f : MvPowerSeries σ α inst✝ : Finite σ n : ℕ ⊢ {f \| degree f ≤ n}.Finite	σ : Type u_1 α : Type ?u.69196 inst✝¹ : Semiring α f : MvPowerSeries σ α inst✝ : Finite σ n : ℕ ⊢ ∀ (x : σ), Function.const σ 1 x ≠ 0	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_1 α : Type ?u.69196 inst✝¹ : Semiring α f : MvPowerSeries σ α inst✝ : Finite σ n : ℕ ⊢ {f \| degree f ≤ n}.Finite TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.finite_of_degree_le	[346, 1]	[350, 74]	intro _	σ : Type u_1 α : Type ?u.69196 inst✝¹ : Semiring α f : MvPowerSeries σ α inst✝ : Finite σ n : ℕ ⊢ ∀ (x : σ), Function.const σ 1 x ≠ 0	σ : Type u_1 α : Type ?u.69196 inst✝¹ : Semiring α f : MvPowerSeries σ α inst✝ : Finite σ n : ℕ x✝ : σ ⊢ Function.const σ 1 x✝ ≠ 0	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_1 α : Type ?u.69196 inst✝¹ : Semiring α f : MvPowerSeries σ α inst✝ : Finite σ n : ℕ ⊢ ∀ (x : σ), Function.const σ 1 x ≠ 0 TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/MvPowerSeries/Order.lean	MvPowerSeries.finite_of_degree_le	[346, 1]	[350, 74]	simp only [Function.const_apply, ne_eq, one_ne_zero, not_false_eq_true]	σ : Type u_1 α : Type ?u.69196 inst✝¹ : Semiring α f : MvPowerSeries σ α inst✝ : Finite σ n : ℕ x✝ : σ ⊢ Function.const σ 1 x✝ ≠ 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: σ : Type u_1 α : Type ?u.69196 inst✝¹ : Semiring α f : MvPowerSeries σ α inst✝ : Finite σ n : ℕ x✝ : σ ⊢ Function.const σ 1 x✝ ≠ 0 TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/Topology/Algebra/Algebra/Basic.lean	ContinuousAlgHom.coe_injective	[142, 1]	[146, 8]	intro f g H	R : Type u_1 inst✝⁹ : CommSemiring R inst✝⁸ : TopologicalSpace R inst✝⁷ : TopologicalSemiring R A : Type u_2 inst✝⁶ : Semiring A inst✝⁵ : TopologicalSpace A inst✝⁴ : TopologicalSemiring A B : Type u_3 inst✝³ : Semiring B inst✝² : TopologicalSpace B inst✝¹ : Algebra R A inst✝ : Algebra R B ⊢ Injective toAlgHom	R : Type u_1 inst✝⁹ : CommSemiring R inst✝⁸ : TopologicalSpace R inst✝⁷ : TopologicalSemiring R A : Type u_2 inst✝⁶ : Semiring A inst✝⁵ : TopologicalSpace A inst✝⁴ : TopologicalSemiring A B : Type u_3 inst✝³ : Semiring B inst✝² : TopologicalSpace B inst✝¹ : Algebra R A inst✝ : Algebra R B f g : A →A[R] B H : ↑f = ↑g ⊢ ...	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝⁹ : CommSemiring R inst✝⁸ : TopologicalSpace R inst✝⁷ : TopologicalSemiring R A : Type u_2 inst✝⁶ : Semiring A inst✝⁵ : TopologicalSpace A inst✝⁴ : TopologicalSemiring A B : Type u_3 inst✝³ : Semiring B inst✝² : TopologicalSpace B inst✝¹ : A...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/Topology/Algebra/Algebra/Basic.lean	ContinuousAlgHom.coe_injective	[142, 1]	[146, 8]	cases f	R : Type u_1 inst✝⁹ : CommSemiring R inst✝⁸ : TopologicalSpace R inst✝⁷ : TopologicalSemiring R A : Type u_2 inst✝⁶ : Semiring A inst✝⁵ : TopologicalSpace A inst✝⁴ : TopologicalSemiring A B : Type u_3 inst✝³ : Semiring B inst✝² : TopologicalSpace B inst✝¹ : Algebra R A inst✝ : Algebra R B f g : A →A[R] B H : ↑f = ↑g ⊢ ...	case mk R : Type u_1 inst✝⁹ : CommSemiring R inst✝⁸ : TopologicalSpace R inst✝⁷ : TopologicalSemiring R A : Type u_2 inst✝⁶ : Semiring A inst✝⁵ : TopologicalSpace A inst✝⁴ : TopologicalSemiring A B : Type u_3 inst✝³ : Semiring B inst✝² : TopologicalSpace B inst✝¹ : Algebra R A inst✝ : Algebra R B g : A →A[R] B toAlgHom...	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝⁹ : CommSemiring R inst✝⁸ : TopologicalSpace R inst✝⁷ : TopologicalSemiring R A : Type u_2 inst✝⁶ : Semiring A inst✝⁵ : TopologicalSpace A inst✝⁴ : TopologicalSemiring A B : Type u_3 inst✝³ : Semiring B inst✝² : TopologicalSpace B inst✝¹ : A...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/Topology/Algebra/Algebra/Basic.lean	ContinuousAlgHom.coe_injective	[142, 1]	[146, 8]	cases g	case mk R : Type u_1 inst✝⁹ : CommSemiring R inst✝⁸ : TopologicalSpace R inst✝⁷ : TopologicalSemiring R A : Type u_2 inst✝⁶ : Semiring A inst✝⁵ : TopologicalSpace A inst✝⁴ : TopologicalSemiring A B : Type u_3 inst✝³ : Semiring B inst✝² : TopologicalSpace B inst✝¹ : Algebra R A inst✝ : Algebra R B g : A →A[R] B toAlgHom...	case mk.mk R : Type u_1 inst✝⁹ : CommSemiring R inst✝⁸ : TopologicalSpace R inst✝⁷ : TopologicalSemiring R A : Type u_2 inst✝⁶ : Semiring A inst✝⁵ : TopologicalSpace A inst✝⁴ : TopologicalSemiring A B : Type u_3 inst✝³ : Semiring B inst✝² : TopologicalSpace B inst✝¹ : Algebra R A inst✝ : Algebra R B toAlgHom✝¹ : A →ₐ[R...	Please generate a tactic in lean4 to solve the state. STATE: case mk R : Type u_1 inst✝⁹ : CommSemiring R inst✝⁸ : TopologicalSpace R inst✝⁷ : TopologicalSemiring R A : Type u_2 inst✝⁶ : Semiring A inst✝⁵ : TopologicalSpace A inst✝⁴ : TopologicalSemiring A B : Type u_3 inst✝³ : Semiring B inst✝² : TopologicalSpace B in...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/Topology/Algebra/Algebra/Basic.lean	ContinuousAlgHom.coe_injective	[142, 1]	[146, 8]	congr	case mk.mk R : Type u_1 inst✝⁹ : CommSemiring R inst✝⁸ : TopologicalSpace R inst✝⁷ : TopologicalSemiring R A : Type u_2 inst✝⁶ : Semiring A inst✝⁵ : TopologicalSpace A inst✝⁴ : TopologicalSemiring A B : Type u_3 inst✝³ : Semiring B inst✝² : TopologicalSpace B inst✝¹ : Algebra R A inst✝ : Algebra R B toAlgHom✝¹ : A →ₐ[R...	no goals	Please generate a tactic in lean4 to solve the state. STATE: case mk.mk R : Type u_1 inst✝⁹ : CommSemiring R inst✝⁸ : TopologicalSpace R inst✝⁷ : TopologicalSemiring R A : Type u_2 inst✝⁶ : Semiring A inst✝⁵ : TopologicalSpace A inst✝⁴ : TopologicalSemiring A B : Type u_3 inst✝³ : Semiring B inst✝² : TopologicalSpace B...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/Topology/Algebra/Algebra/Basic.lean	ContinuousAlgHom.ext_ring	[229, 1]	[231, 23]	apply coe_inj.1	R : Type u_1 inst✝¹⁰ : CommSemiring R inst✝⁹ : TopologicalSpace R inst✝⁸ : TopologicalSemiring R A : Type u_2 inst✝⁷ : Semiring A inst✝⁶ : TopologicalSpace A inst✝⁵ : TopologicalSemiring A B : Type u_3 inst✝⁴ : Semiring B inst✝³ : TopologicalSpace B inst✝² : Algebra R A inst✝¹ : Algebra R B inst✝ : TopologicalSpace R f...	R : Type u_1 inst✝¹⁰ : CommSemiring R inst✝⁹ : TopologicalSpace R inst✝⁸ : TopologicalSemiring R A : Type u_2 inst✝⁷ : Semiring A inst✝⁶ : TopologicalSpace A inst✝⁵ : TopologicalSemiring A B : Type u_3 inst✝⁴ : Semiring B inst✝³ : TopologicalSpace B inst✝² : Algebra R A inst✝¹ : Algebra R B inst✝ : TopologicalSpace R f...	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹⁰ : CommSemiring R inst✝⁹ : TopologicalSpace R inst✝⁸ : TopologicalSemiring R A : Type u_2 inst✝⁷ : Semiring A inst✝⁶ : TopologicalSpace A inst✝⁵ : TopologicalSemiring A B : Type u_3 inst✝⁴ : Semiring B inst✝³ : TopologicalSpace B inst✝² : ...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/Topology/Algebra/Algebra/Basic.lean	ContinuousAlgHom.ext_ring	[229, 1]	[231, 23]	apply Algebra.ext_id	R : Type u_1 inst✝¹⁰ : CommSemiring R inst✝⁹ : TopologicalSpace R inst✝⁸ : TopologicalSemiring R A : Type u_2 inst✝⁷ : Semiring A inst✝⁶ : TopologicalSpace A inst✝⁵ : TopologicalSemiring A B : Type u_3 inst✝⁴ : Semiring B inst✝³ : TopologicalSpace B inst✝² : Algebra R A inst✝¹ : Algebra R B inst✝ : TopologicalSpace R f...	no goals	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹⁰ : CommSemiring R inst✝⁹ : TopologicalSpace R inst✝⁸ : TopologicalSemiring R A : Type u_2 inst✝⁷ : Semiring A inst✝⁶ : TopologicalSpace A inst✝⁵ : TopologicalSemiring A B : Type u_3 inst✝⁴ : Semiring B inst✝³ : TopologicalSpace B inst✝² : ...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/Topology/Algebra/Algebra/Basic.lean	DenseRange.topologicalClosure_map_submodule'	[258, 1]	[263, 40]	rw [SetLike.ext'_iff] at hs ⊢	R : Type u_1 inst✝¹³ : CommSemiring R inst✝¹² : TopologicalSpace R inst✝¹¹ : TopologicalSemiring R A : Type u_2 inst✝¹⁰ : Semiring A inst✝⁹ : TopologicalSpace A inst✝⁸ : TopologicalSemiring A B : Type u_3 inst✝⁷ : Semiring B inst✝⁶ : TopologicalSpace B inst✝⁵ : Algebra R A inst✝⁴ : Algebra R B inst✝³ : TopologicalSpace...	R : Type u_1 inst✝¹³ : CommSemiring R inst✝¹² : TopologicalSpace R inst✝¹¹ : TopologicalSemiring R A : Type u_2 inst✝¹⁰ : Semiring A inst✝⁹ : TopologicalSpace A inst✝⁸ : TopologicalSemiring A B : Type u_3 inst✝⁷ : Semiring B inst✝⁶ : TopologicalSpace B inst✝⁵ : Algebra R A inst✝⁴ : Algebra R B inst✝³ : TopologicalSpace...	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹³ : CommSemiring R inst✝¹² : TopologicalSpace R inst✝¹¹ : TopologicalSemiring R A : Type u_2 inst✝¹⁰ : Semiring A inst✝⁹ : TopologicalSpace A inst✝⁸ : TopologicalSemiring A B : Type u_3 inst✝⁷ : Semiring B inst✝⁶ : TopologicalSpace B inst✝⁵...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/Topology/Algebra/Algebra/Basic.lean	DenseRange.topologicalClosure_map_submodule'	[258, 1]	[263, 40]	simp only [Submodule.topologicalClosure_coe, Submodule.top_coe, ← dense_iff_closure_eq] at hs ⊢	R : Type u_1 inst✝¹³ : CommSemiring R inst✝¹² : TopologicalSpace R inst✝¹¹ : TopologicalSemiring R A : Type u_2 inst✝¹⁰ : Semiring A inst✝⁹ : TopologicalSpace A inst✝⁸ : TopologicalSemiring A B : Type u_3 inst✝⁷ : Semiring B inst✝⁶ : TopologicalSpace B inst✝⁵ : Algebra R A inst✝⁴ : Algebra R B inst✝³ : TopologicalSpace...	R : Type u_1 inst✝¹³ : CommSemiring R inst✝¹² : TopologicalSpace R inst✝¹¹ : TopologicalSemiring R A : Type u_2 inst✝¹⁰ : Semiring A inst✝⁹ : TopologicalSpace A inst✝⁸ : TopologicalSemiring A B : Type u_3 inst✝⁷ : Semiring B inst✝⁶ : TopologicalSpace B inst✝⁵ : Algebra R A inst✝⁴ : Algebra R B inst✝³ : TopologicalSpace...	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹³ : CommSemiring R inst✝¹² : TopologicalSpace R inst✝¹¹ : TopologicalSemiring R A : Type u_2 inst✝¹⁰ : Semiring A inst✝⁹ : TopologicalSpace A inst✝⁸ : TopologicalSemiring A B : Type u_3 inst✝⁷ : Semiring B inst✝⁶ : TopologicalSpace B inst✝⁵...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/Topology/Algebra/Algebra/Basic.lean	DenseRange.topologicalClosure_map_submodule'	[258, 1]	[263, 40]	exact hf'.dense_image f.continuous hs	R : Type u_1 inst✝¹³ : CommSemiring R inst✝¹² : TopologicalSpace R inst✝¹¹ : TopologicalSemiring R A : Type u_2 inst✝¹⁰ : Semiring A inst✝⁹ : TopologicalSpace A inst✝⁸ : TopologicalSemiring A B : Type u_3 inst✝⁷ : Semiring B inst✝⁶ : TopologicalSpace B inst✝⁵ : Algebra R A inst✝⁴ : Algebra R B inst✝³ : TopologicalSpace...	no goals	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹³ : CommSemiring R inst✝¹² : TopologicalSpace R inst✝¹¹ : TopologicalSemiring R A : Type u_2 inst✝¹⁰ : Semiring A inst✝⁹ : TopologicalSpace A inst✝⁸ : TopologicalSemiring A B : Type u_3 inst✝⁷ : Semiring B inst✝⁶ : TopologicalSpace B inst✝⁵...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/ForMathlib/Topology/Algebra/Algebra/Basic.lean	ContinuousAlgHom.coe_eq_id	[287, 1]	[288, 25]	rw [← coe_id, coe_inj]	R : Type u_1 inst✝⁶ : CommSemiring R inst✝⁵ : TopologicalSpace R inst✝⁴ : TopologicalSemiring R A : Type u_2 inst✝³ : Semiring A inst✝² : TopologicalSpace A inst✝¹ : TopologicalSemiring A inst✝ : Algebra R A f : A →A[R] A ⊢ ↑f = AlgHom.id R A ↔ f = id R A	no goals	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝⁶ : CommSemiring R inst✝⁵ : TopologicalSpace R inst✝⁴ : TopologicalSemiring R A : Type u_2 inst✝³ : Semiring A inst✝² : TopologicalSpace A inst✝¹ : TopologicalSemiring A inst✝ : Algebra R A f : A →A[R] A ⊢ ↑f = AlgHom.id R A ↔ f = id R A TAC...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	Ideal.pow_eq_bot	[17, 1]	[24, 53]	induction' n with n ih	R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R n : ℕ hn : n ≠ 0 ⊢ I ^ n = ⊥ ↔ I = ⊥	case zero R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R hn : 0 ≠ 0 ⊢ I ^ 0 = ⊥ ↔ I = ⊥ case succ R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R n : ℕ ih : n ≠ 0 → (I ^ n = ⊥ ↔ I = ⊥) hn : n + 1 ≠ 0 ⊢ I ^ (n + 1) = ⊥ ↔ I = ⊥	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R n : ℕ hn : n ≠ 0 ⊢ I ^ n = ⊥ ↔ I = ⊥ TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	Ideal.pow_eq_bot	[17, 1]	[24, 53]	exfalso	case zero R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R hn : 0 ≠ 0 ⊢ I ^ 0 = ⊥ ↔ I = ⊥	case zero R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R hn : 0 ≠ 0 ⊢ False	Please generate a tactic in lean4 to solve the state. STATE: case zero R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R hn : 0 ≠ 0 ⊢ I ^ 0 = ⊥ ↔ I = ⊥ TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	Ideal.pow_eq_bot	[17, 1]	[24, 53]	exact hn (Eq.refl _)	case zero R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R hn : 0 ≠ 0 ⊢ False	no goals	Please generate a tactic in lean4 to solve the state. STATE: case zero R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R hn : 0 ≠ 0 ⊢ False TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	Ideal.pow_eq_bot	[17, 1]	[24, 53]	by_cases hn0 : n = 0	case succ R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R n : ℕ ih : n ≠ 0 → (I ^ n = ⊥ ↔ I = ⊥) hn : n + 1 ≠ 0 ⊢ I ^ (n + 1) = ⊥ ↔ I = ⊥	case pos R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R n : ℕ ih : n ≠ 0 → (I ^ n = ⊥ ↔ I = ⊥) hn : n + 1 ≠ 0 hn0 : n = 0 ⊢ I ^ (n + 1) = ⊥ ↔ I = ⊥ case neg R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R n : ℕ ih : n ≠ 0 → (I ^ n = ⊥ ↔ I = ⊥) hn : n + 1 ≠ 0 hn0 : ¬...	Please generate a tactic in lean4 to solve the state. STATE: case succ R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R n : ℕ ih : n ≠ 0 → (I ^ n = ⊥ ↔ I = ⊥) hn : n + 1 ≠ 0 ⊢ I ^ (n + 1) = ⊥ ↔ I = ⊥ TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	Ideal.pow_eq_bot	[17, 1]	[24, 53]	rw [hn0, pow_one]	case pos R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R n : ℕ ih : n ≠ 0 → (I ^ n = ⊥ ↔ I = ⊥) hn : n + 1 ≠ 0 hn0 : n = 0 ⊢ I ^ (n + 1) = ⊥ ↔ I = ⊥	no goals	Please generate a tactic in lean4 to solve the state. STATE: case pos R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R n : ℕ ih : n ≠ 0 → (I ^ n = ⊥ ↔ I = ⊥) hn : n + 1 ≠ 0 hn0 : n = 0 ⊢ I ^ (n + 1) = ⊥ ↔ I = ⊥ TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	Ideal.pow_eq_bot	[17, 1]	[24, 53]	rw [pow_succ, mul_eq_bot, ih hn0, or_self_iff]	case neg R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R n : ℕ ih : n ≠ 0 → (I ^ n = ⊥ ↔ I = ⊥) hn : n + 1 ≠ 0 hn0 : ¬n = 0 ⊢ I ^ (n + 1) = ⊥ ↔ I = ⊥	no goals	Please generate a tactic in lean4 to solve the state. STATE: case neg R : Type u_1 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I : Ideal R n : ℕ ih : n ≠ 0 → (I ^ n = ⊥ ↔ I = ⊥) hn : n + 1 ≠ 0 hn0 : ¬n = 0 ⊢ I ^ (n + 1) = ⊥ ↔ I = ⊥ TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	MvPolynomial.eval₂Hom.smul	[32, 1]	[34, 86]	simp only [smul_eq_C_mul, coe_eval₂Hom, eval₂_mul, eval₂_C, Algebra.id.smul_eq_mul]	R : Type u_2 S : Type u_1 σ : Type u_3 inst✝¹ : CommSemiring R inst✝ : CommSemiring S f : R →+* S g : σ → S r : R P : MvPolynomial σ R ⊢ (eval₂Hom f g) (r • P) = f r • (eval₂Hom f g) P	no goals	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 σ : Type u_3 inst✝¹ : CommSemiring R inst✝ : CommSemiring S f : R →+* S g : σ → S r : R P : MvPolynomial σ R ⊢ (eval₂Hom f g) (r • P) = f r • (eval₂Hom f g) P TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	quotient_mk_eq_ofRel	[69, 1]	[75, 89]	suffices hinj : Function.Injective (RingQuot.ringQuotEquivIdealQuotient r).invFun by apply hinj; exact mkRingHom_rel h	A : Type u_1 inst✝ : CommRing A r : A → A → Prop a b : A h : r a b ⊢ (Ideal.Quotient.mk (ofRel r)) a = (Ideal.Quotient.mk (ofRel r)) b	A : Type u_1 inst✝ : CommRing A r : A → A → Prop a b : A h : r a b ⊢ Function.Injective (ringQuotEquivIdealQuotient r).invFun	Please generate a tactic in lean4 to solve the state. STATE: A : Type u_1 inst✝ : CommRing A r : A → A → Prop a b : A h : r a b ⊢ (Ideal.Quotient.mk (ofRel r)) a = (Ideal.Quotient.mk (ofRel r)) b TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	quotient_mk_eq_ofRel	[69, 1]	[75, 89]	rw [Function.injective_iff_hasLeftInverse]	A : Type u_1 inst✝ : CommRing A r : A → A → Prop a b : A h : r a b ⊢ Function.Injective (ringQuotEquivIdealQuotient r).invFun	A : Type u_1 inst✝ : CommRing A r : A → A → Prop a b : A h : r a b ⊢ Function.HasLeftInverse (ringQuotEquivIdealQuotient r).invFun	Please generate a tactic in lean4 to solve the state. STATE: A : Type u_1 inst✝ : CommRing A r : A → A → Prop a b : A h : r a b ⊢ Function.Injective (ringQuotEquivIdealQuotient r).invFun TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	quotient_mk_eq_ofRel	[69, 1]	[75, 89]	exact ⟨(ringQuotEquivIdealQuotient r).toFun, (ringQuotEquivIdealQuotient r).right_inv⟩	A : Type u_1 inst✝ : CommRing A r : A → A → Prop a b : A h : r a b ⊢ Function.HasLeftInverse (ringQuotEquivIdealQuotient r).invFun	no goals	Please generate a tactic in lean4 to solve the state. STATE: A : Type u_1 inst✝ : CommRing A r : A → A → Prop a b : A h : r a b ⊢ Function.HasLeftInverse (ringQuotEquivIdealQuotient r).invFun TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	quotient_mk_eq_ofRel	[69, 1]	[75, 89]	apply hinj	A : Type u_1 inst✝ : CommRing A r : A → A → Prop a b : A h : r a b hinj : Function.Injective (ringQuotEquivIdealQuotient r).invFun ⊢ (Ideal.Quotient.mk (ofRel r)) a = (Ideal.Quotient.mk (ofRel r)) b	case a A : Type u_1 inst✝ : CommRing A r : A → A → Prop a b : A h : r a b hinj : Function.Injective (ringQuotEquivIdealQuotient r).invFun ⊢ (ringQuotEquivIdealQuotient r).invFun ((Ideal.Quotient.mk (ofRel r)) a) = (ringQuotEquivIdealQuotient r).invFun ((Ideal.Quotient.mk (ofRel r)) b)	Please generate a tactic in lean4 to solve the state. STATE: A : Type u_1 inst✝ : CommRing A r : A → A → Prop a b : A h : r a b hinj : Function.Injective (ringQuotEquivIdealQuotient r).invFun ⊢ (Ideal.Quotient.mk (ofRel r)) a = (Ideal.Quotient.mk (ofRel r)) b TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	quotient_mk_eq_ofRel	[69, 1]	[75, 89]	exact mkRingHom_rel h	case a A : Type u_1 inst✝ : CommRing A r : A → A → Prop a b : A h : r a b hinj : Function.Injective (ringQuotEquivIdealQuotient r).invFun ⊢ (ringQuotEquivIdealQuotient r).invFun ((Ideal.Quotient.mk (ofRel r)) a) = (ringQuotEquivIdealQuotient r).invFun ((Ideal.Quotient.mk (ofRel r)) b)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a A : Type u_1 inst✝ : CommRing A r : A → A → Prop a b : A h : r a b hinj : Function.Injective (ringQuotEquivIdealQuotient r).invFun ⊢ (ringQuotEquivIdealQuotient r).invFun ((Ideal.Quotient.mk (ofRel r)) a) = (ringQuotEquivIdealQuotient r).invFun ((I...
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	Ideal.quotient_mk_eq_ringQuot_apply	[80, 1]	[84, 6]	rw [← ringQuotToIdealQuotient_apply r a, ← mkAlgHom_coe R r]	R : Type u_1 inst✝² : CommRing R A : Type u_2 inst✝¹ : CommRing A inst✝ : Algebra R A r : A → A → Prop a : A ⊢ (Quotient.mk (ofRel r)) a = (ringQuotToIdealQuotient r) ((mkAlgHom R r) a)	R : Type u_1 inst✝² : CommRing R A : Type u_2 inst✝¹ : CommRing A inst✝ : Algebra R A r : A → A → Prop a : A ⊢ (ringQuotToIdealQuotient r) (↑(mkAlgHom R r) a) = (ringQuotToIdealQuotient r) ((mkAlgHom R r) a)	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝² : CommRing R A : Type u_2 inst✝¹ : CommRing A inst✝ : Algebra R A r : A → A → Prop a : A ⊢ (Quotient.mk (ofRel r)) a = (ringQuotToIdealQuotient r) ((mkAlgHom R r) a) TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	Ideal.quotient_mk_eq_ringQuot_apply	[80, 1]	[84, 6]	rfl	R : Type u_1 inst✝² : CommRing R A : Type u_2 inst✝¹ : CommRing A inst✝ : Algebra R A r : A → A → Prop a : A ⊢ (ringQuotToIdealQuotient r) (↑(mkAlgHom R r) a) = (ringQuotToIdealQuotient r) ((mkAlgHom R r) a)	no goals	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝² : CommRing R A : Type u_2 inst✝¹ : CommRing A inst✝ : Algebra R A r : A → A → Prop a : A ⊢ (ringQuotToIdealQuotient r) (↑(mkAlgHom R r) a) = (ringQuotToIdealQuotient r) ((mkAlgHom R r) a) TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	Ideal.Quotient.rel_le_ker	[91, 1]	[96, 96]	rw [hr, ofRel, Submodule.span_le]	R : Type u_1 S : Type u_2 inst✝¹ : CommRing R inst✝ : CommRing S I : Ideal R r : R → R → Prop hr : I = ofRel r f : R →+* S hf : ∀ {a b : R}, r a b → f a = f b ⊢ I ≤ RingHom.ker f	R : Type u_1 S : Type u_2 inst✝¹ : CommRing R inst✝ : CommRing S I : Ideal R r : R → R → Prop hr : I = ofRel r f : R →+* S hf : ∀ {a b : R}, r a b → f a = f b ⊢ {x \| ∃ a b, r a b ∧ x + b = a} ⊆ ↑(RingHom.ker f)	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝¹ : CommRing R inst✝ : CommRing S I : Ideal R r : R → R → Prop hr : I = ofRel r f : R →+* S hf : ∀ {a b : R}, r a b → f a = f b ⊢ I ≤ RingHom.ker f TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	Ideal.Quotient.rel_le_ker	[91, 1]	[96, 96]	rintro x ⟨a, b, hx, hab⟩	R : Type u_1 S : Type u_2 inst✝¹ : CommRing R inst✝ : CommRing S I : Ideal R r : R → R → Prop hr : I = ofRel r f : R →+* S hf : ∀ {a b : R}, r a b → f a = f b ⊢ {x \| ∃ a b, r a b ∧ x + b = a} ⊆ ↑(RingHom.ker f)	case intro.intro.intro R : Type u_1 S : Type u_2 inst✝¹ : CommRing R inst✝ : CommRing S I : Ideal R r : R → R → Prop hr : I = ofRel r f : R →+* S hf : ∀ {a b : R}, r a b → f a = f b x a b : R hx : r a b hab : x + b = a ⊢ x ∈ ↑(RingHom.ker f)	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝¹ : CommRing R inst✝ : CommRing S I : Ideal R r : R → R → Prop hr : I = ofRel r f : R →+* S hf : ∀ {a b : R}, r a b → f a = f b ⊢ {x \| ∃ a b, r a b ∧ x + b = a} ⊆ ↑(RingHom.ker f) TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	Ideal.Quotient.rel_le_ker	[91, 1]	[96, 96]	rw [eq_sub_iff_add_eq.mpr hab, SetLike.mem_coe, RingHom.mem_ker, map_sub, sub_eq_zero, hf hx]	case intro.intro.intro R : Type u_1 S : Type u_2 inst✝¹ : CommRing R inst✝ : CommRing S I : Ideal R r : R → R → Prop hr : I = ofRel r f : R →+* S hf : ∀ {a b : R}, r a b → f a = f b x a b : R hx : r a b hab : x + b = a ⊢ x ∈ ↑(RingHom.ker f)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro R : Type u_1 S : Type u_2 inst✝¹ : CommRing R inst✝ : CommRing S I : Ideal R r : R → R → Prop hr : I = ofRel r f : R →+* S hf : ∀ {a b : R}, r a b → f a = f b x a b : R hx : r a b hab : x + b = a ⊢ x ∈ ↑(RingHom.ker f) TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	TrivSqZeroExt.mem_kerIdeal_iff_inr	[122, 1]	[126, 83]	obtain ⟨r, m⟩ := x	R : Type u_1 M : Type u_2 inst✝⁴ : CommSemiring R inst✝³ : AddCommMonoid M inst✝² : Module R M inst✝¹ : Module Rᵐᵒᵖ M inst✝ : IsCentralScalar R M x : TrivSqZeroExt R M ⊢ x ∈ kerIdeal R M ↔ x = inr x.snd	case mk R : Type u_1 M : Type u_2 inst✝⁴ : CommSemiring R inst✝³ : AddCommMonoid M inst✝² : Module R M inst✝¹ : Module Rᵐᵒᵖ M inst✝ : IsCentralScalar R M r : R m : M ⊢ (r, m) ∈ kerIdeal R M ↔ (r, m) = inr (snd (r, m))	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 M : Type u_2 inst✝⁴ : CommSemiring R inst✝³ : AddCommMonoid M inst✝² : Module R M inst✝¹ : Module Rᵐᵒᵖ M inst✝ : IsCentralScalar R M x : TrivSqZeroExt R M ⊢ x ∈ kerIdeal R M ↔ x = inr x.snd TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	TrivSqZeroExt.mem_kerIdeal_iff_inr	[122, 1]	[126, 83]	simp only [kerIdeal, RingHom.mem_ker, fstHom_apply, fst_mk]	case mk R : Type u_1 M : Type u_2 inst✝⁴ : CommSemiring R inst✝³ : AddCommMonoid M inst✝² : Module R M inst✝¹ : Module Rᵐᵒᵖ M inst✝ : IsCentralScalar R M r : R m : M ⊢ (r, m) ∈ kerIdeal R M ↔ (r, m) = inr (snd (r, m))	case mk R : Type u_1 M : Type u_2 inst✝⁴ : CommSemiring R inst✝³ : AddCommMonoid M inst✝² : Module R M inst✝¹ : Module Rᵐᵒᵖ M inst✝ : IsCentralScalar R M r : R m : M ⊢ r = 0 ↔ (r, m) = inr (snd (r, m))	Please generate a tactic in lean4 to solve the state. STATE: case mk R : Type u_1 M : Type u_2 inst✝⁴ : CommSemiring R inst✝³ : AddCommMonoid M inst✝² : Module R M inst✝¹ : Module Rᵐᵒᵖ M inst✝ : IsCentralScalar R M r : R m : M ⊢ (r, m) ∈ kerIdeal R M ↔ (r, m) = inr (snd (r, m)) TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	TrivSqZeroExt.mem_kerIdeal_iff_inr	[122, 1]	[126, 83]	exact ⟨fun hr => by rw [hr]; rfl, fun hrm => by rw [← fst_mk r m, hrm, fst_inr]⟩	case mk R : Type u_1 M : Type u_2 inst✝⁴ : CommSemiring R inst✝³ : AddCommMonoid M inst✝² : Module R M inst✝¹ : Module Rᵐᵒᵖ M inst✝ : IsCentralScalar R M r : R m : M ⊢ r = 0 ↔ (r, m) = inr (snd (r, m))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case mk R : Type u_1 M : Type u_2 inst✝⁴ : CommSemiring R inst✝³ : AddCommMonoid M inst✝² : Module R M inst✝¹ : Module Rᵐᵒᵖ M inst✝ : IsCentralScalar R M r : R m : M ⊢ r = 0 ↔ (r, m) = inr (snd (r, m)) TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	TrivSqZeroExt.mem_kerIdeal_iff_inr	[122, 1]	[126, 83]	rw [hr]	R : Type u_1 M : Type u_2 inst✝⁴ : CommSemiring R inst✝³ : AddCommMonoid M inst✝² : Module R M inst✝¹ : Module Rᵐᵒᵖ M inst✝ : IsCentralScalar R M r : R m : M hr : r = 0 ⊢ (r, m) = inr (snd (r, m))	R : Type u_1 M : Type u_2 inst✝⁴ : CommSemiring R inst✝³ : AddCommMonoid M inst✝² : Module R M inst✝¹ : Module Rᵐᵒᵖ M inst✝ : IsCentralScalar R M r : R m : M hr : r = 0 ⊢ (0, m) = inr (snd (0, m))	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 M : Type u_2 inst✝⁴ : CommSemiring R inst✝³ : AddCommMonoid M inst✝² : Module R M inst✝¹ : Module Rᵐᵒᵖ M inst✝ : IsCentralScalar R M r : R m : M hr : r = 0 ⊢ (r, m) = inr (snd (r, m)) TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	TrivSqZeroExt.mem_kerIdeal_iff_inr	[122, 1]	[126, 83]	rfl	R : Type u_1 M : Type u_2 inst✝⁴ : CommSemiring R inst✝³ : AddCommMonoid M inst✝² : Module R M inst✝¹ : Module Rᵐᵒᵖ M inst✝ : IsCentralScalar R M r : R m : M hr : r = 0 ⊢ (0, m) = inr (snd (0, m))	no goals	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 M : Type u_2 inst✝⁴ : CommSemiring R inst✝³ : AddCommMonoid M inst✝² : Module R M inst✝¹ : Module Rᵐᵒᵖ M inst✝ : IsCentralScalar R M r : R m : M hr : r = 0 ⊢ (0, m) = inr (snd (0, m)) TACTIC:
https://github.com/AntoineChambert-Loir/DividedPowers4.git	18a13603ed0158d2880b6b0b0369d78417040a1d	DividedPowers/DPAlgebra/Misc.lean	TrivSqZeroExt.mem_kerIdeal_iff_inr	[122, 1]	[126, 83]	rw [← fst_mk r m, hrm, fst_inr]	R : Type u_1 M : Type u_2 inst✝⁴ : CommSemiring R inst✝³ : AddCommMonoid M inst✝² : Module R M inst✝¹ : Module Rᵐᵒᵖ M inst✝ : IsCentralScalar R M r : R m : M hrm : (r, m) = inr (snd (r, m)) ⊢ r = 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 M : Type u_2 inst✝⁴ : CommSemiring R inst✝³ : AddCommMonoid M inst✝² : Module R M inst✝¹ : Module Rᵐᵒᵖ M inst✝ : IsCentralScalar R M r : R m : M hrm : (r, m) = inr (snd (r, m)) ⊢ r = 0 TACTIC:

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