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/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
|
EulerSine.integral_cos_mul_cos_pow_aux
|
[
{
"state_after": "z : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) =\n ↑n / (2 * z) * ∫ (x : ℝ) in 0 ..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1)",
"state_before": "z : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\n⊢ (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) =\n ↑n / (2 * z) * ∫ (x : ℝ) in 0 ..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1)",
"tactic": "have der1 :\n ∀ x : ℝ,\n x ∈ uIcc 0 (π / 2) →\n HasDerivAt (fun y : ℝ => (cos y : ℂ) ^ n) (-n * sin x * (cos x : ℂ) ^ (n - 1)) x := by\n intro x _\n have b : HasDerivAt (fun y : ℝ => (cos y : ℂ)) (-sin x) x := by\n simpa using (hasDerivAt_cos x).ofReal_comp\n convert HasDerivAt.comp x (hasDerivAt_pow _ _) b using 1\n ring"
},
{
"state_after": "case h.e'_2.h.e'_5\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ (fun x => Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) = fun x => ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)\n\ncase h.e'_3\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ (↑n / (2 * z) * ∫ (x : ℝ) in 0 ..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1)) =\n ↑(cos (π / 2)) ^ n * (Complex.sin (2 * z * ↑(π / 2)) / (2 * z)) -\n ↑(cos 0) ^ n * (Complex.sin (2 * z * ↑0) / (2 * z)) -\n ∫ (x : ℝ) in 0 ..π / 2, -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1) * (Complex.sin (2 * z * ↑x) / (2 * z))\n\ncase convert_1\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ IntervalIntegrable (fun x => -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) volume 0 (π / 2)\n\ncase convert_2\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ IntervalIntegrable (fun x => Complex.cos (2 * z * ↑x)) volume 0 (π / 2)",
"state_before": "z : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) =\n ↑n / (2 * z) * ∫ (x : ℝ) in 0 ..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1)",
"tactic": "convert (config := { sameFun := true })\n integral_mul_deriv_eq_deriv_mul der1 (fun x _ => antideriv_cos_comp_const_mul hz x) _ _ using 2"
},
{
"state_after": "z : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nx : ℝ\na✝ : x ∈ uIcc 0 (π / 2)\n⊢ HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x",
"state_before": "z : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\n⊢ ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x",
"tactic": "intro x _"
},
{
"state_after": "z : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nx : ℝ\na✝ : x ∈ uIcc 0 (π / 2)\nb : HasDerivAt (fun y => ↑(cos y)) (-↑(sin x)) x\n⊢ HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x",
"state_before": "z : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nx : ℝ\na✝ : x ∈ uIcc 0 (π / 2)\n⊢ HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x",
"tactic": "have b : HasDerivAt (fun y : ℝ => (cos y : ℂ)) (-sin x) x := by\n simpa using (hasDerivAt_cos x).ofReal_comp"
},
{
"state_after": "case h.e'_7\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nx : ℝ\na✝ : x ∈ uIcc 0 (π / 2)\nb : HasDerivAt (fun y => ↑(cos y)) (-↑(sin x)) x\n⊢ -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1) = ↑n * ↑(cos x) ^ (n - 1) * -↑(sin x)",
"state_before": "z : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nx : ℝ\na✝ : x ∈ uIcc 0 (π / 2)\nb : HasDerivAt (fun y => ↑(cos y)) (-↑(sin x)) x\n⊢ HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x",
"tactic": "convert HasDerivAt.comp x (hasDerivAt_pow _ _) b using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nx : ℝ\na✝ : x ∈ uIcc 0 (π / 2)\nb : HasDerivAt (fun y => ↑(cos y)) (-↑(sin x)) x\n⊢ -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1) = ↑n * ↑(cos x) ^ (n - 1) * -↑(sin x)",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "z : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nx : ℝ\na✝ : x ∈ uIcc 0 (π / 2)\n⊢ HasDerivAt (fun y => ↑(cos y)) (-↑(sin x)) x",
"tactic": "simpa using (hasDerivAt_cos x).ofReal_comp"
},
{
"state_after": "case h.e'_2.h.e'_5.h\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\nx : ℝ\n⊢ Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n = ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)",
"state_before": "case h.e'_2.h.e'_5\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ (fun x => Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) = fun x => ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)",
"tactic": "ext1 x"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_5.h\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\nx : ℝ\n⊢ Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n = ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)",
"tactic": "rw [mul_comm]"
},
{
"state_after": "case h.e'_3\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ (∫ (x : ℝ) in 0 ..π / 2, ↑n / (2 * z) * (Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1))) =\n ∫ (x : ℝ) in 0 ..π / 2, -(-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1) * (Complex.sin (2 * z * ↑x) / (2 * z)))",
"state_before": "case h.e'_3\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ (↑n / (2 * z) * ∫ (x : ℝ) in 0 ..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1)) =\n ↑(cos (π / 2)) ^ n * (Complex.sin (2 * z * ↑(π / 2)) / (2 * z)) -\n ↑(cos 0) ^ n * (Complex.sin (2 * z * ↑0) / (2 * z)) -\n ∫ (x : ℝ) in 0 ..π / 2, -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1) * (Complex.sin (2 * z * ↑x) / (2 * z))",
"tactic": "rw [Complex.ofReal_zero, MulZeroClass.mul_zero, Complex.sin_zero, zero_div,\n MulZeroClass.mul_zero, sub_zero, cos_pi_div_two, Complex.ofReal_zero,\n zero_pow (by positivity : 0 < n), MulZeroClass.zero_mul, zero_sub, ← integral_neg, ←\n integral_const_mul]"
},
{
"state_after": "case h.e'_3\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\nx : ℝ\nx✝ : x ∈ uIcc 0 (π / 2)\n⊢ ↑n / (2 * z) * (Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1)) =\n -(-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1) * (Complex.sin (2 * z * ↑x) / (2 * z)))",
"state_before": "case h.e'_3\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ (∫ (x : ℝ) in 0 ..π / 2, ↑n / (2 * z) * (Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1))) =\n ∫ (x : ℝ) in 0 ..π / 2, -(-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1) * (Complex.sin (2 * z * ↑x) / (2 * z)))",
"tactic": "refine' integral_congr fun x _ => _"
},
{
"state_after": "case h.e'_3\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\nx : ℝ\nx✝ : x ∈ uIcc 0 (π / 2)\n⊢ ↑n * (Complex.sin (2 * z * ↑x) * Complex.sin ↑x * Complex.cos ↑x ^ (n - 1)) =\n ↑n * Complex.sin ↑x * Complex.cos ↑x ^ (n - 1) * Complex.sin (2 * z * ↑x)",
"state_before": "case h.e'_3\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\nx : ℝ\nx✝ : x ∈ uIcc 0 (π / 2)\n⊢ ↑n / (2 * z) * (Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1)) =\n -(-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1) * (Complex.sin (2 * z * ↑x) / (2 * z)))",
"tactic": "field_simp"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\nx : ℝ\nx✝ : x ∈ uIcc 0 (π / 2)\n⊢ ↑n * (Complex.sin (2 * z * ↑x) * Complex.sin ↑x * Complex.cos ↑x ^ (n - 1)) =\n ↑n * Complex.sin ↑x * Complex.cos ↑x ^ (n - 1) * Complex.sin (2 * z * ↑x)",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "z : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ 0 < n",
"tactic": "positivity"
},
{
"state_after": "case convert_1.hu\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ Continuous fun x => -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)",
"state_before": "case convert_1\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ IntervalIntegrable (fun x => -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) volume 0 (π / 2)",
"tactic": "apply Continuous.intervalIntegrable"
},
{
"state_after": "no goals",
"state_before": "case convert_1.hu\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ Continuous fun x => -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)",
"tactic": "exact\n (continuous_const.mul (Complex.continuous_ofReal.comp continuous_sin)).mul\n ((Complex.continuous_ofReal.comp continuous_cos).pow (n - 1))"
},
{
"state_after": "case convert_2.hu\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ Continuous fun x => Complex.cos (2 * z * ↑x)",
"state_before": "case convert_2\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ IntervalIntegrable (fun x => Complex.cos (2 * z * ↑x)) volume 0 (π / 2)",
"tactic": "apply Continuous.intervalIntegrable"
},
{
"state_after": "no goals",
"state_before": "case convert_2.hu\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ (x : ℝ), x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y => ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ Continuous fun x => Complex.cos (2 * z * ↑x)",
"tactic": "exact Complex.continuous_cos.comp (continuous_const.mul Complex.continuous_ofReal)"
}
] |
[
92,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.not_mem_range_self
|
[] |
[
3013,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3012,
1
] |
Mathlib/Logic/Unique.lean
|
unique_subtype_iff_exists_unique
|
[
{
"state_after": "case e_val\nα : Sort u_1\np : α → Prop\nx✝¹ : ∃! a, p a\na : α\nha : (fun a => p a) a\nhe : ∀ (y : α), (fun a => p a) y → y = a\nx✝ : Subtype p\nb : α\nhb : p b\n⊢ b = a",
"state_before": "α : Sort u_1\np : α → Prop\nx✝¹ : ∃! a, p a\na : α\nha : (fun a => p a) a\nhe : ∀ (y : α), (fun a => p a) y → y = a\nx✝ : Subtype p\nb : α\nhb : p b\n⊢ { val := b, property := hb } = default",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_val\nα : Sort u_1\np : α → Prop\nx✝¹ : ∃! a, p a\na : α\nha : (fun a => p a) a\nhe : ∀ (y : α), (fun a => p a) y → y = a\nx✝ : Subtype p\nb : α\nhb : p b\n⊢ b = a",
"tactic": "exact he b hb"
}
] |
[
78,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean
|
FreeAbelianGroup.one_def
|
[] |
[
552,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
551,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
tsum_ite_eq
|
[] |
[
537,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.sum_boole
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.817559\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ns : Finset α\np : α → Prop\ninst✝ : NonAssocSemiring β\nhp : DecidablePred p\n⊢ (∑ x in s, if p x then 1 else 0) = ↑(card (filter p s))",
"tactic": "simp only [add_zero, mul_one, Finset.sum_const, nsmul_eq_mul, eq_self_iff_true,\n Finset.sum_const_zero, Finset.sum_ite, mul_zero]"
}
] |
[
1767,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1764,
1
] |
Mathlib/Topology/Algebra/FilterBasis.lean
|
ContinuousSMul.of_basis_zero
|
[
{
"state_after": "case hmul\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\n⊢ Tendsto (fun p => p.fst • p.snd) (𝓝 0 ×ˢ 𝓝 0) (𝓝 0)\n\ncase hmulleft\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\n⊢ ∀ (m : M), Tendsto (fun a => a • m) (𝓝 0) (𝓝 0)\n\ncase hmulright\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\n⊢ ∀ (a : R), Tendsto (fun m => a • m) (𝓝 0) (𝓝 0)",
"state_before": "R : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\n⊢ ContinuousSMul R M",
"tactic": "apply ContinuousSMul.of_nhds_zero"
},
{
"state_after": "case hmul\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\n⊢ ∀ (i : ι), p i → ∀ᶠ (x : R × M) in 𝓝 0 ×ˢ 𝓝 0, x.fst • x.snd ∈ b i",
"state_before": "case hmul\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\n⊢ Tendsto (fun p => p.fst • p.snd) (𝓝 0 ×ˢ 𝓝 0) (𝓝 0)",
"tactic": "rw [h.tendsto_right_iff]"
},
{
"state_after": "case hmul\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\ni : ι\nhi : p i\n⊢ ∀ᶠ (x : R × M) in 𝓝 0 ×ˢ 𝓝 0, x.fst • x.snd ∈ b i",
"state_before": "case hmul\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\n⊢ ∀ (i : ι), p i → ∀ᶠ (x : R × M) in 𝓝 0 ×ˢ 𝓝 0, x.fst • x.snd ∈ b i",
"tactic": "intro i hi"
},
{
"state_after": "case hmul.intro.intro.intro.intro\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\ni : ι\nhi : p i\nV : Set R\nV_in : V ∈ 𝓝 0\nj : ι\nhj : p j\nhVj : V • b j ⊆ b i\n⊢ ∀ᶠ (x : R × M) in 𝓝 0 ×ˢ 𝓝 0, x.fst • x.snd ∈ b i",
"state_before": "case hmul\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\ni : ι\nhi : p i\n⊢ ∀ᶠ (x : R × M) in 𝓝 0 ×ˢ 𝓝 0, x.fst • x.snd ∈ b i",
"tactic": "rcases hsmul hi with ⟨V, V_in, j, hj, hVj⟩"
},
{
"state_after": "case hmul.intro.intro.intro.intro\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\ni : ι\nhi : p i\nV : Set R\nV_in : V ∈ 𝓝 0\nj : ι\nhj : p j\nhVj : V • b j ⊆ b i\n⊢ V ×ˢ b j ⊆ {x | (fun x => x.fst • x.snd ∈ b i) x}",
"state_before": "case hmul.intro.intro.intro.intro\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\ni : ι\nhi : p i\nV : Set R\nV_in : V ∈ 𝓝 0\nj : ι\nhj : p j\nhVj : V • b j ⊆ b i\n⊢ ∀ᶠ (x : R × M) in 𝓝 0 ×ˢ 𝓝 0, x.fst • x.snd ∈ b i",
"tactic": "apply mem_of_superset (prod_mem_prod V_in <| h.mem_of_mem hj)"
},
{
"state_after": "case hmul.intro.intro.intro.intro.mk.intro\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\ni : ι\nhi : p i\nV : Set R\nV_in : V ∈ 𝓝 0\nj : ι\nhj : p j\nhVj : V • b j ⊆ b i\nv : R\nw : M\nv_in : v ∈ V\nw_in : w ∈ b j\n⊢ (v, w) ∈ {x | (fun x => x.fst • x.snd ∈ b i) x}",
"state_before": "case hmul.intro.intro.intro.intro\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\ni : ι\nhi : p i\nV : Set R\nV_in : V ∈ 𝓝 0\nj : ι\nhj : p j\nhVj : V • b j ⊆ b i\n⊢ V ×ˢ b j ⊆ {x | (fun x => x.fst • x.snd ∈ b i) x}",
"tactic": "rintro ⟨v, w⟩ ⟨v_in : v ∈ V, w_in : w ∈ b j⟩"
},
{
"state_after": "no goals",
"state_before": "case hmul.intro.intro.intro.intro.mk.intro\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\ni : ι\nhi : p i\nV : Set R\nV_in : V ∈ 𝓝 0\nj : ι\nhj : p j\nhVj : V • b j ⊆ b i\nv : R\nw : M\nv_in : v ∈ V\nw_in : w ∈ b j\n⊢ (v, w) ∈ {x | (fun x => x.fst • x.snd ∈ b i) x}",
"tactic": "exact hVj (Set.smul_mem_smul v_in w_in)"
},
{
"state_after": "case hmulleft\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\nm₀ : M\n⊢ Tendsto (fun a => a • m₀) (𝓝 0) (𝓝 0)",
"state_before": "case hmulleft\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\n⊢ ∀ (m : M), Tendsto (fun a => a • m) (𝓝 0) (𝓝 0)",
"tactic": "intro m₀"
},
{
"state_after": "case hmulleft\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\nm₀ : M\n⊢ ∀ (i : ι), p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i",
"state_before": "case hmulleft\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\nm₀ : M\n⊢ Tendsto (fun a => a • m₀) (𝓝 0) (𝓝 0)",
"tactic": "rw [h.tendsto_right_iff]"
},
{
"state_after": "case hmulleft\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\nm₀ : M\ni : ι\nhi : p i\n⊢ ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i",
"state_before": "case hmulleft\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\nm₀ : M\n⊢ ∀ (i : ι), p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i",
"tactic": "intro i hi"
},
{
"state_after": "no goals",
"state_before": "case hmulleft\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\nm₀ : M\ni : ι\nhi : p i\n⊢ ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i",
"tactic": "exact hsmul_right m₀ hi"
},
{
"state_after": "case hmulright\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\nx₀ : R\n⊢ Tendsto (fun m => x₀ • m) (𝓝 0) (𝓝 0)",
"state_before": "case hmulright\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\n⊢ ∀ (a : R), Tendsto (fun m => a • m) (𝓝 0) (𝓝 0)",
"tactic": "intro x₀"
},
{
"state_after": "case hmulright\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\nx₀ : R\n⊢ ∀ (i : ι), p i → ∀ᶠ (x : M) in 𝓝 0, x₀ • x ∈ b i",
"state_before": "case hmulright\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\nx₀ : R\n⊢ Tendsto (fun m => x₀ • m) (𝓝 0) (𝓝 0)",
"tactic": "rw [h.tendsto_right_iff]"
},
{
"state_after": "case hmulright\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\nx₀ : R\ni : ι\nhi : p i\n⊢ ∀ᶠ (x : M) in 𝓝 0, x₀ • x ∈ b i",
"state_before": "case hmulright\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\nx₀ : R\n⊢ ∀ (i : ι), p i → ∀ᶠ (x : M) in 𝓝 0, x₀ • x ∈ b i",
"tactic": "intro i hi"
},
{
"state_after": "case hmulright.intro.intro\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\nx₀ : R\ni : ι\nhi : p i\nj : ι\nhj : p j\nhji : b j ⊆ (fun x => x₀ • x) ⁻¹' b i\n⊢ ∀ᶠ (x : M) in 𝓝 0, x₀ • x ∈ b i",
"state_before": "case hmulright\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\nx₀ : R\ni : ι\nhi : p i\n⊢ ∀ᶠ (x : M) in 𝓝 0, x₀ • x ∈ b i",
"tactic": "rcases hsmul_left x₀ hi with ⟨j, hj, hji⟩"
},
{
"state_after": "no goals",
"state_before": "case hmulright.intro.intro\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nB : ModuleFilterBasis R M\nι : Type u_1\ninst✝² : TopologicalRing R\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalAddGroup M\np : ι → Prop\nb : ι → Set M\nh : HasBasis (𝓝 0) p b\nhsmul : ∀ {i : ι}, p i → ∃ V, V ∈ 𝓝 0 ∧ ∃ j x, V • b j ⊆ b i\nhsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j x, b j ⊆ (fun x => x₀ • x) ⁻¹' b i\nhsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i\nx₀ : R\ni : ι\nhi : p i\nj : ι\nhj : p j\nhji : b j ⊆ (fun x => x₀ • x) ⁻¹' b i\n⊢ ∀ᶠ (x : M) in 𝓝 0, x₀ • x ∈ b i",
"tactic": "exact mem_of_superset (h.mem_of_mem hj) hji"
}
] |
[
433,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
413,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.arg_coe_angle_toReal_eq_arg
|
[
{
"state_after": "z : ℂ\n⊢ arg z ∈ Set.Ioc (-π) π",
"state_before": "z : ℂ\n⊢ Angle.toReal ↑(arg z) = arg z",
"tactic": "rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc]"
},
{
"state_after": "no goals",
"state_before": "z : ℂ\n⊢ arg z ∈ Set.Ioc (-π) π",
"tactic": "exact arg_mem_Ioc _"
}
] |
[
494,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
492,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.ae_restrict_of_ae_eq_of_ae_restrict
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.589734\nγ : Type ?u.589737\nδ : Type ?u.589740\nι : Type ?u.589743\nR : Type ?u.589746\nR' : Type ?u.589749\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\ns t : α → Prop\nhst : s =ᶠ[ae μ] t\np : α → Prop\n⊢ (∀ᵐ (x : α) ∂Measure.restrict μ s, p x) → ∀ᵐ (x : α) ∂Measure.restrict μ t, p x",
"tactic": "simp [Measure.restrict_congr_set hst]"
}
] |
[
2927,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2926,
1
] |
Mathlib/Topology/Algebra/Ring/Basic.lean
|
TopologicalRing.of_nhds_zero
|
[] |
[
219,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean
|
Nat.factorial_one
|
[] |
[
57,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
|
TopCat.openEmbedding_of_pullback_open_embeddings
|
[
{
"state_after": "case h.e'_5.h\nJ : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\nH₁ : OpenEmbedding ((forget TopCat).map f)\nH₂ : OpenEmbedding ((forget TopCat).map g)\ne_2✝ : (forget TopCat).obj ((cospan f g).obj WalkingCospan.one) = (forget TopCat).obj S\n⊢ (forget TopCat).map (limit.π (cospan f g) WalkingCospan.one) =\n (forget TopCat).map g ∘ (forget TopCat).map pullback.snd",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\nH₁ : OpenEmbedding ((forget TopCat).map f)\nH₂ : OpenEmbedding ((forget TopCat).map g)\n⊢ OpenEmbedding ((forget TopCat).map (limit.π (cospan f g) WalkingCospan.one))",
"tactic": "convert H₂.comp (snd_openEmbedding_of_left_openEmbedding H₁ g)"
},
{
"state_after": "case h.e'_5.h\nJ : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\nH₁ : OpenEmbedding ((forget TopCat).map f)\nH₂ : OpenEmbedding ((forget TopCat).map g)\ne_2✝ : (forget TopCat).obj ((cospan f g).obj WalkingCospan.one) = (forget TopCat).obj S\n⊢ (forget TopCat).map (limit.π (cospan f g) WalkingCospan.one) = (forget TopCat).map (pullback.snd ≫ g)",
"state_before": "case h.e'_5.h\nJ : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\nH₁ : OpenEmbedding ((forget TopCat).map f)\nH₂ : OpenEmbedding ((forget TopCat).map g)\ne_2✝ : (forget TopCat).obj ((cospan f g).obj WalkingCospan.one) = (forget TopCat).obj S\n⊢ (forget TopCat).map (limit.π (cospan f g) WalkingCospan.one) =\n (forget TopCat).map g ∘ (forget TopCat).map pullback.snd",
"tactic": "erw [← coe_comp]"
},
{
"state_after": "case h.e'_5.h.e_a\nJ : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\nH₁ : OpenEmbedding ((forget TopCat).map f)\nH₂ : OpenEmbedding ((forget TopCat).map g)\ne_2✝ : (forget TopCat).obj ((cospan f g).obj WalkingCospan.one) = (forget TopCat).obj S\n⊢ limit.π (cospan f g) WalkingCospan.one = pullback.snd ≫ g",
"state_before": "case h.e'_5.h\nJ : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\nH₁ : OpenEmbedding ((forget TopCat).map f)\nH₂ : OpenEmbedding ((forget TopCat).map g)\ne_2✝ : (forget TopCat).obj ((cospan f g).obj WalkingCospan.one) = (forget TopCat).obj S\n⊢ (forget TopCat).map (limit.π (cospan f g) WalkingCospan.one) = (forget TopCat).map (pullback.snd ≫ g)",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.e_a\nJ : Type v\ninst✝ : SmallCategory J\nX✝ Y✝ Z : TopCat\nX Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\nH₁ : OpenEmbedding ((forget TopCat).map f)\nH₂ : OpenEmbedding ((forget TopCat).map g)\ne_2✝ : (forget TopCat).obj ((cospan f g).obj WalkingCospan.one) = (forget TopCat).obj S\n⊢ limit.π (cospan f g) WalkingCospan.one = pullback.snd ≫ g",
"tactic": "exact (limit.w _ WalkingCospan.Hom.inr).symm"
}
] |
[
341,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
integral_cos_pow_three
|
[
{
"state_after": "a b : ℝ\nn : ℕ\nthis : (∫ (x : ℝ) in a..b, sin x ^ 0 * cos x ^ (2 * 1 + 1)) = ∫ (u : ℝ) in sin a..sin b, u ^ 0 * (1 - u ^ 2) ^ 1\n⊢ (∫ (x : ℝ) in a..b, cos x ^ 3) = sin b - sin a - (sin b ^ 3 - sin a ^ 3) / 3",
"state_before": "a b : ℝ\nn : ℕ\n⊢ (∫ (x : ℝ) in a..b, cos x ^ 3) = sin b - sin a - (sin b ^ 3 - sin a ^ 3) / 3",
"tactic": "have := @integral_sin_pow_mul_cos_pow_odd a b 0 1"
},
{
"state_after": "a b : ℝ\nn : ℕ\nthis : (∫ (x : ℝ) in a..b, cos x ^ 3) = sin b - sin a - (sin b ^ 3 - sin a ^ 3) / 3\n⊢ (∫ (x : ℝ) in a..b, cos x ^ 3) = sin b - sin a - (sin b ^ 3 - sin a ^ 3) / 3",
"state_before": "a b : ℝ\nn : ℕ\nthis : (∫ (x : ℝ) in a..b, sin x ^ 0 * cos x ^ (2 * 1 + 1)) = ∫ (u : ℝ) in sin a..sin b, u ^ 0 * (1 - u ^ 2) ^ 1\n⊢ (∫ (x : ℝ) in a..b, cos x ^ 3) = sin b - sin a - (sin b ^ 3 - sin a ^ 3) / 3",
"tactic": "norm_num at this"
},
{
"state_after": "no goals",
"state_before": "a b : ℝ\nn : ℕ\nthis : (∫ (x : ℝ) in a..b, cos x ^ 3) = sin b - sin a - (sin b ^ 3 - sin a ^ 3) / 3\n⊢ (∫ (x : ℝ) in a..b, cos x ^ 3) = sin b - sin a - (sin b ^ 3 - sin a ^ 3) / 3",
"tactic": "exact this"
}
] |
[
783,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
780,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.zero
|
[] |
[
361,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
360,
11
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biprod.inl_desc
|
[] |
[
1377,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1375,
1
] |
Mathlib/Data/Rat/Defs.lean
|
Rat.coe_int_eq_divInt
|
[] |
[
112,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.restrict_restrict
|
[] |
[
1639,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1638,
1
] |
Mathlib/CategoryTheory/Functor/Category.lean
|
CategoryTheory.NatTrans.ext'
|
[] |
[
62,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Topology/Sheaves/Presheaf.lean
|
TopCat.Presheaf.pushforwardObj_obj
|
[] |
[
144,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Algebra/Group/Defs.lean
|
mul_comm
|
[] |
[
310,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
309,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.RightInvOn.mono
|
[] |
[
1143,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1142,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
Left.mul_lt_one_of_lt_of_le
|
[] |
[
626,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
623,
1
] |
Mathlib/Data/Part.lean
|
Part.div_def
|
[] |
[
702,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
702,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean
|
AddSubmonoid.bot_mul
|
[
{
"state_after": "α : Type ?u.250853\nG : Type ?u.250856\nM : Type ?u.250859\nR : Type u_1\nA : Type ?u.250865\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS : AddSubmonoid R\nm : R\nhm : m = 0\nn : R\nhn : n ∈ S\n⊢ m * n = 0",
"state_before": "α : Type ?u.250853\nG : Type ?u.250856\nM : Type ?u.250859\nR : Type u_1\nA : Type ?u.250865\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS : AddSubmonoid R\nm : R\nhm : m ∈ ⊥\nn : R\nhn : n ∈ S\n⊢ m * n ∈ ⊥",
"tactic": "rw [AddSubmonoid.mem_bot] at hm ⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.250853\nG : Type ?u.250856\nM : Type ?u.250859\nR : Type u_1\nA : Type ?u.250865\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS : AddSubmonoid R\nm : R\nhm : m = 0\nn : R\nhn : n ∈ S\n⊢ m * n = 0",
"tactic": "rw [hm, zero_mul]"
}
] |
[
582,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
IsEquivalent.summable_iff
|
[] |
[
712,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
709,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioc_diff_Iic
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.210763\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\na a₁ a₂ b b₁ b₂ c d : α\n⊢ Ioc a b \\ Iic c = Ioc (max a c) b",
"tactic": "rw [diff_eq, compl_Iic, Ioc_inter_Ioi, sup_eq_max]"
}
] |
[
1837,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1836,
1
] |
Mathlib/LinearAlgebra/Isomorphisms.lean
|
LinearMap.quotientInfEquivSupQuotient_symm_apply_left
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\nM₂ : Type ?u.100895\nM₃ : Type ?u.100898\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup M₃\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module R M₃\nf : M →ₗ[R] M₂\np p' : Submodule R M\nx : { x // x ∈ p ⊔ p' }\nhx : ↑x ∈ p\n⊢ Submodule.Quotient.mk x = ↑(quotientInfEquivSupQuotient p p') (Submodule.Quotient.mk { val := ↑x, property := hx })",
"tactic": "rw [quotientInfEquivSupQuotient_apply_mk, ofLe_apply]"
}
] |
[
133,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Data/Polynomial/Laurent.lean
|
LaurentPolynomial.support_C_mul_T
|
[
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\na : R\nn : ℤ\n⊢ (Finsupp.single n a).support ⊆ {n}",
"state_before": "R : Type u_1\ninst✝ : Semiring R\na : R\nn : ℤ\n⊢ (↑C a * T n).support ⊆ {n}",
"tactic": "rw [← single_eq_C_mul_T]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\na : R\nn : ℤ\n⊢ (Finsupp.single n a).support ⊆ {n}",
"tactic": "exact support_single_subset"
}
] |
[
438,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.uncurrySum_apply
|
[] |
[
1805,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1802,
1
] |
Mathlib/Analysis/NormedSpace/Star/Basic.lean
|
CstarRing.nnnorm_star_mul_self
|
[] |
[
134,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/RingTheory/FreeCommRing.lean
|
FreeRing.coe_sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nx y : FreeRing α\n⊢ ↑(x - y) = ↑x - ↑y",
"tactic": "rw [castFreeCommRing, map_sub]"
}
] |
[
355,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
11
] |
Mathlib/Order/RelClasses.lean
|
transitive_gt
|
[] |
[
913,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
912,
1
] |
Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
|
DiscreteValuationRing.unit_mul_pow_congr_pow
|
[
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\nkey : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))\n⊢ m = n",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\n⊢ m = n",
"tactic": "have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by\n rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]\n refine' ⟨u * v⁻¹, _⟩\n simp only [Units.val_mul]\n rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\nkey : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))\nthis : ↑Multiset.card (Multiset.replicate m p) = ↑Multiset.card (Multiset.replicate n q)\n⊢ m = n",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\nkey : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))\n⊢ m = n",
"tactic": "have := by\n refine' Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique _ _ key)\n all_goals\n intro x hx\n obtain rfl := Multiset.eq_of_mem_replicate hx\n assumption"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\nkey : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))\nthis : ↑Multiset.card (Multiset.replicate m p) = ↑Multiset.card (Multiset.replicate n q)\n⊢ m = n",
"tactic": "simpa only [Multiset.card_replicate]"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\n⊢ ∃ u, p ^ m * ↑u = q ^ n",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\n⊢ Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))",
"tactic": "rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\n⊢ p ^ m * ↑(u * v⁻¹) = q ^ n",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\n⊢ ∃ u, p ^ m * ↑u = q ^ n",
"tactic": "refine' ⟨u * v⁻¹, _⟩"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\n⊢ p ^ m * (↑u * ↑v⁻¹) = q ^ n",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\n⊢ p ^ m * ↑(u * v⁻¹) = q ^ n",
"tactic": "simp only [Units.val_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\n⊢ p ^ m * (↑u * ↑v⁻¹) = q ^ n",
"tactic": "rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]"
},
{
"state_after": "case refine'_1\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\nkey : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))\n⊢ ∀ (x : R), x ∈ Multiset.replicate m p → Irreducible x\n\ncase refine'_2\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\nkey : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))\n⊢ ∀ (x : R), x ∈ Multiset.replicate n q → Irreducible x",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\nkey : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))\n⊢ ?m.265183",
"tactic": "refine' Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique _ _ key)"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\nkey : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))\n⊢ ∀ (x : R), x ∈ Multiset.replicate m p → Irreducible x\n\ncase refine'_2\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\nkey : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))\n⊢ ∀ (x : R), x ∈ Multiset.replicate n q → Irreducible x",
"tactic": "all_goals\n intro x hx\n obtain rfl := Multiset.eq_of_mem_replicate hx\n assumption"
},
{
"state_after": "case refine'_2\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\nkey : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))\nx : R\nhx : x ∈ Multiset.replicate n q\n⊢ Irreducible x",
"state_before": "case refine'_2\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\nkey : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))\n⊢ ∀ (x : R), x ∈ Multiset.replicate n q → Irreducible x",
"tactic": "intro x hx"
},
{
"state_after": "case refine'_2\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np : R\nhp : Irreducible p\nu v : Rˣ\nm n : ℕ\nx : R\nhq : Irreducible x\nh : ↑u * p ^ m = ↑v * x ^ n\nkey : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n x))\nhx : x ∈ Multiset.replicate n x\n⊢ Irreducible x",
"state_before": "case refine'_2\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\nkey : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))\nx : R\nhx : x ∈ Multiset.replicate n q\n⊢ Irreducible x",
"tactic": "obtain rfl := Multiset.eq_of_mem_replicate hx"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\np : R\nhp : Irreducible p\nu v : Rˣ\nm n : ℕ\nx : R\nhq : Irreducible x\nh : ↑u * p ^ m = ↑v * x ^ n\nkey : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n x))\nhx : x ∈ Multiset.replicate n x\n⊢ Irreducible x",
"tactic": "assumption"
}
] |
[
382,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
inner_neg_right
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1737773\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ↑(starRingEnd 𝕜) (-inner y x) = -inner x y",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1737773\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ inner x (-y) = -inner x y",
"tactic": "rw [← inner_conj_symm, inner_neg_left]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1737773\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ↑(starRingEnd 𝕜) (-inner y x) = -inner x y",
"tactic": "simp only [RingHom.map_neg, inner_conj_symm]"
}
] |
[
641,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
640,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.lf_of_mk_le
|
[] |
[
494,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
493,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.ord_proj_le
|
[] |
[
389,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
388,
1
] |
Mathlib/Data/Nat/Cast/Defs.lean
|
two_add_one_eq_three
|
[
{
"state_after": "α : Type u_1\ninst✝ : AddMonoidWithOne α\n⊢ ↑(1 + 1 + 1) = 3",
"state_before": "α : Type u_1\ninst✝ : AddMonoidWithOne α\n⊢ 2 + 1 = 3",
"tactic": "rw [←one_add_one_eq_two, ←Nat.cast_one, ←Nat.cast_add, ←Nat.cast_add]"
},
{
"state_after": "case h\nα : Type u_1\ninst✝ : AddMonoidWithOne α\n⊢ 1 + 1 + 1 = 3",
"state_before": "α : Type u_1\ninst✝ : AddMonoidWithOne α\n⊢ ↑(1 + 1 + 1) = 3",
"tactic": "apply congrArg"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ninst✝ : AddMonoidWithOne α\n⊢ 1 + 1 + 1 = 3",
"tactic": "decide"
}
] |
[
236,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean
|
isometry_id
|
[] |
[
99,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/Data/Finsupp/BigOperators.lean
|
Finset.mem_sup_support_iff
|
[] |
[
80,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Data/Nat/Size.lean
|
Nat.size_eq_bits_len
|
[
{
"state_after": "case z\n\n⊢ List.length (bits 0) = size 0\n\ncase f\nb : Bool\nn : ℕ\nh : n = 0 → b = true\nih : List.length (bits n) = size n\n⊢ List.length (bits (bit b n)) = size (bit b n)",
"state_before": "n : ℕ\n⊢ List.length (bits n) = size n",
"tactic": "induction' n using Nat.binaryRec' with b n h ih"
},
{
"state_after": "case f\nb : Bool\nn : ℕ\nh : n = 0 → b = true\nih : List.length (bits n) = size n\n⊢ List.length (b :: bits n) = succ (size n)\n\ncase f\nb : Bool\nn : ℕ\nh : n = 0 → b = true\nih : List.length (bits n) = size n\n⊢ bit b n ≠ 0",
"state_before": "case f\nb : Bool\nn : ℕ\nh : n = 0 → b = true\nih : List.length (bits n) = size n\n⊢ List.length (bits (bit b n)) = size (bit b n)",
"tactic": "rw [size_bit, bits_append_bit _ _ h]"
},
{
"state_after": "no goals",
"state_before": "case z\n\n⊢ List.length (bits 0) = size 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case f\nb : Bool\nn : ℕ\nh : n = 0 → b = true\nih : List.length (bits n) = size n\n⊢ List.length (b :: bits n) = succ (size n)",
"tactic": "simp [ih]"
},
{
"state_after": "no goals",
"state_before": "case f\nb : Bool\nn : ℕ\nh : n = 0 → b = true\nih : List.length (bits n) = size n\n⊢ bit b n ≠ 0",
"tactic": "simpa [bit_eq_zero_iff]"
}
] |
[
182,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/CategoryTheory/StructuredArrow.lean
|
CategoryTheory.StructuredArrow.ext
|
[] |
[
143,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/CategoryTheory/IsConnected.lean
|
CategoryTheory.any_functor_const_on_obj
|
[
{
"state_after": "case as\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : IsPreconnected J\nα : Type u₁\nF : J ⥤ Discrete α\nj j' : J\n⊢ (F.obj j).as = (F.obj j').as",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : IsPreconnected J\nα : Type u₁\nF : J ⥤ Discrete α\nj j' : J\n⊢ F.obj j = F.obj j'",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case as\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\ninst✝ : IsPreconnected J\nα : Type u₁\nF : J ⥤ Discrete α\nj j' : J\n⊢ (F.obj j).as = (F.obj j').as",
"tactic": "exact ((isoConstant F j').hom.app j).down.1"
}
] |
[
103,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
StrictConvexOn.add
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.296238\nα : Type ?u.296241\nβ : Type u_3\nι : Type ?u.296247\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedCancelAddCommMonoid β\ninst✝¹ : SMul 𝕜 E\ninst✝ : DistribMulAction 𝕜 β\ns : Set E\nf g : E → β\nhf : StrictConvexOn 𝕜 s f\nhg : StrictConvexOn 𝕜 s g\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • f x + b • f y + (a • g x + b • g y) = a • (f x + g x) + b • (f y + g y)",
"tactic": "rw [smul_add, smul_add, add_add_add_comm]"
}
] |
[
522,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
516,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
nhdsWithin_hasBasis
|
[] |
[
86,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
continuous_piecewise
|
[] |
[
1205,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1202,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.disjoint_sup_right
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.140859\nα : Type u_1\nβ : Type ?u.140865\nγ : Type ?u.140868\nι : Type u_2\nκ : Type ?u.140874\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\ns : Finset ι\nt : Finset κ\nf : ι → α\ng : κ → α\na : α\n⊢ _root_.Disjoint a (sup s f) ↔ ∀ ⦃i : ι⦄, i ∈ s → _root_.Disjoint a (f i)",
"tactic": "simp only [disjoint_iff, sup_inf_distrib_left, Finset.sup_eq_bot_iff]"
}
] |
[
528,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
11
] |
Mathlib/Order/Basic.lean
|
lt_of_le_of_ne'
|
[] |
[
100,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Analysis/InnerProductSpace/Symmetric.lean
|
LinearMap.isSymmetric_iff_sesqForm
|
[] |
[
73,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Data/Polynomial/Splits.lean
|
Polynomial.splits_of_splits_gcd_left
|
[] |
[
261,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Mathlib/Analysis/NormedSpace/AddTorsor.lean
|
dist_midpoint_midpoint_le'
|
[
{
"state_after": "α : Type ?u.73606\nV : Type u_3\nP : Type u_1\nW : Type ?u.73615\nQ : Type ?u.73618\ninst✝⁹ : SeminormedAddCommGroup V\ninst✝⁸ : PseudoMetricSpace P\ninst✝⁷ : NormedAddTorsor V P\ninst✝⁶ : NormedAddCommGroup W\ninst✝⁵ : MetricSpace Q\ninst✝⁴ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : Invertible 2\np₁ p₂ p₃ p₄ : P\n⊢ ‖midpoint 𝕜 (p₁ -ᵥ p₃) (p₂ -ᵥ p₄)‖ ≤ (‖p₁ -ᵥ p₃‖ + ‖p₂ -ᵥ p₄‖) / ‖2‖",
"state_before": "α : Type ?u.73606\nV : Type u_3\nP : Type u_1\nW : Type ?u.73615\nQ : Type ?u.73618\ninst✝⁹ : SeminormedAddCommGroup V\ninst✝⁸ : PseudoMetricSpace P\ninst✝⁷ : NormedAddTorsor V P\ninst✝⁶ : NormedAddCommGroup W\ninst✝⁵ : MetricSpace Q\ninst✝⁴ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : Invertible 2\np₁ p₂ p₃ p₄ : P\n⊢ dist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / ‖2‖",
"tactic": "rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, midpoint_vsub_midpoint]"
},
{
"state_after": "α : Type ?u.73606\nV : Type u_3\nP : Type u_1\nW : Type ?u.73615\nQ : Type ?u.73618\ninst✝⁹ : SeminormedAddCommGroup V\ninst✝⁸ : PseudoMetricSpace P\ninst✝⁷ : NormedAddTorsor V P\ninst✝⁶ : NormedAddCommGroup W\ninst✝⁵ : MetricSpace Q\ninst✝⁴ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : Invertible 2\np₁ p₂ p₃ p₄ : P\n⊢ ‖midpoint 𝕜 (p₁ -ᵥ p₃) (p₂ -ᵥ p₄)‖ ≤ (‖p₁ -ᵥ p₃‖ + ‖p₂ -ᵥ p₄‖) / ‖2‖",
"state_before": "α : Type ?u.73606\nV : Type u_3\nP : Type u_1\nW : Type ?u.73615\nQ : Type ?u.73618\ninst✝⁹ : SeminormedAddCommGroup V\ninst✝⁸ : PseudoMetricSpace P\ninst✝⁷ : NormedAddTorsor V P\ninst✝⁶ : NormedAddCommGroup W\ninst✝⁵ : MetricSpace Q\ninst✝⁴ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : Invertible 2\np₁ p₂ p₃ p₄ : P\n⊢ ‖midpoint 𝕜 (p₁ -ᵥ p₃) (p₂ -ᵥ p₄)‖ ≤ (‖p₁ -ᵥ p₃‖ + ‖p₂ -ᵥ p₄‖) / ‖2‖",
"tactic": "try infer_instance"
},
{
"state_after": "α : Type ?u.73606\nV : Type u_3\nP : Type u_1\nW : Type ?u.73615\nQ : Type ?u.73618\ninst✝⁹ : SeminormedAddCommGroup V\ninst✝⁸ : PseudoMetricSpace P\ninst✝⁷ : NormedAddTorsor V P\ninst✝⁶ : NormedAddCommGroup W\ninst✝⁵ : MetricSpace Q\ninst✝⁴ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : Invertible 2\np₁ p₂ p₃ p₄ : P\n⊢ ‖p₁ -ᵥ p₃ + (p₂ -ᵥ p₄)‖ / ‖2‖ ≤ (‖p₁ -ᵥ p₃‖ + ‖p₂ -ᵥ p₄‖) / ‖2‖",
"state_before": "α : Type ?u.73606\nV : Type u_3\nP : Type u_1\nW : Type ?u.73615\nQ : Type ?u.73618\ninst✝⁹ : SeminormedAddCommGroup V\ninst✝⁸ : PseudoMetricSpace P\ninst✝⁷ : NormedAddTorsor V P\ninst✝⁶ : NormedAddCommGroup W\ninst✝⁵ : MetricSpace Q\ninst✝⁴ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : Invertible 2\np₁ p₂ p₃ p₄ : P\n⊢ ‖midpoint 𝕜 (p₁ -ᵥ p₃) (p₂ -ᵥ p₄)‖ ≤ (‖p₁ -ᵥ p₃‖ + ‖p₂ -ᵥ p₄‖) / ‖2‖",
"tactic": "rw [midpoint_eq_smul_add, norm_smul, invOf_eq_inv, norm_inv, ← div_eq_inv_mul]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.73606\nV : Type u_3\nP : Type u_1\nW : Type ?u.73615\nQ : Type ?u.73618\ninst✝⁹ : SeminormedAddCommGroup V\ninst✝⁸ : PseudoMetricSpace P\ninst✝⁷ : NormedAddTorsor V P\ninst✝⁶ : NormedAddCommGroup W\ninst✝⁵ : MetricSpace Q\ninst✝⁴ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : Invertible 2\np₁ p₂ p₃ p₄ : P\n⊢ ‖p₁ -ᵥ p₃ + (p₂ -ᵥ p₄)‖ / ‖2‖ ≤ (‖p₁ -ᵥ p₃‖ + ‖p₂ -ᵥ p₄‖) / ‖2‖",
"tactic": "exact div_le_div_of_le_of_nonneg (norm_add_le _ _) (norm_nonneg _)"
},
{
"state_after": "α : Type ?u.73606\nV : Type u_3\nP : Type u_1\nW : Type ?u.73615\nQ : Type ?u.73618\ninst✝⁹ : SeminormedAddCommGroup V\ninst✝⁸ : PseudoMetricSpace P\ninst✝⁷ : NormedAddTorsor V P\ninst✝⁶ : NormedAddCommGroup W\ninst✝⁵ : MetricSpace Q\ninst✝⁴ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : Invertible 2\np₁ p₂ p₃ p₄ : P\n⊢ ‖midpoint 𝕜 (p₁ -ᵥ p₃) (p₂ -ᵥ p₄)‖ ≤ (‖p₁ -ᵥ p₃‖ + ‖p₂ -ᵥ p₄‖) / ‖2‖",
"state_before": "α : Type ?u.73606\nV : Type u_3\nP : Type u_1\nW : Type ?u.73615\nQ : Type ?u.73618\ninst✝⁹ : SeminormedAddCommGroup V\ninst✝⁸ : PseudoMetricSpace P\ninst✝⁷ : NormedAddTorsor V P\ninst✝⁶ : NormedAddCommGroup W\ninst✝⁵ : MetricSpace Q\ninst✝⁴ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedSpace 𝕜 W\ninst✝ : Invertible 2\np₁ p₂ p₃ p₄ : P\n⊢ ‖midpoint 𝕜 (p₁ -ᵥ p₃) (p₂ -ᵥ p₄)‖ ≤ (‖p₁ -ᵥ p₃‖ + ‖p₂ -ᵥ p₄‖) / ‖2‖",
"tactic": "infer_instance"
}
] |
[
220,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Algebra/Homology/ImageToKernel.lean
|
subobject_ofLE_as_imageToKernel
|
[] |
[
60,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.lsub_pos
|
[] |
[
1669,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1668,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
FractionalIdeal.div_eq_mul_inv
|
[
{
"state_after": "case pos\nR : Type ?u.709535\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI J : FractionalIdeal A⁰ K\nhJ : J = 0\n⊢ I / J = I * J⁻¹\n\ncase neg\nR : Type ?u.709535\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI J : FractionalIdeal A⁰ K\nhJ : ¬J = 0\n⊢ I / J = I * J⁻¹",
"state_before": "R : Type ?u.709535\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI J : FractionalIdeal A⁰ K\n⊢ I / J = I * J⁻¹",
"tactic": "by_cases hJ : J = 0"
},
{
"state_after": "case neg.refine'_1\nR : Type ?u.709535\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI J : FractionalIdeal A⁰ K\nhJ : ¬J = 0\n⊢ I / J * J ≤ I * J⁻¹ * J\n\ncase neg.refine'_2\nR : Type ?u.709535\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI J : FractionalIdeal A⁰ K\nhJ : ¬J = 0\n⊢ I * J⁻¹ * J ≤ I",
"state_before": "case neg\nR : Type ?u.709535\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI J : FractionalIdeal A⁰ K\nhJ : ¬J = 0\n⊢ I / J = I * J⁻¹",
"tactic": "refine' le_antisymm ((mul_right_le_iff hJ).mp _) ((le_div_iff_mul_le hJ).mpr _)"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_2\nR : Type ?u.709535\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI J : FractionalIdeal A⁰ K\nhJ : ¬J = 0\n⊢ I * J⁻¹ * J ≤ I",
"tactic": "rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type ?u.709535\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI J : FractionalIdeal A⁰ K\nhJ : J = 0\n⊢ I / J = I * J⁻¹",
"tactic": "rw [hJ, div_zero, inv_zero', MulZeroClass.mul_zero]"
},
{
"state_after": "case neg.refine'_1\nR : Type ?u.709535\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI J : FractionalIdeal A⁰ K\nhJ : ¬J = 0\n⊢ ∀ (i : K), i ∈ I / J → ∀ (j : K), j ∈ J → i * j ∈ I",
"state_before": "case neg.refine'_1\nR : Type ?u.709535\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI J : FractionalIdeal A⁰ K\nhJ : ¬J = 0\n⊢ I / J * J ≤ I * J⁻¹ * J",
"tactic": "rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one, mul_le]"
},
{
"state_after": "case neg.refine'_1\nR : Type ?u.709535\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI J : FractionalIdeal A⁰ K\nhJ : ¬J = 0\nx : K\nhx : x ∈ I / J\ny : K\nhy : y ∈ J\n⊢ x * y ∈ I",
"state_before": "case neg.refine'_1\nR : Type ?u.709535\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI J : FractionalIdeal A⁰ K\nhJ : ¬J = 0\n⊢ ∀ (i : K), i ∈ I / J → ∀ (j : K), j ∈ J → i * j ∈ I",
"tactic": "intro x hx y hy"
},
{
"state_after": "case neg.refine'_1\nR : Type ?u.709535\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI J : FractionalIdeal A⁰ K\nhJ : ¬J = 0\nx : K\nhx : ∀ (y : K), y ∈ J → x * y ∈ I\ny : K\nhy : y ∈ J\n⊢ x * y ∈ I",
"state_before": "case neg.refine'_1\nR : Type ?u.709535\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI J : FractionalIdeal A⁰ K\nhJ : ¬J = 0\nx : K\nhx : x ∈ I / J\ny : K\nhy : y ∈ J\n⊢ x * y ∈ I",
"tactic": "rw [mem_div_iff_of_nonzero hJ] at hx"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_1\nR : Type ?u.709535\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI J : FractionalIdeal A⁰ K\nhJ : ¬J = 0\nx : K\nhx : ∀ (y : K), y ∈ J → x * y ∈ I\ny : K\nhy : y ∈ J\n⊢ x * y ∈ I",
"tactic": "exact hx y hy"
}
] |
[
587,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
578,
11
] |
Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean
|
Path.Homotopy.transAssocReparamAux_mem_I
|
[
{
"state_after": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nt : ↑I\n⊢ (if ↑t ≤ 1 / 4 then 2 * ↑t else if ↑t ≤ 1 / 2 then ↑t + 1 / 4 else 1 / 2 * (↑t + 1)) ∈ I",
"state_before": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nt : ↑I\n⊢ transAssocReparamAux t ∈ I",
"tactic": "unfold transAssocReparamAux"
},
{
"state_after": "no goals",
"state_before": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\nt : ↑I\n⊢ (if ↑t ≤ 1 / 4 then 2 * ↑t else if ↑t ≤ 1 / 2 then ↑t + 1 / 4 else 1 / 2 * (↑t + 1)) ∈ I",
"tactic": "split_ifs <;> constructor <;> linarith [unitInterval.le_one t, unitInterval.nonneg t]"
}
] |
[
215,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Std/Data/List/Lemmas.lean
|
List.erase_subset
|
[] |
[
1077,
88
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1077,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
mem_nhds_right
|
[] |
[
801,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
800,
1
] |
Mathlib/Topology/Basic.lean
|
interior_eq_univ
|
[] |
[
339,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Order/Antisymmetrization.lean
|
toAntisymmetrization_le_toAntisymmetrization_iff
|
[] |
[
185,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Topology/LocallyConstant/Basic.lean
|
IsLocallyConstant.of_discrete
|
[] |
[
71,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Algebra/DirectSum/Decomposition.lean
|
DirectSum.decompose_sub
|
[] |
[
219,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
tsum_iSup_decode₂
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\n⊢ (∑' (i : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)) = ∑' (b : γ), m (s b)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\n⊢ (∑' (i : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)) = ∑' (b : γ), m (s b)",
"tactic": "have H : ∀ n, m (⨆ b ∈ decode₂ γ n, s b) ≠ 0 → (decode₂ γ n).isSome :=by\n intro n h\n generalize decode₂ γ n = foo at *\n cases' foo with b\n . refine' (h <| by simp [m0]).elim\n . exact rfl"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\n⊢ (∑' (b : γ), m (s b)) = ∑' (i : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\n⊢ (∑' (i : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)) = ∑' (b : γ), m (s b)",
"tactic": "symm"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\n⊢ ∀ ⦃x y : ↑(support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b))⦄,\n (fun a => Option.get (decode₂ γ ↑a) (_ : Option.isSome (decode₂ γ ↑a) = true)) x =\n (fun a => Option.get (decode₂ γ ↑a) (_ : Option.isSome (decode₂ γ ↑a) = true)) y →\n ↑x = ↑y\n\ncase refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\n⊢ (support fun b => m (s b)) ⊆ Set.range fun a => Option.get (decode₂ γ ↑a) (_ : Option.isSome (decode₂ γ ↑a) = true)\n\ncase refine'_3\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\n⊢ ∀ (x : ↑(support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b))),\n m (s ((fun a => Option.get (decode₂ γ ↑a) (_ : Option.isSome (decode₂ γ ↑a) = true)) x)) =\n m (⨆ (b : γ) (_ : b ∈ decode₂ γ ↑x), s b)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\n⊢ (∑' (b : γ), m (s b)) = ∑' (i : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)",
"tactic": "refine' tsum_eq_tsum_of_ne_zero_bij (fun a => Option.get _ (H a.1 a.2)) _ _ _"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nn : ℕ\nh : m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0\n⊢ Option.isSome (decode₂ γ n) = true",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\n⊢ ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true",
"tactic": "intro n h"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nn : ℕ\nfoo : Option γ\nh : m (⨆ (b : γ) (_ : b ∈ foo), s b) ≠ 0\n⊢ Option.isSome foo = true",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nn : ℕ\nh : m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0\n⊢ Option.isSome (decode₂ γ n) = true",
"tactic": "generalize decode₂ γ n = foo at *"
},
{
"state_after": "case none\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nn : ℕ\nh : m (⨆ (b : γ) (_ : b ∈ none), s b) ≠ 0\n⊢ Option.isSome none = true\n\ncase some\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nn : ℕ\nb : γ\nh : m (⨆ (b_1 : γ) (_ : b_1 ∈ some b), s b_1) ≠ 0\n⊢ Option.isSome (some b) = true",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nn : ℕ\nfoo : Option γ\nh : m (⨆ (b : γ) (_ : b ∈ foo), s b) ≠ 0\n⊢ Option.isSome foo = true",
"tactic": "cases' foo with b"
},
{
"state_after": "case some\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nn : ℕ\nb : γ\nh : m (⨆ (b_1 : γ) (_ : b_1 ∈ some b), s b_1) ≠ 0\n⊢ Option.isSome (some b) = true",
"state_before": "case none\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nn : ℕ\nh : m (⨆ (b : γ) (_ : b ∈ none), s b) ≠ 0\n⊢ Option.isSome none = true\n\ncase some\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nn : ℕ\nb : γ\nh : m (⨆ (b_1 : γ) (_ : b_1 ∈ some b), s b_1) ≠ 0\n⊢ Option.isSome (some b) = true",
"tactic": ". refine' (h <| by simp [m0]).elim"
},
{
"state_after": "no goals",
"state_before": "case some\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nn : ℕ\nb : γ\nh : m (⨆ (b_1 : γ) (_ : b_1 ∈ some b), s b_1) ≠ 0\n⊢ Option.isSome (some b) = true",
"tactic": ". exact rfl"
},
{
"state_after": "no goals",
"state_before": "case none\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nn : ℕ\nh : m (⨆ (b : γ) (_ : b ∈ none), s b) ≠ 0\n⊢ Option.isSome none = true",
"tactic": "refine' (h <| by simp [m0]).elim"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nn : ℕ\nh : m (⨆ (b : γ) (_ : b ∈ none), s b) ≠ 0\n⊢ m (⨆ (b : γ) (_ : b ∈ none), s b) = 0",
"tactic": "simp [m0]"
},
{
"state_after": "no goals",
"state_before": "case some\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nn : ℕ\nb : γ\nh : m (⨆ (b_1 : γ) (_ : b_1 ∈ some b), s b_1) ≠ 0\n⊢ Option.isSome (some b) = true",
"tactic": "exact rfl"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\n⊢ ∀ ⦃x y : ↑(support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b))⦄,\n Option.get (decode₂ γ ↑x) (_ : Option.isSome (decode₂ γ ↑x) = true) =\n Option.get (decode₂ γ ↑y) (_ : Option.isSome (decode₂ γ ↑y) = true) →\n ↑x = ↑y",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\n⊢ ∀ ⦃x y : ↑(support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b))⦄,\n (fun a => Option.get (decode₂ γ ↑a) (_ : Option.isSome (decode₂ γ ↑a) = true)) x =\n (fun a => Option.get (decode₂ γ ↑a) (_ : Option.isSome (decode₂ γ ↑a) = true)) y →\n ↑x = ↑y",
"tactic": "dsimp only []"
},
{
"state_after": "case refine'_1.mk.mk\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm✝ : β → α\nm0 : m✝ ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m✝ (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nm : ℕ\nhm : m ∈ support fun i => m✝ (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\nn : ℕ\nhn : n ∈ support fun i => m✝ (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\ne :\n Option.get (decode₂ γ ↑{ val := m, property := hm })\n (_ : Option.isSome (decode₂ γ ↑{ val := m, property := hm }) = true) =\n Option.get (decode₂ γ ↑{ val := n, property := hn })\n (_ : Option.isSome (decode₂ γ ↑{ val := n, property := hn }) = true)\n⊢ ↑{ val := m, property := hm } = ↑{ val := n, property := hn }",
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"tactic": "rintro ⟨m, hm⟩ ⟨n, hn⟩ e"
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{
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"tactic": "have := mem_decode₂.1 (Option.get_mem (H n hn))"
},
{
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"tactic": "rwa [← e, mem_decode₂.1 (Option.get_mem (H m hm))] at this"
},
{
"state_after": "case refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nb : γ\nh : b ∈ support fun b => m (s b)\n⊢ b ∈ Set.range fun a => Option.get (decode₂ γ ↑a) (_ : Option.isSome (decode₂ γ ↑a) = true)",
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"tactic": "intro b h"
},
{
"state_after": "case refine'_2.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nb : γ\nh : b ∈ support fun b => m (s b)\n⊢ encode b ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n\ncase refine'_2.refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nb : γ\nh : b ∈ support fun b => m (s b)\n⊢ (fun a => Option.get (decode₂ γ ↑a) (_ : Option.isSome (decode₂ γ ↑a) = true))\n { val := encode b, property := ?refine'_2.refine'_1 } =\n b",
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"tactic": "refine' ⟨⟨encode b, _⟩, _⟩"
},
{
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"tactic": "simp only [mem_support, encodek₂] at h⊢"
},
{
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"tactic": "convert h"
},
{
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"tactic": "simp [Set.ext_iff, encodek₂]"
},
{
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"state_before": "case refine'_2.refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nb : γ\nh : b ∈ support fun b => m (s b)\n⊢ (fun a => Option.get (decode₂ γ ↑a) (_ : Option.isSome (decode₂ γ ↑a) = true))\n { val := encode b, property := (_ : encode b ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)) } =\n b",
"tactic": "exact Option.get_of_mem _ (encodek₂ _)"
},
{
"state_after": "case refine'_3.mk\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ m (s ((fun a => Option.get (decode₂ γ ↑a) (_ : Option.isSome (decode₂ γ ↑a) = true)) { val := n, property := h })) =\n m (⨆ (b : γ) (_ : b ∈ decode₂ γ ↑{ val := n, property := h }), s b)",
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"tactic": "rintro ⟨n, h⟩"
},
{
"state_after": "case refine'_3.mk\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ m (s (Option.get (decode₂ γ n) (_ : Option.isSome (decode₂ γ n) = true))) = m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b)",
"state_before": "case refine'_3.mk\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ m (s ((fun a => Option.get (decode₂ γ ↑a) (_ : Option.isSome (decode₂ γ ↑a) = true)) { val := n, property := h })) =\n m (⨆ (b : γ) (_ : b ∈ decode₂ γ ↑{ val := n, property := h }), s b)",
"tactic": "dsimp only [Subtype.coe_mk]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ m (s (Option.get (decode₂ γ n) (_ : Option.isSome (decode₂ γ n) = true))) = ?m.371147\n\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ ?m.371147 = m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b)\n\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ α",
"state_before": "case refine'_3.mk\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ m (s (Option.get (decode₂ γ n) (_ : Option.isSome (decode₂ γ n) = true))) = m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b)",
"tactic": "trans"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ ?m.371147 = m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b)\n\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ m (s (Option.get (decode₂ γ n) (_ : Option.isSome (decode₂ γ n) = true))) = ?m.371147\n\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ α",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ m (s (Option.get (decode₂ γ n) (_ : Option.isSome (decode₂ γ n) = true))) = ?m.371147\n\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ ?m.371147 = m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b)\n\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ α",
"tactic": "swap"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ m (s (Option.get (decode₂ γ n) (_ : Option.isSome (decode₂ γ n) = true))) =\n m (⨆ (b : γ) (_ : b ∈ some (Option.get (decode₂ γ n) (_ : Option.isSome (decode₂ γ n) = true))), s b)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ ?m.371147 = m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b)\n\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ m (s (Option.get (decode₂ γ n) (_ : Option.isSome (decode₂ γ n) = true))) = ?m.371147\n\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ α",
"tactic": "rw [show decode₂ γ n = _ from Option.get_mem (H n h)]"
},
{
"state_after": "case e_a\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ s (Option.get (decode₂ γ n) (_ : Option.isSome (decode₂ γ n) = true)) =\n ⨆ (b : γ) (_ : b ∈ some (Option.get (decode₂ γ n) (_ : Option.isSome (decode₂ γ n) = true))), s b",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ m (s (Option.get (decode₂ γ n) (_ : Option.isSome (decode₂ γ n) = true))) =\n m (⨆ (b : γ) (_ : b ∈ some (Option.get (decode₂ γ n) (_ : Option.isSome (decode₂ γ n) = true))), s b)",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_a\nα : Type u_2\nβ : Type u_1\nγ : Type u_3\nδ : Type ?u.362154\ninst✝⁴ : AddCommMonoid α\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\nf g : β → α\na a₁ a₂ : α\ninst✝¹ : Encodable γ\ninst✝ : CompleteLattice β\nm : β → α\nm0 : m ⊥ = 0\ns : γ → β\nH : ∀ (n : ℕ), m (⨆ (b : γ) (_ : b ∈ decode₂ γ n), s b) ≠ 0 → Option.isSome (decode₂ γ n) = true\nn : ℕ\nh : n ∈ support fun i => m (⨆ (b : γ) (_ : b ∈ decode₂ γ i), s b)\n⊢ s (Option.get (decode₂ γ n) (_ : Option.isSome (decode₂ γ n) = true)) =\n ⨆ (b : γ) (_ : b ∈ some (Option.get (decode₂ γ n) (_ : Option.isSome (decode₂ γ n) = true))), s b",
"tactic": "simp [ext_iff, -Option.some_get]"
}
] |
[
720,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
694,
1
] |
Mathlib/Analysis/Normed/Group/AddTorsor.lean
|
dist_vsub_cancel_right
|
[] |
[
164,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.range_le
|
[] |
[
145,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/Analysis/NormedSpace/Pointwise.lean
|
ball_sub_ball
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.798220\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y z : E\nδ ε : ℝ\nhε : 0 < ε\nhδ : 0 < δ\na b : E\n⊢ Metric.ball a ε - Metric.ball b δ = Metric.ball (a - b) (ε + δ)",
"tactic": "simp_rw [sub_eq_add_neg, neg_ball, ball_add_ball hε hδ]"
}
] |
[
355,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
MonoidWithZeroHom.toMonoidHom_coe
|
[] |
[
639,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
638,
1
] |
Mathlib/CategoryTheory/Sites/Sheafification.lean
|
CategoryTheory.GrothendieckTopology.Plus.toPlus_mk
|
[
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\n⊢ (forget D).map (Cover.toMultiequalizer ⊤ P ≫ colimit.ι (diagram J P X) ⊤.op) x =\n (forget D).map (colimit.ι (diagram J P X) S.op) (↑(Meq.equiv P S).symm (Meq.mk S x))",
"state_before": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\n⊢ (forget D).map ((toPlus J P).app X.op) x = mk (Meq.mk S x)",
"tactic": "dsimp [mk, toPlus]"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\n⊢ (forget D).map (Cover.toMultiequalizer ⊤ P ≫ colimit.ι (diagram J P X) ⊤.op) x =\n (forget D).map (colimit.ι (diagram J P X) S.op) (↑(Meq.equiv P S).symm (Meq.mk S x))",
"state_before": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\n⊢ (forget D).map (Cover.toMultiequalizer ⊤ P ≫ colimit.ι (diagram J P X) ⊤.op) x =\n (forget D).map (colimit.ι (diagram J P X) S.op) (↑(Meq.equiv P S).symm (Meq.mk S x))",
"tactic": "let e : S ⟶ ⊤ := homOfLE (OrderTop.le_top _)"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\n⊢ (forget D).map (Cover.toMultiequalizer ⊤ P ≫ (diagram J P X).map e.op ≫ colimit.ι (diagram J P X) S.op) x =\n (forget D).map (colimit.ι (diagram J P X) S.op) (↑(Meq.equiv P S).symm (Meq.mk S x))",
"state_before": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\n⊢ (forget D).map (Cover.toMultiequalizer ⊤ P ≫ colimit.ι (diagram J P X) ⊤.op) x =\n (forget D).map (colimit.ι (diagram J P X) S.op) (↑(Meq.equiv P S).symm (Meq.mk S x))",
"tactic": "rw [← colimit.w _ e.op]"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\n⊢ (forget D).map\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj X.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I) ≫\n (diagram J P X).map e.op ≫ colimit.ι (diagram J P X) S.op)\n x =\n (forget D).map (colimit.ι (diagram J P X) S.op) (↑(Meq.equiv P S).symm (Meq.mk S x))",
"state_before": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\n⊢ (forget D).map (Cover.toMultiequalizer ⊤ P ≫ (diagram J P X).map e.op ≫ colimit.ι (diagram J P X) S.op) x =\n (forget D).map (colimit.ι (diagram J P X) S.op) (↑(Meq.equiv P S).symm (Meq.mk S x))",
"tactic": "delta Cover.toMultiequalizer"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\n⊢ (forget D).map (colimit.ι (diagram J P X) S.op)\n ((forget D).map ((diagram J P X).map (homOfLE (_ : S ≤ ⊤)).op)\n ((forget D).map\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj X.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I))\n x)) =\n (forget D).map (colimit.ι (diagram J P X) S.op) (↑(Meq.equiv P S).symm (Meq.mk S x))",
"state_before": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\n⊢ (forget D).map\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj X.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I) ≫\n (diagram J P X).map e.op ≫ colimit.ι (diagram J P X) S.op)\n x =\n (forget D).map (colimit.ι (diagram J P X) S.op) (↑(Meq.equiv P S).symm (Meq.mk S x))",
"tactic": "simp only [comp_apply]"
},
{
"state_after": "case h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\n⊢ (forget D).map ((diagram J P X).map (homOfLE (_ : S ≤ ⊤)).op)\n ((forget D).map\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj X.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I))\n x) =\n ↑(Meq.equiv P S).symm (Meq.mk S x)",
"state_before": "C : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\n⊢ (forget D).map (colimit.ι (diagram J P X) S.op)\n ((forget D).map ((diagram J P X).map (homOfLE (_ : S ≤ ⊤)).op)\n ((forget D).map\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj X.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I))\n x)) =\n (forget D).map (colimit.ι (diagram J P X) S.op) (↑(Meq.equiv P S).symm (Meq.mk S x))",
"tactic": "apply congr_arg"
},
{
"state_after": "case h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\n⊢ (forget D).map\n (Multiequalizer.lift (Cover.index S P) (multiequalizer (Cover.index ⊤ P))\n (fun I => Multiequalizer.ι (Cover.index ⊤ P) (Cover.Arrow.map I (homOfLE (_ : S ≤ ⊤))))\n (_ :\n ∀ (I : (Cover.index S.op.unop P).R),\n Multiequalizer.ι (Cover.index ⊤.op.unop P)\n (MulticospanIndex.fstTo (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)) ≫\n MulticospanIndex.fst (Cover.index ⊤.op.unop P) (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop) =\n Multiequalizer.ι (Cover.index ⊤.op.unop P)\n (MulticospanIndex.sndTo (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)) ≫\n MulticospanIndex.snd (Cover.index ⊤.op.unop P) (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)))\n ((forget D).map\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj X.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I))\n x) =\n ↑(Meq.equiv P S).symm (Meq.mk S x)",
"state_before": "case h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\n⊢ (forget D).map ((diagram J P X).map (homOfLE (_ : S ≤ ⊤)).op)\n ((forget D).map\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj X.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I))\n x) =\n ↑(Meq.equiv P S).symm (Meq.mk S x)",
"tactic": "dsimp [diagram]"
},
{
"state_after": "case h.h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\n⊢ ∀ (t : (Cover.index S P).L),\n (forget D).map (Multiequalizer.ι (Cover.index S P) t)\n ((forget D).map\n (Multiequalizer.lift (Cover.index S P) (multiequalizer (Cover.index ⊤ P))\n (fun I => Multiequalizer.ι (Cover.index ⊤ P) (Cover.Arrow.map I (homOfLE (_ : S ≤ ⊤))))\n (_ :\n ∀ (I : (Cover.index S.op.unop P).R),\n Multiequalizer.ι (Cover.index ⊤.op.unop P)\n (MulticospanIndex.fstTo (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)) ≫\n MulticospanIndex.fst (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop) =\n Multiequalizer.ι (Cover.index ⊤.op.unop P)\n (MulticospanIndex.sndTo (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)) ≫\n MulticospanIndex.snd (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)))\n ((forget D).map\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj X.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I))\n x)) =\n (forget D).map (Multiequalizer.ι (Cover.index S P) t) (↑(Meq.equiv P S).symm (Meq.mk S x))",
"state_before": "case h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\n⊢ (forget D).map\n (Multiequalizer.lift (Cover.index S P) (multiequalizer (Cover.index ⊤ P))\n (fun I => Multiequalizer.ι (Cover.index ⊤ P) (Cover.Arrow.map I (homOfLE (_ : S ≤ ⊤))))\n (_ :\n ∀ (I : (Cover.index S.op.unop P).R),\n Multiequalizer.ι (Cover.index ⊤.op.unop P)\n (MulticospanIndex.fstTo (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)) ≫\n MulticospanIndex.fst (Cover.index ⊤.op.unop P) (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop) =\n Multiequalizer.ι (Cover.index ⊤.op.unop P)\n (MulticospanIndex.sndTo (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)) ≫\n MulticospanIndex.snd (Cover.index ⊤.op.unop P) (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)))\n ((forget D).map\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj X.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I))\n x) =\n ↑(Meq.equiv P S).symm (Meq.mk S x)",
"tactic": "apply Concrete.multiequalizer_ext"
},
{
"state_after": "case h.h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\ni : (Cover.index S P).L\n⊢ (forget D).map (Multiequalizer.ι (Cover.index S P) i)\n ((forget D).map\n (Multiequalizer.lift (Cover.index S P) (multiequalizer (Cover.index ⊤ P))\n (fun I => Multiequalizer.ι (Cover.index ⊤ P) (Cover.Arrow.map I (homOfLE (_ : S ≤ ⊤))))\n (_ :\n ∀ (I : (Cover.index S.op.unop P).R),\n Multiequalizer.ι (Cover.index ⊤.op.unop P)\n (MulticospanIndex.fstTo (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)) ≫\n MulticospanIndex.fst (Cover.index ⊤.op.unop P) (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop) =\n Multiequalizer.ι (Cover.index ⊤.op.unop P)\n (MulticospanIndex.sndTo (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)) ≫\n MulticospanIndex.snd (Cover.index ⊤.op.unop P) (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)))\n ((forget D).map\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj X.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I))\n x)) =\n (forget D).map (Multiequalizer.ι (Cover.index S P) i) (↑(Meq.equiv P S).symm (Meq.mk S x))",
"state_before": "case h.h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\n⊢ ∀ (t : (Cover.index S P).L),\n (forget D).map (Multiequalizer.ι (Cover.index S P) t)\n ((forget D).map\n (Multiequalizer.lift (Cover.index S P) (multiequalizer (Cover.index ⊤ P))\n (fun I => Multiequalizer.ι (Cover.index ⊤ P) (Cover.Arrow.map I (homOfLE (_ : S ≤ ⊤))))\n (_ :\n ∀ (I : (Cover.index S.op.unop P).R),\n Multiequalizer.ι (Cover.index ⊤.op.unop P)\n (MulticospanIndex.fstTo (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)) ≫\n MulticospanIndex.fst (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop) =\n Multiequalizer.ι (Cover.index ⊤.op.unop P)\n (MulticospanIndex.sndTo (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)) ≫\n MulticospanIndex.snd (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)))\n ((forget D).map\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj X.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I))\n x)) =\n (forget D).map (Multiequalizer.ι (Cover.index S P) t) (↑(Meq.equiv P S).symm (Meq.mk S x))",
"tactic": "intro i"
},
{
"state_after": "case h.h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\ni : (Cover.index S P).L\n⊢ (forget D).map (P.map (Cover.Arrow.map i (homOfLE (_ : S ≤ ⊤))).f.op) x = ↑(Meq.mk S x) i",
"state_before": "case h.h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\ni : (Cover.index S P).L\n⊢ (forget D).map (Multiequalizer.ι (Cover.index S P) i)\n ((forget D).map\n (Multiequalizer.lift (Cover.index S P) (multiequalizer (Cover.index ⊤ P))\n (fun I => Multiequalizer.ι (Cover.index ⊤ P) (Cover.Arrow.map I (homOfLE (_ : S ≤ ⊤))))\n (_ :\n ∀ (I : (Cover.index S.op.unop P).R),\n Multiequalizer.ι (Cover.index ⊤.op.unop P)\n (MulticospanIndex.fstTo (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)) ≫\n MulticospanIndex.fst (Cover.index ⊤.op.unop P) (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop) =\n Multiequalizer.ι (Cover.index ⊤.op.unop P)\n (MulticospanIndex.sndTo (Cover.index ⊤.op.unop P)\n (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)) ≫\n MulticospanIndex.snd (Cover.index ⊤.op.unop P) (Cover.Relation.map I (homOfLE (_ : S ≤ ⊤)).op.unop)))\n ((forget D).map\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj X.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I))\n x)) =\n (forget D).map (Multiequalizer.ι (Cover.index S P) i) (↑(Meq.equiv P S).symm (Meq.mk S x))",
"tactic": "simp only [← comp_apply, Category.assoc, Multiequalizer.lift_ι, Category.comp_id,\n Meq.equiv_symm_eq_apply]"
},
{
"state_after": "no goals",
"state_before": "case h.h\nC : Type u\ninst✝⁵ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category D\ninst✝³ : ConcreteCategory D\ninst✝² : PreservesLimits (forget D)\ninst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx : (forget D).obj (P.obj X.op)\ne : S ⟶ ⊤ := homOfLE (_ : S ≤ ⊤)\ni : (Cover.index S P).L\n⊢ (forget D).map (P.map (Cover.Arrow.map i (homOfLE (_ : S ≤ ⊤))).f.op) x = ↑(Meq.mk S x) i",
"tactic": "rfl"
}
] |
[
193,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Data/Multiset/Sum.lean
|
Multiset.disjSum_zero
|
[] |
[
43,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
|
CategoryTheory.Limits.kernelIsoOfEq_refl
|
[
{
"state_after": "case w.h\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : HasKernel f\nh : f = f\n⊢ (kernelIsoOfEq h).hom ≫ equalizer.ι f 0 = (Iso.refl (kernel f)).hom ≫ equalizer.ι f 0",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : HasKernel f\nh : f = f\n⊢ kernelIsoOfEq h = Iso.refl (kernel f)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case w.h\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : HasKernel f\nh : f = f\n⊢ (kernelIsoOfEq h).hom ≫ equalizer.ι f 0 = (Iso.refl (kernel f)).hom ≫ equalizer.ι f 0",
"tactic": "simp [kernelIsoOfEq]"
}
] |
[
352,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.ncard_le_ncard_insert
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.11853\ns✝ t : Set α\na✝ b x y : α\nf : α → β\na : α\ns : Set α\n⊢ ncard s ≤ ncard (insert a s)",
"tactic": "classical\nrefine'\n s.finite_or_infinite.elim (fun h ↦ _) (fun h ↦ by (rw [h.ncard]; exact Nat.zero_le _))\nrw [ncard_insert_eq_ite h]; split_ifs <;> simp"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.11853\ns✝ t : Set α\na✝ b x y : α\nf : α → β\na : α\ns : Set α\nh : Set.Finite s\n⊢ ncard s ≤ ncard (insert a s)",
"state_before": "α : Type u_1\nβ : Type ?u.11853\ns✝ t : Set α\na✝ b x y : α\nf : α → β\na : α\ns : Set α\n⊢ ncard s ≤ ncard (insert a s)",
"tactic": "refine'\n s.finite_or_infinite.elim (fun h ↦ _) (fun h ↦ by (rw [h.ncard]; exact Nat.zero_le _))"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.11853\ns✝ t : Set α\na✝ b x y : α\nf : α → β\na : α\ns : Set α\nh : Set.Finite s\n⊢ ncard s ≤ if a ∈ s then ncard s else ncard s + 1",
"state_before": "α : Type u_1\nβ : Type ?u.11853\ns✝ t : Set α\na✝ b x y : α\nf : α → β\na : α\ns : Set α\nh : Set.Finite s\n⊢ ncard s ≤ ncard (insert a s)",
"tactic": "rw [ncard_insert_eq_ite h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.11853\ns✝ t : Set α\na✝ b x y : α\nf : α → β\na : α\ns : Set α\nh : Set.Finite s\n⊢ ncard s ≤ if a ∈ s then ncard s else ncard s + 1",
"tactic": "split_ifs <;> simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.11853\ns✝ t : Set α\na✝ b x y : α\nf : α → β\na : α\ns : Set α\nh : Set.Infinite s\n⊢ ncard s ≤ ncard (insert a s)",
"tactic": "(rw [h.ncard]; exact Nat.zero_le _)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.11853\ns✝ t : Set α\na✝ b x y : α\nf : α → β\na : α\ns : Set α\nh : Set.Infinite s\n⊢ 0 ≤ ncard (insert a s)",
"state_before": "α : Type u_1\nβ : Type ?u.11853\ns✝ t : Set α\na✝ b x y : α\nf : α → β\na : α\ns : Set α\nh : Set.Infinite s\n⊢ ncard s ≤ ncard (insert a s)",
"tactic": "rw [h.ncard]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.11853\ns✝ t : Set α\na✝ b x y : α\nf : α → β\na : α\ns : Set α\nh : Set.Infinite s\n⊢ 0 ≤ ncard (insert a s)",
"tactic": "exact Nat.zero_le _"
}
] |
[
177,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Algebra/Order/Interval.lean
|
Interval.length_pure
|
[] |
[
696,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
695,
1
] |
Mathlib/Algebra/DirectLimit.lean
|
Field.DirectLimit.inv_mul_cancel
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁴ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : Nonempty ι\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : (i : ι) → Field (G i)\nf : (i j : ι) → i ≤ j → G i → G j\nf' : (i j : ι) → i ≤ j → G i →+* G j\np : Ring.DirectLimit G f\nhp : p ≠ 0\n⊢ inv G f p * p = 1",
"tactic": "rw [_root_.mul_comm, DirectLimit.mul_inv_cancel G f hp]"
}
] |
[
736,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
735,
11
] |
Mathlib/Analysis/Calculus/Deriv/Star.lean
|
HasDerivAtFilter.star
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nf : 𝕜 → F\ninst✝⁴ : StarRing 𝕜\ninst✝³ : TrivialStar 𝕜\ninst✝² : StarAddMonoid F\ninst✝¹ : ContinuousStar F\ninst✝ : StarModule 𝕜 F\nf' : F\nx : 𝕜\nL : Filter 𝕜\nh : HasDerivAtFilter f f' x L\n⊢ HasDerivAtFilter (fun x => star (f x)) (star f') x L",
"tactic": "simpa using h.star.hasDerivAtFilter"
}
] |
[
40,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
18
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
Basis.exists_op_nnnorm_le
|
[
{
"state_after": "case intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nv : Basis ι 𝕜 E\nval✝ : Fintype ι\n⊢ ∃ C, C > 0 ∧ ∀ {u : E →L[𝕜] F} (M : ℝ≥0), (∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M) → ‖u‖₊ ≤ C * M",
"state_before": "𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nv : Basis ι 𝕜 E\n⊢ ∃ C, C > 0 ∧ ∀ {u : E →L[𝕜] F} (M : ℝ≥0), (∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M) → ‖u‖₊ ≤ C * M",
"tactic": "cases nonempty_fintype ι"
},
{
"state_after": "no goals",
"state_before": "case intro\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nv : Basis ι 𝕜 E\nval✝ : Fintype ι\n⊢ ∃ C, C > 0 ∧ ∀ {u : E →L[𝕜] F} (M : ℝ≥0), (∀ (i : ι), ‖↑u (↑v i)‖₊ ≤ M) → ‖u‖₊ ≤ C * M",
"tactic": "exact\n ⟨max (Fintype.card ι • ‖v.equivFunL.toContinuousLinearMap‖₊) 1,\n zero_lt_one.trans_le (le_max_right _ _), fun {u} M hu =>\n (v.op_nnnorm_le M hu).trans <| mul_le_mul_of_nonneg_right (le_max_left _ _) (zero_le M)⟩"
}
] |
[
306,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.trans_self_symm
|
[] |
[
996,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
995,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.mem_map_rangeS
|
[
{
"state_after": "case mp\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\n⊢ p ∈ RingHom.rangeS (mapRingHom f) → ∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f\n\ncase mpr\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\n⊢ (∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f) → p ∈ RingHom.rangeS (mapRingHom f)",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\n⊢ p ∈ RingHom.rangeS (mapRingHom f) ↔ ∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\nn : ℕ\n⊢ coeff (↑(mapRingHom f) p) n ∈ RingHom.rangeS f",
"state_before": "case mp\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\n⊢ p ∈ RingHom.rangeS (mapRingHom f) → ∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f",
"tactic": "rintro ⟨p, rfl⟩ n"
},
{
"state_after": "case mp.intro\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\nn : ℕ\n⊢ ↑f (coeff p n) ∈ RingHom.rangeS f",
"state_before": "case mp.intro\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\nn : ℕ\n⊢ coeff (↑(mapRingHom f) p) n ∈ RingHom.rangeS f",
"tactic": "rw [coe_mapRingHom, coeff_map]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\nn : ℕ\n⊢ ↑f (coeff p n) ∈ RingHom.rangeS f",
"tactic": "exact Set.mem_range_self _"
},
{
"state_after": "case mpr\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nh : ∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f\n⊢ p ∈ RingHom.rangeS (mapRingHom f)",
"state_before": "case mpr\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\n⊢ (∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f) → p ∈ RingHom.rangeS (mapRingHom f)",
"tactic": "intro h"
},
{
"state_after": "case mpr\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nh : ∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f\n⊢ ∑ i in range (natDegree p + 1), ↑C (coeff p i) * X ^ i ∈ RingHom.rangeS (mapRingHom f)",
"state_before": "case mpr\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nh : ∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f\n⊢ p ∈ RingHom.rangeS (mapRingHom f)",
"tactic": "rw [p.as_sum_range_C_mul_X_pow]"
},
{
"state_after": "case mpr\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nh : ∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f\n⊢ ∀ (c : ℕ), c ∈ range (natDegree p + 1) → ↑C (coeff p c) * X ^ c ∈ RingHom.rangeS (mapRingHom f)",
"state_before": "case mpr\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nh : ∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f\n⊢ ∑ i in range (natDegree p + 1), ↑C (coeff p i) * X ^ i ∈ RingHom.rangeS (mapRingHom f)",
"tactic": "refine' (mapRingHom f).rangeS.sum_mem _"
},
{
"state_after": "case mpr\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nh : ∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f\ni : ℕ\n_hi : i ∈ range (natDegree p + 1)\n⊢ ↑C (coeff p i) * X ^ i ∈ RingHom.rangeS (mapRingHom f)",
"state_before": "case mpr\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nh : ∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f\n⊢ ∀ (c : ℕ), c ∈ range (natDegree p + 1) → ↑C (coeff p c) * X ^ c ∈ RingHom.rangeS (mapRingHom f)",
"tactic": "intro i _hi"
},
{
"state_after": "case mpr.intro\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nh : ∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f\ni : ℕ\n_hi : i ∈ range (natDegree p + 1)\nc : R\nhc : ↑f c = coeff p i\n⊢ ↑C (coeff p i) * X ^ i ∈ RingHom.rangeS (mapRingHom f)",
"state_before": "case mpr\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nh : ∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f\ni : ℕ\n_hi : i ∈ range (natDegree p + 1)\n⊢ ↑C (coeff p i) * X ^ i ∈ RingHom.rangeS (mapRingHom f)",
"tactic": "rcases h i with ⟨c, hc⟩"
},
{
"state_after": "case mpr.intro\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nh : ∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f\ni : ℕ\n_hi : i ∈ range (natDegree p + 1)\nc : R\nhc : ↑f c = coeff p i\n⊢ ↑(mapRingHom f) (↑C c * X ^ i) = ↑C (coeff p i) * X ^ i",
"state_before": "case mpr.intro\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nh : ∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f\ni : ℕ\n_hi : i ∈ range (natDegree p + 1)\nc : R\nhc : ↑f c = coeff p i\n⊢ ↑C (coeff p i) * X ^ i ∈ RingHom.rangeS (mapRingHom f)",
"tactic": "use C c * X ^ i"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nh : ∀ (n : ℕ), coeff p n ∈ RingHom.rangeS f\ni : ℕ\n_hi : i ∈ range (natDegree p + 1)\nc : R\nhc : ↑f c = coeff p i\n⊢ ↑(mapRingHom f) (↑C c * X ^ i) = ↑C (coeff p i) * X ^ i",
"tactic": "rw [coe_mapRingHom, Polynomial.map_mul, map_C, hc, Polynomial.map_pow, map_X]"
}
] |
[
926,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
915,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.coeFn_inv
|
[] |
[
733,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
732,
1
] |
Mathlib/Algebra/Tropical/Basic.lean
|
Tropical.mul_eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "R✝ : Type u\ninst✝¹ : LinearOrderedAddCommMonoidWithTop R✝\nR : Type u_1\ninst✝ : LinearOrderedAddCommMonoid R\na b : Tropical (WithTop R)\n⊢ a * b = 0 ↔ a = 0 ∨ b = 0",
"tactic": "simp [← untrop_inj_iff, WithTop.add_eq_top]"
}
] |
[
580,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
579,
1
] |
Mathlib/Topology/UniformSpace/Equiv.lean
|
UniformEquiv.uniformEquiv_mk_coe_symm
|
[] |
[
131,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Topology/FiberBundle/Constructions.lean
|
Trivialization.Prod.continuous_to_fun
|
[
{
"state_after": "B : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\n⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))",
"state_before": "B : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\n⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))",
"tactic": "let f₁ : TotalSpace (E₁ ×ᵇ E₂) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p ↦ ((⟨p.1, p.2.1⟩ : TotalSpace E₁), (⟨p.1, p.2.2⟩ : TotalSpace E₂))"
},
{
"state_after": "B : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\n⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))",
"state_before": "B : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\n⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))",
"tactic": "let f₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p ↦ ⟨e₁ p.1, e₂ p.2⟩"
},
{
"state_after": "B : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\n⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))",
"state_before": "B : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\n⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))",
"tactic": "let f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p ↦ ⟨p.1.1, p.1.2, p.2.2⟩"
},
{
"state_after": "B : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\n⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))",
"state_before": "B : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\n⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))",
"tactic": "have hf₁ : Continuous f₁ := (Prod.inducing_diag E₁ E₂).continuous"
},
{
"state_after": "B : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\n⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))",
"state_before": "B : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\n⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))",
"tactic": "have hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) :=\n e₁.toLocalHomeomorph.continuousOn.prod_map e₂.toLocalHomeomorph.continuousOn"
},
{
"state_after": "B : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\n⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))",
"state_before": "B : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\n⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))",
"tactic": "have hf₃ : Continuous f₃ :=\n (continuous_fst.comp continuous_fst).prod_mk (continuous_snd.prod_map continuous_snd)"
},
{
"state_after": "case refine'_1\nB : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\n⊢ MapsTo f₁ (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) (e₁.source ×ˢ e₂.source)\n\ncase refine'_2\nB : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\n⊢ EqOn (toFun' e₁ e₂) ((f₃ ∘ f₂) ∘ f₁) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))",
"state_before": "B : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\n⊢ ContinuousOn (toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))",
"tactic": "refine' ((hf₃.comp_continuousOn hf₂).comp hf₁.continuousOn _).congr _"
},
{
"state_after": "case refine'_2.mk.mk.intro\nB : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\nb : B\nv₁ : E₁ b\nv₂ : E₂ b\nhb₁ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₁.baseSet\nright✝ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₂.baseSet\n⊢ toFun' e₁ e₂ { fst := b, snd := (v₁, v₂) } = ((f₃ ∘ f₂) ∘ f₁) { fst := b, snd := (v₁, v₂) }",
"state_before": "case refine'_2\nB : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\n⊢ EqOn (toFun' e₁ e₂) ((f₃ ∘ f₂) ∘ f₁) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))",
"tactic": "rintro ⟨b, v₁, v₂⟩ ⟨hb₁, _⟩"
},
{
"state_after": "case refine'_2.mk.mk.intro\nB : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\nb : B\nv₁ : E₁ b\nv₂ : E₂ b\nhb₁ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₁.baseSet\nright✝ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₂.baseSet\n⊢ b = (↑e₁ { fst := b, snd := v₁ }).fst",
"state_before": "case refine'_2.mk.mk.intro\nB : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\nb : B\nv₁ : E₁ b\nv₂ : E₂ b\nhb₁ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₁.baseSet\nright✝ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₂.baseSet\n⊢ toFun' e₁ e₂ { fst := b, snd := (v₁, v₂) } = ((f₃ ∘ f₂) ∘ f₁) { fst := b, snd := (v₁, v₂) }",
"tactic": "simp only [Prod.toFun', Prod.mk.inj_iff, Function.comp_apply, and_true_iff]"
},
{
"state_after": "case refine'_2.mk.mk.intro\nB : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\nb : B\nv₁ : E₁ b\nv₂ : E₂ b\nhb₁ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₁.baseSet\nright✝ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₂.baseSet\n⊢ { fst := b, snd := v₁ } ∈ e₁.source",
"state_before": "case refine'_2.mk.mk.intro\nB : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\nb : B\nv₁ : E₁ b\nv₂ : E₂ b\nhb₁ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₁.baseSet\nright✝ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₂.baseSet\n⊢ b = (↑e₁ { fst := b, snd := v₁ }).fst",
"tactic": "rw [e₁.coe_fst]"
},
{
"state_after": "case refine'_2.mk.mk.intro\nB : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\nb : B\nv₁ : E₁ b\nv₂ : E₂ b\nhb₁ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₁.baseSet\nright✝ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₂.baseSet\n⊢ TotalSpace.proj { fst := b, snd := v₁ } ∈ e₁.baseSet",
"state_before": "case refine'_2.mk.mk.intro\nB : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\nb : B\nv₁ : E₁ b\nv₂ : E₂ b\nhb₁ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₁.baseSet\nright✝ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₂.baseSet\n⊢ { fst := b, snd := v₁ } ∈ e₁.source",
"tactic": "rw [e₁.source_eq, mem_preimage]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.mk.mk.intro\nB : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\nb : B\nv₁ : E₁ b\nv₂ : E₂ b\nhb₁ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₁.baseSet\nright✝ : TotalSpace.proj { fst := b, snd := (v₁, v₂) } ∈ e₂.baseSet\n⊢ TotalSpace.proj { fst := b, snd := v₁ } ∈ e₁.baseSet",
"tactic": "exact hb₁"
},
{
"state_after": "case refine'_1\nB : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\n⊢ MapsTo f₁ (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))\n ((TotalSpace.proj ⁻¹' e₁.baseSet) ×ˢ (TotalSpace.proj ⁻¹' e₂.baseSet))",
"state_before": "case refine'_1\nB : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\n⊢ MapsTo f₁ (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) (e₁.source ×ˢ e₂.source)",
"tactic": "rw [e₁.source_eq, e₂.source_eq]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nB : Type u_3\ninst✝⁴ : TopologicalSpace B\nF₁ : Type u_5\ninst✝³ : TopologicalSpace F₁\nE₁ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹ : TopologicalSpace F₂\nE₂ : B → Type u_1\ninst✝ : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\nf₁ : (TotalSpace fun x => E₁ x × E₂ x) → TotalSpace E₁ × TotalSpace E₂ :=\n fun p => ({ fst := p.fst, snd := p.snd.fst }, { fst := p.fst, snd := p.snd.snd })\nf₂ : TotalSpace E₁ × TotalSpace E₂ → (B × F₁) × B × F₂ := fun p => (↑e₁ p.fst, ↑e₂ p.snd)\nf₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p => (p.fst.fst, p.fst.snd, p.snd.snd)\nhf₁ : Continuous f₁\nhf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source)\nhf₃ : Continuous f₃\n⊢ MapsTo f₁ (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))\n ((TotalSpace.proj ⁻¹' e₁.baseSet) ×ˢ (TotalSpace.proj ⁻¹' e₂.baseSet))",
"tactic": "exact mapsTo_preimage _ _"
}
] |
[
186,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.ball_mem_nhds
|
[] |
[
701,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
700,
1
] |
Mathlib/LinearAlgebra/Matrix/Trace.lean
|
Matrix.trace_transpose
|
[] |
[
70,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Topology/Algebra/Module/Multilinear.lean
|
ContinuousMultilinearMap.zero_apply
|
[] |
[
149,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Topology/MetricSpace/MetricSeparated.lean
|
IsMetricSeparated.finset_iUnion_left_iff
|
[] |
[
126,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Topology/UnitInterval.lean
|
unitInterval.fract_mem
|
[] |
[
63,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Topology/Instances/Real.lean
|
Real.uniformContinuous_neg
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nε : ℝ\nε0 : ε > 0\na✝ b✝ : ℝ\nh : dist b✝ a✝ < ε\n⊢ dist (-a✝) (-b✝) < ε",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nε : ℝ\nε0 : ε > 0\na✝ b✝ : ℝ\nh : dist a✝ b✝ < ε\n⊢ dist (-a✝) (-b✝) < ε",
"tactic": "rw [dist_comm] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nε : ℝ\nε0 : ε > 0\na✝ b✝ : ℝ\nh : dist b✝ a✝ < ε\n⊢ dist (-a✝) (-b✝) < ε",
"tactic": "simpa only [Real.dist_eq, neg_sub_neg] using h"
}
] |
[
49,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
Submodule.eq_top_of_nonempty_interior'
|
[
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : NeBot (𝓝[{x | IsUnit x}] 0)\ns : Submodule R M\ny : M\nhy : y ∈ interior ↑s\n⊢ s = ⊤",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : NeBot (𝓝[{x | IsUnit x}] 0)\ns : Submodule R M\nhs : Set.Nonempty (interior ↑s)\n⊢ s = ⊤",
"tactic": "rcases hs with ⟨y, hy⟩"
},
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : NeBot (𝓝[{x | IsUnit x}] 0)\ns : Submodule R M\ny : M\nhy : y ∈ interior ↑s\nx : M\n⊢ x ∈ s",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : NeBot (𝓝[{x | IsUnit x}] 0)\ns : Submodule R M\ny : M\nhy : y ∈ interior ↑s\n⊢ s = ⊤",
"tactic": "refine' Submodule.eq_top_iff'.2 fun x => _"
},
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : NeBot (𝓝[{x | IsUnit x}] 0)\ns : Submodule R M\ny : M\nhy : ↑s ∈ 𝓝 y\nx : M\n⊢ x ∈ s",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : NeBot (𝓝[{x | IsUnit x}] 0)\ns : Submodule R M\ny : M\nhy : y ∈ interior ↑s\nx : M\n⊢ x ∈ s",
"tactic": "rw [mem_interior_iff_mem_nhds] at hy"
},
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : NeBot (𝓝[{x | IsUnit x}] 0)\ns : Submodule R M\ny : M\nhy : ↑s ∈ 𝓝 y\nx : M\nthis : Tendsto (fun c => y + c • x) (𝓝[{x | IsUnit x}] 0) (𝓝 (y + 0 • x))\n⊢ x ∈ s",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : NeBot (𝓝[{x | IsUnit x}] 0)\ns : Submodule R M\ny : M\nhy : ↑s ∈ 𝓝 y\nx : M\n⊢ x ∈ s",
"tactic": "have : Tendsto (fun c : R => y + c • x) (𝓝[{ x : R | IsUnit x }] 0) (𝓝 (y + (0 : R) • x)) :=\n tendsto_const_nhds.add ((tendsto_nhdsWithin_of_tendsto_nhds tendsto_id).smul tendsto_const_nhds)"
},
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : NeBot (𝓝[{x | IsUnit x}] 0)\ns : Submodule R M\ny : M\nhy : ↑s ∈ 𝓝 y\nx : M\nthis : Tendsto (fun c => y + c • x) (𝓝[{x | IsUnit x}] 0) (𝓝 y)\n⊢ x ∈ s",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : NeBot (𝓝[{x | IsUnit x}] 0)\ns : Submodule R M\ny : M\nhy : ↑s ∈ 𝓝 y\nx : M\nthis : Tendsto (fun c => y + c • x) (𝓝[{x | IsUnit x}] 0) (𝓝 (y + 0 • x))\n⊢ x ∈ s",
"tactic": "rw [zero_smul, add_zero] at this"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : NeBot (𝓝[{x | IsUnit x}] 0)\ns : Submodule R M\ny : M\nhy : ↑s ∈ 𝓝 y\nx : M\nthis : Tendsto (fun c => y + c • x) (𝓝[{x | IsUnit x}] 0) (𝓝 y)\nu : Rˣ\nhu : y + ↑u • x ∈ s\n⊢ x ∈ s",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : NeBot (𝓝[{x | IsUnit x}] 0)\ns : Submodule R M\ny : M\nhy : ↑s ∈ 𝓝 y\nx : M\nthis : Tendsto (fun c => y + c • x) (𝓝[{x | IsUnit x}] 0) (𝓝 y)\n⊢ x ∈ s",
"tactic": "obtain ⟨_, hu : y + _ • _ ∈ s, u, rfl⟩ :=\n nonempty_of_mem (inter_mem (Filter.mem_map.1 (this hy)) self_mem_nhdsWithin)"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : NeBot (𝓝[{x | IsUnit x}] 0)\ns : Submodule R M\ny : M\nhy : ↑s ∈ 𝓝 y\nx : M\nthis : Tendsto (fun c => y + c • x) (𝓝[{x | IsUnit x}] 0) (𝓝 y)\nu : Rˣ\nhu : y + ↑u • x ∈ s\nhy' : y ∈ ↑s\n⊢ x ∈ s",
"state_before": "case intro.intro.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : NeBot (𝓝[{x | IsUnit x}] 0)\ns : Submodule R M\ny : M\nhy : ↑s ∈ 𝓝 y\nx : M\nthis : Tendsto (fun c => y + c • x) (𝓝[{x | IsUnit x}] 0) (𝓝 y)\nu : Rˣ\nhu : y + ↑u • x ∈ s\n⊢ x ∈ s",
"tactic": "have hy' : y ∈ ↑s := mem_of_mem_nhds hy"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommGroup M\ninst✝³ : ContinuousAdd M\ninst✝² : Module R M\ninst✝¹ : ContinuousSMul R M\ninst✝ : NeBot (𝓝[{x | IsUnit x}] 0)\ns : Submodule R M\ny : M\nhy : ↑s ∈ 𝓝 y\nx : M\nthis : Tendsto (fun c => y + c • x) (𝓝[{x | IsUnit x}] 0) (𝓝 y)\nu : Rˣ\nhu : y + ↑u • x ∈ s\nhy' : y ∈ ↑s\n⊢ x ∈ s",
"tactic": "rwa [s.add_mem_iff_right hy', ← Units.smul_def, s.smul_mem_iff' u] at hu"
}
] |
[
74,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
padicNormE.is_rat
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ_[p]\nh : q = 0\n⊢ ‖q‖ = ↑0",
"tactic": "simp [h]"
}
] |
[
883,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
879,
11
] |
Mathlib/FieldTheory/Finite/Basic.lean
|
FiniteField.even_card_iff_char_two
|
[
{
"state_after": "case intro.intro\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\n⊢ ringChar F = 2 ↔ Fintype.card F % 2 = 0",
"state_before": "K : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\n⊢ ringChar F = 2 ↔ Fintype.card F % 2 = 0",
"tactic": "rcases FiniteField.card F (ringChar F) with ⟨n, hp, h⟩"
},
{
"state_after": "case intro.intro\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\n⊢ ringChar F = 2 ↔ (ringChar F % 2) ^ ↑n % 2 = 0",
"state_before": "case intro.intro\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\n⊢ ringChar F = 2 ↔ Fintype.card F % 2 = 0",
"tactic": "rw [h, Nat.pow_mod]"
},
{
"state_after": "case intro.intro.mp\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\n⊢ ringChar F = 2 → (ringChar F % 2) ^ ↑n % 2 = 0\n\ncase intro.intro.mpr\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\n⊢ (ringChar F % 2) ^ ↑n % 2 = 0 → ringChar F = 2",
"state_before": "case intro.intro\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\n⊢ ringChar F = 2 ↔ (ringChar F % 2) ^ ↑n % 2 = 0",
"tactic": "constructor"
},
{
"state_after": "case intro.intro.mp\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\nhF : ringChar F = 2\n⊢ (ringChar F % 2) ^ ↑n % 2 = 0",
"state_before": "case intro.intro.mp\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\n⊢ ringChar F = 2 → (ringChar F % 2) ^ ↑n % 2 = 0",
"tactic": "intro hF"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.mp\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\nhF : ringChar F = 2\n⊢ (ringChar F % 2) ^ ↑n % 2 = 0",
"tactic": "simp [hF]"
},
{
"state_after": "case intro.intro.mpr\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\n⊢ Even (ringChar F % 2) ∧ ↑n ≠ 0 → ringChar F = 2",
"state_before": "case intro.intro.mpr\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\n⊢ (ringChar F % 2) ^ ↑n % 2 = 0 → ringChar F = 2",
"tactic": "rw [← Nat.even_iff, Nat.even_pow]"
},
{
"state_after": "case intro.intro.mpr.intro\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\nhev : Even (ringChar F % 2)\nhnz : ↑n ≠ 0\n⊢ ringChar F = 2",
"state_before": "case intro.intro.mpr\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\n⊢ Even (ringChar F % 2) ∧ ↑n ≠ 0 → ringChar F = 2",
"tactic": "rintro ⟨hev, hnz⟩"
},
{
"state_after": "case intro.intro.mpr.intro\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\nhev : ringChar F % 2 = 0\nhnz : ↑n ≠ 0\n⊢ ringChar F = 2",
"state_before": "case intro.intro.mpr.intro\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\nhev : Even (ringChar F % 2)\nhnz : ↑n ≠ 0\n⊢ ringChar F = 2",
"tactic": "rw [Nat.even_iff, Nat.mod_mod] at hev"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.mpr.intro\nK : Type ?u.1097070\nR : Type ?u.1097073\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ ↑n\nhev : ringChar F % 2 = 0\nhnz : ↑n ≠ 0\n⊢ ringChar F = 2",
"tactic": "exact (Nat.Prime.eq_two_or_odd hp).resolve_right (ne_of_eq_of_ne hev zero_ne_one)"
}
] |
[
513,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
504,
1
] |
Mathlib/Analysis/Complex/ReImTopology.lean
|
Complex.frontier_setOf_le_re_and_im_le
|
[
{
"state_after": "no goals",
"state_before": "a b : ℝ\n⊢ frontier {z | a ≤ z.re ∧ z.im ≤ b} = {z | a ≤ z.re ∧ z.im = b ∨ z.re = a ∧ z.im ≤ b}",
"tactic": "simpa only [closure_Ici, closure_Iic, frontier_Ici, frontier_Iic] using\n frontier_reProdIm (Ici a) (Iic b)"
}
] |
[
199,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
MeasurableSpace.comap_comp
|
[] |
[
123,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
|
DiscreteValuationRing.unit_mul_pow_congr_unit
|
[
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\nϖ : R\nhirr : Irreducible ϖ\nu v : Rˣ\nm : ℕ\nh : ↑u * ϖ ^ m = ↑v * ϖ ^ m\n⊢ u = v",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\nϖ : R\nhirr : Irreducible ϖ\nu v : Rˣ\nm n : ℕ\nh : ↑u * ϖ ^ m = ↑v * ϖ ^ n\n⊢ u = v",
"tactic": "obtain rfl : m = n := unit_mul_pow_congr_pow hirr hirr u v m n h"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\nϖ : R\nhirr : Irreducible ϖ\nu v : Rˣ\nm : ℕ\nh✝ : ↑u * ϖ ^ m = ↑v * ϖ ^ m\nh : ↑u * ϖ ^ m - ↑v * ϖ ^ m = 0\n⊢ u = v",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\nϖ : R\nhirr : Irreducible ϖ\nu v : Rˣ\nm : ℕ\nh : ↑u * ϖ ^ m = ↑v * ϖ ^ m\n⊢ u = v",
"tactic": "rw [← sub_eq_zero] at h"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\nϖ : R\nhirr : Irreducible ϖ\nu v : Rˣ\nm : ℕ\nh✝ : ↑u * ϖ ^ m = ↑v * ϖ ^ m\nh : ↑u - ↑v = 0 ∨ ϖ ^ m = 0\n⊢ u = v",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\nϖ : R\nhirr : Irreducible ϖ\nu v : Rˣ\nm : ℕ\nh✝ : ↑u * ϖ ^ m = ↑v * ϖ ^ m\nh : ↑u * ϖ ^ m - ↑v * ϖ ^ m = 0\n⊢ u = v",
"tactic": "rw [← sub_mul, mul_eq_zero] at h"
},
{
"state_after": "case inl\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\nϖ : R\nhirr : Irreducible ϖ\nu v : Rˣ\nm : ℕ\nh✝ : ↑u * ϖ ^ m = ↑v * ϖ ^ m\nh : ↑u - ↑v = 0\n⊢ u = v\n\ncase inr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\nϖ : R\nhirr : Irreducible ϖ\nu v : Rˣ\nm : ℕ\nh✝ : ↑u * ϖ ^ m = ↑v * ϖ ^ m\nh : ϖ ^ m = 0\n⊢ u = v",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\nϖ : R\nhirr : Irreducible ϖ\nu v : Rˣ\nm : ℕ\nh✝ : ↑u * ϖ ^ m = ↑v * ϖ ^ m\nh : ↑u - ↑v = 0 ∨ ϖ ^ m = 0\n⊢ u = v",
"tactic": "cases' h with h h"
},
{
"state_after": "case inl\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\nϖ : R\nhirr : Irreducible ϖ\nu v : Rˣ\nm : ℕ\nh✝ : ↑u * ϖ ^ m = ↑v * ϖ ^ m\nh : ↑u = ↑v\n⊢ u = v",
"state_before": "case inl\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\nϖ : R\nhirr : Irreducible ϖ\nu v : Rˣ\nm : ℕ\nh✝ : ↑u * ϖ ^ m = ↑v * ϖ ^ m\nh : ↑u - ↑v = 0\n⊢ u = v",
"tactic": "rw [sub_eq_zero] at h"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\nϖ : R\nhirr : Irreducible ϖ\nu v : Rˣ\nm : ℕ\nh✝ : ↑u * ϖ ^ m = ↑v * ϖ ^ m\nh : ↑u = ↑v\n⊢ u = v",
"tactic": "exact_mod_cast h"
},
{
"state_after": "no goals",
"state_before": "case inr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DiscreteValuationRing R\nϖ : R\nhirr : Irreducible ϖ\nu v : Rˣ\nm : ℕ\nh✝ : ↑u * ϖ ^ m = ↑v * ϖ ^ m\nh : ϖ ^ m = 0\n⊢ u = v",
"tactic": "apply (hirr.ne_zero (pow_eq_zero h)).elim"
}
] |
[
393,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
FractionalIdeal.invertible_iff_generator_nonzero
|
[
{
"state_after": "case mp\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\n⊢ I * I⁻¹ = 1 → generator ↑I ≠ 0\n\ncase mpr\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\n⊢ generator ↑I ≠ 0 → I * I⁻¹ = 1",
"state_before": "R : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\n⊢ I * I⁻¹ = 1 ↔ generator ↑I ≠ 0",
"tactic": "constructor"
},
{
"state_after": "case mp\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhI : I * I⁻¹ = 1\nhg : generator ↑I = 0\n⊢ False",
"state_before": "case mp\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\n⊢ I * I⁻¹ = 1 → generator ↑I ≠ 0",
"tactic": "intro hI hg"
},
{
"state_after": "case mp\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhI : I * I⁻¹ = 1\nhg : generator ↑I = 0\n⊢ I = 0",
"state_before": "case mp\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhI : I * I⁻¹ = 1\nhg : generator ↑I = 0\n⊢ False",
"tactic": "apply ne_zero_of_mul_eq_one _ _ hI"
},
{
"state_after": "no goals",
"state_before": "case mp\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhI : I * I⁻¹ = 1\nhg : generator ↑I = 0\n⊢ I = 0",
"tactic": "rw [eq_spanSingleton_of_principal I, hg, spanSingleton_zero]"
},
{
"state_after": "case mpr\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhg : generator ↑I ≠ 0\n⊢ I * I⁻¹ = 1",
"state_before": "case mpr\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\n⊢ generator ↑I ≠ 0 → I * I⁻¹ = 1",
"tactic": "intro hg"
},
{
"state_after": "case mpr.h\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhg : generator ↑I ≠ 0\n⊢ I ≠ 0",
"state_before": "case mpr\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhg : generator ↑I ≠ 0\n⊢ I * I⁻¹ = 1",
"tactic": "apply invertible_of_principal"
},
{
"state_after": "case mpr.h\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhg : generator ↑I ≠ 0\n⊢ spanSingleton R₁⁰ (generator ↑I) ≠ 0",
"state_before": "case mpr.h\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhg : generator ↑I ≠ 0\n⊢ I ≠ 0",
"tactic": "rw [eq_spanSingleton_of_principal I]"
},
{
"state_after": "case mpr.h\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhg : generator ↑I ≠ 0\nhI : spanSingleton R₁⁰ (generator ↑I) = 0\n⊢ False",
"state_before": "case mpr.h\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhg : generator ↑I ≠ 0\n⊢ spanSingleton R₁⁰ (generator ↑I) ≠ 0",
"tactic": "intro hI"
},
{
"state_after": "case mpr.h\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhg : generator ↑I ≠ 0\nhI : spanSingleton R₁⁰ (generator ↑I) = 0\nthis : generator ↑I ∈ spanSingleton R₁⁰ (generator ↑I)\n⊢ False",
"state_before": "case mpr.h\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhg : generator ↑I ≠ 0\nhI : spanSingleton R₁⁰ (generator ↑I) = 0\n⊢ False",
"tactic": "have := mem_spanSingleton_self R₁⁰ (generator (I : Submodule R₁ K))"
},
{
"state_after": "case mpr.h\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhg : generator ↑I ≠ 0\nhI : spanSingleton R₁⁰ (generator ↑I) = 0\nthis : generator ↑I = 0\n⊢ False",
"state_before": "case mpr.h\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhg : generator ↑I ≠ 0\nhI : spanSingleton R₁⁰ (generator ↑I) = 0\nthis : generator ↑I ∈ spanSingleton R₁⁰ (generator ↑I)\n⊢ False",
"tactic": "rw [hI, mem_zero_iff] at this"
},
{
"state_after": "no goals",
"state_before": "case mpr.h\nR : Type ?u.77862\nA : Type ?u.77865\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.78876\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nhg : generator ↑I ≠ 0\nhI : spanSingleton R₁⁰ (generator ↑I) = 0\nthis : generator ↑I = 0\n⊢ False",
"tactic": "contradiction"
}
] |
[
227,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
AddMonoidAlgebra.mul_single_zero_apply
|
[
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.2056792\ninst✝¹ : Semiring k\ninst✝ : AddZeroClass G\nf : AddMonoidAlgebra k G\nr : k\nx a : G\n⊢ a + 0 = x ↔ a = x",
"tactic": "rw [add_zero]"
}
] |
[
1632,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1630,
1
] |
Mathlib/CategoryTheory/Sites/Sieves.lean
|
CategoryTheory.Sieve.pullback_comp
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf✝ : Y ⟶ X\nS✝ R : Sieve X\nf : Y ⟶ X\ng : Z ⟶ Y\nS : Sieve X\n⊢ pullback (g ≫ f) S = pullback g (pullback f S)",
"tactic": "simp [Sieve.ext_iff]"
}
] |
[
473,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
Differentiable.prod
|
[] |
[
116,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.coe_equivMapOfInjective_apply
|
[] |
[
781,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
779,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.opNorm_eq_of_bounds
|
[] |
[
290,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.head?_cons
|
[] |
[
25,
61
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
25,
9
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean
|
CategoryTheory.Presieve.FamilyOfElements.compPresheafMap_comp
|
[] |
[
361,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/Data/List/Destutter.lean
|
List.destutter'_sublist
|
[
{
"state_after": "case nil\nα : Type u_1\nl : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝¹ b a✝ a : α\n⊢ destutter' R a [] <+ [a]\n\ncase cons\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝¹ b✝ a✝ b : α\nl : List α\nhl : ∀ (a : α), destutter' R a l <+ a :: l\na : α\n⊢ destutter' R a (b :: l) <+ a :: b :: l",
"state_before": "α : Type u_1\nl : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝ b a : α\n⊢ destutter' R a l <+ a :: l",
"tactic": "induction' l with b l hl generalizing a"
},
{
"state_after": "case cons\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝¹ b✝ a✝ b : α\nl : List α\nhl : ∀ (a : α), destutter' R a l <+ a :: l\na : α\n⊢ (if R a b then a :: destutter' R b l else destutter' R a l) <+ a :: b :: l",
"state_before": "case cons\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝¹ b✝ a✝ b : α\nl : List α\nhl : ∀ (a : α), destutter' R a l <+ a :: l\na : α\n⊢ destutter' R a (b :: l) <+ a :: b :: l",
"tactic": "rw [destutter']"
},
{
"state_after": "case cons.inl\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝¹ b✝ a✝ b : α\nl : List α\nhl : ∀ (a : α), destutter' R a l <+ a :: l\na : α\nh✝ : R a b\n⊢ a :: destutter' R b l <+ a :: b :: l\n\ncase cons.inr\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝¹ b✝ a✝ b : α\nl : List α\nhl : ∀ (a : α), destutter' R a l <+ a :: l\na : α\nh✝ : ¬R a b\n⊢ destutter' R a l <+ a :: b :: l",
"state_before": "case cons\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝¹ b✝ a✝ b : α\nl : List α\nhl : ∀ (a : α), destutter' R a l <+ a :: l\na : α\n⊢ (if R a b then a :: destutter' R b l else destutter' R a l) <+ a :: b :: l",
"tactic": "split_ifs"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nl : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝¹ b a✝ a : α\n⊢ destutter' R a [] <+ [a]",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons.inl\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝¹ b✝ a✝ b : α\nl : List α\nhl : ∀ (a : α), destutter' R a l <+ a :: l\na : α\nh✝ : R a b\n⊢ a :: destutter' R b l <+ a :: b :: l",
"tactic": "exact Sublist.cons₂ a (hl b)"
},
{
"state_after": "no goals",
"state_before": "case cons.inr\nα : Type u_1\nl✝ : List α\nR : α → α → Prop\ninst✝ : DecidableRel R\na✝¹ b✝ a✝ b : α\nl : List α\nhl : ∀ (a : α), destutter' R a l <+ a :: l\na : α\nh✝ : ¬R a b\n⊢ destutter' R a l <+ a :: b :: l",
"tactic": "exact (hl a).trans ((l.sublist_cons b).cons_cons a)"
}
] |
[
73,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
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