file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/LinearAlgebra/AffineSpace/Ordered.lean
|
map_lt_lineMap_iff_slope_lt_slope
|
[] |
[
296,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/Data/Finset/Powerset.lean
|
Finset.powersetLen_zero
|
[
{
"state_after": "case a\nα : Type u_1\ns✝ t s a✝ : Finset α\n⊢ a✝ ∈ powersetLen 0 s ↔ a✝ ∈ {∅}",
"state_before": "α : Type u_1\ns✝ t s : Finset α\n⊢ powersetLen 0 s = {∅}",
"tactic": "ext"
},
{
"state_after": "case a\nα : Type u_1\ns✝ t s a✝ : Finset α\n⊢ a✝ ⊆ s ∧ a✝ = ∅ ↔ a✝ = ∅",
"state_before": "case a\nα : Type u_1\ns✝ t s a✝ : Finset α\n⊢ a✝ ∈ powersetLen 0 s ↔ a✝ ∈ {∅}",
"tactic": "rw [mem_powersetLen, mem_singleton, card_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\ns✝ t s a✝ : Finset α\n⊢ a✝ ⊆ s ∧ a✝ = ∅ ↔ a✝ = ∅",
"tactic": "refine'\n ⟨fun h => h.2, fun h => by\n rw [h]\n exact ⟨empty_subset s, rfl⟩⟩"
},
{
"state_after": "α : Type u_1\ns✝ t s a✝ : Finset α\nh : a✝ = ∅\n⊢ ∅ ⊆ s ∧ ∅ = ∅",
"state_before": "α : Type u_1\ns✝ t s a✝ : Finset α\nh : a✝ = ∅\n⊢ a✝ ⊆ s ∧ a✝ = ∅",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns✝ t s a✝ : Finset α\nh : a✝ = ∅\n⊢ ∅ ⊆ s ∧ ∅ = ∅",
"tactic": "exact ⟨empty_subset s, rfl⟩"
}
] |
[
227,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.monomial_eq_monomial_iff
|
[] |
[
387,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.ne_zero_of_out_nonempty
|
[] |
[
289,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.coeff_monomial
|
[
{
"state_after": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : DecidableEq σ\nm n : σ →₀ ℕ\na : R\n⊢ ↑(LinearMap.stdBasis R (fun x => R) n) a m = if m = n then a else 0",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : DecidableEq σ\nm n : σ →₀ ℕ\na : R\n⊢ ↑(coeff R m) (↑(monomial R n) a) = if m = n then a else 0",
"tactic": "rw [coeff, monomial_def, LinearMap.proj_apply]"
},
{
"state_after": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : DecidableEq σ\nm n : σ →₀ ℕ\na : R\n⊢ ↑(LinearMap.stdBasis R (fun x => R) n) a m = if m = n then a else 0",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : DecidableEq σ\nm n : σ →₀ ℕ\na : R\n⊢ ↑(LinearMap.stdBasis R (fun x => R) n) a m = if m = n then a else 0",
"tactic": "dsimp only"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : DecidableEq σ\nm n : σ →₀ ℕ\na : R\n⊢ ↑(LinearMap.stdBasis R (fun x => R) n) a m = if m = n then a else 0",
"tactic": "rw [LinearMap.stdBasis_apply, Function.update_apply, Pi.zero_apply]"
}
] |
[
157,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Algebra.coe_top
|
[] |
[
782,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
782,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.scalar_apply_eq
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.432380\nm : Type ?u.432383\nn : Type u_1\no : Type ?u.432389\nm' : o → Type ?u.432394\nn' : o → Type ?u.432399\nR : Type ?u.432402\nS : Type ?u.432405\nα : Type v\nβ : Type w\nγ : Type ?u.432412\ninst✝² : Semiring α\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\na : α\ni : n\n⊢ ↑(scalar n) a i i = a",
"tactic": "simp only [coe_scalar, Matrix.smul_apply, one_apply_eq, smul_eq_mul, mul_one]"
}
] |
[
1255,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1253,
1
] |
Std/Logic.lean
|
and_iff_left
|
[] |
[
204,
77
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
204,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.lf_iff_exists_le
|
[
{
"state_after": "x y : PGame\n⊢ ((¬∀ (i : LeftMoves y), moveLeft y i ⧏ x) ∨ ¬∀ (j : RightMoves x), y ⧏ moveRight x j) ↔\n (∃ i, x ≤ moveLeft y i) ∨ ∃ j, moveRight x j ≤ y",
"state_before": "x y : PGame\n⊢ x ⧏ y ↔ (∃ i, x ≤ moveLeft y i) ∨ ∃ j, moveRight x j ≤ y",
"tactic": "rw [Lf, le_iff_forall_lf, not_and_or]"
},
{
"state_after": "no goals",
"state_before": "x y : PGame\n⊢ ((¬∀ (i : LeftMoves y), moveLeft y i ⧏ x) ∨ ¬∀ (j : RightMoves x), y ⧏ moveRight x j) ↔\n (∃ i, x ≤ moveLeft y i) ∨ ∃ j, moveRight x j ≤ y",
"tactic": "simp"
}
] |
[
452,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/Data/Polynomial/Coeff.lean
|
Polynomial.support_smul
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝² : Semiring R\np✝ q r✝ : R[X]\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nr : S\np : R[X]\ni : ℕ\nhi : i ∈ support (r • p)\n⊢ i ∈ support p",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝² : Semiring R\np✝ q r✝ : R[X]\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nr : S\np : R[X]\n⊢ support (r • p) ⊆ support p",
"tactic": "intro i hi"
},
{
"state_after": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝² : Semiring R\np✝ q r✝ : R[X]\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nr : S\np : R[X]\ni : ℕ\nhi : ¬r • coeff p i = 0\n⊢ ¬coeff p i = 0",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝² : Semiring R\np✝ q r✝ : R[X]\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nr : S\np : R[X]\ni : ℕ\nhi : i ∈ support (r • p)\n⊢ i ∈ support p",
"tactic": "simp [mem_support_iff] at hi⊢"
},
{
"state_after": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝² : Semiring R\np✝ q r✝ : R[X]\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nr : S\np : R[X]\ni : ℕ\nhi : coeff p i = 0\n⊢ r • coeff p i = 0",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝² : Semiring R\np✝ q r✝ : R[X]\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nr : S\np : R[X]\ni : ℕ\nhi : ¬r • coeff p i = 0\n⊢ ¬coeff p i = 0",
"tactic": "contrapose! hi"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝² : Semiring R\np✝ q r✝ : R[X]\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nr : S\np : R[X]\ni : ℕ\nhi : coeff p i = 0\n⊢ r • coeff p i = 0",
"tactic": "simp [hi]"
}
] |
[
72,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Lp.cauchy_tendsto_of_tendsto
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\n⊢ ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - f_lim) p μ ≤ ε",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\n⊢ Tendsto (fun n => snorm (f n - f_lim) p μ) atTop (𝓝 0)",
"tactic": "rw [ENNReal.tendsto_atTop_zero]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - f_lim) p μ ≤ ε",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\n⊢ ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - f_lim) p μ ≤ ε",
"tactic": "intro ε hε"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nh_B : ∃ N, B N ≤ ε\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - f_lim) p μ ≤ ε",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - f_lim) p μ ≤ ε",
"tactic": "have h_B : ∃ N : ℕ, B N ≤ ε := by\n suffices h_tendsto_zero : ∃ N : ℕ, ∀ n : ℕ, N ≤ n → B n ≤ ε\n exact ⟨h_tendsto_zero.choose, h_tendsto_zero.choose_spec _ le_rfl⟩\n exact (ENNReal.tendsto_atTop_zero.mp (ENNReal.tendsto_atTop_zero_of_tsum_ne_top hB)) ε hε"
},
{
"state_after": "case intro\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - f_lim) p μ ≤ ε",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nh_B : ∃ N, B N ≤ ε\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - f_lim) p μ ≤ ε",
"tactic": "cases' h_B with N h_B"
},
{
"state_after": "case intro\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\n⊢ snorm (f n - f_lim) p μ ≤ ε",
"state_before": "case intro\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → snorm (f n - f_lim) p μ ≤ ε",
"tactic": "refine' ⟨N, fun n hn => _⟩"
},
{
"state_after": "case intro\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\nh_sub : snorm (f n - f_lim) p μ ≤ liminf (fun m => snorm (f n - f m) p μ) atTop\n⊢ snorm (f n - f_lim) p μ ≤ ε",
"state_before": "case intro\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\n⊢ snorm (f n - f_lim) p μ ≤ ε",
"tactic": "have h_sub : snorm (f n - f_lim) p μ ≤ atTop.liminf fun m => snorm (f n - f m) p μ := by\n refine' snorm_lim_le_liminf_snorm (fun m => (hf n).sub (hf m)) (f n - f_lim) _\n refine' h_lim.mono fun x hx => _\n simp_rw [sub_eq_add_neg]\n exact Tendsto.add tendsto_const_nhds (Tendsto.neg hx)"
},
{
"state_after": "case intro\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\nh_sub : snorm (f n - f_lim) p μ ≤ liminf (fun m => snorm (f n - f m) p μ) atTop\n⊢ liminf (fun m => snorm (f n - f m) p μ) atTop ≤ ε",
"state_before": "case intro\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\nh_sub : snorm (f n - f_lim) p μ ≤ liminf (fun m => snorm (f n - f m) p μ) atTop\n⊢ snorm (f n - f_lim) p μ ≤ ε",
"tactic": "refine' h_sub.trans _"
},
{
"state_after": "case intro\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\nh_sub : snorm (f n - f_lim) p μ ≤ liminf (fun m => snorm (f n - f m) p μ) atTop\n⊢ ∀ (a : ℕ), ∃ b, b ≥ a ∧ snorm (f n - f b) p μ ≤ ε",
"state_before": "case intro\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\nh_sub : snorm (f n - f_lim) p μ ≤ liminf (fun m => snorm (f n - f m) p μ) atTop\n⊢ liminf (fun m => snorm (f n - f m) p μ) atTop ≤ ε",
"tactic": "refine' liminf_le_of_frequently_le' (frequently_atTop.mpr _)"
},
{
"state_after": "case intro\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\nh_sub : snorm (f n - f_lim) p μ ≤ liminf (fun m => snorm (f n - f m) p μ) atTop\nN1 : ℕ\n⊢ snorm (f n - f (max N N1)) p μ ≤ ε",
"state_before": "case intro\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\nh_sub : snorm (f n - f_lim) p μ ≤ liminf (fun m => snorm (f n - f m) p μ) atTop\n⊢ ∀ (a : ℕ), ∃ b, b ≥ a ∧ snorm (f n - f b) p μ ≤ ε",
"tactic": "refine' fun N1 => ⟨max N N1, le_max_right _ _, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\nh_sub : snorm (f n - f_lim) p μ ≤ liminf (fun m => snorm (f n - f m) p μ) atTop\nN1 : ℕ\n⊢ snorm (f n - f (max N N1)) p μ ≤ ε",
"tactic": "exact (h_cau N n (max N N1) hn (le_max_left _ _)).le.trans h_B"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nh_tendsto_zero : ∃ N, ∀ (n : ℕ), N ≤ n → B n ≤ ε\n⊢ ∃ N, B N ≤ ε\n\ncase h_tendsto_zero\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\n⊢ ∃ N, ∀ (n : ℕ), N ≤ n → B n ≤ ε",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\n⊢ ∃ N, B N ≤ ε",
"tactic": "suffices h_tendsto_zero : ∃ N : ℕ, ∀ n : ℕ, N ≤ n → B n ≤ ε"
},
{
"state_after": "case h_tendsto_zero\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\n⊢ ∃ N, ∀ (n : ℕ), N ≤ n → B n ≤ ε",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nh_tendsto_zero : ∃ N, ∀ (n : ℕ), N ≤ n → B n ≤ ε\n⊢ ∃ N, B N ≤ ε\n\ncase h_tendsto_zero\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\n⊢ ∃ N, ∀ (n : ℕ), N ≤ n → B n ≤ ε",
"tactic": "exact ⟨h_tendsto_zero.choose, h_tendsto_zero.choose_spec _ le_rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h_tendsto_zero\nα : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\n⊢ ∃ N, ∀ (n : ℕ), N ≤ n → B n ≤ ε",
"tactic": "exact (ENNReal.tendsto_atTop_zero.mp (ENNReal.tendsto_atTop_zero_of_tsum_ne_top hB)) ε hε"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\n⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n_1 => (f n - f n_1) x) atTop (𝓝 ((f n - f_lim) x))",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\n⊢ snorm (f n - f_lim) p μ ≤ liminf (fun m => snorm (f n - f m) p μ) atTop",
"tactic": "refine' snorm_lim_le_liminf_snorm (fun m => (hf n).sub (hf m)) (f n - f_lim) _"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\nx : α\nhx : Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\n⊢ Tendsto (fun n_1 => (f n - f n_1) x) atTop (𝓝 ((f n - f_lim) x))",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\n⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n_1 => (f n - f n_1) x) atTop (𝓝 ((f n - f_lim) x))",
"tactic": "refine' h_lim.mono fun x hx => _"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\nx : α\nhx : Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\n⊢ Tendsto (fun n_1 => (f n + -f n_1) x) atTop (𝓝 ((f n + -f_lim) x))",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\nx : α\nhx : Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\n⊢ Tendsto (fun n_1 => (f n - f n_1) x) atTop (𝓝 ((f n - f_lim) x))",
"tactic": "simp_rw [sub_eq_add_neg]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.8211817\nG : Type ?u.8211820\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : (∑' (i : ℕ), B i) ≠ ⊤\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\nε : ℝ≥0∞\nhε : ε > 0\nN : ℕ\nh_B : B N ≤ ε\nn : ℕ\nhn : n ≥ N\nx : α\nhx : Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))\n⊢ Tendsto (fun n_1 => (f n + -f n_1) x) atTop (𝓝 ((f n + -f_lim) x))",
"tactic": "exact Tendsto.add tendsto_const_nhds (Tendsto.neg hx)"
}
] |
[
1510,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1489,
1
] |
Mathlib/Algebra/Order/LatticeGroup.lean
|
LatticeOrderedCommGroup.sup_div_inf_eq_abs_div
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b : α\n⊢ (a ⊔ b) / (a ⊓ b) = abs (b / a)",
"tactic": "rw [sup_eq_mul_pos_div, inf_comm, inf_eq_div_pos_div, div_eq_mul_inv, div_eq_mul_inv b ((b / a)⁺),\n mul_inv_rev, inv_inv, mul_comm, ← mul_assoc, inv_mul_cancel_right, pos_eq_neg_inv (a / b),\n div_eq_mul_inv a b, mul_inv_rev, ← div_eq_mul_inv, inv_inv, ← pos_mul_neg]"
}
] |
[
392,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
388,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
mul_div_mul_left_eq_div
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.65645\nβ : Type ?u.65648\nG : Type u_1\ninst✝ : CommGroup G\na✝ b✝ c✝ d a b c : G\n⊢ c * a / (c * b) = a / b",
"tactic": "rw [div_eq_mul_inv, mul_inv_rev, mul_comm b⁻¹ c⁻¹, mul_comm c a, mul_assoc, ←mul_assoc c,\n mul_right_inv, one_mul, div_eq_mul_inv]"
}
] |
[
910,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
908,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.coprime_of_dvd_coprime
|
[
{
"state_after": "case nonzero\nd : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\n⊢ ¬(b.re = 0 ∧ b.im = 0)\n\ncase H\nd : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\n⊢ ∀ (z : ℤ), z ∈ nonunits ℤ → z ≠ 0 → z ∣ b.re → ¬z ∣ b.im",
"state_before": "d : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\n⊢ IsCoprime b.re b.im",
"tactic": "apply isCoprime_of_dvd"
},
{
"state_after": "case nonzero.intro\nd : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nhre : b.re = 0\nhim : b.im = 0\n⊢ False",
"state_before": "case nonzero\nd : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\n⊢ ¬(b.re = 0 ∧ b.im = 0)",
"tactic": "rintro ⟨hre, him⟩"
},
{
"state_after": "case nonzero.intro\nd : ℤ\na : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : 0 ∣ a\nhre : 0.re = 0\nhim : 0.im = 0\n⊢ False",
"state_before": "case nonzero.intro\nd : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nhre : b.re = 0\nhim : b.im = 0\n⊢ False",
"tactic": "obtain rfl : b = 0 := by\n simp only [ext, hre, eq_self_iff_true, zero_im, him, and_self_iff, zero_re]"
},
{
"state_after": "case nonzero.intro\nd : ℤ\na : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : a = 0\nhre : 0.re = 0\nhim : 0.im = 0\n⊢ False",
"state_before": "case nonzero.intro\nd : ℤ\na : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : 0 ∣ a\nhre : 0.re = 0\nhim : 0.im = 0\n⊢ False",
"tactic": "rw [zero_dvd_iff] at hdvd"
},
{
"state_after": "no goals",
"state_before": "case nonzero.intro\nd : ℤ\na : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : a = 0\nhre : 0.re = 0\nhim : 0.im = 0\n⊢ False",
"tactic": "simp [hdvd, zero_im, zero_re, not_isCoprime_zero_zero] at hcoprime"
},
{
"state_after": "no goals",
"state_before": "d : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nhre : b.re = 0\nhim : b.im = 0\n⊢ b = 0",
"tactic": "simp only [ext, hre, eq_self_iff_true, zero_im, him, and_self_iff, zero_re]"
},
{
"state_after": "case H\nd : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nz : ℤ\nhz : z ∈ nonunits ℤ\nhzdvdu : z ∣ b.re\nhzdvdv : z ∣ b.im\n⊢ False",
"state_before": "case H\nd : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\n⊢ ∀ (z : ℤ), z ∈ nonunits ℤ → z ≠ 0 → z ∣ b.re → ¬z ∣ b.im",
"tactic": "rintro z hz - hzdvdu hzdvdv"
},
{
"state_after": "case H\nd : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nz : ℤ\nhz : z ∈ nonunits ℤ\nhzdvdu : z ∣ b.re\nhzdvdv : z ∣ b.im\n⊢ IsUnit z",
"state_before": "case H\nd : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nz : ℤ\nhz : z ∈ nonunits ℤ\nhzdvdu : z ∣ b.re\nhzdvdv : z ∣ b.im\n⊢ False",
"tactic": "apply hz"
},
{
"state_after": "case H.intro\nd : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nz : ℤ\nhz : z ∈ nonunits ℤ\nhzdvdu : z ∣ b.re\nhzdvdv : z ∣ b.im\nha : z ∣ a.re\nhb : z ∣ a.im\n⊢ IsUnit z",
"state_before": "case H\nd : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nz : ℤ\nhz : z ∈ nonunits ℤ\nhzdvdu : z ∣ b.re\nhzdvdv : z ∣ b.im\n⊢ IsUnit z",
"tactic": "obtain ⟨ha, hb⟩ : z ∣ a.re ∧ z ∣ a.im := by\n rw [← coe_int_dvd_iff]\n apply dvd_trans _ hdvd\n rw [coe_int_dvd_iff]\n exact ⟨hzdvdu, hzdvdv⟩"
},
{
"state_after": "no goals",
"state_before": "case H.intro\nd : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nz : ℤ\nhz : z ∈ nonunits ℤ\nhzdvdu : z ∣ b.re\nhzdvdv : z ∣ b.im\nha : z ∣ a.re\nhb : z ∣ a.im\n⊢ IsUnit z",
"tactic": "exact hcoprime.isUnit_of_dvd' ha hb"
},
{
"state_after": "d : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nz : ℤ\nhz : z ∈ nonunits ℤ\nhzdvdu : z ∣ b.re\nhzdvdv : z ∣ b.im\n⊢ ↑z ∣ a",
"state_before": "d : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nz : ℤ\nhz : z ∈ nonunits ℤ\nhzdvdu : z ∣ b.re\nhzdvdv : z ∣ b.im\n⊢ z ∣ a.re ∧ z ∣ a.im",
"tactic": "rw [← coe_int_dvd_iff]"
},
{
"state_after": "d : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nz : ℤ\nhz : z ∈ nonunits ℤ\nhzdvdu : z ∣ b.re\nhzdvdv : z ∣ b.im\n⊢ ↑z ∣ b",
"state_before": "d : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nz : ℤ\nhz : z ∈ nonunits ℤ\nhzdvdu : z ∣ b.re\nhzdvdv : z ∣ b.im\n⊢ ↑z ∣ a",
"tactic": "apply dvd_trans _ hdvd"
},
{
"state_after": "d : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nz : ℤ\nhz : z ∈ nonunits ℤ\nhzdvdu : z ∣ b.re\nhzdvdv : z ∣ b.im\n⊢ z ∣ b.re ∧ z ∣ b.im",
"state_before": "d : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nz : ℤ\nhz : z ∈ nonunits ℤ\nhzdvdu : z ∣ b.re\nhzdvdv : z ∣ b.im\n⊢ ↑z ∣ b",
"tactic": "rw [coe_int_dvd_iff]"
},
{
"state_after": "no goals",
"state_before": "d : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nz : ℤ\nhz : z ∈ nonunits ℤ\nhzdvdu : z ∣ b.re\nhzdvdv : z ∣ b.im\n⊢ z ∣ b.re ∧ z ∣ b.im",
"tactic": "exact ⟨hzdvdu, hzdvdv⟩"
}
] |
[
412,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
397,
1
] |
Mathlib/Analysis/Asymptotics/Theta.lean
|
Asymptotics.isTheta_of_norm_eventuallyEq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.23422\nE : Type u_2\nF : Type u_3\nG : Type ?u.23431\nE' : Type ?u.23434\nF' : Type ?u.23437\nG' : Type ?u.23440\nE'' : Type ?u.23443\nF'' : Type ?u.23446\nG'' : Type ?u.23449\nR : Type ?u.23452\nR' : Type ?u.23455\n𝕜 : Type ?u.23458\n𝕜' : Type ?u.23461\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nl l' : Filter α\nh : (fun x => ‖f x‖) =ᶠ[l] fun x => ‖g x‖\n⊢ ∀ᶠ (x : α) in l, ‖f x‖ ≤ 1 * ‖g x‖",
"tactic": "simpa only [one_mul] using h.le"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.23422\nE : Type u_2\nF : Type u_3\nG : Type ?u.23431\nE' : Type ?u.23434\nF' : Type ?u.23437\nG' : Type ?u.23440\nE'' : Type ?u.23443\nF'' : Type ?u.23446\nG'' : Type ?u.23449\nR : Type ?u.23452\nR' : Type ?u.23455\n𝕜 : Type ?u.23458\n𝕜' : Type ?u.23461\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nl l' : Filter α\nh : (fun x => ‖f x‖) =ᶠ[l] fun x => ‖g x‖\n⊢ ∀ᶠ (x : α) in l, ‖g x‖ ≤ 1 * ‖f x‖",
"tactic": "simpa only [one_mul] using h.symm.le"
}
] |
[
167,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
AffineIsometry.toAffineIsometryEquiv_apply
|
[] |
[
118,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.ElementarilyEquivalent.symm
|
[] |
[
1102,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1101,
8
] |
Mathlib/CategoryTheory/Simple.lean
|
CategoryTheory.cokernel_zero_of_nonzero_to_simple
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\nX Y : C\ninst✝ : Simple Y\nf : X ⟶ Y\nw : f ≠ 0\n⊢ cokernel.π f = 0",
"tactic": "classical\n by_contra h\n haveI := isIso_of_epi_of_nonzero h\n exact w (eq_zero_of_mono_cokernel f)"
},
{
"state_after": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\nX Y : C\ninst✝ : Simple Y\nf : X ⟶ Y\nw : f ≠ 0\nh : ¬cokernel.π f = 0\n⊢ False",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\nX Y : C\ninst✝ : Simple Y\nf : X ⟶ Y\nw : f ≠ 0\n⊢ cokernel.π f = 0",
"tactic": "by_contra h"
},
{
"state_after": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\nX Y : C\ninst✝ : Simple Y\nf : X ⟶ Y\nw : f ≠ 0\nh : ¬cokernel.π f = 0\nthis : IsIso (cokernel.π f)\n⊢ False",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\nX Y : C\ninst✝ : Simple Y\nf : X ⟶ Y\nw : f ≠ 0\nh : ¬cokernel.π f = 0\n⊢ False",
"tactic": "haveI := isIso_of_epi_of_nonzero h"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\nX Y : C\ninst✝ : Simple Y\nf : X ⟶ Y\nw : f ≠ 0\nh : ¬cokernel.π f = 0\nthis : IsIso (cokernel.π f)\n⊢ False",
"tactic": "exact w (eq_zero_of_mono_cokernel f)"
}
] |
[
179,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/Data/Sym/Basic.lean
|
SymOptionSuccEquiv.encode_decode
|
[
{
"state_after": "case inl\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym (Option α) n\n⊢ encode (decode (Sum.inl s)) = Sum.inl s\n\ncase inr\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\n⊢ encode (decode (Sum.inr s)) = Sum.inr s",
"state_before": "α : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym (Option α) n ⊕ Sym α (Nat.succ n)\n⊢ encode (decode s) = s",
"tactic": "obtain s | s := s"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym (Option α) n\n⊢ encode (decode (Sum.inl s)) = Sum.inl s",
"tactic": "simp"
},
{
"state_after": "case inr\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\n⊢ (if h : none ∈ decode (Sum.inr s) then Sum.inl (erase (decode (Sum.inr s)) none h)\n else Sum.inr (map (fun o => Option.get ↑o (_ : Option.isSome ↑o = true)) (attach (decode (Sum.inr s))))) =\n Sum.inr s",
"state_before": "case inr\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\n⊢ encode (decode (Sum.inr s)) = Sum.inr s",
"tactic": "unfold SymOptionSuccEquiv.encode"
},
{
"state_after": "case inr.inl\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : none ∈ decode (Sum.inr s)\n⊢ False\n\ncase inr.inr\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : ¬none ∈ decode (Sum.inr s)\n⊢ Sum.inr (map (fun o => Option.get ↑o (_ : Option.isSome ↑o = true)) (attach (decode (Sum.inr s)))) = Sum.inr s",
"state_before": "case inr\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\n⊢ (if h : none ∈ decode (Sum.inr s) then Sum.inl (erase (decode (Sum.inr s)) none h)\n else Sum.inr (map (fun o => Option.get ↑o (_ : Option.isSome ↑o = true)) (attach (decode (Sum.inr s))))) =\n Sum.inr s",
"tactic": "split_ifs with h"
},
{
"state_after": "case inr.inl.intro.intro\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : none ∈ decode (Sum.inr s)\na : α\nleft✝ : a ∈ ↑s\nha : ↑Embedding.some a = none\n⊢ False",
"state_before": "case inr.inl\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : none ∈ decode (Sum.inr s)\n⊢ False",
"tactic": "obtain ⟨a, _, ha⟩ := Multiset.mem_map.mp h"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.intro.intro\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : none ∈ decode (Sum.inr s)\na : α\nleft✝ : a ∈ ↑s\nha : ↑Embedding.some a = none\n⊢ False",
"tactic": "exact Option.some_ne_none _ ha"
},
{
"state_after": "case inr.inr\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : ¬none ∈ decode (Sum.inr s)\n⊢ map (fun o => Option.get ↑o (_ : Option.isSome ↑o = true)) (attach (decode (Sum.inr s))) = s",
"state_before": "case inr.inr\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : ¬none ∈ decode (Sum.inr s)\n⊢ Sum.inr (map (fun o => Option.get ↑o (_ : Option.isSome ↑o = true)) (attach (decode (Sum.inr s)))) = Sum.inr s",
"tactic": "refine' congr_arg Sum.inr _"
},
{
"state_after": "case inr.inr\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : ¬none ∈ decode (Sum.inr s)\n⊢ map some (map (fun o => Option.get ↑o (_ : Option.isSome ↑o = true)) (attach (decode (Sum.inr s)))) = map some s",
"state_before": "case inr.inr\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : ¬none ∈ decode (Sum.inr s)\n⊢ map (fun o => Option.get ↑o (_ : Option.isSome ↑o = true)) (attach (decode (Sum.inr s))) = s",
"tactic": "refine' map_injective (Option.some_injective _) _ _"
},
{
"state_after": "case inr.inr.refine'_1\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : ¬none ∈ decode (Sum.inr s)\n⊢ map some (map (fun o => Option.get ↑o (_ : Option.isSome ↑o = true)) (attach (decode (Sum.inr s)))) =\n map Subtype.val (attach (decode (Sum.inr s)))\n\ncase inr.inr.refine'_2\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : ¬none ∈ decode (Sum.inr s)\n⊢ decode (Sum.inr s) = map some s",
"state_before": "case inr.inr\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : ¬none ∈ decode (Sum.inr s)\n⊢ map some (map (fun o => Option.get ↑o (_ : Option.isSome ↑o = true)) (attach (decode (Sum.inr s)))) = map some s",
"tactic": "refine' Eq.trans _ (Eq.trans (SymOptionSuccEquiv.decode (Sum.inr s)).attach_map_coe _)"
},
{
"state_after": "case inr.inr.refine'_2\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : ¬none ∈ decode (Sum.inr s)\n⊢ decode (Sum.inr s) = map some s",
"state_before": "case inr.inr.refine'_1\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : ¬none ∈ decode (Sum.inr s)\n⊢ map some (map (fun o => Option.get ↑o (_ : Option.isSome ↑o = true)) (attach (decode (Sum.inr s)))) =\n map Subtype.val (attach (decode (Sum.inr s)))\n\ncase inr.inr.refine'_2\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : ¬none ∈ decode (Sum.inr s)\n⊢ decode (Sum.inr s) = map some s",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.refine'_2\nα : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\ns : Sym α (Nat.succ n)\nh : ¬none ∈ decode (Sum.inr s)\n⊢ decode (Sum.inr s) = map some s",
"tactic": "simp"
}
] |
[
665,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
654,
1
] |
Mathlib/Topology/Instances/Matrix.lean
|
Continuous.matrix_trace
|
[] |
[
193,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean
|
Polynomial.map_cyclotomic
|
[
{
"state_after": "n : ℕ\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\n⊢ map (RingHom.comp f (Int.castRingHom R)) (cyclotomic n ℤ) = map (Int.castRingHom S) (cyclotomic n ℤ)",
"state_before": "n : ℕ\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\n⊢ map f (cyclotomic n R) = cyclotomic n S",
"tactic": "rw [← map_cyclotomic_int n R, ← map_cyclotomic_int n S, map_map]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\n⊢ map (RingHom.comp f (Int.castRingHom R)) (cyclotomic n ℤ) = map (Int.castRingHom S) (cyclotomic n ℤ)",
"tactic": "congr!"
}
] |
[
297,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
norm_pos_iff'''
|
[
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.501471\n𝕜 : Type ?u.501474\nα : Type ?u.501477\nι : Type ?u.501480\nκ : Type ?u.501483\nE : Type u_1\nF : Type ?u.501489\nG : Type ?u.501492\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : SeminormedGroup G\ns : Set E\na✝ a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : T0Space E\na : E\n⊢ 0 < ‖a‖ ↔ a ≠ 1",
"tactic": "rw [← not_le, norm_le_zero_iff''']"
}
] |
[
1287,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1286,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.mem_find_iff
|
[
{
"state_after": "case intro\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\ni : Fin n\nhpi : p i\nhj : ∀ (j : Fin n), p j → i ≤ j\n⊢ i ∈ find p",
"state_before": "m n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\ni : Fin n\n⊢ (p i ∧ ∀ (j : Fin n), p j → i ≤ j) → i ∈ find p",
"tactic": "rintro ⟨hpi, hj⟩"
},
{
"state_after": "case intro.none\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\ni : Fin n\nhpi : p i\nhj : ∀ (j : Fin n), p j → i ≤ j\nhfp : find p = none\n⊢ i ∈ none\n\ncase intro.some\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\ni : Fin n\nhpi : p i\nhj : ∀ (j : Fin n), p j → i ≤ j\nval✝ : Fin n\nhfp : find p = some val✝\n⊢ i ∈ some val✝",
"state_before": "case intro\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\ni : Fin n\nhpi : p i\nhj : ∀ (j : Fin n), p j → i ≤ j\n⊢ i ∈ find p",
"tactic": "cases hfp : Fin.find p"
},
{
"state_after": "case intro.none\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\ni : Fin n\nhpi : p i\nhj : ∀ (j : Fin n), p j → i ≤ j\nhfp : ∀ (i : Fin n), ¬p i\n⊢ i ∈ none",
"state_before": "case intro.none\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\ni : Fin n\nhpi : p i\nhj : ∀ (j : Fin n), p j → i ≤ j\nhfp : find p = none\n⊢ i ∈ none",
"tactic": "rw [find_eq_none_iff] at hfp"
},
{
"state_after": "no goals",
"state_before": "case intro.none\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\ni : Fin n\nhpi : p i\nhj : ∀ (j : Fin n), p j → i ≤ j\nhfp : ∀ (i : Fin n), ¬p i\n⊢ i ∈ none",
"tactic": "exact (hfp _ hpi).elim"
},
{
"state_after": "no goals",
"state_before": "case intro.some\nm n : ℕ\np : Fin n → Prop\ninst✝ : DecidablePred p\ni : Fin n\nhpi : p i\nhj : ∀ (j : Fin n), p j → i ≤ j\nval✝ : Fin n\nhfp : find p = some val✝\n⊢ i ∈ some val✝",
"tactic": "exact Option.some_inj.2 (le_antisymm (find_min' hfp hpi) (hj _ (find_spec _ hfp)))"
}
] |
[
937,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
930,
1
] |
Mathlib/Analysis/LocallyConvex/Bounded.lean
|
TotallyBounded.isVonNBounded
|
[
{
"state_after": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝ : TotallyBounded s\nhs : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\n⊢ Bornology.IsVonNBounded 𝕜 s",
"state_before": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs : TotallyBounded s\n⊢ Bornology.IsVonNBounded 𝕜 s",
"tactic": "rw [totallyBounded_iff_subset_finite_iUnion_nhds_zero] at hs"
},
{
"state_after": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝ : TotallyBounded s\nhs : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\n⊢ Absorbs 𝕜 U s",
"state_before": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝ : TotallyBounded s\nhs : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\n⊢ Bornology.IsVonNBounded 𝕜 s",
"tactic": "intro U hU"
},
{
"state_after": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝ : TotallyBounded s\nhs : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 (0 + 0))\n⊢ Absorbs 𝕜 U s",
"state_before": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝ : TotallyBounded s\nhs : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\n⊢ Absorbs 𝕜 U s",
"tactic": "have h : Filter.Tendsto (fun x : E × E => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 ((0 : E) + (0 : E))) :=\n tendsto_add"
},
{
"state_after": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝ : TotallyBounded s\nhs : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\n⊢ Absorbs 𝕜 U s",
"state_before": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝ : TotallyBounded s\nhs : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 (0 + 0))\n⊢ Absorbs 𝕜 U s",
"tactic": "rw [add_zero] at h"
},
{
"state_after": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝ : TotallyBounded s\nhs : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n id i.fst ×ˢ id i.snd\n⊢ Absorbs 𝕜 U s",
"state_before": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝ : TotallyBounded s\nhs : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\n⊢ Absorbs 𝕜 U s",
"tactic": "have h' := (nhds_basis_balanced 𝕜 E).prod (nhds_basis_balanced 𝕜 E)"
},
{
"state_after": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝ : TotallyBounded s\nhs : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\n⊢ Absorbs 𝕜 U s",
"state_before": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝ : TotallyBounded s\nhs : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n id i.fst ×ˢ id i.snd\n⊢ Absorbs 𝕜 U s",
"tactic": "simp_rw [← nhds_prod_eq, id.def] at h'"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝ : TotallyBounded s\nhs : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\n⊢ Absorbs 𝕜 U s",
"state_before": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝ : TotallyBounded s\nhs : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\n⊢ Absorbs 𝕜 U s",
"tactic": "rcases h.basis_left h' U hU with ⟨x, hx, h''⟩"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\n⊢ Absorbs 𝕜 U s",
"state_before": "case intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝ : TotallyBounded s\nhs : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\n⊢ Absorbs 𝕜 U s",
"tactic": "rcases hs x.snd hx.2.1 with ⟨t, ht, hs⟩"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\n⊢ Absorbs 𝕜 U (⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd)",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\n⊢ Absorbs 𝕜 U s",
"tactic": "refine' Absorbs.mono_right _ hs"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\n⊢ ∀ (i : E), i ∈ t → Absorbs 𝕜 U (i +ᵥ x.snd)",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\n⊢ Absorbs 𝕜 U (⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd)",
"tactic": "rw [ht.absorbs_iUnion]"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\nhx_fstsnd : x.fst + x.snd ⊆ U\n⊢ ∀ (i : E), i ∈ t → Absorbs 𝕜 U (i +ᵥ x.snd)",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\n⊢ ∀ (i : E), i ∈ t → Absorbs 𝕜 U (i +ᵥ x.snd)",
"tactic": "have hx_fstsnd : x.fst + x.snd ⊆ U := by\n intro z hz\n rcases Set.mem_add.mp hz with ⟨z1, z2, hz1, hz2, hz⟩\n have hz' : (z1, z2) ∈ x.fst ×ˢ x.snd := ⟨hz1, hz2⟩\n simpa only [hz] using h'' hz'"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\nhx_fstsnd : x.fst + x.snd ⊆ U\ny : E\nx✝ : y ∈ t\n⊢ Absorbs 𝕜 (x.fst + x.snd) (y +ᵥ x.snd)",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\nhx_fstsnd : x.fst + x.snd ⊆ U\n⊢ ∀ (i : E), i ∈ t → Absorbs 𝕜 U (i +ᵥ x.snd)",
"tactic": "refine' fun y _ => Absorbs.mono_left _ hx_fstsnd"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\nhx_fstsnd : x.fst + x.snd ⊆ U\ny : E\nx✝ : y ∈ t\n⊢ Absorbs 𝕜 (x.fst + x.snd) ({y} + x.snd)",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\nhx_fstsnd : x.fst + x.snd ⊆ U\ny : E\nx✝ : y ∈ t\n⊢ Absorbs 𝕜 (x.fst + x.snd) (y +ᵥ x.snd)",
"tactic": "rw [← Set.singleton_vadd, vadd_eq_add]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\nhx_fstsnd : x.fst + x.snd ⊆ U\ny : E\nx✝ : y ∈ t\n⊢ Absorbs 𝕜 (x.fst + x.snd) ({y} + x.snd)",
"tactic": "exact (absorbent_nhds_zero hx.1.1).absorbs.add hx.2.2.absorbs_self"
},
{
"state_after": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\nz : E\nhz : z ∈ x.fst + x.snd\n⊢ z ∈ U",
"state_before": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\n⊢ x.fst + x.snd ⊆ U",
"tactic": "intro z hz"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\nz : E\nhz✝ : z ∈ x.fst + x.snd\nz1 z2 : E\nhz1 : z1 ∈ x.fst\nhz2 : z2 ∈ x.snd\nhz : z1 + z2 = z\n⊢ z ∈ U",
"state_before": "𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\nz : E\nhz : z ∈ x.fst + x.snd\n⊢ z ∈ U",
"tactic": "rcases Set.mem_add.mp hz with ⟨z1, z2, hz1, hz2, hz⟩"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\nz : E\nhz✝ : z ∈ x.fst + x.snd\nz1 z2 : E\nhz1 : z1 ∈ x.fst\nhz2 : z2 ∈ x.snd\nhz : z1 + z2 = z\nhz' : (z1, z2) ∈ x.fst ×ˢ x.snd\n⊢ z ∈ U",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\nz : E\nhz✝ : z ∈ x.fst + x.snd\nz1 z2 : E\nhz1 : z1 ∈ x.fst\nhz2 : z2 ∈ x.snd\nhz : z1 + z2 = z\n⊢ z ∈ U",
"tactic": "have hz' : (z1, z2) ∈ x.fst ×ˢ x.snd := ⟨hz1, hz2⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜' : Type ?u.221754\nE : Type u_1\nE' : Type ?u.221760\nF : Type ?u.221763\nι : Type ?u.221766\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : UniformSpace E\ninst✝¹ : UniformAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\ns : Set E\nhs✝¹ : TotallyBounded s\nhs✝ : ∀ (U : Set E), U ∈ 𝓝 0 → ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ U\nU : Set E\nhU : U ∈ 𝓝 0\nh : Tendsto (fun x => x.fst + x.snd) (𝓝 (0, 0)) (𝓝 0)\nh' :\n HasBasis (𝓝 (0, 0)) (fun i => (i.fst ∈ 𝓝 0 ∧ Balanced 𝕜 i.fst) ∧ i.snd ∈ 𝓝 0 ∧ Balanced 𝕜 i.snd) fun i =>\n i.fst ×ˢ i.snd\nx : Set E × Set E\nhx : (x.fst ∈ 𝓝 0 ∧ Balanced 𝕜 x.fst) ∧ x.snd ∈ 𝓝 0 ∧ Balanced 𝕜 x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst ×ˢ x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s ⊆ ⋃ (y : E) (_ : y ∈ t), y +ᵥ x.snd\nz : E\nhz✝ : z ∈ x.fst + x.snd\nz1 z2 : E\nhz1 : z1 ∈ x.fst\nhz2 : z2 ∈ x.snd\nhz : z1 + z2 = z\nhz' : (z1, z2) ∈ x.fst ×ˢ x.snd\n⊢ z ∈ U",
"tactic": "simpa only [hz] using h'' hz'"
}
] |
[
266,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.single_coeff_same
|
[] |
[
161,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
eq_mul_inv_of_mul_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.49889\nβ : Type ?u.49892\nG : Type u_1\ninst✝ : Group G\na b c d : G\nh : a * c = b\n⊢ a = b * c⁻¹",
"tactic": "simp [h.symm]"
}
] |
[
628,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
628,
1
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
TopologicalSpace.Opens.infLELeft_apply
|
[] |
[
96,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
|
StructureGroupoid.liftPropWithinAt_self_source
|
[] |
[
230,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/Order/Atoms.lean
|
IsAtom.disjoint_of_ne
|
[] |
[
216,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.IsEquiv.comap
|
[] |
[
401,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
1
] |
Mathlib/RingTheory/Adjoin/Tower.lean
|
Algebra.adjoin_restrictScalars
|
[
{
"state_after": "R : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\n⊢ Set.range ↑(algebraMap D E) = Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E)",
"state_before": "R : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\n⊢ Subalgebra.restrictScalars C (adjoin D S) =\n Subalgebra.restrictScalars C (adjoin { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } S)",
"tactic": "suffices\n Set.range (algebraMap D E) =\n Set.range (algebraMap ((⊤ : Subalgebra C D).map (IsScalarTower.toAlgHom C D E)) E) by\n ext x\n change x ∈ Subsemiring.closure (_ ∪ S) ↔ x ∈ Subsemiring.closure (_ ∪ S)\n rw [this]"
},
{
"state_after": "case h\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\nx : E\n⊢ x ∈ Set.range ↑(algebraMap D E) ↔\n x ∈ Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E)",
"state_before": "R : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\n⊢ Set.range ↑(algebraMap D E) = Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E)",
"tactic": "ext x"
},
{
"state_after": "case h.mp\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\nx : E\n⊢ x ∈ Set.range ↑(algebraMap D E) →\n x ∈ Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E)\n\ncase h.mpr\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\nx : E\n⊢ x ∈ Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E) →\n x ∈ Set.range ↑(algebraMap D E)",
"state_before": "case h\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\nx : E\n⊢ x ∈ Set.range ↑(algebraMap D E) ↔\n x ∈ Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E)",
"tactic": "constructor"
},
{
"state_after": "case h\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\nthis :\n Set.range ↑(algebraMap D E) = Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E)\nx : E\n⊢ x ∈ Subalgebra.restrictScalars C (adjoin D S) ↔\n x ∈ Subalgebra.restrictScalars C (adjoin { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } S)",
"state_before": "R : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\nthis :\n Set.range ↑(algebraMap D E) = Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E)\n⊢ Subalgebra.restrictScalars C (adjoin D S) =\n Subalgebra.restrictScalars C (adjoin { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } S)",
"tactic": "ext x"
},
{
"state_after": "case h\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\nthis :\n Set.range ↑(algebraMap D E) = Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E)\nx : E\n⊢ x ∈ Subsemiring.closure (Set.range ↑(algebraMap D E) ∪ S) ↔\n x ∈ Subsemiring.closure (Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E) ∪ S)",
"state_before": "case h\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\nthis :\n Set.range ↑(algebraMap D E) = Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E)\nx : E\n⊢ x ∈ Subalgebra.restrictScalars C (adjoin D S) ↔\n x ∈ Subalgebra.restrictScalars C (adjoin { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } S)",
"tactic": "change x ∈ Subsemiring.closure (_ ∪ S) ↔ x ∈ Subsemiring.closure (_ ∪ S)"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\nthis :\n Set.range ↑(algebraMap D E) = Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E)\nx : E\n⊢ x ∈ Subsemiring.closure (Set.range ↑(algebraMap D E) ∪ S) ↔\n x ∈ Subsemiring.closure (Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E) ∪ S)",
"tactic": "rw [this]"
},
{
"state_after": "case h.mp.intro\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\nx : E\ny : D\nhy : ↑(algebraMap D E) y = x\n⊢ x ∈ Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E)",
"state_before": "case h.mp\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\nx : E\n⊢ x ∈ Set.range ↑(algebraMap D E) →\n x ∈ Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E)",
"tactic": "rintro ⟨y, hy⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\nx : E\ny : D\nhy : ↑(algebraMap D E) y = x\n⊢ x ∈ Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E)",
"tactic": "exact ⟨⟨algebraMap D E y, ⟨y, ⟨Algebra.mem_top, rfl⟩⟩⟩, hy⟩"
},
{
"state_after": "case h.mpr.intro.mk.intro.intro\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\nx y : E\nz : D\nh0 : z ∈ ↑⊤.toSubsemiring\nh1 : ↑↑(IsScalarTower.toAlgHom C D E) z = y\nh2 :\n ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E)\n { val := y, property := (_ : ∃ a, a ∈ ↑⊤.toSubsemiring ∧ ↑↑(IsScalarTower.toAlgHom C D E) a = y) } =\n x\n⊢ x ∈ Set.range ↑(algebraMap D E)",
"state_before": "case h.mpr\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\nx : E\n⊢ x ∈ Set.range ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E) →\n x ∈ Set.range ↑(algebraMap D E)",
"tactic": "rintro ⟨⟨y, ⟨z, ⟨h0, h1⟩⟩⟩, h2⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro.mk.intro.intro\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁶ : CommSemiring C\ninst✝⁵ : CommSemiring D\ninst✝⁴ : CommSemiring E\ninst✝³ : Algebra C D\ninst✝² : Algebra C E\ninst✝¹ : Algebra D E\ninst✝ : IsScalarTower C D E\nS : Set E\nx y : E\nz : D\nh0 : z ∈ ↑⊤.toSubsemiring\nh1 : ↑↑(IsScalarTower.toAlgHom C D E) z = y\nh2 :\n ↑(algebraMap { x // x ∈ Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤ } E)\n { val := y, property := (_ : ∃ a, a ∈ ↑⊤.toSubsemiring ∧ ↑↑(IsScalarTower.toAlgHom C D E) a = y) } =\n x\n⊢ x ∈ Set.range ↑(algebraMap D E)",
"tactic": "exact ⟨z, Eq.trans h1 h2⟩"
}
] |
[
56,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.erase_cons
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.216030\nγ : Type ?u.216033\ninst✝ : DecidableEq α\ns✝ t u v : Finset α\na✝ b : α\ns : Finset α\na : α\nh : ¬a ∈ s\n⊢ erase (cons a s h) a = s",
"tactic": "rw [cons_eq_insert, erase_insert_eq_erase, erase_eq_of_not_mem h]"
}
] |
[
1977,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1976,
1
] |
Mathlib/Tactic/Linarith/Lemmas.lean
|
Linarith.eq_of_eq_of_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedSemiring α\na b : α\nha : a = 0\nhb : b = 0\n⊢ a + b = 0",
"tactic": "simp [*]"
}
] |
[
28,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
27,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Split.lean
|
BoxIntegral.Prepartition.mem_split_iff'
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nM : Type ?u.25531\nn : ℕ\nI J : Box ι\ni : ι\nx : ℝ\n⊢ J ∈ split I i x ↔ ↑J = ↑I ∩ {y | y i ≤ x} ∨ ↑J = ↑I ∩ {y | x < y i}",
"tactic": "simp [mem_split_iff, ← Box.withBotCoe_inj]"
}
] |
[
184,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Topology/UniformSpace/UniformEmbedding.lean
|
UniformInducing.uniformEmbedding
|
[] |
[
197,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
11
] |
Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean
|
MeasureTheory.Measure.withDensityᵥ_absolutelyContinuous
|
[
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.59120\nm : MeasurableSpace α\nμ✝ ν : Measure α\nE : Type ?u.59139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nμ : Measure α\nf : α → ℝ\nhf : Integrable f\n⊢ withDensityᵥ μ f ≪ᵥ toENNRealVectorMeasure μ\n\ncase neg\nα : Type u_1\nβ : Type ?u.59120\nm : MeasurableSpace α\nμ✝ ν : Measure α\nE : Type ?u.59139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nμ : Measure α\nf : α → ℝ\nhf : ¬Integrable f\n⊢ withDensityᵥ μ f ≪ᵥ toENNRealVectorMeasure μ",
"state_before": "α : Type u_1\nβ : Type ?u.59120\nm : MeasurableSpace α\nμ✝ ν : Measure α\nE : Type ?u.59139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nμ : Measure α\nf : α → ℝ\n⊢ withDensityᵥ μ f ≪ᵥ toENNRealVectorMeasure μ",
"tactic": "by_cases hf : Integrable f μ"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.59120\nm : MeasurableSpace α\nμ✝ ν : Measure α\nE : Type ?u.59139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nμ : Measure α\nf : α → ℝ\nhf : Integrable f\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : ↑(toENNRealVectorMeasure μ) i = 0\n⊢ ↑(withDensityᵥ μ f) i = 0",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.59120\nm : MeasurableSpace α\nμ✝ ν : Measure α\nE : Type ?u.59139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nμ : Measure α\nf : α → ℝ\nhf : Integrable f\n⊢ withDensityᵥ μ f ≪ᵥ toENNRealVectorMeasure μ",
"tactic": "refine' VectorMeasure.AbsolutelyContinuous.mk fun i hi₁ hi₂ => _"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.59120\nm : MeasurableSpace α\nμ✝ ν : Measure α\nE : Type ?u.59139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nμ : Measure α\nf : α → ℝ\nhf : Integrable f\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : ↑↑μ i = 0\n⊢ ↑(withDensityᵥ μ f) i = 0",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.59120\nm : MeasurableSpace α\nμ✝ ν : Measure α\nE : Type ?u.59139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nμ : Measure α\nf : α → ℝ\nhf : Integrable f\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : ↑(toENNRealVectorMeasure μ) i = 0\n⊢ ↑(withDensityᵥ μ f) i = 0",
"tactic": "rw [toENNRealVectorMeasure_apply_measurable hi₁] at hi₂"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.59120\nm : MeasurableSpace α\nμ✝ ν : Measure α\nE : Type ?u.59139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nμ : Measure α\nf : α → ℝ\nhf : Integrable f\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : ↑↑μ i = 0\n⊢ ↑(withDensityᵥ μ f) i = 0",
"tactic": "rw [withDensityᵥ_apply hf hi₁, Measure.restrict_zero_set hi₂, integral_zero_measure]"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.59120\nm : MeasurableSpace α\nμ✝ ν : Measure α\nE : Type ?u.59139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nμ : Measure α\nf : α → ℝ\nhf : ¬Integrable f\n⊢ 0 ≪ᵥ toENNRealVectorMeasure μ",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.59120\nm : MeasurableSpace α\nμ✝ ν : Measure α\nE : Type ?u.59139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nμ : Measure α\nf : α → ℝ\nhf : ¬Integrable f\n⊢ withDensityᵥ μ f ≪ᵥ toENNRealVectorMeasure μ",
"tactic": "rw [withDensityᵥ, dif_neg hf]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.59120\nm : MeasurableSpace α\nμ✝ ν : Measure α\nE : Type ?u.59139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nμ : Measure α\nf : α → ℝ\nhf : ¬Integrable f\n⊢ 0 ≪ᵥ toENNRealVectorMeasure μ",
"tactic": "exact VectorMeasure.AbsolutelyContinuous.zero _"
}
] |
[
142,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Data/Nat/Pow.lean
|
Nat.pow_dvd_pow_iff_pow_le_pow
|
[
{
"state_after": "case mp\nk l x : ℕ\nw : 0 < x + 1\n⊢ (x + 1) ^ k ∣ (x + 1) ^ l → (x + 1) ^ k ≤ (x + 1) ^ l\n\ncase mpr\nk l x : ℕ\nw : 0 < x + 1\n⊢ (x + 1) ^ k ≤ (x + 1) ^ l → (x + 1) ^ k ∣ (x + 1) ^ l",
"state_before": "k l x : ℕ\nw : 0 < x + 1\n⊢ (x + 1) ^ k ∣ (x + 1) ^ l ↔ (x + 1) ^ k ≤ (x + 1) ^ l",
"tactic": "constructor"
},
{
"state_after": "case mp\nk l x : ℕ\nw : 0 < x + 1\na : (x + 1) ^ k ∣ (x + 1) ^ l\n⊢ (x + 1) ^ k ≤ (x + 1) ^ l",
"state_before": "case mp\nk l x : ℕ\nw : 0 < x + 1\n⊢ (x + 1) ^ k ∣ (x + 1) ^ l → (x + 1) ^ k ≤ (x + 1) ^ l",
"tactic": "intro a"
},
{
"state_after": "no goals",
"state_before": "case mp\nk l x : ℕ\nw : 0 < x + 1\na : (x + 1) ^ k ∣ (x + 1) ^ l\n⊢ (x + 1) ^ k ≤ (x + 1) ^ l",
"tactic": "exact le_of_dvd (pow_pos (succ_pos x) l) a"
},
{
"state_after": "case mpr\nk l x : ℕ\nw : 0 < x + 1\na : (x + 1) ^ k ≤ (x + 1) ^ l\n⊢ (x + 1) ^ k ∣ (x + 1) ^ l",
"state_before": "case mpr\nk l x : ℕ\nw : 0 < x + 1\n⊢ (x + 1) ^ k ≤ (x + 1) ^ l → (x + 1) ^ k ∣ (x + 1) ^ l",
"tactic": "intro a"
},
{
"state_after": "case mpr.zero\nk l : ℕ\nw : 0 < zero + 1\na : (zero + 1) ^ k ≤ (zero + 1) ^ l\n⊢ (zero + 1) ^ k ∣ (zero + 1) ^ l\n\ncase mpr.succ\nk l x : ℕ\nw : 0 < succ x + 1\na : (succ x + 1) ^ k ≤ (succ x + 1) ^ l\n⊢ (succ x + 1) ^ k ∣ (succ x + 1) ^ l",
"state_before": "case mpr\nk l x : ℕ\nw : 0 < x + 1\na : (x + 1) ^ k ≤ (x + 1) ^ l\n⊢ (x + 1) ^ k ∣ (x + 1) ^ l",
"tactic": "cases' x with x"
},
{
"state_after": "no goals",
"state_before": "case mpr.zero\nk l : ℕ\nw : 0 < zero + 1\na : (zero + 1) ^ k ≤ (zero + 1) ^ l\n⊢ (zero + 1) ^ k ∣ (zero + 1) ^ l",
"tactic": "simp"
},
{
"state_after": "case mpr.succ\nk l x : ℕ\nw : 0 < succ x + 1\na : (succ x + 1) ^ k ≤ (succ x + 1) ^ l\nle : k ≤ l\n⊢ (succ x + 1) ^ k ∣ (succ x + 1) ^ l",
"state_before": "case mpr.succ\nk l x : ℕ\nw : 0 < succ x + 1\na : (succ x + 1) ^ k ≤ (succ x + 1) ^ l\n⊢ (succ x + 1) ^ k ∣ (succ x + 1) ^ l",
"tactic": "have le := (pow_le_iff_le_right (Nat.le_add_left _ _)).mp a"
},
{
"state_after": "case mpr.succ\nk l x : ℕ\nw : 0 < succ x + 1\na : (succ x + 1) ^ k ≤ (succ x + 1) ^ l\nle : k ≤ l\n⊢ (succ x + 1) ^ l = (succ x + 1) ^ k * (x + 2) ^ (l - k)",
"state_before": "case mpr.succ\nk l x : ℕ\nw : 0 < succ x + 1\na : (succ x + 1) ^ k ≤ (succ x + 1) ^ l\nle : k ≤ l\n⊢ (succ x + 1) ^ k ∣ (succ x + 1) ^ l",
"tactic": "use (x + 2) ^ (l - k)"
},
{
"state_after": "no goals",
"state_before": "case mpr.succ\nk l x : ℕ\nw : 0 < succ x + 1\na : (succ x + 1) ^ k ≤ (succ x + 1) ^ l\nle : k ≤ l\n⊢ (succ x + 1) ^ l = (succ x + 1) ^ k * (x + 2) ^ (l - k)",
"tactic": "rw [← pow_add, add_comm k, tsub_add_cancel_of_le le]"
}
] |
[
207,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/CategoryTheory/Monad/Limits.lean
|
CategoryTheory.hasLimit_of_reflective
|
[] |
[
359,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.arrowCongr_trans
|
[
{
"state_after": "case H.H\nR : Type u\nA₁ : Type v\nA₂ : Type w\nA₃ : Type u₁\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring A₁\ninst✝¹⁰ : Semiring A₂\ninst✝⁹ : Semiring A₃\ninst✝⁸ : Algebra R A₁\ninst✝⁷ : Algebra R A₂\ninst✝⁶ : Algebra R A₃\ne : A₁ ≃ₐ[R] A₂\nA₁' : Type u_1\nA₂' : Type u_2\nA₃' : Type u_3\ninst✝⁵ : Semiring A₁'\ninst✝⁴ : Semiring A₂'\ninst✝³ : Semiring A₃'\ninst✝² : Algebra R A₁'\ninst✝¹ : Algebra R A₂'\ninst✝ : Algebra R A₃'\ne₁ : A₁ ≃ₐ[R] A₂\ne₁' : A₁' ≃ₐ[R] A₂'\ne₂ : A₂ ≃ₐ[R] A₃\ne₂' : A₂' ≃ₐ[R] A₃'\nx✝¹ : A₁ →ₐ[R] A₁'\nx✝ : A₃\n⊢ ↑(↑(arrowCongr (trans e₁ e₂) (trans e₁' e₂')) x✝¹) x✝ = ↑(↑((arrowCongr e₁ e₁').trans (arrowCongr e₂ e₂')) x✝¹) x✝",
"state_before": "R : Type u\nA₁ : Type v\nA₂ : Type w\nA₃ : Type u₁\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring A₁\ninst✝¹⁰ : Semiring A₂\ninst✝⁹ : Semiring A₃\ninst✝⁸ : Algebra R A₁\ninst✝⁷ : Algebra R A₂\ninst✝⁶ : Algebra R A₃\ne : A₁ ≃ₐ[R] A₂\nA₁' : Type u_1\nA₂' : Type u_2\nA₃' : Type u_3\ninst✝⁵ : Semiring A₁'\ninst✝⁴ : Semiring A₂'\ninst✝³ : Semiring A₃'\ninst✝² : Algebra R A₁'\ninst✝¹ : Algebra R A₂'\ninst✝ : Algebra R A₃'\ne₁ : A₁ ≃ₐ[R] A₂\ne₁' : A₁' ≃ₐ[R] A₂'\ne₂ : A₂ ≃ₐ[R] A₃\ne₂' : A₂' ≃ₐ[R] A₃'\n⊢ arrowCongr (trans e₁ e₂) (trans e₁' e₂') = (arrowCongr e₁ e₁').trans (arrowCongr e₂ e₂')",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H.H\nR : Type u\nA₁ : Type v\nA₂ : Type w\nA₃ : Type u₁\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring A₁\ninst✝¹⁰ : Semiring A₂\ninst✝⁹ : Semiring A₃\ninst✝⁸ : Algebra R A₁\ninst✝⁷ : Algebra R A₂\ninst✝⁶ : Algebra R A₃\ne : A₁ ≃ₐ[R] A₂\nA₁' : Type u_1\nA₂' : Type u_2\nA₃' : Type u_3\ninst✝⁵ : Semiring A₁'\ninst✝⁴ : Semiring A₂'\ninst✝³ : Semiring A₃'\ninst✝² : Algebra R A₁'\ninst✝¹ : Algebra R A₂'\ninst✝ : Algebra R A₃'\ne₁ : A₁ ≃ₐ[R] A₂\ne₁' : A₁' ≃ₐ[R] A₂'\ne₂ : A₂ ≃ₐ[R] A₃\ne₂' : A₂' ≃ₐ[R] A₃'\nx✝¹ : A₁ →ₐ[R] A₁'\nx✝ : A₃\n⊢ ↑(↑(arrowCongr (trans e₁ e₂) (trans e₁' e₂')) x✝¹) x✝ = ↑(↑((arrowCongr e₁ e₁').trans (arrowCongr e₂ e₂')) x✝¹) x✝",
"tactic": "rfl"
}
] |
[
477,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/AlgebraicTopology/DoldKan/Projections.lean
|
AlgebraicTopology.DoldKan.P_idem
|
[
{
"state_after": "case h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nq n : ℕ\n⊢ HomologicalComplex.Hom.f (P q ≫ P q) n = HomologicalComplex.Hom.f (P q) n",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nq : ℕ\n⊢ P q ≫ P q = P q",
"tactic": "ext n"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nq n : ℕ\n⊢ HomologicalComplex.Hom.f (P q ≫ P q) n = HomologicalComplex.Hom.f (P q) n",
"tactic": "exact P_f_idem q n"
}
] |
[
173,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/CategoryTheory/Functor/EpiMono.lean
|
CategoryTheory.Functor.preservesEpimorphisms.iso_iff
|
[] |
[
147,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.sum_equivFun
|
[
{
"state_after": "ι : Type u_2\nι' : Type ?u.659117\nR : Type u_3\nR₂ : Type ?u.659123\nK : Type ?u.659126\nM : Type u_1\nM' : Type ?u.659132\nM'' : Type ?u.659135\nV : Type u\nV' : Type ?u.659140\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nu : M\n⊢ ∑ i : ι, ↑(equivFun b) u i • ↑b i = ↑(Finsupp.total ι M R ↑b) (↑b.repr u)",
"state_before": "ι : Type u_2\nι' : Type ?u.659117\nR : Type u_3\nR₂ : Type ?u.659123\nK : Type ?u.659126\nM : Type u_1\nM' : Type ?u.659132\nM'' : Type ?u.659135\nV : Type u\nV' : Type ?u.659140\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nu : M\n⊢ ∑ i : ι, ↑(equivFun b) u i • ↑b i = u",
"tactic": "conv_rhs => rw [← b.total_repr u]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nι' : Type ?u.659117\nR : Type u_3\nR₂ : Type ?u.659123\nK : Type ?u.659126\nM : Type u_1\nM' : Type ?u.659132\nM'' : Type ?u.659135\nV : Type u\nV' : Type ?u.659140\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nu : M\n⊢ ∑ i : ι, ↑(equivFun b) u i • ↑b i = ↑(Finsupp.total ι M R ↑b) (↑b.repr u)",
"tactic": "simp [Finsupp.total_apply, Finsupp.sum_fintype, b.equivFun_apply]"
}
] |
[
929,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
927,
1
] |
Mathlib/Algebra/ModEq.lean
|
AddCommGroup.neg_modEq_neg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddCommGroup α\np a a₁ a₂ b b₁ b₂ c : α\nn : ℕ\nz : ℤ\n⊢ -b ≡ -a [PMOD p] ↔ a ≡ b [PMOD p]",
"tactic": "simp [ModEq, neg_add_eq_sub]"
}
] |
[
83,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Tactic/Sat/FromLRAT.lean
|
Sat.Clause.reify_zero
|
[] |
[
203,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.floor_one
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.128362\nα : Type u_1\nβ : Type ?u.128368\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na : α\n⊢ ⌊1⌋ = 1",
"tactic": "rw [← cast_one, floor_intCast]"
}
] |
[
710,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
710,
1
] |
Mathlib/Order/Filter/SmallSets.lean
|
Filter.eventually_smallSets
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.1327\nι : Sort ?u.1330\nl l' la : Filter α\nlb : Filter β\np : Set α → Prop\n⊢ (∃ t, t ∈ l ∧ ∀ (y : Set α), y ∈ 𝒫 t → p y) ↔ ∃ s, s ∈ l ∧ ∀ (t : Set α), t ⊆ s → p t\n\ncase hh\nα : Type u_1\nβ : Type ?u.1327\nι : Sort ?u.1330\nl l' la : Filter α\nlb : Filter β\np : Set α → Prop\n⊢ Monotone powerset",
"state_before": "α : Type u_1\nβ : Type ?u.1327\nι : Sort ?u.1330\nl l' la : Filter α\nlb : Filter β\np : Set α → Prop\n⊢ (∀ᶠ (s : Set α) in smallSets l, p s) ↔ ∃ s, s ∈ l ∧ ∀ (t : Set α), t ⊆ s → p t",
"tactic": "rw [smallSets, eventually_lift'_iff]"
},
{
"state_after": "case hh\nα : Type u_1\nβ : Type ?u.1327\nι : Sort ?u.1330\nl l' la : Filter α\nlb : Filter β\np : Set α → Prop\n⊢ Monotone powerset",
"state_before": "α : Type u_1\nβ : Type ?u.1327\nι : Sort ?u.1330\nl l' la : Filter α\nlb : Filter β\np : Set α → Prop\n⊢ (∃ t, t ∈ l ∧ ∀ (y : Set α), y ∈ 𝒫 t → p y) ↔ ∃ s, s ∈ l ∧ ∀ (t : Set α), t ⊆ s → p t\n\ncase hh\nα : Type u_1\nβ : Type ?u.1327\nι : Sort ?u.1330\nl l' la : Filter α\nlb : Filter β\np : Set α → Prop\n⊢ Monotone powerset",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case hh\nα : Type u_1\nβ : Type ?u.1327\nι : Sort ?u.1330\nl l' la : Filter α\nlb : Filter β\np : Set α → Prop\n⊢ Monotone powerset",
"tactic": "exact monotone_powerset"
}
] |
[
69,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/MeasureTheory/Constructions/Polish.lean
|
MeasureTheory.measurableEquiv_range_coe_nat_of_infinite_of_countable
|
[
{
"state_after": "α : Type u_1\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.398049\ninst✝⁴ : MeasurableSpace α\ninst✝³ : PolishSpace α\ninst✝² : BorelSpace α\ninst✝¹ : Infinite α\ninst✝ : Countable α\nthis : PolishSpace ↑(range Nat.cast)\n⊢ Nonempty (α ≃ᵐ ↑(range Nat.cast))",
"state_before": "α : Type u_1\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.398049\ninst✝⁴ : MeasurableSpace α\ninst✝³ : PolishSpace α\ninst✝² : BorelSpace α\ninst✝¹ : Infinite α\ninst✝ : Countable α\n⊢ Nonempty (α ≃ᵐ ↑(range Nat.cast))",
"tactic": "have : PolishSpace (range ((↑) : ℕ → ℝ)) :=\n Nat.closedEmbedding_coe_real.isClosedMap.closed_range.polishSpace"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.398049\ninst✝⁴ : MeasurableSpace α\ninst✝³ : PolishSpace α\ninst✝² : BorelSpace α\ninst✝¹ : Infinite α\ninst✝ : Countable α\nthis : PolishSpace ↑(range Nat.cast)\n⊢ α ≃ ↑(range Nat.cast)",
"state_before": "α : Type u_1\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.398049\ninst✝⁴ : MeasurableSpace α\ninst✝³ : PolishSpace α\ninst✝² : BorelSpace α\ninst✝¹ : Infinite α\ninst✝ : Countable α\nthis : PolishSpace ↑(range Nat.cast)\n⊢ Nonempty (α ≃ᵐ ↑(range Nat.cast))",
"tactic": "refine' ⟨PolishSpace.Equiv.measurableEquiv _⟩"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.398049\ninst✝⁴ : MeasurableSpace α\ninst✝³ : PolishSpace α\ninst✝² : BorelSpace α\ninst✝¹ : Infinite α\ninst✝ : Countable α\nthis : PolishSpace ↑(range Nat.cast)\n⊢ ℕ ≃ ↑(range Nat.cast)",
"state_before": "α : Type u_1\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.398049\ninst✝⁴ : MeasurableSpace α\ninst✝³ : PolishSpace α\ninst✝² : BorelSpace α\ninst✝¹ : Infinite α\ninst✝ : Countable α\nthis : PolishSpace ↑(range Nat.cast)\n⊢ α ≃ ↑(range Nat.cast)",
"tactic": "refine' (nonempty_equiv_of_countable.some : α ≃ ℕ).trans _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : TopologicalSpace α\nι : Type ?u.398049\ninst✝⁴ : MeasurableSpace α\ninst✝³ : PolishSpace α\ninst✝² : BorelSpace α\ninst✝¹ : Infinite α\ninst✝ : Countable α\nthis : PolishSpace ↑(range Nat.cast)\n⊢ ℕ ≃ ↑(range Nat.cast)",
"tactic": "exact Equiv.ofInjective ((↑) : ℕ → ℝ) Nat.cast_injective"
}
] |
[
797,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
791,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.coe_lt_top
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.88258\nγ : Type ?u.88261\nδ : Type ?u.88264\ninst✝ : LT α\na✝ b a : α\n⊢ ↑a < ⊤",
"tactic": "simp [← toDual_lt_toDual_iff, WithBot.bot_lt_coe]"
}
] |
[
1065,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1064,
1
] |
Mathlib/Data/PFunctor/Univariate/Basic.lean
|
PFunctor.W.dest_mk
|
[
{
"state_after": "case mk\nP : PFunctor\nα β : Type u\nfst✝ : P.A\nsnd✝ : B P fst✝ → W P\n⊢ dest (mk { fst := fst✝, snd := snd✝ }) = { fst := fst✝, snd := snd✝ }",
"state_before": "P : PFunctor\nα β : Type u\np : Obj P (W P)\n⊢ dest (mk p) = p",
"tactic": "cases p"
},
{
"state_after": "no goals",
"state_before": "case mk\nP : PFunctor\nα β : Type u\nfst✝ : P.A\nsnd✝ : B P fst✝ → W P\n⊢ dest (mk { fst := fst✝, snd := snd✝ }) = { fst := fst✝, snd := snd✝ }",
"tactic": "rfl"
}
] |
[
114,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Order/Disjoint.lean
|
disjoint_iff_inf_le
|
[] |
[
125,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.restrict_lintegral
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.885602\nγ : Type ?u.885605\nδ : Type ?u.885608\nm : MeasurableSpace α\nμ ν : Measure α\nf : α →ₛ ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nx : α\nhx : ↑(restrict f s) x ≠ 0\nhxs : x ∈ s\nx✝ : ↑↑μ (↑(restrict f s) ⁻¹' {↑(restrict f s) x}) ≠ 0\n⊢ ↑(restrict f s) x ∈ SimpleFunc.range f",
"tactic": "simp only [f.restrict_apply hs, indicator_of_mem hxs, mem_range_self]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.885602\nγ : Type ?u.885605\nδ : Type ?u.885608\nm : MeasurableSpace α\nμ ν : Measure α\nf : α →ₛ ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nx : α\nhx : ↑(restrict f s) x ≠ 0\nhxs : ¬x ∈ s\n⊢ ↑(restrict f s) x = 0",
"tactic": "simp [*]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.885602\nγ : Type ?u.885605\nδ : Type ?u.885608\nm : MeasurableSpace α\nμ ν : Measure α\nf : α →ₛ ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nb : α\nhb : ↑f b = 0\n⊢ ↑f b * ↑↑μ (↑(restrict f s) ⁻¹' {↑f b}) = ↑f b * ↑↑μ (↑f ⁻¹' {↑f b} ∩ s)",
"tactic": "simp only [hb, MulZeroClass.zero_mul]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.885602\nγ : Type ?u.885605\nδ : Type ?u.885608\nm : MeasurableSpace α\nμ ν : Measure α\nf : α →ₛ ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nb : α\nhb : ¬↑f b = 0\n⊢ ↑f b * ↑↑μ (↑(restrict f s) ⁻¹' {↑f b}) = ↑f b * ↑↑μ (↑f ⁻¹' {↑f b} ∩ s)",
"tactic": "rw [restrict_preimage_singleton _ hs hb, inter_comm]"
}
] |
[
1077,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1065,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.setToFun_eq
|
[] |
[
1284,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1282,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.empty'
|
[] |
[
97,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/CategoryTheory/Functor/FullyFaithful.lean
|
CategoryTheory.Functor.map_surjective
|
[] |
[
100,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
src/lean/Init/Prelude.lean
|
Nat.le_succ_of_le
|
[] |
[
1603,
29
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
1602,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.cos_pi_div_four
|
[
{
"state_after": "x : ℝ\n⊢ cos (π / 4) = cos (π / 2 ^ 2)\n\nx : ℝ\n⊢ cos (π / 2 ^ 2) = sqrt 2 / 2",
"state_before": "x : ℝ\n⊢ cos (π / 4) = sqrt 2 / 2",
"tactic": "trans cos (π / 2 ^ 2)"
},
{
"state_after": "case e_x.e_a\nx : ℝ\n⊢ 4 = 2 ^ 2\n\nx : ℝ\n⊢ cos (π / 2 ^ 2) = sqrt 2 / 2",
"state_before": "x : ℝ\n⊢ cos (π / 4) = cos (π / 2 ^ 2)\n\nx : ℝ\n⊢ cos (π / 2 ^ 2) = sqrt 2 / 2",
"tactic": "congr"
},
{
"state_after": "x : ℝ\n⊢ cos (π / 2 ^ 2) = sqrt 2 / 2",
"state_before": "case e_x.e_a\nx : ℝ\n⊢ 4 = 2 ^ 2\n\nx : ℝ\n⊢ cos (π / 2 ^ 2) = sqrt 2 / 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ cos (π / 2 ^ 2) = sqrt 2 / 2",
"tactic": "simp"
}
] |
[
785,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
781,
1
] |
Mathlib/Analysis/NormedSpace/Pointwise.lean
|
diam_smul₀
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\nc : 𝕜\nx : Set E\n⊢ diam (c • x) = ‖c‖ * diam x",
"tactic": "simp_rw [diam, ediam_smul₀, ENNReal.toReal_smul, NNReal.smul_def, coe_nnnorm, smul_eq_mul]"
}
] |
[
60,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Analysis/SpecialFunctions/Polynomials.lean
|
Polynomial.isBigO_of_degree_le
|
[
{
"state_after": "case pos\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : degree P ≤ degree Q\nhp : P = 0\n⊢ (fun x => eval x P) =O[atTop] fun x => eval x Q\n\ncase neg\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : degree P ≤ degree Q\nhp : ¬P = 0\n⊢ (fun x => eval x P) =O[atTop] fun x => eval x Q",
"state_before": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : degree P ≤ degree Q\n⊢ (fun x => eval x P) =O[atTop] fun x => eval x Q",
"tactic": "by_cases hp : P = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : degree P ≤ degree Q\nhp : P = 0\n⊢ (fun x => eval x P) =O[atTop] fun x => eval x Q",
"tactic": "simpa [hp] using isBigO_zero (fun x => eval x Q) atTop"
},
{
"state_after": "case neg\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : degree P ≤ degree Q\nhp : ¬P = 0\nhq : Q ≠ 0\n⊢ (fun x => eval x P) =O[atTop] fun x => eval x Q",
"state_before": "case neg\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : degree P ≤ degree Q\nhp : ¬P = 0\n⊢ (fun x => eval x P) =O[atTop] fun x => eval x Q",
"tactic": "have hq : Q ≠ 0 := ne_zero_of_degree_ge_degree h hp"
},
{
"state_after": "case neg\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : degree P ≤ degree Q\nhp : ¬P = 0\nhq : Q ≠ 0\nhPQ : ∀ᶠ (x : 𝕜) in atTop, eval x Q = 0 → eval x P = 0\n⊢ (fun x => eval x P) =O[atTop] fun x => eval x Q",
"state_before": "case neg\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : degree P ≤ degree Q\nhp : ¬P = 0\nhq : Q ≠ 0\n⊢ (fun x => eval x P) =O[atTop] fun x => eval x Q",
"tactic": "have hPQ : ∀ᶠ x : 𝕜 in atTop, eval x Q = 0 → eval x P = 0 :=\n Filter.mem_of_superset (Polynomial.eventually_no_roots Q hq) fun x h h' => absurd h' h"
},
{
"state_after": "case neg.inl\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh✝ : degree P ≤ degree Q\nhp : ¬P = 0\nhq : Q ≠ 0\nhPQ : ∀ᶠ (x : 𝕜) in atTop, eval x Q = 0 → eval x P = 0\nh : degree P < degree Q\n⊢ (fun x => eval x P) =O[atTop] fun x => eval x Q\n\ncase neg.inr\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh✝ : degree P ≤ degree Q\nhp : ¬P = 0\nhq : Q ≠ 0\nhPQ : ∀ᶠ (x : 𝕜) in atTop, eval x Q = 0 → eval x P = 0\nh : degree P = degree Q\n⊢ (fun x => eval x P) =O[atTop] fun x => eval x Q",
"state_before": "case neg\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : degree P ≤ degree Q\nhp : ¬P = 0\nhq : Q ≠ 0\nhPQ : ∀ᶠ (x : 𝕜) in atTop, eval x Q = 0 → eval x P = 0\n⊢ (fun x => eval x P) =O[atTop] fun x => eval x Q",
"tactic": "cases' le_iff_lt_or_eq.mp h with h h"
},
{
"state_after": "no goals",
"state_before": "case neg.inl\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh✝ : degree P ≤ degree Q\nhp : ¬P = 0\nhq : Q ≠ 0\nhPQ : ∀ᶠ (x : 𝕜) in atTop, eval x Q = 0 → eval x P = 0\nh : degree P < degree Q\n⊢ (fun x => eval x P) =O[atTop] fun x => eval x Q",
"tactic": "exact isBigO_of_div_tendsto_nhds hPQ 0 (div_tendsto_zero_of_degree_lt P Q h)"
},
{
"state_after": "no goals",
"state_before": "case neg.inr\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh✝ : degree P ≤ degree Q\nhp : ¬P = 0\nhq : Q ≠ 0\nhPQ : ∀ᶠ (x : 𝕜) in atTop, eval x Q = 0 → eval x P = 0\nh : degree P = degree Q\n⊢ (fun x => eval x P) =O[atTop] fun x => eval x Q",
"tactic": "exact isBigO_of_div_tendsto_nhds hPQ _ (div_tendsto_leadingCoeff_div_of_degree_eq P Q h)"
}
] |
[
240,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.cof_lsub_def_nonempty
|
[] |
[
227,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Algebra/Star/SelfAdjoint.lean
|
selfAdjoint.val_zpow
|
[] |
[
399,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
398,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
uniformity_lift_le_comp
|
[] |
[
564,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
557,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.rangeFactorization_coe
|
[] |
[
1054,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1053,
1
] |
Mathlib/Algebra/Polynomial/BigOperators.lean
|
Polynomial.coeff_prod_of_natDegree_le
|
[
{
"state_after": "case mk\nR : Type u\nι : Type w\ninst✝ : CommSemiring R\nf✝ : ι → R[X]\nt : Multiset R[X]\nf : ι → R[X]\nn : ℕ\nl : Multiset ι\nhl : Nodup l\nh : ∀ (p : ι), p ∈ { val := l, nodup := hl } → natDegree (f p) ≤ n\n⊢ coeff (∏ i in { val := l, nodup := hl }, f i) (Finset.card { val := l, nodup := hl } * n) =\n ∏ i in { val := l, nodup := hl }, coeff (f i) n",
"state_before": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf✝ : ι → R[X]\nt : Multiset R[X]\nf : ι → R[X]\nn : ℕ\nh : ∀ (p : ι), p ∈ s → natDegree (f p) ≤ n\n⊢ coeff (∏ i in s, f i) (Finset.card s * n) = ∏ i in s, coeff (f i) n",
"tactic": "cases' s with l hl"
},
{
"state_after": "case h.e'_2.h.e'_4.h.e'_5\nR : Type u\nι : Type w\ninst✝ : CommSemiring R\nf✝ : ι → R[X]\nt : Multiset R[X]\nf : ι → R[X]\nn : ℕ\nl : Multiset ι\nhl : Nodup l\nh : ∀ (p : ι), p ∈ { val := l, nodup := hl } → natDegree (f p) ≤ n\n⊢ Finset.card { val := l, nodup := hl } = ↑Multiset.card (Multiset.map f l)\n\ncase h.e'_3\nR : Type u\nι : Type w\ninst✝ : CommSemiring R\nf✝ : ι → R[X]\nt : Multiset R[X]\nf : ι → R[X]\nn : ℕ\nl : Multiset ι\nhl : Nodup l\nh : ∀ (p : ι), p ∈ { val := l, nodup := hl } → natDegree (f p) ≤ n\n⊢ ∏ i in { val := l, nodup := hl }, coeff (f i) n = prod (Multiset.map (fun p => coeff p n) (Multiset.map f l))\n\ncase mk\nR : Type u\nι : Type w\ninst✝ : CommSemiring R\nf✝ : ι → R[X]\nt : Multiset R[X]\nf : ι → R[X]\nn : ℕ\nl : Multiset ι\nhl : Nodup l\nh : ∀ (p : ι), p ∈ { val := l, nodup := hl } → natDegree (f p) ≤ n\n⊢ ∀ (p : R[X]), p ∈ Multiset.map f l → natDegree p ≤ n",
"state_before": "case mk\nR : Type u\nι : Type w\ninst✝ : CommSemiring R\nf✝ : ι → R[X]\nt : Multiset R[X]\nf : ι → R[X]\nn : ℕ\nl : Multiset ι\nhl : Nodup l\nh : ∀ (p : ι), p ∈ { val := l, nodup := hl } → natDegree (f p) ≤ n\n⊢ coeff (∏ i in { val := l, nodup := hl }, f i) (Finset.card { val := l, nodup := hl } * n) =\n ∏ i in { val := l, nodup := hl }, coeff (f i) n",
"tactic": "convert coeff_multiset_prod_of_natDegree_le (l.map f) n ?_"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_4.h.e'_5\nR : Type u\nι : Type w\ninst✝ : CommSemiring R\nf✝ : ι → R[X]\nt : Multiset R[X]\nf : ι → R[X]\nn : ℕ\nl : Multiset ι\nhl : Nodup l\nh : ∀ (p : ι), p ∈ { val := l, nodup := hl } → natDegree (f p) ≤ n\n⊢ Finset.card { val := l, nodup := hl } = ↑Multiset.card (Multiset.map f l)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nR : Type u\nι : Type w\ninst✝ : CommSemiring R\nf✝ : ι → R[X]\nt : Multiset R[X]\nf : ι → R[X]\nn : ℕ\nl : Multiset ι\nhl : Nodup l\nh : ∀ (p : ι), p ∈ { val := l, nodup := hl } → natDegree (f p) ≤ n\n⊢ ∏ i in { val := l, nodup := hl }, coeff (f i) n = prod (Multiset.map (fun p => coeff p n) (Multiset.map f l))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case mk\nR : Type u\nι : Type w\ninst✝ : CommSemiring R\nf✝ : ι → R[X]\nt : Multiset R[X]\nf : ι → R[X]\nn : ℕ\nl : Multiset ι\nhl : Nodup l\nh : ∀ (p : ι), p ∈ { val := l, nodup := hl } → natDegree (f p) ≤ n\n⊢ ∀ (p : R[X]), p ∈ Multiset.map f l → natDegree p ≤ n",
"tactic": "simpa using h"
}
] |
[
226,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/Order/Lattice.lean
|
sup_sup_distrib_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : SemilatticeSup α\na✝ b✝ c✝ d a b c : α\n⊢ a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c)",
"tactic": "rw [sup_sup_sup_comm, sup_idem]"
}
] |
[
290,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/Combinatorics/Composition.lean
|
List.join_splitWrtComposition
|
[] |
[
768,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
766,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.NodupKeys.kerase
|
[] |
[
477,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
476,
1
] |
Mathlib/RingTheory/WittVector/MulP.lean
|
WittVector.bind₁_wittMulN_wittPolynomial
|
[
{
"state_after": "case zero\np : ℕ\nR : Type ?u.51070\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nk : ℕ\n⊢ ↑(bind₁ (wittMulN p Nat.zero)) (wittPolynomial p ℤ k) = ↑Nat.zero * wittPolynomial p ℤ k\n\ncase succ\np : ℕ\nR : Type ?u.51070\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nk n : ℕ\nih : ↑(bind₁ (wittMulN p n)) (wittPolynomial p ℤ k) = ↑n * wittPolynomial p ℤ k\n⊢ ↑(bind₁ (wittMulN p (Nat.succ n))) (wittPolynomial p ℤ k) = ↑(Nat.succ n) * wittPolynomial p ℤ k",
"state_before": "p : ℕ\nR : Type ?u.51070\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nn k : ℕ\n⊢ ↑(bind₁ (wittMulN p n)) (wittPolynomial p ℤ k) = ↑n * wittPolynomial p ℤ k",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\np : ℕ\nR : Type ?u.51070\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nk : ℕ\n⊢ ↑(bind₁ (wittMulN p Nat.zero)) (wittPolynomial p ℤ k) = ↑Nat.zero * wittPolynomial p ℤ k",
"tactic": "simp [wittMulN, Nat.cast_zero, MulZeroClass.zero_mul, bind₁_zero_wittPolynomial]"
},
{
"state_after": "case succ\np : ℕ\nR : Type ?u.51070\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nk n : ℕ\nih : ↑(bind₁ (wittMulN p n)) (wittPolynomial p ℤ k) = ↑n * wittPolynomial p ℤ k\n⊢ ↑(bind₁ (Function.uncurry ![wittMulN p n, X]))\n (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (wittPolynomial p ℤ k)) (X 0 + X 1)) =\n ↑(Nat.succ n) * wittPolynomial p ℤ k",
"state_before": "case succ\np : ℕ\nR : Type ?u.51070\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nk n : ℕ\nih : ↑(bind₁ (wittMulN p n)) (wittPolynomial p ℤ k) = ↑n * wittPolynomial p ℤ k\n⊢ ↑(bind₁ (wittMulN p (Nat.succ n))) (wittPolynomial p ℤ k) = ↑(Nat.succ n) * wittPolynomial p ℤ k",
"tactic": "rw [wittMulN, ← bind₁_bind₁, wittAdd, wittStructureInt_prop]"
},
{
"state_after": "case succ\np : ℕ\nR : Type ?u.51070\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nk n : ℕ\nih : ↑(bind₁ (wittMulN p n)) (wittPolynomial p ℤ k) = ↑n * wittPolynomial p ℤ k\n⊢ ↑(bind₁ (Function.uncurry ![wittMulN p n, X])) (↑(rename (Prod.mk 0)) (wittPolynomial p ℤ k)) +\n ↑(bind₁ (Function.uncurry ![wittMulN p n, X])) (↑(rename (Prod.mk 1)) (wittPolynomial p ℤ k)) =\n (↑n + 1) * wittPolynomial p ℤ k",
"state_before": "case succ\np : ℕ\nR : Type ?u.51070\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nk n : ℕ\nih : ↑(bind₁ (wittMulN p n)) (wittPolynomial p ℤ k) = ↑n * wittPolynomial p ℤ k\n⊢ ↑(bind₁ (Function.uncurry ![wittMulN p n, X]))\n (↑(bind₁ fun i => ↑(rename (Prod.mk i)) (wittPolynomial p ℤ k)) (X 0 + X 1)) =\n ↑(Nat.succ n) * wittPolynomial p ℤ k",
"tactic": "simp only [AlgHom.map_add, Nat.cast_succ, bind₁_X_right]"
},
{
"state_after": "case succ\np : ℕ\nR : Type ?u.51070\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nk n : ℕ\nih : ↑(bind₁ (wittMulN p n)) (wittPolynomial p ℤ k) = ↑n * wittPolynomial p ℤ k\n⊢ ↑(bind₁ (Function.uncurry ![wittMulN p n, X] ∘ Prod.mk 0)) (wittPolynomial p ℤ k) +\n ↑(bind₁ (Function.uncurry ![wittMulN p n, X] ∘ Prod.mk 1)) (wittPolynomial p ℤ k) =\n ↑n * wittPolynomial p ℤ k + wittPolynomial p ℤ k",
"state_before": "case succ\np : ℕ\nR : Type ?u.51070\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nk n : ℕ\nih : ↑(bind₁ (wittMulN p n)) (wittPolynomial p ℤ k) = ↑n * wittPolynomial p ℤ k\n⊢ ↑(bind₁ (Function.uncurry ![wittMulN p n, X])) (↑(rename (Prod.mk 0)) (wittPolynomial p ℤ k)) +\n ↑(bind₁ (Function.uncurry ![wittMulN p n, X])) (↑(rename (Prod.mk 1)) (wittPolynomial p ℤ k)) =\n (↑n + 1) * wittPolynomial p ℤ k",
"tactic": "rw [add_mul, one_mul, bind₁_rename, bind₁_rename]"
},
{
"state_after": "no goals",
"state_before": "case succ\np : ℕ\nR : Type ?u.51070\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nk n : ℕ\nih : ↑(bind₁ (wittMulN p n)) (wittPolynomial p ℤ k) = ↑n * wittPolynomial p ℤ k\n⊢ ↑(bind₁ (Function.uncurry ![wittMulN p n, X] ∘ Prod.mk 0)) (wittPolynomial p ℤ k) +\n ↑(bind₁ (Function.uncurry ![wittMulN p n, X] ∘ Prod.mk 1)) (wittPolynomial p ℤ k) =\n ↑n * wittPolynomial p ℤ k + wittPolynomial p ℤ k",
"tactic": "simp only [ih, Function.uncurry, Function.comp, bind₁_X_left, AlgHom.id_apply,\n Matrix.cons_val_zero, Matrix.head_cons, Matrix.cons_val_one]"
}
] |
[
84,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Topology/MetricSpace/ThickenedIndicator.lean
|
thickenedIndicatorAux_mono
|
[] |
[
110,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
ContinuousLinearMap.comp_analyticOn
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g✝ : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx✝ : E\nr r' : ℝ≥0∞\ns : Set E\ng : F →L[𝕜] G\nh : AnalyticOn 𝕜 f s\nx : E\nhx : x ∈ s\n⊢ AnalyticAt 𝕜 (↑g ∘ f) x",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g✝ : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\ns : Set E\ng : F →L[𝕜] G\nh : AnalyticOn 𝕜 f s\n⊢ AnalyticOn 𝕜 (↑g ∘ f) s",
"tactic": "rintro x hx"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g✝ : E → F\np✝ pf pg : FormalMultilinearSeries 𝕜 E F\nx✝ : E\nr✝ r' : ℝ≥0∞\ns : Set E\ng : F →L[𝕜] G\nh : AnalyticOn 𝕜 f s\nx : E\nhx : x ∈ s\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nhp : HasFPowerSeriesOnBall f p x r\n⊢ AnalyticAt 𝕜 (↑g ∘ f) x",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g✝ : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx✝ : E\nr r' : ℝ≥0∞\ns : Set E\ng : F →L[𝕜] G\nh : AnalyticOn 𝕜 f s\nx : E\nhx : x ∈ s\n⊢ AnalyticAt 𝕜 (↑g ∘ f) x",
"tactic": "rcases h x hx with ⟨p, r, hp⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g✝ : E → F\np✝ pf pg : FormalMultilinearSeries 𝕜 E F\nx✝ : E\nr✝ r' : ℝ≥0∞\ns : Set E\ng : F →L[𝕜] G\nh : AnalyticOn 𝕜 f s\nx : E\nhx : x ∈ s\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nhp : HasFPowerSeriesOnBall f p x r\n⊢ AnalyticAt 𝕜 (↑g ∘ f) x",
"tactic": "exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩"
}
] |
[
641,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
637,
1
] |
Mathlib/Topology/UniformSpace/Equiv.lean
|
UniformEquiv.symm_comp_self
|
[] |
[
210,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.ball_antitone
|
[] |
[
746,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
745,
1
] |
Mathlib/CategoryTheory/Limits/Constructions/Pullbacks.lean
|
CategoryTheory.Limits.hasPushouts_of_hasBinaryCoproducts_of_hasCoequalizers
|
[] |
[
100,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.infinitesimal_zero
|
[] |
[
685,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
685,
1
] |
Mathlib/Combinatorics/Catalan.lean
|
Tree.coe_treesOfNumNodesEq
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ∀ (x : Tree Unit), x ∈ ↑(treesOfNumNodesEq n) ↔ x ∈ {x | numNodes x = n}",
"tactic": "simp"
}
] |
[
209,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.filter_def
|
[
{
"state_after": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\np : ι → Prop\ninst✝ : DecidablePred p\nf : Π₀ (i : ι), β i\ni : ι\n⊢ ↑(filter p f) i = ↑(mk (Finset.filter p (support f)) fun i => ↑f ↑i) i",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\np : ι → Prop\ninst✝ : DecidablePred p\nf : Π₀ (i : ι), β i\n⊢ filter p f = mk (Finset.filter p (support f)) fun i => ↑f ↑i",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\np : ι → Prop\ninst✝ : DecidablePred p\nf : Π₀ (i : ι), β i\ni : ι\n⊢ ↑(filter p f) i = ↑(mk (Finset.filter p (support f)) fun i => ↑f ↑i) i",
"tactic": "by_cases h1 : p i <;> by_cases h2 : f i ≠ 0 <;> simp at h2 <;> simp [h1, h2]"
}
] |
[
1251,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1250,
1
] |
Mathlib/Data/Nat/Squarefree.lean
|
Nat.sq_mul_squarefree_of_pos'
|
[
{
"state_after": "case intro.intro.intro.intro.intro\nn : ℕ\nh : 0 < n\na₁ b₁ : ℕ\nha₁ : 0 < a₁\nhb₁ : 0 < b₁\nhab₁ : b₁ ^ 2 * a₁ = n\nhab₂ : Squarefree a₁\n⊢ ∃ a b, (b + 1) ^ 2 * (a + 1) = n ∧ Squarefree (a + 1)",
"state_before": "n : ℕ\nh : 0 < n\n⊢ ∃ a b, (b + 1) ^ 2 * (a + 1) = n ∧ Squarefree (a + 1)",
"tactic": "obtain ⟨a₁, b₁, ha₁, hb₁, hab₁, hab₂⟩ := sq_mul_squarefree_of_pos h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nn : ℕ\nh : 0 < n\na₁ b₁ : ℕ\nha₁ : 0 < a₁\nhb₁ : 0 < b₁\nhab₁ : b₁ ^ 2 * a₁ = n\nhab₂ : Squarefree a₁\n⊢ ∃ a b, (b + 1) ^ 2 * (a + 1) = n ∧ Squarefree (a + 1)",
"tactic": "refine' ⟨a₁.pred, b₁.pred, _, _⟩ <;> simpa only [add_one, succ_pred_eq_of_pos, ha₁, hb₁]"
}
] |
[
358,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
differentiableAt_id
|
[] |
[
1008,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1007,
1
] |
Mathlib/RingTheory/FiniteType.lean
|
Subalgebra.fg_iff_finiteType
|
[] |
[
184,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
deriv_zero_of_not_differentiableAt
|
[
{
"state_after": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nh : ¬DifferentiableAt 𝕜 f x\n⊢ ↑(fderiv 𝕜 f x) 1 = 0",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nh : ¬DifferentiableAt 𝕜 f x\n⊢ deriv f x = 0",
"tactic": "unfold deriv"
},
{
"state_after": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nh : ¬DifferentiableAt 𝕜 f x\n⊢ ↑0 1 = 0",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nh : ¬DifferentiableAt 𝕜 f x\n⊢ ↑(fderiv 𝕜 f x) 1 = 0",
"tactic": "rw [fderiv_zero_of_not_differentiableAt h]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nh : ¬DifferentiableAt 𝕜 f x\n⊢ ↑0 1 = 0",
"tactic": "simp"
}
] |
[
244,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/Logic/Basic.lean
|
exists_unique_false
|
[] |
[
863,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
863,
9
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
image_le_of_deriv_right_lt_deriv_boundary
|
[] |
[
213,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Topology/Support.lean
|
HasCompactMulSupport.isCompact_range
|
[
{
"state_after": "case inl\nX : Type ?u.12625\nα : Type u_2\nα' : Type ?u.12631\nβ : Type u_1\nγ : Type ?u.12637\nδ : Type ?u.12640\nM : Type ?u.12643\nE : Type ?u.12646\nR : Type ?u.12649\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace α'\ninst✝³ : One β\ninst✝² : One γ\ninst✝¹ : One δ\ng : β → γ\nf : α → β\nf₂ : α → γ\nm : β → γ → δ\nx : α\ninst✝ : TopologicalSpace β\nh : HasCompactMulSupport f\nhf : Continuous f\nh2 : range f = f '' mulTSupport f\n⊢ IsCompact (f '' mulTSupport f)\n\ncase inr\nX : Type ?u.12625\nα : Type u_2\nα' : Type ?u.12631\nβ : Type u_1\nγ : Type ?u.12637\nδ : Type ?u.12640\nM : Type ?u.12643\nE : Type ?u.12646\nR : Type ?u.12649\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace α'\ninst✝³ : One β\ninst✝² : One γ\ninst✝¹ : One δ\ng : β → γ\nf : α → β\nf₂ : α → γ\nm : β → γ → δ\nx : α\ninst✝ : TopologicalSpace β\nh : HasCompactMulSupport f\nhf : Continuous f\nh2 : range f = insert 1 (f '' mulTSupport f)\n⊢ IsCompact (insert 1 (f '' mulTSupport f))",
"state_before": "X : Type ?u.12625\nα : Type u_2\nα' : Type ?u.12631\nβ : Type u_1\nγ : Type ?u.12637\nδ : Type ?u.12640\nM : Type ?u.12643\nE : Type ?u.12646\nR : Type ?u.12649\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace α'\ninst✝³ : One β\ninst✝² : One γ\ninst✝¹ : One δ\ng : β → γ\nf : α → β\nf₂ : α → γ\nm : β → γ → δ\nx : α\ninst✝ : TopologicalSpace β\nh : HasCompactMulSupport f\nhf : Continuous f\n⊢ IsCompact (range f)",
"tactic": "cases' range_eq_image_mulTSupport_or f with h2 h2 <;> rw [h2]"
},
{
"state_after": "no goals",
"state_before": "case inl\nX : Type ?u.12625\nα : Type u_2\nα' : Type ?u.12631\nβ : Type u_1\nγ : Type ?u.12637\nδ : Type ?u.12640\nM : Type ?u.12643\nE : Type ?u.12646\nR : Type ?u.12649\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace α'\ninst✝³ : One β\ninst✝² : One γ\ninst✝¹ : One δ\ng : β → γ\nf : α → β\nf₂ : α → γ\nm : β → γ → δ\nx : α\ninst✝ : TopologicalSpace β\nh : HasCompactMulSupport f\nhf : Continuous f\nh2 : range f = f '' mulTSupport f\n⊢ IsCompact (f '' mulTSupport f)\n\ncase inr\nX : Type ?u.12625\nα : Type u_2\nα' : Type ?u.12631\nβ : Type u_1\nγ : Type ?u.12637\nδ : Type ?u.12640\nM : Type ?u.12643\nE : Type ?u.12646\nR : Type ?u.12649\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace α'\ninst✝³ : One β\ninst✝² : One γ\ninst✝¹ : One δ\ng : β → γ\nf : α → β\nf₂ : α → γ\nm : β → γ → δ\nx : α\ninst✝ : TopologicalSpace β\nh : HasCompactMulSupport f\nhf : Continuous f\nh2 : range f = insert 1 (f '' mulTSupport f)\n⊢ IsCompact (insert 1 (f '' mulTSupport f))",
"tactic": "exacts [h.image hf, (h.image hf).insert 1]"
}
] |
[
184,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.lift'_map_le
|
[] |
[
302,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Algebra/Category/Ring/Basic.lean
|
SemiRingCat.ofHom_apply
|
[] |
[
130,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Algebra/Ring/CompTypeclasses.lean
|
RingHom.surjective
|
[] |
[
168,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Algebra/Module/Torsion.lean
|
Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInf
|
[
{
"state_after": "case inl\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : S = ∅\n⊢ (⨆ (i : ι) (_ : i ∈ S), torsionBySet R M ↑(p i)) = torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\n\ncase inr\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\n⊢ (⨆ (i : ι) (_ : i ∈ S), torsionBySet R M ↑(p i)) = torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)",
"state_before": "R : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\n⊢ (⨆ (i : ι) (_ : i ∈ S), torsionBySet R M ↑(p i)) = torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)",
"tactic": "cases' S.eq_empty_or_nonempty with h h"
},
{
"state_after": "case inr.a\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\n⊢ (⨆ (i : ι) (_ : i ∈ S), torsionBySet R M ↑(p i)) ≤ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\n\ncase inr.a\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\n⊢ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i) ≤ ⨆ (i : ι) (_ : i ∈ S), torsionBySet R M ↑(p i)",
"state_before": "case inr\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\n⊢ (⨆ (i : ι) (_ : i ∈ S), torsionBySet R M ↑(p i)) = torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)",
"tactic": "apply le_antisymm"
},
{
"state_after": "case inl\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : S = ∅\n⊢ (⨆ (i : ι) (_ : i ∈ ∅), torsionBySet R M ↑(p i)) = torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ ∅), p i)",
"state_before": "case inl\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : S = ∅\n⊢ (⨆ (i : ι) (_ : i ∈ S), torsionBySet R M ↑(p i)) = torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)",
"tactic": "simp only [h]"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : S = ∅\n⊢ (⨆ (i : ι) (_ : i ∈ ∅), torsionBySet R M ↑(p i)) = torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ ∅), p i)",
"tactic": "convert iSup_emptyset (f := fun i => torsionBySet R M (p i)) <;> simp"
},
{
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"tactic": "apply iSup_le _"
},
{
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"state_before": "R : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\n⊢ ∀ (i : ι), (⨆ (_ : i ∈ S), torsionBySet R M ↑(p i)) ≤ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)",
"tactic": "intro i"
},
{
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"tactic": "apply iSup_le _"
},
{
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"state_before": "R : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\ni : ι\n⊢ i ∈ S → torsionBySet R M ↑(p i) ≤ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)",
"tactic": "intro is"
},
{
"state_after": "case st\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\ni : ι\nis : i ∈ S\n⊢ ↑(⨅ (i : ι) (_ : i ∈ S), p i) ⊆ ↑(p i)",
"state_before": "R : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\ni : ι\nis : i ∈ S\n⊢ torsionBySet R M ↑(p i) ≤ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)",
"tactic": "apply torsionBySet_le_torsionBySet_of_subset"
},
{
"state_after": "no goals",
"state_before": "case st\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\ni : ι\nis : i ∈ S\n⊢ ↑(⨅ (i : ι) (_ : i ∈ S), p i) ⊆ ↑(p i)",
"tactic": "exact (iInf_le (fun i => ⨅ _ : i ∈ S, p i) i).trans (iInf_le _ is)"
},
{
"state_after": "case inr.a\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : x ∈ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\n⊢ x ∈ ⨆ (i : ι) (_ : i ∈ S), torsionBySet R M ↑(p i)",
"state_before": "case inr.a\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\n⊢ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i) ≤ ⨆ (i : ι) (_ : i ∈ S), torsionBySet R M ↑(p i)",
"tactic": "intro x hx"
},
{
"state_after": "case inr.a\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : x ∈ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\n⊢ ∃ μ, ∑ i in S, ↑(μ i) = x",
"state_before": "case inr.a\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : x ∈ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\n⊢ x ∈ ⨆ (i : ι) (_ : i ∈ S), torsionBySet R M ↑(p i)",
"tactic": "rw [mem_iSup_finset_iff_exists_sum]"
},
{
"state_after": "case inr.a.intro\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : x ∈ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\n⊢ ∃ μ, ∑ i in S, ↑(μ i) = x",
"state_before": "case inr.a\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : x ∈ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\n⊢ ∃ μ, ∑ i in S, ↑(μ i) = x",
"tactic": "obtain ⟨μ, hμ⟩ :=\n (mem_iSup_finset_iff_exists_sum _ _).mp\n ((Ideal.eq_top_iff_one _).mp <| (Ideal.iSup_iInf_eq_top_iff_pairwise h _).mpr hp)"
},
{
"state_after": "case inr.a.intro.refine'_1\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : x ∈ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\n⊢ ↑(μ i) • x ∈ torsionBySet R M ↑(p i)\n\ncase inr.a.intro.refine'_2\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : x ∈ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\n⊢ ∑ i in S, ↑((fun i => { val := ↑(μ i) • x, property := (_ : ↑(μ i) • x ∈ torsionBySet R M ↑(p i)) }) i) = x",
"state_before": "case inr.a.intro\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : x ∈ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\n⊢ ∃ μ, ∑ i in S, ↑(μ i) = x",
"tactic": "refine' ⟨fun i => ⟨(μ i : R) • x, _⟩, _⟩"
},
{
"state_after": "case inr.a.intro.refine'_1\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\n⊢ ∀ (a : ↑↑(p i)), ↑a • ↑(μ i) • x = 0",
"state_before": "case inr.a.intro.refine'_1\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : x ∈ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\n⊢ ↑(μ i) • x ∈ torsionBySet R M ↑(p i)",
"tactic": "rw [mem_torsionBySet_iff] at hx⊢"
},
{
"state_after": "case inr.a.intro.refine'_1.mk\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\n⊢ ↑{ val := a, property := ha } • ↑(μ i) • x = 0",
"state_before": "case inr.a.intro.refine'_1\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\n⊢ ∀ (a : ↑↑(p i)), ↑a • ↑(μ i) • x = 0",
"tactic": "rintro ⟨a, ha⟩"
},
{
"state_after": "case inr.a.intro.refine'_1.mk\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\n⊢ (↑{ val := a, property := ha } * ↑(μ i)) • x = 0",
"state_before": "case inr.a.intro.refine'_1.mk\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\n⊢ ↑{ val := a, property := ha } • ↑(μ i) • x = 0",
"tactic": "rw [smul_smul]"
},
{
"state_after": "case inr.a.intro.refine'_1.mk\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nthis : a * ↑(μ i) ∈ ⨅ (i : ι) (_ : i ∈ S), p i\n⊢ (↑{ val := a, property := ha } * ↑(μ i)) • x = 0\n\ncase this\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\n⊢ a * ↑(μ i) ∈ ⨅ (i : ι) (_ : i ∈ S), p i",
"state_before": "case inr.a.intro.refine'_1.mk\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\n⊢ (↑{ val := a, property := ha } * ↑(μ i)) • x = 0",
"tactic": "suffices : a * μ i ∈ ⨅ i ∈ S, p i"
},
{
"state_after": "case this\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\n⊢ a * ↑(μ i) ∈ ⨅ (i : ι) (_ : i ∈ S), p i",
"state_before": "case inr.a.intro.refine'_1.mk\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nthis : a * ↑(μ i) ∈ ⨅ (i : ι) (_ : i ∈ S), p i\n⊢ (↑{ val := a, property := ha } * ↑(μ i)) • x = 0\n\ncase this\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\n⊢ a * ↑(μ i) ∈ ⨅ (i : ι) (_ : i ∈ S), p i",
"tactic": "exact hx ⟨_, this⟩"
},
{
"state_after": "case this\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\n⊢ ∀ (i_1 : ι), a * ↑(μ i) ∈ ⨅ (_ : i_1 ∈ S), p i_1",
"state_before": "case this\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\n⊢ a * ↑(μ i) ∈ ⨅ (i : ι) (_ : i ∈ S), p i",
"tactic": "rw [mem_iInf]"
},
{
"state_after": "case this\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\n⊢ a * ↑(μ i) ∈ ⨅ (_ : j ∈ S), p j",
"state_before": "case this\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\n⊢ ∀ (i_1 : ι), a * ↑(μ i) ∈ ⨅ (_ : i_1 ∈ S), p i_1",
"tactic": "intro j"
},
{
"state_after": "case this\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\n⊢ j ∈ S → a * ↑(μ i) ∈ p j",
"state_before": "case this\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\n⊢ a * ↑(μ i) ∈ ⨅ (_ : j ∈ S), p j",
"tactic": "rw [mem_iInf]"
},
{
"state_after": "case this\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\nhj : j ∈ S\n⊢ a * ↑(μ i) ∈ p j",
"state_before": "case this\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\n⊢ j ∈ S → a * ↑(μ i) ∈ p j",
"tactic": "intro hj"
},
{
"state_after": "case pos\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\nhj : j ∈ S\nij : j = i\n⊢ a * ↑(μ i) ∈ p j\n\ncase neg\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\nhj : j ∈ S\nij : ¬j = i\n⊢ a * ↑(μ i) ∈ p j",
"state_before": "case this\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\nhj : j ∈ S\n⊢ a * ↑(μ i) ∈ p j",
"tactic": "by_cases ij : j = i"
},
{
"state_after": "case pos\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\nhj : j ∈ S\nij : j = i\n⊢ a * ↑(μ i) ∈ p i",
"state_before": "case pos\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\nhj : j ∈ S\nij : j = i\n⊢ a * ↑(μ i) ∈ p j",
"tactic": "rw [ij]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\nhj : j ∈ S\nij : j = i\n⊢ a * ↑(μ i) ∈ p i",
"tactic": "exact Ideal.mul_mem_right _ _ ha"
},
{
"state_after": "case neg\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\nhj : j ∈ S\nij : ¬j = i\nthis : ↑(μ i) ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j\n⊢ a * ↑(μ i) ∈ p j",
"state_before": "case neg\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\nhj : j ∈ S\nij : ¬j = i\n⊢ a * ↑(μ i) ∈ p j",
"tactic": "have := coe_mem (μ i)"
},
{
"state_after": "case neg\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\nhj : j ∈ S\nij : ¬j = i\nthis : ∀ (i_1 : ι), i_1 ∈ S → i_1 ≠ i → ↑(μ i) ∈ p i_1\n⊢ a * ↑(μ i) ∈ p j",
"state_before": "case neg\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\nhj : j ∈ S\nij : ¬j = i\nthis : ↑(μ i) ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j\n⊢ a * ↑(μ i) ∈ p j",
"tactic": "simp only [mem_iInf] at this"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na✝ : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : ∀ (a : ↑↑(⨅ (i : ι) (_ : i ∈ S), p i)), ↑a • x = 0\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\ni : ι\na : R\nha : a ∈ ↑(p i)\nj : ι\nhj : j ∈ S\nij : ¬j = i\nthis : ∀ (i_1 : ι), i_1 ∈ S → i_1 ≠ i → ↑(μ i) ∈ p i_1\n⊢ a * ↑(μ i) ∈ p j",
"tactic": "exact Ideal.mul_mem_left _ _ (this j hj ij)"
},
{
"state_after": "case inr.a.intro.refine'_2\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : x ∈ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\n⊢ ∑ x_1 in S, ↑(μ x_1) • x = x",
"state_before": "case inr.a.intro.refine'_2\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : x ∈ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\n⊢ ∑ i in S, ↑((fun i => { val := ↑(μ i) • x, property := (_ : ↑(μ i) • x ∈ torsionBySet R M ↑(p i)) }) i) = x",
"tactic": "simp_rw [coe_mk]"
},
{
"state_after": "no goals",
"state_before": "case inr.a.intro.refine'_2\nR : Type u_3\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set R\na : R\nι : Type u_1\np : ι → Ideal R\nS : Finset ι\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\ninst✝ : DecidableEq ι\nh : Finset.Nonempty S\nx : M\nhx : x ∈ torsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\nμ : (i : ι) → { x // x ∈ ⨅ (j : ι) (_ : j ∈ S) (_ : j ≠ i), p j }\nhμ : ∑ i in S, ↑(μ i) = 1\n⊢ ∑ x_1 in S, ↑(μ x_1) • x = x",
"tactic": "rw [← Finset.sum_smul, hμ, one_smul]"
}
] |
[
431,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.norm_mkPiField
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nz : G\n⊢ ‖z‖ ≤\n ‖MultilinearMap.mkContinuous (MultilinearMap.mkPiRing 𝕜 ι z) ‖z‖\n (_ : ∀ (m : ι → 𝕜), ‖↑(MultilinearMap.mkPiRing 𝕜 ι z) m‖ ≤ ‖z‖ * ∏ i : ι, ‖m i‖)‖",
"tactic": "simpa using (ContinuousMultilinearMap.mkPiField 𝕜 ι z).le_op_norm fun _ => 1"
}
] |
[
908,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
906,
1
] |
Mathlib/LinearAlgebra/ProjectiveSpace/Basic.lean
|
Projectivization.map_comp
|
[
{
"state_after": "case h.mk\nK : Type u_3\nV : Type u_5\ninst✝⁹ : DivisionRing K\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module K V\nL : Type u_4\nW : Type u_6\ninst✝⁶ : DivisionRing L\ninst✝⁵ : AddCommGroup W\ninst✝⁴ : Module L W\nF : Type u_1\nU : Type u_2\ninst✝³ : Field F\ninst✝² : AddCommGroup U\ninst✝¹ : Module F U\nσ : K →+* L\nτ : L →+* F\nγ : K →+* F\ninst✝ : RingHomCompTriple σ τ γ\nf : V →ₛₗ[σ] W\nhf : Function.Injective ↑f\ng : W →ₛₗ[τ] U\nhg : Function.Injective ↑g\nx✝ : ℙ K V\nv : { v // v ≠ 0 }\n⊢ map (LinearMap.comp g f) (_ : Function.Injective (↑g ∘ fun x => ↑f x)) (Quot.mk Setoid.r v) =\n (map g hg ∘ map f hf) (Quot.mk Setoid.r v)",
"state_before": "K : Type u_3\nV : Type u_5\ninst✝⁹ : DivisionRing K\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module K V\nL : Type u_4\nW : Type u_6\ninst✝⁶ : DivisionRing L\ninst✝⁵ : AddCommGroup W\ninst✝⁴ : Module L W\nF : Type u_1\nU : Type u_2\ninst✝³ : Field F\ninst✝² : AddCommGroup U\ninst✝¹ : Module F U\nσ : K →+* L\nτ : L →+* F\nγ : K →+* F\ninst✝ : RingHomCompTriple σ τ γ\nf : V →ₛₗ[σ] W\nhf : Function.Injective ↑f\ng : W →ₛₗ[τ] U\nhg : Function.Injective ↑g\n⊢ map (LinearMap.comp g f) (_ : Function.Injective (↑g ∘ fun x => ↑f x)) = map g hg ∘ map f hf",
"tactic": "ext ⟨v⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nK : Type u_3\nV : Type u_5\ninst✝⁹ : DivisionRing K\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module K V\nL : Type u_4\nW : Type u_6\ninst✝⁶ : DivisionRing L\ninst✝⁵ : AddCommGroup W\ninst✝⁴ : Module L W\nF : Type u_1\nU : Type u_2\ninst✝³ : Field F\ninst✝² : AddCommGroup U\ninst✝¹ : Module F U\nσ : K →+* L\nτ : L →+* F\nγ : K →+* F\ninst✝ : RingHomCompTriple σ τ γ\nf : V →ₛₗ[σ] W\nhf : Function.Injective ↑f\ng : W →ₛₗ[τ] U\nhg : Function.Injective ↑g\nx✝ : ℙ K V\nv : { v // v ≠ 0 }\n⊢ map (LinearMap.comp g f) (_ : Function.Injective (↑g ∘ fun x => ↑f x)) (Quot.mk Setoid.r v) =\n (map g hg ∘ map f hf) (Quot.mk Setoid.r v)",
"tactic": "rfl"
}
] |
[
232,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/CategoryTheory/Sites/Whiskering.lean
|
CategoryTheory.Presheaf.IsSheaf.comp
|
[
{
"state_after": "C : Type u₁\ninst✝³ : Category C\nA : Type u₂\ninst✝² : Category A\nB : Type u₃\ninst✝¹ : Category B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF : A ⥤ B\ninst✝ :\n (X : C) →\n (S : GrothendieckTopology.Cover J X) →\n (P : Cᵒᵖ ⥤ A) → PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\nP : Cᵒᵖ ⥤ A\nhP : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\n⊢ ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S (P ⋙ F)))",
"state_before": "C : Type u₁\ninst✝³ : Category C\nA : Type u₂\ninst✝² : Category A\nB : Type u₃\ninst✝¹ : Category B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF : A ⥤ B\ninst✝ :\n (X : C) →\n (S : GrothendieckTopology.Cover J X) →\n (P : Cᵒᵖ ⥤ A) → PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\nP : Cᵒᵖ ⥤ A\nhP : IsSheaf J P\n⊢ IsSheaf J (P ⋙ F)",
"tactic": "rw [Presheaf.isSheaf_iff_multifork] at hP⊢"
},
{
"state_after": "C : Type u₁\ninst✝³ : Category C\nA : Type u₂\ninst✝² : Category A\nB : Type u₃\ninst✝¹ : Category B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF : A ⥤ B\ninst✝ :\n (X : C) →\n (S : GrothendieckTopology.Cover J X) →\n (P : Cᵒᵖ ⥤ A) → PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\nP : Cᵒᵖ ⥤ A\nhP : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nX : C\nS : GrothendieckTopology.Cover J X\n⊢ Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S (P ⋙ F)))",
"state_before": "C : Type u₁\ninst✝³ : Category C\nA : Type u₂\ninst✝² : Category A\nB : Type u₃\ninst✝¹ : Category B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF : A ⥤ B\ninst✝ :\n (X : C) →\n (S : GrothendieckTopology.Cover J X) →\n (P : Cᵒᵖ ⥤ A) → PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\nP : Cᵒᵖ ⥤ A\nhP : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\n⊢ ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S (P ⋙ F)))",
"tactic": "intro X S"
},
{
"state_after": "case intro\nC : Type u₁\ninst✝³ : Category C\nA : Type u₂\ninst✝² : Category A\nB : Type u₃\ninst✝¹ : Category B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF : A ⥤ B\ninst✝ :\n (X : C) →\n (S : GrothendieckTopology.Cover J X) →\n (P : Cᵒᵖ ⥤ A) → PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\nP : Cᵒᵖ ⥤ A\nhP : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nX : C\nS : GrothendieckTopology.Cover J X\nh : IsLimit (GrothendieckTopology.Cover.multifork S P)\n⊢ Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S (P ⋙ F)))",
"state_before": "C : Type u₁\ninst✝³ : Category C\nA : Type u₂\ninst✝² : Category A\nB : Type u₃\ninst✝¹ : Category B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF : A ⥤ B\ninst✝ :\n (X : C) →\n (S : GrothendieckTopology.Cover J X) →\n (P : Cᵒᵖ ⥤ A) → PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\nP : Cᵒᵖ ⥤ A\nhP : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nX : C\nS : GrothendieckTopology.Cover J X\n⊢ Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S (P ⋙ F)))",
"tactic": "obtain ⟨h⟩ := hP X S"
},
{
"state_after": "case intro\nC : Type u₁\ninst✝³ : Category C\nA : Type u₂\ninst✝² : Category A\nB : Type u₃\ninst✝¹ : Category B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF : A ⥤ B\ninst✝ :\n (X : C) →\n (S : GrothendieckTopology.Cover J X) →\n (P : Cᵒᵖ ⥤ A) → PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\nP : Cᵒᵖ ⥤ A\nhP : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nX : C\nS : GrothendieckTopology.Cover J X\nh : IsLimit (F.mapCone (GrothendieckTopology.Cover.multifork S P))\n⊢ Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S (P ⋙ F)))",
"state_before": "case intro\nC : Type u₁\ninst✝³ : Category C\nA : Type u₂\ninst✝² : Category A\nB : Type u₃\ninst✝¹ : Category B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF : A ⥤ B\ninst✝ :\n (X : C) →\n (S : GrothendieckTopology.Cover J X) →\n (P : Cᵒᵖ ⥤ A) → PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\nP : Cᵒᵖ ⥤ A\nhP : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nX : C\nS : GrothendieckTopology.Cover J X\nh : IsLimit (GrothendieckTopology.Cover.multifork S P)\n⊢ Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S (P ⋙ F)))",
"tactic": "replace h := isLimitOfPreserves F h"
},
{
"state_after": "case intro\nC : Type u₁\ninst✝³ : Category C\nA : Type u₂\ninst✝² : Category A\nB : Type u₃\ninst✝¹ : Category B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF : A ⥤ B\ninst✝ :\n (X : C) →\n (S : GrothendieckTopology.Cover J X) →\n (P : Cᵒᵖ ⥤ A) → PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\nP : Cᵒᵖ ⥤ A\nhP : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nX : C\nS : GrothendieckTopology.Cover J X\nh :\n IsLimit\n ((Cones.postcompose (GrothendieckTopology.Cover.multicospanComp F P S).hom).obj\n (GrothendieckTopology.Cover.multifork S (P ⋙ F)))\n⊢ Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S (P ⋙ F)))",
"state_before": "case intro\nC : Type u₁\ninst✝³ : Category C\nA : Type u₂\ninst✝² : Category A\nB : Type u₃\ninst✝¹ : Category B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF : A ⥤ B\ninst✝ :\n (X : C) →\n (S : GrothendieckTopology.Cover J X) →\n (P : Cᵒᵖ ⥤ A) → PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\nP : Cᵒᵖ ⥤ A\nhP : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nX : C\nS : GrothendieckTopology.Cover J X\nh : IsLimit (F.mapCone (GrothendieckTopology.Cover.multifork S P))\n⊢ Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S (P ⋙ F)))",
"tactic": "replace h := Limits.IsLimit.ofIsoLimit h (S.mapMultifork F P)"
},
{
"state_after": "no goals",
"state_before": "case intro\nC : Type u₁\ninst✝³ : Category C\nA : Type u₂\ninst✝² : Category A\nB : Type u₃\ninst✝¹ : Category B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF : A ⥤ B\ninst✝ :\n (X : C) →\n (S : GrothendieckTopology.Cover J X) →\n (P : Cᵒᵖ ⥤ A) → PreservesLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) F\nP : Cᵒᵖ ⥤ A\nhP : ∀ (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nX : C\nS : GrothendieckTopology.Cover J X\nh :\n IsLimit\n ((Cones.postcompose (GrothendieckTopology.Cover.multicospanComp F P S).hom).obj\n (GrothendieckTopology.Cover.multifork S (P ⋙ F)))\n⊢ Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S (P ⋙ F)))",
"tactic": "exact ⟨Limits.IsLimit.postcomposeHomEquiv (S.multicospanComp F P) _ h⟩"
}
] |
[
129,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
LinearIsometryEquiv.coe_toAffineIsometryEquiv
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\nV : Type u_1\nV₁ : Type ?u.380007\nV₂ : Type u_2\nV₃ : Type ?u.380013\nV₄ : Type ?u.380016\nP₁ : Type ?u.380019\nP : Type ?u.380022\nP₂ : Type ?u.380025\nP₃ : Type ?u.380028\nP₄ : Type ?u.380031\ninst✝²⁰ : NormedField 𝕜\ninst✝¹⁹ : SeminormedAddCommGroup V\ninst✝¹⁸ : SeminormedAddCommGroup V₁\ninst✝¹⁷ : SeminormedAddCommGroup V₂\ninst✝¹⁶ : SeminormedAddCommGroup V₃\ninst✝¹⁵ : SeminormedAddCommGroup V₄\ninst✝¹⁴ : NormedSpace 𝕜 V\ninst✝¹³ : NormedSpace 𝕜 V₁\ninst✝¹² : NormedSpace 𝕜 V₂\ninst✝¹¹ : NormedSpace 𝕜 V₃\ninst✝¹⁰ : NormedSpace 𝕜 V₄\ninst✝⁹ : PseudoMetricSpace P\ninst✝⁸ : MetricSpace P₁\ninst✝⁷ : PseudoMetricSpace P₂\ninst✝⁶ : PseudoMetricSpace P₃\ninst✝⁵ : PseudoMetricSpace P₄\ninst✝⁴ : NormedAddTorsor V P\ninst✝³ : NormedAddTorsor V₁ P₁\ninst✝² : NormedAddTorsor V₂ P₂\ninst✝¹ : NormedAddTorsor V₃ P₃\ninst✝ : NormedAddTorsor V₄ P₄\ne : V ≃ₗᵢ[𝕜] V₂\n⊢ ↑(toAffineIsometryEquiv e) = ↑e",
"tactic": "rfl"
}
] |
[
410,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
409,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
one_div_le_one_div_of_neg
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.179667\nα : Type u_1\nβ : Type ?u.179673\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nha : a < 0\nhb : b < 0\n⊢ 1 / a ≤ 1 / b ↔ b ≤ a",
"tactic": "simpa [one_div] using inv_le_inv_of_neg ha hb"
}
] |
[
885,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
884,
1
] |
Mathlib/Algebra/GroupWithZero/Defs.lean
|
mul_left_injective₀
|
[] |
[
90,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/CategoryTheory/MorphismProperty.lean
|
CategoryTheory.MorphismProperty.RespectsIso.diagonal
|
[
{
"state_after": "case left\nC : Type u\ninst✝² : Category C\nD : Type ?u.69245\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\n⊢ ∀ {X Y Z : C} (e : X ≅ Y) (f : Y ⟶ Z), MorphismProperty.diagonal P f → MorphismProperty.diagonal P (e.hom ≫ f)\n\ncase right\nC : Type u\ninst✝² : Category C\nD : Type ?u.69245\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\n⊢ ∀ {X Y Z : C} (e : Y ≅ Z) (f : X ⟶ Y), MorphismProperty.diagonal P f → MorphismProperty.diagonal P (f ≫ e.hom)",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.69245\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\n⊢ RespectsIso (MorphismProperty.diagonal P)",
"tactic": "constructor"
},
{
"state_after": "case left\nC : Type u\ninst✝² : Category C\nD : Type ?u.69245\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\nX Y Z : C\ne : X ≅ Y\nf : Y ⟶ Z\nH : MorphismProperty.diagonal P f\n⊢ MorphismProperty.diagonal P (e.hom ≫ f)",
"state_before": "case left\nC : Type u\ninst✝² : Category C\nD : Type ?u.69245\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\n⊢ ∀ {X Y Z : C} (e : X ≅ Y) (f : Y ⟶ Z), MorphismProperty.diagonal P f → MorphismProperty.diagonal P (e.hom ≫ f)",
"tactic": "introv H"
},
{
"state_after": "no goals",
"state_before": "case left\nC : Type u\ninst✝² : Category C\nD : Type ?u.69245\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\nX Y Z : C\ne : X ≅ Y\nf : Y ⟶ Z\nH : MorphismProperty.diagonal P f\n⊢ MorphismProperty.diagonal P (e.hom ≫ f)",
"tactic": "rwa [diagonal_iff, pullback.diagonal_comp, hP.cancel_left_isIso, hP.cancel_left_isIso,\n ← hP.cancel_right_isIso _\n (pullback.map (e.hom ≫ f) (e.hom ≫ f) f f e.hom e.hom (𝟙 Z) (by simp) (by simp)),\n ← pullback.condition, hP.cancel_left_isIso]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.69245\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\nX Y Z : C\ne : X ≅ Y\nf : Y ⟶ Z\nH : MorphismProperty.diagonal P f\n⊢ (e.hom ≫ f) ≫ 𝟙 Z = e.hom ≫ f",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.69245\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\nX Y Z : C\ne : X ≅ Y\nf : Y ⟶ Z\nH : MorphismProperty.diagonal P f\n⊢ (e.hom ≫ f) ≫ 𝟙 Z = e.hom ≫ f",
"tactic": "simp"
},
{
"state_after": "case right\nC : Type u\ninst✝² : Category C\nD : Type ?u.69245\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\nX Y Z : C\ne : Y ≅ Z\nf : X ⟶ Y\nH : MorphismProperty.diagonal P f\n⊢ MorphismProperty.diagonal P (f ≫ e.hom)",
"state_before": "case right\nC : Type u\ninst✝² : Category C\nD : Type ?u.69245\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\n⊢ ∀ {X Y Z : C} (e : Y ≅ Z) (f : X ⟶ Y), MorphismProperty.diagonal P f → MorphismProperty.diagonal P (f ≫ e.hom)",
"tactic": "introv H"
},
{
"state_after": "case right\nC : Type u\ninst✝² : Category C\nD : Type ?u.69245\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\nX Y Z : C\ne : Y ≅ Z\nf : X ⟶ Y\nH : MorphismProperty.diagonal P f\n⊢ P (pullback.diagonal (f ≫ e.hom))",
"state_before": "case right\nC : Type u\ninst✝² : Category C\nD : Type ?u.69245\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\nX Y Z : C\ne : Y ≅ Z\nf : X ⟶ Y\nH : MorphismProperty.diagonal P f\n⊢ MorphismProperty.diagonal P (f ≫ e.hom)",
"tactic": "delta diagonal"
},
{
"state_after": "no goals",
"state_before": "case right\nC : Type u\ninst✝² : Category C\nD : Type ?u.69245\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\nX Y Z : C\ne : Y ≅ Z\nf : X ⟶ Y\nH : MorphismProperty.diagonal P f\n⊢ P (pullback.diagonal (f ≫ e.hom))",
"tactic": "rwa [pullback.diagonal_comp, hP.cancel_right_isIso]"
}
] |
[
539,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
530,
1
] |
Mathlib/Topology/Order.lean
|
TopologicalSpace.isOpen_top_iff
|
[
{
"state_after": "case basic\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU s✝ : Set α\na✝ : s✝ ∈ ↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)\n⊢ s✝ = ∅ ∨ s✝ = univ\n\ncase univ\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\n⊢ univ = ∅ ∨ univ = univ\n\ncase inter\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU s✝ t✝ : Set α\na✝¹ : GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s✝\na✝ : GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) t✝\na_ih✝¹ : s✝ = ∅ ∨ s✝ = univ\na_ih✝ : t✝ = ∅ ∨ t✝ = univ\n⊢ s✝ ∩ t✝ = ∅ ∨ s✝ ∩ t✝ = univ\n\ncase sUnion\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → s = ∅ ∨ s = univ\n⊢ ⋃₀ S✝ = ∅ ∨ ⋃₀ S✝ = univ",
"state_before": "α✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\nh : IsOpen U\n⊢ U = ∅ ∨ U = univ",
"tactic": "induction h"
},
{
"state_after": "case univ\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\n⊢ univ = ∅ ∨ univ = univ\n\ncase inter\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU s✝ t✝ : Set α\na✝¹ : GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s✝\na✝ : GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) t✝\na_ih✝¹ : s✝ = ∅ ∨ s✝ = univ\na_ih✝ : t✝ = ∅ ∨ t✝ = univ\n⊢ s✝ ∩ t✝ = ∅ ∨ s✝ ∩ t✝ = univ\n\ncase sUnion\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → s = ∅ ∨ s = univ\n⊢ ⋃₀ S✝ = ∅ ∨ ⋃₀ S✝ = univ",
"state_before": "case basic\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU s✝ : Set α\na✝ : s✝ ∈ ↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)\n⊢ s✝ = ∅ ∨ s✝ = univ\n\ncase univ\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\n⊢ univ = ∅ ∨ univ = univ\n\ncase inter\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU s✝ t✝ : Set α\na✝¹ : GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s✝\na✝ : GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) t✝\na_ih✝¹ : s✝ = ∅ ∨ s✝ = univ\na_ih✝ : t✝ = ∅ ∨ t✝ = univ\n⊢ s✝ ∩ t✝ = ∅ ∨ s✝ ∩ t✝ = univ\n\ncase sUnion\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → s = ∅ ∨ s = univ\n⊢ ⋃₀ S✝ = ∅ ∨ ⋃₀ S✝ = univ",
"tactic": "case basic h => exact False.elim h"
},
{
"state_after": "case inter\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU s✝ t✝ : Set α\na✝¹ : GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s✝\na✝ : GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) t✝\na_ih✝¹ : s✝ = ∅ ∨ s✝ = univ\na_ih✝ : t✝ = ∅ ∨ t✝ = univ\n⊢ s✝ ∩ t✝ = ∅ ∨ s✝ ∩ t✝ = univ\n\ncase sUnion\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → s = ∅ ∨ s = univ\n⊢ ⋃₀ S✝ = ∅ ∨ ⋃₀ S✝ = univ",
"state_before": "case univ\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\n⊢ univ = ∅ ∨ univ = univ\n\ncase inter\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU s✝ t✝ : Set α\na✝¹ : GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s✝\na✝ : GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) t✝\na_ih✝¹ : s✝ = ∅ ∨ s✝ = univ\na_ih✝ : t✝ = ∅ ∨ t✝ = univ\n⊢ s✝ ∩ t✝ = ∅ ∨ s✝ ∩ t✝ = univ\n\ncase sUnion\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → s = ∅ ∨ s = univ\n⊢ ⋃₀ S✝ = ∅ ∨ ⋃₀ S✝ = univ",
"tactic": "case univ => exact .inr rfl"
},
{
"state_after": "case sUnion\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → s = ∅ ∨ s = univ\n⊢ ⋃₀ S✝ = ∅ ∨ ⋃₀ S✝ = univ",
"state_before": "case inter\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU s✝ t✝ : Set α\na✝¹ : GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s✝\na✝ : GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) t✝\na_ih✝¹ : s✝ = ∅ ∨ s✝ = univ\na_ih✝ : t✝ = ∅ ∨ t✝ = univ\n⊢ s✝ ∩ t✝ = ∅ ∨ s✝ ∩ t✝ = univ\n\ncase sUnion\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → s = ∅ ∨ s = univ\n⊢ ⋃₀ S✝ = ∅ ∨ ⋃₀ S✝ = univ",
"tactic": "case inter h₁ h₂ =>\n rcases h₁ with (rfl | rfl) <;> rcases h₂ with (rfl | rfl) <;> simp"
},
{
"state_after": "no goals",
"state_before": "case sUnion\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → s = ∅ ∨ s = univ\n⊢ ⋃₀ S✝ = ∅ ∨ ⋃₀ S✝ = univ",
"tactic": "case sUnion _ ih => exact sUnion_mem_empty_univ ih"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU s✝ : Set α\nh : s✝ ∈ ↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)\n⊢ s✝ = ∅ ∨ s✝ = univ",
"tactic": "exact False.elim h"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\n⊢ univ = ∅ ∨ univ = univ",
"tactic": "exact .inr rfl"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU s✝ t✝ : Set α\na✝¹ : GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s✝\na✝ : GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) t✝\nh₁ : s✝ = ∅ ∨ s✝ = univ\nh₂ : t✝ = ∅ ∨ t✝ = univ\n⊢ s✝ ∩ t✝ = ∅ ∨ s✝ ∩ t✝ = univ",
"tactic": "rcases h₁ with (rfl | rfl) <;> rcases h₂ with (rfl | rfl) <;> simp"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen (↑OrderDual.ofDual (↑OrderDual.ofDual ⊥)) s\nih : ∀ (s : Set α), s ∈ S✝ → s = ∅ ∨ s = univ\n⊢ ⋃₀ S✝ = ∅ ∨ ⋃₀ S✝ = univ",
"tactic": "exact sUnion_mem_empty_univ ih"
},
{
"state_after": "case inl\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\n⊢ IsOpen ∅\n\ncase inr\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\n⊢ IsOpen univ",
"state_before": "α✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\nU : Set α\n⊢ U = ∅ ∨ U = univ → IsOpen U",
"tactic": "rintro (rfl | rfl)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\n⊢ IsOpen ∅\n\ncase inr\nα✝ : Type ?u.11392\nt t₁ t₂ : TopologicalSpace α✝\ns : Set α✝\nα : Type u_1\n⊢ IsOpen univ",
"tactic": "exacts [@isOpen_empty _ ⊤, @isOpen_univ _ ⊤]"
}
] |
[
265,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
Real.volume_pi_Ioc_toReal
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝ : Fintype ι\na b : ι → ℝ\nh : a ≤ b\n⊢ ENNReal.toReal (↑↑volume (Set.pi univ fun i => Ioc (a i) (b i))) = ∏ i : ι, (b i - a i)",
"tactic": "simp only [volume_pi_Ioc, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))]"
}
] |
[
245,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/Topology/Algebra/GroupCompletion.lean
|
UniformSpace.Completion.continuous_toCompl
|
[] |
[
192,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean
|
HomogeneousLocalization.NumDenSameDeg.num_smul
|
[] |
[
271,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/NumberTheory/VonMangoldt.lean
|
Nat.ArithmeticFunction.vonMangoldt_apply_prime
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Prime p\n⊢ ↑Λ p = Real.log ↑p",
"tactic": "rw [vonMangoldt_apply, Prime.minFac_eq hp, if_pos hp.prime.isPrimePow]"
}
] |
[
94,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
RingHom.charP_iff_charP
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.80816\nK : Type u_1\nL : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : Semiring L\ninst✝ : Nontrivial L\nf : K →+* L\np : ℕ\n⊢ CharP K p ↔ CharP L p",
"tactic": "simp only [charP_iff, ← f.injective.eq_iff, map_natCast f, f.map_zero]"
}
] |
[
327,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.cast_pow
|
[] |
[
350,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
349,
1
] |
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