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/Computability/TMToPartrec.lean
|
Turing.PartrecToTM2.tr_ret_cons₁
|
[] |
[
1137,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1134,
1
] |
Mathlib/Data/PFun.lean
|
PFun.coe_comp
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type ?u.59817\nε : Type ?u.59820\nι : Type ?u.59823\nf✝ : α →. β\ng : β → γ\nf : α → β\nx✝¹ : α\nx✝ : γ\n⊢ x✝ ∈ ↑(g ∘ f) x✝¹ ↔ x✝ ∈ comp (↑g) (↑f) x✝¹",
"tactic": "simp only [coe_val, comp_apply, Function.comp, Part.bind_some]"
}
] |
[
629,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
628,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.exp_bound_div_one_sub_of_interval'
|
[
{
"state_after": "case H\nx : ℝ\nh1 : 0 < x\nh2 : x < 1\n⊢ 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n\nx : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ exp x < 1 / (1 - x)",
"state_before": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\n⊢ exp x < 1 / (1 - x)",
"tactic": "have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x)"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ exp x < 1 / (1 - x)",
"tactic": "calc\n exp x ≤ _ := exp_bound' h1.le h2.le zero_lt_three\n _ ≤ 1 + x + x ^ 2 := by\n rw [Finset.sum, range_val]\n nth_rw 1 [← two_add_one_eq_three]\n rw [← Nat.succ_eq_add_one, Multiset.range_succ, Multiset.map_cons, Multiset.sum_cons]\n nth_rw 3 [← one_add_one_eq_two]\n rw [← Nat.succ_eq_add_one, Multiset.range_succ, Multiset.map_cons, Multiset.sum_cons]\n nth_rw 3 [← zero_add 1]\n rw [← Nat.succ_eq_add_one, Multiset.range_succ, Multiset.map_cons, Multiset.sum_cons]\n rw [Multiset.range_zero, Multiset.map_zero, Multiset.sum_zero]\n norm_num\n nlinarith\n _ < 1 / (1 - x) := by rw [lt_div_iff] <;> nlinarith"
},
{
"state_after": "no goals",
"state_before": "case H\nx : ℝ\nh1 : 0 < x\nh2 : x < 1\n⊢ 0 < 1 - (1 + x + x ^ 2) * (1 - x)",
"tactic": "calc\n 0 < x ^ 3 := by positivity\n _ = 1 - (1 + x + x ^ 2) * (1 - x) := by ring"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\n⊢ 0 < x ^ 3",
"tactic": "positivity"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\n⊢ x ^ 3 = 1 - (1 + x + x ^ 2) * (1 - x)",
"tactic": "ring"
},
{
"state_after": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ Multiset.sum (Multiset.map (fun m => x ^ m / ↑(Nat.factorial m)) (Multiset.range 3)) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"state_before": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ ∑ m in range 3, x ^ m / ↑(Nat.factorial m) + x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤ 1 + x + x ^ 2",
"tactic": "rw [Finset.sum, range_val]"
},
{
"state_after": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ Multiset.sum (Multiset.map (fun m => x ^ m / ↑(Nat.factorial m)) (Multiset.range (2 + 1))) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"state_before": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ Multiset.sum (Multiset.map (fun m => x ^ m / ↑(Nat.factorial m)) (Multiset.range 3)) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"tactic": "nth_rw 1 [← two_add_one_eq_three]"
},
{
"state_after": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ x ^ 2 / ↑(Nat.factorial 2) + Multiset.sum (Multiset.map (fun m => x ^ m / ↑(Nat.factorial m)) (Multiset.range 2)) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"state_before": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ Multiset.sum (Multiset.map (fun m => x ^ m / ↑(Nat.factorial m)) (Multiset.range (2 + 1))) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"tactic": "rw [← Nat.succ_eq_add_one, Multiset.range_succ, Multiset.map_cons, Multiset.sum_cons]"
},
{
"state_after": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ x ^ 2 / ↑(Nat.factorial 2) +\n Multiset.sum (Multiset.map (fun m => x ^ m / ↑(Nat.factorial m)) (Multiset.range (1 + 1))) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"state_before": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ x ^ 2 / ↑(Nat.factorial 2) + Multiset.sum (Multiset.map (fun m => x ^ m / ↑(Nat.factorial m)) (Multiset.range 2)) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"tactic": "nth_rw 3 [← one_add_one_eq_two]"
},
{
"state_after": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ x ^ 2 / ↑(Nat.factorial 2) +\n (x ^ 1 / ↑(Nat.factorial 1) +\n Multiset.sum (Multiset.map (fun m => x ^ m / ↑(Nat.factorial m)) (Multiset.range 1))) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"state_before": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ x ^ 2 / ↑(Nat.factorial 2) +\n Multiset.sum (Multiset.map (fun m => x ^ m / ↑(Nat.factorial m)) (Multiset.range (1 + 1))) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"tactic": "rw [← Nat.succ_eq_add_one, Multiset.range_succ, Multiset.map_cons, Multiset.sum_cons]"
},
{
"state_after": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ x ^ 2 / ↑(Nat.factorial 2) +\n (x ^ 1 / ↑(Nat.factorial 1) +\n Multiset.sum (Multiset.map (fun m => x ^ m / ↑(Nat.factorial m)) (Multiset.range (0 + 1)))) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"state_before": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ x ^ 2 / ↑(Nat.factorial 2) +\n (x ^ 1 / ↑(Nat.factorial 1) +\n Multiset.sum (Multiset.map (fun m => x ^ m / ↑(Nat.factorial m)) (Multiset.range 1))) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"tactic": "nth_rw 3 [← zero_add 1]"
},
{
"state_after": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ x ^ 2 / ↑(Nat.factorial 2) +\n (x ^ 1 / ↑(Nat.factorial 1) +\n (x ^ 0 / ↑(Nat.factorial 0) +\n Multiset.sum (Multiset.map (fun m => x ^ m / ↑(Nat.factorial m)) (Multiset.range 0)))) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"state_before": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ x ^ 2 / ↑(Nat.factorial 2) +\n (x ^ 1 / ↑(Nat.factorial 1) +\n Multiset.sum (Multiset.map (fun m => x ^ m / ↑(Nat.factorial m)) (Multiset.range (0 + 1)))) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"tactic": "rw [← Nat.succ_eq_add_one, Multiset.range_succ, Multiset.map_cons, Multiset.sum_cons]"
},
{
"state_after": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ x ^ 2 / ↑(Nat.factorial 2) + (x ^ 1 / ↑(Nat.factorial 1) + (x ^ 0 / ↑(Nat.factorial 0) + 0)) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"state_before": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ x ^ 2 / ↑(Nat.factorial 2) +\n (x ^ 1 / ↑(Nat.factorial 1) +\n (x ^ 0 / ↑(Nat.factorial 0) +\n Multiset.sum (Multiset.map (fun m => x ^ m / ↑(Nat.factorial m)) (Multiset.range 0)))) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"tactic": "rw [Multiset.range_zero, Multiset.map_zero, Multiset.sum_zero]"
},
{
"state_after": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ x ^ 2 / 2 + (x + 1) + x ^ 3 * 4 / 18 ≤ 1 + x + x ^ 2",
"state_before": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ x ^ 2 / ↑(Nat.factorial 2) + (x ^ 1 / ↑(Nat.factorial 1) + (x ^ 0 / ↑(Nat.factorial 0) + 0)) +\n x ^ 3 * (↑3 + 1) / (↑(Nat.factorial 3) * ↑3) ≤\n 1 + x + x ^ 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ x ^ 2 / 2 + (x + 1) + x ^ 3 * 4 / 18 ≤ 1 + x + x ^ 2",
"tactic": "nlinarith"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh1 : 0 < x\nh2 : x < 1\nH : 0 < 1 - (1 + x + x ^ 2) * (1 - x)\n⊢ 1 + x + x ^ 2 < 1 / (1 - x)",
"tactic": "rw [lt_div_iff] <;> nlinarith"
}
] |
[
1952,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1930,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.coe_mul
|
[] |
[
172,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Data/Rat/NNRat.lean
|
NNRat.coe_injective
|
[] |
[
77,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
11
] |
Mathlib/Algebra/Periodic.lean
|
Function.Periodic.eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.108782\nf g : α → β\nc c₁ c₂ x : α\ninst✝ : AddZeroClass α\nh : Periodic f c\n⊢ f c = f 0",
"tactic": "simpa only [zero_add] using h 0"
}
] |
[
262,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
11
] |
Mathlib/Tactic/Ring/Basic.lean
|
Mathlib.Tactic.Ring.mul_congr
|
[
{
"state_after": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na' b' : R\n⊢ a' * b' = a' * b'",
"state_before": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na a' b b' c : R\nx✝² : a = a'\nx✝¹ : b = b'\nx✝ : a' * b' = c\n⊢ a * b = c",
"tactic": "subst_vars"
},
{
"state_after": "no goals",
"state_before": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na' b' : R\n⊢ a' * b' = a' * b'",
"tactic": "rfl"
}
] |
[
965,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
964,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.le_log_iff_exp_le
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhy : 0 < y\n⊢ x ≤ log y ↔ exp x ≤ y",
"tactic": "rw [← exp_le_exp, exp_log hy]"
}
] |
[
164,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Std/Data/String/Lemmas.lean
|
String.append_assoc
|
[] |
[
492,
37
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
491,
1
] |
Mathlib/RingTheory/Polynomial/Bernstein.lean
|
bernsteinPolynomial.iterate_derivative_at_0
|
[
{
"state_after": "case pos\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : ν ≤ n\n⊢ eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\n\ncase neg\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : ¬ν ≤ n\n⊢ eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\n⊢ eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)",
"tactic": "by_cases h : ν ≤ n"
},
{
"state_after": "case pos.zero\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν : ℕ\nh✝ : ν ≤ n✝\nn : ℕ\nh : Nat.zero ≤ n\n⊢ eval 0 ((↑derivative^[Nat.zero]) (bernsteinPolynomial R n Nat.zero)) =\n eval (↑(n - (Nat.zero - 1))) (pochhammer R Nat.zero)\n\ncase pos.succ\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν✝ : ℕ\nh✝ : ν✝ ≤ n✝\nν : ℕ\nih : ∀ (n : ℕ), ν ≤ n → eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\nn : ℕ\nh : Nat.succ ν ≤ n\n⊢ eval 0 ((↑derivative^[Nat.succ ν]) (bernsteinPolynomial R n (Nat.succ ν))) =\n eval (↑(n - (Nat.succ ν - 1))) (pochhammer R (Nat.succ ν))",
"state_before": "case pos\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : ν ≤ n\n⊢ eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)",
"tactic": "induction' ν with ν ih generalizing n"
},
{
"state_after": "no goals",
"state_before": "case pos.zero\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν : ℕ\nh✝ : ν ≤ n✝\nn : ℕ\nh : Nat.zero ≤ n\n⊢ eval 0 ((↑derivative^[Nat.zero]) (bernsteinPolynomial R n Nat.zero)) =\n eval (↑(n - (Nat.zero - 1))) (pochhammer R Nat.zero)",
"tactic": "simp [eval_at_0]"
},
{
"state_after": "case pos.succ\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν✝ : ℕ\nh✝ : ν✝ ≤ n✝\nν : ℕ\nih : ∀ (n : ℕ), ν ≤ n → eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\nn : ℕ\nh : Nat.succ ν ≤ n\nh' : ν ≤ n - 1\n⊢ eval 0 ((↑derivative^[Nat.succ ν]) (bernsteinPolynomial R n (Nat.succ ν))) =\n eval (↑(n - (Nat.succ ν - 1))) (pochhammer R (Nat.succ ν))",
"state_before": "case pos.succ\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν✝ : ℕ\nh✝ : ν✝ ≤ n✝\nν : ℕ\nih : ∀ (n : ℕ), ν ≤ n → eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\nn : ℕ\nh : Nat.succ ν ≤ n\n⊢ eval 0 ((↑derivative^[Nat.succ ν]) (bernsteinPolynomial R n (Nat.succ ν))) =\n eval (↑(n - (Nat.succ ν - 1))) (pochhammer R (Nat.succ ν))",
"tactic": "have h' : ν ≤ n - 1 := le_tsub_of_add_le_right h"
},
{
"state_after": "case pos.succ\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν✝ : ℕ\nh✝ : ν✝ ≤ n✝\nν : ℕ\nih : ∀ (n : ℕ), ν ≤ n → eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\nn : ℕ\nh : Nat.succ ν ≤ n\nh' : ν ≤ n - 1\n⊢ ↑n * eval (↑(n - 1 - (ν - 1))) (pochhammer R ν) = ↑(n - ν) * eval (↑(n - ν) + 1) (pochhammer R ν)",
"state_before": "case pos.succ\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν✝ : ℕ\nh✝ : ν✝ ≤ n✝\nν : ℕ\nih : ∀ (n : ℕ), ν ≤ n → eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\nn : ℕ\nh : Nat.succ ν ≤ n\nh' : ν ≤ n - 1\n⊢ eval 0 ((↑derivative^[Nat.succ ν]) (bernsteinPolynomial R n (Nat.succ ν))) =\n eval (↑(n - (Nat.succ ν - 1))) (pochhammer R (Nat.succ ν))",
"tactic": "simp only [derivative_succ, ih (n - 1) h', iterate_derivative_succ_at_0_eq_zero,\n Nat.succ_sub_succ_eq_sub, tsub_zero, sub_zero, iterate_derivative_sub,\n iterate_derivative_nat_cast_mul, eval_one, eval_mul, eval_add, eval_sub, eval_X, eval_comp,\n eval_nat_cast, Function.comp_apply, Function.iterate_succ, pochhammer_succ_left]"
},
{
"state_after": "case pos.succ.inl\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν : ℕ\nh✝ : ν ≤ n✝\nn : ℕ\nih : ∀ (n : ℕ), 0 ≤ n → eval 0 ((↑derivative^[0]) (bernsteinPolynomial R n 0)) = eval (↑(n - (0 - 1))) (pochhammer R 0)\nh : Nat.succ 0 ≤ n\nh' : 0 ≤ n - 1\n⊢ ↑n * eval (↑(n - 1 - (0 - 1))) (pochhammer R 0) = ↑(n - 0) * eval (↑(n - 0) + 1) (pochhammer R 0)\n\ncase pos.succ.inr\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν✝ : ℕ\nh✝ : ν✝ ≤ n✝\nν : ℕ\nih : ∀ (n : ℕ), ν ≤ n → eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\nn : ℕ\nh : Nat.succ ν ≤ n\nh' : ν ≤ n - 1\nh'' : ν > 0\n⊢ ↑n * eval (↑(n - 1 - (ν - 1))) (pochhammer R ν) = ↑(n - ν) * eval (↑(n - ν) + 1) (pochhammer R ν)",
"state_before": "case pos.succ\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν✝ : ℕ\nh✝ : ν✝ ≤ n✝\nν : ℕ\nih : ∀ (n : ℕ), ν ≤ n → eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\nn : ℕ\nh : Nat.succ ν ≤ n\nh' : ν ≤ n - 1\n⊢ ↑n * eval (↑(n - 1 - (ν - 1))) (pochhammer R ν) = ↑(n - ν) * eval (↑(n - ν) + 1) (pochhammer R ν)",
"tactic": "obtain rfl | h'' := ν.eq_zero_or_pos"
},
{
"state_after": "no goals",
"state_before": "case pos.succ.inl\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν : ℕ\nh✝ : ν ≤ n✝\nn : ℕ\nih : ∀ (n : ℕ), 0 ≤ n → eval 0 ((↑derivative^[0]) (bernsteinPolynomial R n 0)) = eval (↑(n - (0 - 1))) (pochhammer R 0)\nh : Nat.succ 0 ≤ n\nh' : 0 ≤ n - 1\n⊢ ↑n * eval (↑(n - 1 - (0 - 1))) (pochhammer R 0) = ↑(n - 0) * eval (↑(n - 0) + 1) (pochhammer R 0)",
"tactic": "simp"
},
{
"state_after": "case pos.succ.inr\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν✝ : ℕ\nh✝ : ν✝ ≤ n✝\nν : ℕ\nih : ∀ (n : ℕ), ν ≤ n → eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\nn : ℕ\nh : Nat.succ ν ≤ n\nh' : ν ≤ n - 1\nh'' : ν > 0\nthis : n - 1 - (ν - 1) = n - ν\n⊢ ↑n * eval (↑(n - 1 - (ν - 1))) (pochhammer R ν) = ↑(n - ν) * eval (↑(n - ν) + 1) (pochhammer R ν)",
"state_before": "case pos.succ.inr\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν✝ : ℕ\nh✝ : ν✝ ≤ n✝\nν : ℕ\nih : ∀ (n : ℕ), ν ≤ n → eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\nn : ℕ\nh : Nat.succ ν ≤ n\nh' : ν ≤ n - 1\nh'' : ν > 0\n⊢ ↑n * eval (↑(n - 1 - (ν - 1))) (pochhammer R ν) = ↑(n - ν) * eval (↑(n - ν) + 1) (pochhammer R ν)",
"tactic": "have : n - 1 - (ν - 1) = n - ν := by\n rw [gt_iff_lt, ← Nat.succ_le_iff] at h''\n rw [← tsub_add_eq_tsub_tsub, add_comm, tsub_add_cancel_of_le h'']"
},
{
"state_after": "case pos.succ.inr\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν✝ : ℕ\nh✝ : ν✝ ≤ n✝\nν : ℕ\nih : ∀ (n : ℕ), ν ≤ n → eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\nn : ℕ\nh : Nat.succ ν ≤ n\nh' : ν ≤ n - 1\nh'' : ν > 0\nthis : n - 1 - (ν - 1) = n - ν\n⊢ ↑n * eval (↑(n - ν)) (pochhammer R ν) = (↑(n - ν) + ↑ν) * eval (↑(n - ν)) (pochhammer R ν)",
"state_before": "case pos.succ.inr\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν✝ : ℕ\nh✝ : ν✝ ≤ n✝\nν : ℕ\nih : ∀ (n : ℕ), ν ≤ n → eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\nn : ℕ\nh : Nat.succ ν ≤ n\nh' : ν ≤ n - 1\nh'' : ν > 0\nthis : n - 1 - (ν - 1) = n - ν\n⊢ ↑n * eval (↑(n - 1 - (ν - 1))) (pochhammer R ν) = ↑(n - ν) * eval (↑(n - ν) + 1) (pochhammer R ν)",
"tactic": "rw [this, pochhammer_eval_succ]"
},
{
"state_after": "no goals",
"state_before": "case pos.succ.inr\nR : Type u_1\ninst✝ : CommRing R\nn✝ ν✝ : ℕ\nh✝ : ν✝ ≤ n✝\nν : ℕ\nih : ∀ (n : ℕ), ν ≤ n → eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\nn : ℕ\nh : Nat.succ ν ≤ n\nh' : ν ≤ n - 1\nh'' : ν > 0\nthis : n - 1 - (ν - 1) = n - ν\n⊢ ↑n * eval (↑(n - ν)) (pochhammer R ν) = (↑(n - ν) + ↑ν) * eval (↑(n - ν)) (pochhammer R ν)",
"tactic": "rw_mod_cast [tsub_add_cancel_of_le (h'.trans n.pred_le)]"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn✝ ν✝ : ℕ\nh✝ : ν✝ ≤ n✝\nν : ℕ\nih : ∀ (n : ℕ), ν ≤ n → eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\nn : ℕ\nh : Nat.succ ν ≤ n\nh' : ν ≤ n - 1\nh'' : Nat.succ 0 ≤ ν\n⊢ n - 1 - (ν - 1) = n - ν",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ ν✝ : ℕ\nh✝ : ν✝ ≤ n✝\nν : ℕ\nih : ∀ (n : ℕ), ν ≤ n → eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\nn : ℕ\nh : Nat.succ ν ≤ n\nh' : ν ≤ n - 1\nh'' : ν > 0\n⊢ n - 1 - (ν - 1) = n - ν",
"tactic": "rw [gt_iff_lt, ← Nat.succ_le_iff] at h''"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn✝ ν✝ : ℕ\nh✝ : ν✝ ≤ n✝\nν : ℕ\nih : ∀ (n : ℕ), ν ≤ n → eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)\nn : ℕ\nh : Nat.succ ν ≤ n\nh' : ν ≤ n - 1\nh'' : Nat.succ 0 ≤ ν\n⊢ n - 1 - (ν - 1) = n - ν",
"tactic": "rw [← tsub_add_eq_tsub_tsub, add_comm, tsub_add_cancel_of_le h'']"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : n < ν\n⊢ eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)",
"state_before": "case neg\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : ¬ν ≤ n\n⊢ eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)",
"tactic": "simp only [not_le] at h"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : n < ν\n⊢ eval 0 ((↑derivative^[ν]) 0) = eval (↑0) (pochhammer R ν)",
"state_before": "case neg\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : n < ν\n⊢ eval 0 ((↑derivative^[ν]) (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (pochhammer R ν)",
"tactic": "rw [tsub_eq_zero_iff_le.mpr (Nat.le_pred_of_lt h), eq_zero_of_lt R h]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : n < ν\n⊢ eval 0 ((↑derivative^[ν]) 0) = eval (↑0) (pochhammer R ν)",
"tactic": "simp [pos_iff_ne_zero.mp (pos_of_gt h)]"
}
] |
[
199,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/CategoryTheory/EqToHom.lean
|
CategoryTheory.comp_eqToHom_iff
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝ : Category C\nX Y Y' : C\np : Y = Y'\nf : X ⟶ Y\ng : X ⟶ Y'\nh : f ≫ eqToHom p = g\n⊢ f = (f ≫ eqToHom p) ≫ eqToHom (_ : Y' = Y)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝ : Category C\nX Y Y' : C\np : Y = Y'\nf : X ⟶ Y\ng : X ⟶ Y'\nh : f = g ≫ eqToHom (_ : Y' = Y)\n⊢ f ≫ eqToHom p = g",
"tactic": "simp [eq_whisker h (eqToHom p)]"
}
] |
[
66,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Data/List/Basic.lean
|
List.removeNth_insertNth
|
[
{
"state_after": "ι : Type ?u.113196\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nn : ℕ\nl : List α\n⊢ modifyNthTail (tail ∘ cons a) n l = l",
"state_before": "ι : Type ?u.113196\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nn : ℕ\nl : List α\n⊢ removeNth (insertNth n a l) n = l",
"tactic": "rw [removeNth_eq_nth_tail, insertNth, modifyNthTail_modifyNthTail_same]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.113196\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nn : ℕ\nl : List α\n⊢ modifyNthTail (tail ∘ cons a) n l = l",
"tactic": "exact modifyNthTail_id _ _"
}
] |
[
1618,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1616,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.Nonempty.of_image₂_left
|
[] |
[
131,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.neg_coe_pi
|
[
{
"state_after": "⊢ ∃ k, -π - π = 2 * π * ↑k",
"state_before": "⊢ -↑π = ↑π",
"tactic": "rw [← coe_neg, angle_eq_iff_two_pi_dvd_sub]"
},
{
"state_after": "⊢ -π - π = 2 * π * ↑(-1)",
"state_before": "⊢ ∃ k, -π - π = 2 * π * ↑k",
"tactic": "use -1"
},
{
"state_after": "no goals",
"state_before": "⊢ -π - π = 2 * π * ↑(-1)",
"tactic": "simp [two_mul, sub_eq_add_neg]"
}
] |
[
137,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.forall_mem_cons
|
[] |
[
251,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/Topology/Order/Basic.lean
|
IsGLB.mem_lowerBounds_of_tendsto
|
[] |
[
2050,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2047,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.disjoint_singleton_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.69965\nγ : Type ?u.69968\nf : α → β\ns t u : Finset α\na b : α\n⊢ _root_.Disjoint {a} s ↔ ¬a ∈ s",
"tactic": "simp only [disjoint_left, mem_singleton, forall_eq]"
}
] |
[
959,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
958,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.lift_umax
|
[] |
[
204,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.add_moveRight_inr
|
[
{
"state_after": "case mk\ny : PGame\ni : RightMoves y\nα✝ β✝ : Type u_1\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\n⊢ moveRight (mk α✝ β✝ a✝¹ a✝ + y) (↑toRightMovesAdd (Sum.inr i)) = mk α✝ β✝ a✝¹ a✝ + moveRight y i",
"state_before": "x y : PGame\ni : RightMoves y\n⊢ moveRight (x + y) (↑toRightMovesAdd (Sum.inr i)) = x + moveRight y i",
"tactic": "cases x"
},
{
"state_after": "case mk.mk\nα✝¹ β✝¹ : Type u_1\na✝³ : α✝¹ → PGame\na✝² : β✝¹ → PGame\nα✝ β✝ : Type u_1\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\ni : RightMoves (mk α✝ β✝ a✝¹ a✝)\n⊢ moveRight (mk α✝¹ β✝¹ a✝³ a✝² + mk α✝ β✝ a✝¹ a✝) (↑toRightMovesAdd (Sum.inr i)) =\n mk α✝¹ β✝¹ a✝³ a✝² + moveRight (mk α✝ β✝ a✝¹ a✝) i",
"state_before": "case mk\ny : PGame\ni : RightMoves y\nα✝ β✝ : Type u_1\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\n⊢ moveRight (mk α✝ β✝ a✝¹ a✝ + y) (↑toRightMovesAdd (Sum.inr i)) = mk α✝ β✝ a✝¹ a✝ + moveRight y i",
"tactic": "cases y"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nα✝¹ β✝¹ : Type u_1\na✝³ : α✝¹ → PGame\na✝² : β✝¹ → PGame\nα✝ β✝ : Type u_1\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\ni : RightMoves (mk α✝ β✝ a✝¹ a✝)\n⊢ moveRight (mk α✝¹ β✝¹ a✝³ a✝² + mk α✝ β✝ a✝¹ a✝) (↑toRightMovesAdd (Sum.inr i)) =\n mk α✝¹ β✝¹ a✝³ a✝² + moveRight (mk α✝ β✝ a✝¹ a✝) i",
"tactic": "rfl"
}
] |
[
1556,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1552,
1
] |
Mathlib/CategoryTheory/Monoidal/OfHasFiniteProducts.lean
|
CategoryTheory.monoidalOfHasFiniteCoproducts.tensorObj
|
[] |
[
181,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.coe_smul
|
[] |
[
1287,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1286,
1
] |
Mathlib/Order/SymmDiff.lean
|
Pi.bihimp_apply
|
[] |
[
871,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
869,
1
] |
Mathlib/Analysis/NormedSpace/Ray.lean
|
norm_injOn_ray_right
|
[
{
"state_after": "no goals",
"state_before": "E : Type ?u.122924\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : F\nhy : y ≠ 0\n⊢ Set.InjOn Norm.norm {x | SameRay ℝ x y}",
"tactic": "simpa only [SameRay.sameRay_comm] using norm_injOn_ray_left hy"
}
] |
[
73,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.not_isUnit_X_pow_sub_one
|
[
{
"state_after": "R✝ : Type u\nS : Type v\na b : R✝\nm n✝ : ℕ\nι : Type y\ninst✝² : Ring R✝\np : R✝[X]\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nn : ℕ\nh : IsUnit (X ^ n - 1)\n⊢ False",
"state_before": "R✝ : Type u\nS : Type v\na b : R✝\nm n✝ : ℕ\nι : Type y\ninst✝² : Ring R✝\np : R✝[X]\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nn : ℕ\n⊢ ¬IsUnit (X ^ n - 1)",
"tactic": "intro h"
},
{
"state_after": "case inl\nR✝ : Type u\nS : Type v\na b : R✝\nm n : ℕ\nι : Type y\ninst✝² : Ring R✝\np : R✝[X]\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nh : IsUnit (X ^ 0 - 1)\n⊢ False\n\ncase inr\nR✝ : Type u\nS : Type v\na b : R✝\nm n✝ : ℕ\nι : Type y\ninst✝² : Ring R✝\np : R✝[X]\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nn : ℕ\nh : IsUnit (X ^ n - 1)\nhn : n ≠ 0\n⊢ False",
"state_before": "R✝ : Type u\nS : Type v\na b : R✝\nm n✝ : ℕ\nι : Type y\ninst✝² : Ring R✝\np : R✝[X]\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nn : ℕ\nh : IsUnit (X ^ n - 1)\n⊢ False",
"tactic": "rcases eq_or_ne n 0 with (rfl | hn)"
},
{
"state_after": "case inr\nR✝ : Type u\nS : Type v\na b : R✝\nm n✝ : ℕ\nι : Type y\ninst✝² : Ring R✝\np : R✝[X]\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nn : ℕ\nh : IsUnit (X ^ n - 1)\nhn : n ≠ 0\n⊢ n = 0",
"state_before": "case inr\nR✝ : Type u\nS : Type v\na b : R✝\nm n✝ : ℕ\nι : Type y\ninst✝² : Ring R✝\np : R✝[X]\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nn : ℕ\nh : IsUnit (X ^ n - 1)\nhn : n ≠ 0\n⊢ False",
"tactic": "apply hn"
},
{
"state_after": "no goals",
"state_before": "case inr\nR✝ : Type u\nS : Type v\na b : R✝\nm n✝ : ℕ\nι : Type y\ninst✝² : Ring R✝\np : R✝[X]\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nn : ℕ\nh : IsUnit (X ^ n - 1)\nhn : n ≠ 0\n⊢ n = 0",
"tactic": "rw [← @natDegree_one R, ← (monic_X_pow_sub_C _ hn).eq_one_of_isUnit h, natDegree_X_pow_sub_C]"
},
{
"state_after": "no goals",
"state_before": "case inl\nR✝ : Type u\nS : Type v\na b : R✝\nm n : ℕ\nι : Type y\ninst✝² : Ring R✝\np : R✝[X]\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nh : IsUnit (X ^ 0 - 1)\n⊢ False",
"tactic": "simp at h"
}
] |
[
399,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.le_succ_of_pred_le
|
[] |
[
342,
24
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
339,
1
] |
Mathlib/Topology/UniformSpace/Separation.lean
|
separated_def
|
[
{
"state_after": "α✝ : Type u\nβ : Type v\nγ : Type w\ninst✝³ : UniformSpace α✝\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\nα : Type u\ninst✝ : UniformSpace α\n⊢ (∀ (a b : α), (∀ (t : Set (α × α)), t ∈ (𝓤 α).sets → (a, b) ∈ t) ↔ a = b) ↔\n ∀ (x y : α), (∀ (r : Set (α × α)), r ∈ 𝓤 α → (x, y) ∈ r) → x = y",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\ninst✝³ : UniformSpace α✝\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\nα : Type u\ninst✝ : UniformSpace α\n⊢ SeparatedSpace α ↔ ∀ (x y : α), (∀ (r : Set (α × α)), r ∈ 𝓤 α → (x, y) ∈ r) → x = y",
"tactic": "simp only [separatedSpace_iff, Set.ext_iff, Prod.forall, mem_idRel, separationRel, mem_sInter]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\ninst✝³ : UniformSpace α✝\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\nα : Type u\ninst✝ : UniformSpace α\n⊢ (∀ (a b : α), (∀ (t : Set (α × α)), t ∈ (𝓤 α).sets → (a, b) ∈ t) ↔ a = b) ↔\n ∀ (x y : α), (∀ (r : Set (α × α)), r ∈ 𝓤 α → (x, y) ∈ r) → x = y",
"tactic": "exact forall₂_congr fun _ _ => ⟨Iff.mp, fun h => ⟨h, fun H U hU => H ▸ refl_mem_uniformity hU⟩⟩"
}
] |
[
139,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/FieldTheory/Subfield.lean
|
RingHom.fieldRange_eq_map
|
[
{
"state_after": "case h\nK : Type u\nL : Type v\nM : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field M\ng : L →+* M\nf : K →+* L\nx✝ : L\n⊢ x✝ ∈ fieldRange f ↔ x✝ ∈ Subfield.map f ⊤",
"state_before": "K : Type u\nL : Type v\nM : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field M\ng : L →+* M\nf : K →+* L\n⊢ fieldRange f = Subfield.map f ⊤",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nK : Type u\nL : Type v\nM : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field M\ng : L →+* M\nf : K →+* L\nx✝ : L\n⊢ x✝ ∈ fieldRange f ↔ x✝ ∈ Subfield.map f ⊤",
"tactic": "simp"
}
] |
[
565,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
563,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
MonoidAlgebra.mul_single_one_apply
|
[
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.504877\ninst✝¹ : Semiring k\ninst✝ : MulOneClass G\nf : MonoidAlgebra k G\nr : k\nx a : G\n⊢ a * 1 = x ↔ a = x",
"tactic": "rw [mul_one]"
}
] |
[
558,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
556,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.PullbackCone.mono_fst_of_is_pullback_of_mono
|
[
{
"state_after": "C : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW✝ X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nt : PullbackCone f g\nht : IsLimit t\ninst✝ : Mono g\nW : C\nh k : W ⟶ t.pt\ni : h ≫ fst t = k ≫ fst t\n⊢ h ≫ snd t = k ≫ snd t",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nt : PullbackCone f g\nht : IsLimit t\ninst✝ : Mono g\n⊢ Mono (fst t)",
"tactic": "refine ⟨fun {W} h k i => IsLimit.hom_ext ht i ?_⟩"
},
{
"state_after": "C : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW✝ X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nt : PullbackCone f g\nht : IsLimit t\ninst✝ : Mono g\nW : C\nh k : W ⟶ t.pt\ni : h ≫ fst t = k ≫ fst t\n⊢ h ≫ fst t ≫ f = k ≫ fst t ≫ f",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW✝ X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nt : PullbackCone f g\nht : IsLimit t\ninst✝ : Mono g\nW : C\nh k : W ⟶ t.pt\ni : h ≫ fst t = k ≫ fst t\n⊢ h ≫ snd t = k ≫ snd t",
"tactic": "rw [← cancel_mono g, Category.assoc, Category.assoc, ←condition]"
},
{
"state_after": "C : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW✝ X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nt : PullbackCone f g\nht : IsLimit t\ninst✝ : Mono g\nW : C\nh k : W ⟶ t.pt\ni : h ≫ fst t = k ≫ fst t\nthis : (h ≫ fst t) ≫ f = (k ≫ fst t) ≫ f\n⊢ h ≫ fst t ≫ f = k ≫ fst t ≫ f",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW✝ X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nt : PullbackCone f g\nht : IsLimit t\ninst✝ : Mono g\nW : C\nh k : W ⟶ t.pt\ni : h ≫ fst t = k ≫ fst t\nthis : (fun x => x ≫ f) (h ≫ fst t) = (fun x => x ≫ f) (k ≫ fst t)\n⊢ h ≫ fst t ≫ f = k ≫ fst t ≫ f",
"tactic": "dsimp at this"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW✝ X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nt : PullbackCone f g\nht : IsLimit t\ninst✝ : Mono g\nW : C\nh k : W ⟶ t.pt\ni : h ≫ fst t = k ≫ fst t\nthis : (h ≫ fst t) ≫ f = (k ≫ fst t) ≫ f\n⊢ h ≫ fst t ≫ f = k ≫ fst t ≫ f",
"tactic": "rwa [Category.assoc, Category.assoc] at this"
}
] |
[
641,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
636,
1
] |
Mathlib/Data/Bool/Basic.lean
|
Bool.left_le_or
|
[
{
"state_after": "no goals",
"state_before": "⊢ ∀ (x y : Bool), x ≤ (x || y)",
"tactic": "decide"
}
] |
[
364,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
364,
1
] |
Mathlib/GroupTheory/GroupAction/Defs.lean
|
smul_zero
|
[] |
[
702,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
701,
1
] |
Mathlib/Order/RelClasses.lean
|
eq_or_ssubset_of_subset
|
[] |
[
769,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
768,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
sbtw_iff_right_ne_and_left_mem_image_Ioi
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.511588\nP : Type u_3\nP' : Type ?u.511594\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\n⊢ Sbtw R x y z ↔ z ≠ y ∧ x ∈ ↑(lineMap z y) '' Set.Ioi 1",
"tactic": "rw [sbtw_comm, sbtw_iff_left_ne_and_right_mem_image_Ioi]"
}
] |
[
754,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
752,
1
] |
Mathlib/CategoryTheory/Adjunction/Basic.lean
|
CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_left
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : CoreHomEquiv F G\nX' X : C\nY Y' : D\nf : X' ⟶ X\ng : F.obj X ⟶ Y\n⊢ F.map f ≫ g = ↑(homEquiv adj X' Y).symm (f ≫ ↑(homEquiv adj X Y) g)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : CoreHomEquiv F G\nX' X : C\nY Y' : D\nf : X' ⟶ X\ng : F.obj X ⟶ Y\n⊢ ↑(homEquiv adj X' Y) (F.map f ≫ g) = f ≫ ↑(homEquiv adj X Y) g",
"tactic": "rw [← Equiv.eq_symm_apply]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : CoreHomEquiv F G\nX' X : C\nY Y' : D\nf : X' ⟶ X\ng : F.obj X ⟶ Y\n⊢ F.map f ≫ g = ↑(homEquiv adj X' Y).symm (f ≫ ↑(homEquiv adj X Y) g)",
"tactic": "simp"
}
] |
[
287,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/Computability/Partrec.lean
|
Computable.option_getD
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.191844\nσ : Type ?u.191847\ninst✝³ : Primcodable α\ninst✝² : Primcodable β\ninst✝¹ : Primcodable γ\ninst✝ : Primcodable σ\nf : α → Option β\ng : α → β\nhf : Computable f\nhg : Computable g\na : α\n⊢ (Option.casesOn (f a) (g a) fun b => b) = Option.getD (f a) (g a)",
"tactic": "cases f a <;> rfl"
}
] |
[
706,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
703,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.succ_sub_one
|
[] |
[
143,
55
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
143,
1
] |
Mathlib/Algebra/Ring/Idempotents.lean
|
IsIdempotentElem.one_sub
|
[
{
"state_after": "no goals",
"state_before": "M : Type ?u.2741\nN : Type ?u.2744\nS : Type ?u.2747\nM₀ : Type ?u.2750\nM₁ : Type ?u.2753\nR : Type u_1\nG : Type ?u.2759\nG₀ : Type ?u.2762\ninst✝⁷ : Mul M\ninst✝⁶ : Monoid N\ninst✝⁵ : Semigroup S\ninst✝⁴ : MulZeroClass M₀\ninst✝³ : MulOneClass M₁\ninst✝² : NonAssocRing R\ninst✝¹ : Group G\ninst✝ : CancelMonoidWithZero G₀\np : R\nh : IsIdempotentElem p\n⊢ IsIdempotentElem (1 - p)",
"tactic": "rw [IsIdempotentElem, mul_sub, mul_one, sub_mul, one_mul, h.eq, sub_self, sub_zero]"
}
] |
[
69,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Topology/Homotopy/HSpaces.lean
|
unitInterval.qRight_zero_right
|
[
{
"state_after": "t : ↑I\n⊢ ↑(Set.projIcc 0 1 qRight.proof_1 (2 * ↑t)) = if ↑t ≤ 1 / 2 then 2 * ↑t else 1",
"state_before": "t : ↑I\n⊢ ↑(qRight (t, 0)) = if ↑t ≤ 1 / 2 then 2 * ↑t else 1",
"tactic": "simp only [qRight, coe_zero, add_zero, div_one]"
},
{
"state_after": "case inl\nt : ↑I\nh✝ : ↑t ≤ 1 / 2\n⊢ ↑(Set.projIcc 0 1 qRight.proof_1 (2 * ↑t)) = 2 * ↑t\n\ncase inr\nt : ↑I\nh✝ : ¬↑t ≤ 1 / 2\n⊢ ↑(Set.projIcc 0 1 qRight.proof_1 (2 * ↑t)) = 1",
"state_before": "t : ↑I\n⊢ ↑(Set.projIcc 0 1 qRight.proof_1 (2 * ↑t)) = if ↑t ≤ 1 / 2 then 2 * ↑t else 1",
"tactic": "split_ifs"
},
{
"state_after": "t : ↑I\nh✝ : ↑t ≤ 1 / 2\n⊢ ↑t ∈ Set.Icc 0 (1 / 2)",
"state_before": "case inl\nt : ↑I\nh✝ : ↑t ≤ 1 / 2\n⊢ ↑(Set.projIcc 0 1 qRight.proof_1 (2 * ↑t)) = 2 * ↑t",
"tactic": "rw [Set.projIcc_of_mem _ ((mul_pos_mem_iff zero_lt_two).2 _)]"
},
{
"state_after": "t : ↑I\nh✝ : ↑t ≤ 1 / 2\n⊢ ↑t ≤ 1 / 2",
"state_before": "t : ↑I\nh✝ : ↑t ≤ 1 / 2\n⊢ ↑t ∈ Set.Icc 0 (1 / 2)",
"tactic": "refine' ⟨t.2.1, _⟩"
},
{
"state_after": "no goals",
"state_before": "t : ↑I\nh✝ : ↑t ≤ 1 / 2\n⊢ ↑t ≤ 1 / 2",
"tactic": "tauto"
},
{
"state_after": "case inr\nt : ↑I\nh✝ : ¬↑t ≤ 1 / 2\n⊢ 1 ≤ 2 * ↑t\n\nt : ↑I\nh✝ : ¬↑t ≤ 1 / 2\n⊢ 0 < 1",
"state_before": "case inr\nt : ↑I\nh✝ : ¬↑t ≤ 1 / 2\n⊢ ↑(Set.projIcc 0 1 qRight.proof_1 (2 * ↑t)) = 1",
"tactic": "rw [(Set.projIcc_eq_right _).2]"
},
{
"state_after": "no goals",
"state_before": "case inr\nt : ↑I\nh✝ : ¬↑t ≤ 1 / 2\n⊢ 1 ≤ 2 * ↑t",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "t : ↑I\nh✝ : ¬↑t ≤ 1 / 2\n⊢ 0 < 1",
"tactic": "exact zero_lt_one"
}
] |
[
214,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Int.addLeft_one_isCycle
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.3065754\nα : Type ?u.3065757\nβ : Type ?u.3065760\ninst✝ : DecidableEq α\nn : ℤ\nx✝ : ↑(Equiv.addLeft 1) n ≠ n\n⊢ ↑(Equiv.addLeft 1 ^ n) 0 = n",
"tactic": "simp"
}
] |
[
1860,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1859,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
BilinForm.isSymm_zero
|
[] |
[
935,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
935,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_finset_sum'
|
[
{
"state_after": "case empty\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.929424\nδ : Type ?u.929427\nm : MeasurableSpace α\nμ ν : Measure α\ns : Finset β\nf : β → α → ℝ≥0∞\nhf✝ : ∀ (b : β), b ∈ s → AEMeasurable (f b)\nhf : ∀ (b : β), b ∈ ∅ → AEMeasurable (f b)\n⊢ (∫⁻ (a : α), ∑ b in ∅, f b a ∂μ) = ∑ b in ∅, ∫⁻ (a : α), f b a ∂μ\n\ncase insert\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.929424\nδ : Type ?u.929427\nm : MeasurableSpace α\nμ ν : Measure α\ns✝ : Finset β\nf : β → α → ℝ≥0∞\nhf✝ : ∀ (b : β), b ∈ s✝ → AEMeasurable (f b)\na : β\ns : Finset β\nhas : ¬a ∈ s\nih : (∀ (b : β), b ∈ s → AEMeasurable (f b)) → (∫⁻ (a : α), ∑ b in s, f b a ∂μ) = ∑ b in s, ∫⁻ (a : α), f b a ∂μ\nhf : ∀ (b : β), b ∈ insert a s → AEMeasurable (f b)\n⊢ (∫⁻ (a_1 : α), ∑ b in insert a s, f b a_1 ∂μ) = ∑ b in insert a s, ∫⁻ (a : α), f b a ∂μ",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.929424\nδ : Type ?u.929427\nm : MeasurableSpace α\nμ ν : Measure α\ns : Finset β\nf : β → α → ℝ≥0∞\nhf : ∀ (b : β), b ∈ s → AEMeasurable (f b)\n⊢ (∫⁻ (a : α), ∑ b in s, f b a ∂μ) = ∑ b in s, ∫⁻ (a : α), f b a ∂μ",
"tactic": "induction' s using Finset.induction_on with a s has ih"
},
{
"state_after": "no goals",
"state_before": "case empty\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.929424\nδ : Type ?u.929427\nm : MeasurableSpace α\nμ ν : Measure α\ns : Finset β\nf : β → α → ℝ≥0∞\nhf✝ : ∀ (b : β), b ∈ s → AEMeasurable (f b)\nhf : ∀ (b : β), b ∈ ∅ → AEMeasurable (f b)\n⊢ (∫⁻ (a : α), ∑ b in ∅, f b a ∂μ) = ∑ b in ∅, ∫⁻ (a : α), f b a ∂μ",
"tactic": "simp"
},
{
"state_after": "case insert\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.929424\nδ : Type ?u.929427\nm : MeasurableSpace α\nμ ν : Measure α\ns✝ : Finset β\nf : β → α → ℝ≥0∞\nhf✝ : ∀ (b : β), b ∈ s✝ → AEMeasurable (f b)\na : β\ns : Finset β\nhas : ¬a ∈ s\nih : (∀ (b : β), b ∈ s → AEMeasurable (f b)) → (∫⁻ (a : α), ∑ b in s, f b a ∂μ) = ∑ b in s, ∫⁻ (a : α), f b a ∂μ\nhf : ∀ (b : β), b ∈ insert a s → AEMeasurable (f b)\n⊢ (∫⁻ (a_1 : α), f a a_1 + ∑ b in s, f b a_1 ∂μ) = (∫⁻ (a_1 : α), f a a_1 ∂μ) + ∑ b in s, ∫⁻ (a : α), f b a ∂μ",
"state_before": "case insert\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.929424\nδ : Type ?u.929427\nm : MeasurableSpace α\nμ ν : Measure α\ns✝ : Finset β\nf : β → α → ℝ≥0∞\nhf✝ : ∀ (b : β), b ∈ s✝ → AEMeasurable (f b)\na : β\ns : Finset β\nhas : ¬a ∈ s\nih : (∀ (b : β), b ∈ s → AEMeasurable (f b)) → (∫⁻ (a : α), ∑ b in s, f b a ∂μ) = ∑ b in s, ∫⁻ (a : α), f b a ∂μ\nhf : ∀ (b : β), b ∈ insert a s → AEMeasurable (f b)\n⊢ (∫⁻ (a_1 : α), ∑ b in insert a s, f b a_1 ∂μ) = ∑ b in insert a s, ∫⁻ (a : α), f b a ∂μ",
"tactic": "simp only [Finset.sum_insert has]"
},
{
"state_after": "case insert\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.929424\nδ : Type ?u.929427\nm : MeasurableSpace α\nμ ν : Measure α\ns✝ : Finset β\nf : β → α → ℝ≥0∞\nhf✝ : ∀ (b : β), b ∈ s✝ → AEMeasurable (f b)\na : β\ns : Finset β\nhas : ¬a ∈ s\nih : (∀ (b : β), b ∈ s → AEMeasurable (f b)) → (∫⁻ (a : α), ∑ b in s, f b a ∂μ) = ∑ b in s, ∫⁻ (a : α), f b a ∂μ\nhf : AEMeasurable (f a) ∧ ∀ (x : β), x ∈ s → AEMeasurable (f x)\n⊢ (∫⁻ (a_1 : α), f a a_1 + ∑ b in s, f b a_1 ∂μ) = (∫⁻ (a_1 : α), f a a_1 ∂μ) + ∑ b in s, ∫⁻ (a : α), f b a ∂μ",
"state_before": "case insert\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.929424\nδ : Type ?u.929427\nm : MeasurableSpace α\nμ ν : Measure α\ns✝ : Finset β\nf : β → α → ℝ≥0∞\nhf✝ : ∀ (b : β), b ∈ s✝ → AEMeasurable (f b)\na : β\ns : Finset β\nhas : ¬a ∈ s\nih : (∀ (b : β), b ∈ s → AEMeasurable (f b)) → (∫⁻ (a : α), ∑ b in s, f b a ∂μ) = ∑ b in s, ∫⁻ (a : α), f b a ∂μ\nhf : ∀ (b : β), b ∈ insert a s → AEMeasurable (f b)\n⊢ (∫⁻ (a_1 : α), f a a_1 + ∑ b in s, f b a_1 ∂μ) = (∫⁻ (a_1 : α), f a a_1 ∂μ) + ∑ b in s, ∫⁻ (a : α), f b a ∂μ",
"tactic": "rw [Finset.forall_mem_insert] at hf"
},
{
"state_after": "no goals",
"state_before": "case insert\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.929424\nδ : Type ?u.929427\nm : MeasurableSpace α\nμ ν : Measure α\ns✝ : Finset β\nf : β → α → ℝ≥0∞\nhf✝ : ∀ (b : β), b ∈ s✝ → AEMeasurable (f b)\na : β\ns : Finset β\nhas : ¬a ∈ s\nih : (∀ (b : β), b ∈ s → AEMeasurable (f b)) → (∫⁻ (a : α), ∑ b in s, f b a ∂μ) = ∑ b in s, ∫⁻ (a : α), f b a ∂μ\nhf : AEMeasurable (f a) ∧ ∀ (x : β), x ∈ s → AEMeasurable (f x)\n⊢ (∫⁻ (a_1 : α), f a a_1 + ∑ b in s, f b a_1 ∂μ) = (∫⁻ (a_1 : α), f a a_1 ∂μ) + ∑ b in s, ∫⁻ (a : α), f b a ∂μ",
"tactic": "rw [lintegral_add_left' hf.1, ih hf.2]"
}
] |
[
670,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
663,
1
] |
Std/Data/Option/Lemmas.lean
|
Option.forall
|
[] |
[
18,
70
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
17,
11
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.inv_two_add_inv_two
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.312605\nβ : Type ?u.312608\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ 2⁻¹ + 2⁻¹ = 1",
"tactic": "rw [← two_mul, ← div_eq_mul_inv, ENNReal.div_self two_ne_zero two_ne_top]"
}
] |
[
1720,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1719,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
IsUnit.filter
|
[] |
[
724,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
723,
11
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
Cubic.d_eq_three_roots
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.1111262\nS : Type ?u.1111265\nF : Type u_1\nK : Type u_2\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nx y z : K\nha : P.a ≠ 0\nh3 : roots (map φ P) = {x, y, z}\n⊢ ↑φ P.d = ↑φ P.a * -(x * y * z)",
"tactic": "injection eq_sum_three_roots ha h3"
}
] |
[
554,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
552,
1
] |
Mathlib/LinearAlgebra/FreeModule/Rank.lean
|
rank_directSum
|
[
{
"state_after": "R : Type u\nM✝ : Type v\nN : Type w\ninst✝¹⁰ : Ring R\ninst✝⁹ : StrongRankCondition R\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R M✝\ninst✝⁶ : Module.Free R M✝\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Module.Free R N\nι : Type v\nM : ι → Type w\ninst✝² : (i : ι) → AddCommGroup (M i)\ninst✝¹ : (i : ι) → Module R (M i)\ninst✝ : ∀ (i : ι), Module.Free R (M i)\nB : (i : ι) → Basis (ChooseBasisIndex R (M i)) R (M i) := fun i => chooseBasis R (M i)\n⊢ Module.rank R (⨁ (i : ι), M i) = sum fun i => Module.rank R (M i)",
"state_before": "R : Type u\nM✝ : Type v\nN : Type w\ninst✝¹⁰ : Ring R\ninst✝⁹ : StrongRankCondition R\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R M✝\ninst✝⁶ : Module.Free R M✝\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Module.Free R N\nι : Type v\nM : ι → Type w\ninst✝² : (i : ι) → AddCommGroup (M i)\ninst✝¹ : (i : ι) → Module R (M i)\ninst✝ : ∀ (i : ι), Module.Free R (M i)\n⊢ Module.rank R (⨁ (i : ι), M i) = sum fun i => Module.rank R (M i)",
"tactic": "let B i := chooseBasis R (M i)"
},
{
"state_after": "R : Type u\nM✝ : Type v\nN : Type w\ninst✝¹⁰ : Ring R\ninst✝⁹ : StrongRankCondition R\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R M✝\ninst✝⁶ : Module.Free R M✝\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Module.Free R N\nι : Type v\nM : ι → Type w\ninst✝² : (i : ι) → AddCommGroup (M i)\ninst✝¹ : (i : ι) → Module R (M i)\ninst✝ : ∀ (i : ι), Module.Free R (M i)\nB : (i : ι) → Basis (ChooseBasisIndex R (M i)) R (M i) := fun i => chooseBasis R (M i)\nb : Basis ((i : ι) × ChooseBasisIndex R (M i)) R (⨁ (i : ι), M i) := Dfinsupp.basis fun i => B i\n⊢ Module.rank R (⨁ (i : ι), M i) = sum fun i => Module.rank R (M i)",
"state_before": "R : Type u\nM✝ : Type v\nN : Type w\ninst✝¹⁰ : Ring R\ninst✝⁹ : StrongRankCondition R\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R M✝\ninst✝⁶ : Module.Free R M✝\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Module.Free R N\nι : Type v\nM : ι → Type w\ninst✝² : (i : ι) → AddCommGroup (M i)\ninst✝¹ : (i : ι) → Module R (M i)\ninst✝ : ∀ (i : ι), Module.Free R (M i)\nB : (i : ι) → Basis (ChooseBasisIndex R (M i)) R (M i) := fun i => chooseBasis R (M i)\n⊢ Module.rank R (⨁ (i : ι), M i) = sum fun i => Module.rank R (M i)",
"tactic": "let b : Basis _ R (⨁ i, M i) := Dfinsupp.basis fun i => B i"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nM✝ : Type v\nN : Type w\ninst✝¹⁰ : Ring R\ninst✝⁹ : StrongRankCondition R\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R M✝\ninst✝⁶ : Module.Free R M✝\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Module.Free R N\nι : Type v\nM : ι → Type w\ninst✝² : (i : ι) → AddCommGroup (M i)\ninst✝¹ : (i : ι) → Module R (M i)\ninst✝ : ∀ (i : ι), Module.Free R (M i)\nB : (i : ι) → Basis (ChooseBasisIndex R (M i)) R (M i) := fun i => chooseBasis R (M i)\nb : Basis ((i : ι) × ChooseBasisIndex R (M i)) R (⨁ (i : ι), M i) := Dfinsupp.basis fun i => B i\n⊢ Module.rank R (⨁ (i : ι), M i) = sum fun i => Module.rank R (M i)",
"tactic": "simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']"
}
] |
[
70,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
IntervalIntegrable.mono_set_ae
|
[] |
[
208,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/LinearAlgebra/PiTensorProduct.lean
|
PiTensorProduct.ext
|
[
{
"state_after": "ι : Type u_3\nι₂ : Type ?u.234224\nι₃ : Type ?u.234227\nR : Type u_1\ninst✝⁷ : CommSemiring R\nR₁ : Type ?u.234236\nR₂ : Type ?u.234239\ns : ι → Type u_2\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type ?u.234429\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_4\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type ?u.234697\ninst✝ : AddCommMonoid F\nφ₁ φ₂ : (⨂[R] (i : ι), s i) →ₗ[R] E\nH : LinearMap.compMultilinearMap φ₁ (tprod R) = LinearMap.compMultilinearMap φ₂ (tprod R)\n⊢ ∀ (x : ⨂[R] (i : ι), s i), ↑φ₁ x = ↑φ₂ x",
"state_before": "ι : Type u_3\nι₂ : Type ?u.234224\nι₃ : Type ?u.234227\nR : Type u_1\ninst✝⁷ : CommSemiring R\nR₁ : Type ?u.234236\nR₂ : Type ?u.234239\ns : ι → Type u_2\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type ?u.234429\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_4\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type ?u.234697\ninst✝ : AddCommMonoid F\nφ₁ φ₂ : (⨂[R] (i : ι), s i) →ₗ[R] E\nH : LinearMap.compMultilinearMap φ₁ (tprod R) = LinearMap.compMultilinearMap φ₂ (tprod R)\n⊢ φ₁ = φ₂",
"tactic": "refine' LinearMap.ext _"
},
{
"state_after": "ι : Type u_3\nι₂ : Type ?u.234224\nι₃ : Type ?u.234227\nR : Type u_1\ninst✝⁷ : CommSemiring R\nR₁ : Type ?u.234236\nR₂ : Type ?u.234239\ns : ι → Type u_2\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type ?u.234429\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_4\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type ?u.234697\ninst✝ : AddCommMonoid F\nφ₁ φ₂ : (⨂[R] (i : ι), s i) →ₗ[R] E\nH : LinearMap.compMultilinearMap φ₁ (tprod R) = LinearMap.compMultilinearMap φ₂ (tprod R)\nz : ⨂[R] (i : ι), s i\n⊢ ∀ {r : R} {f : (i : ι) → s i}, ↑φ₁ (tprodCoeff R r f) = ↑φ₂ (tprodCoeff R r f)",
"state_before": "ι : Type u_3\nι₂ : Type ?u.234224\nι₃ : Type ?u.234227\nR : Type u_1\ninst✝⁷ : CommSemiring R\nR₁ : Type ?u.234236\nR₂ : Type ?u.234239\ns : ι → Type u_2\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type ?u.234429\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_4\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type ?u.234697\ninst✝ : AddCommMonoid F\nφ₁ φ₂ : (⨂[R] (i : ι), s i) →ₗ[R] E\nH : LinearMap.compMultilinearMap φ₁ (tprod R) = LinearMap.compMultilinearMap φ₂ (tprod R)\n⊢ ∀ (x : ⨂[R] (i : ι), s i), ↑φ₁ x = ↑φ₂ x",
"tactic": "refine' fun z ↦\n PiTensorProduct.induction_on' z _ fun {x y} hx hy ↦ by rw [φ₁.map_add, φ₂.map_add, hx, hy]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_3\nι₂ : Type ?u.234224\nι₃ : Type ?u.234227\nR : Type u_1\ninst✝⁷ : CommSemiring R\nR₁ : Type ?u.234236\nR₂ : Type ?u.234239\ns : ι → Type u_2\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type ?u.234429\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_4\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type ?u.234697\ninst✝ : AddCommMonoid F\nφ₁ φ₂ : (⨂[R] (i : ι), s i) →ₗ[R] E\nH : LinearMap.compMultilinearMap φ₁ (tprod R) = LinearMap.compMultilinearMap φ₂ (tprod R)\nz x y : ⨂[R] (i : ι), s i\nhx : ↑φ₁ x = ↑φ₂ x\nhy : ↑φ₁ y = ↑φ₂ y\n⊢ ↑φ₁ (x + y) = ↑φ₂ (x + y)",
"tactic": "rw [φ₁.map_add, φ₂.map_add, hx, hy]"
},
{
"state_after": "ι : Type u_3\nι₂ : Type ?u.234224\nι₃ : Type ?u.234227\nR : Type u_1\ninst✝⁷ : CommSemiring R\nR₁ : Type ?u.234236\nR₂ : Type ?u.234239\ns : ι → Type u_2\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type ?u.234429\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_4\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type ?u.234697\ninst✝ : AddCommMonoid F\nφ₁ φ₂ : (⨂[R] (i : ι), s i) →ₗ[R] E\nH : LinearMap.compMultilinearMap φ₁ (tprod R) = LinearMap.compMultilinearMap φ₂ (tprod R)\nz : ⨂[R] (i : ι), s i\nr : R\nf : (i : ι) → s i\n⊢ ↑φ₁ (tprodCoeff R r f) = ↑φ₂ (tprodCoeff R r f)",
"state_before": "ι : Type u_3\nι₂ : Type ?u.234224\nι₃ : Type ?u.234227\nR : Type u_1\ninst✝⁷ : CommSemiring R\nR₁ : Type ?u.234236\nR₂ : Type ?u.234239\ns : ι → Type u_2\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type ?u.234429\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_4\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type ?u.234697\ninst✝ : AddCommMonoid F\nφ₁ φ₂ : (⨂[R] (i : ι), s i) →ₗ[R] E\nH : LinearMap.compMultilinearMap φ₁ (tprod R) = LinearMap.compMultilinearMap φ₂ (tprod R)\nz : ⨂[R] (i : ι), s i\n⊢ ∀ {r : R} {f : (i : ι) → s i}, ↑φ₁ (tprodCoeff R r f) = ↑φ₂ (tprodCoeff R r f)",
"tactic": "intro r f"
},
{
"state_after": "ι : Type u_3\nι₂ : Type ?u.234224\nι₃ : Type ?u.234227\nR : Type u_1\ninst✝⁷ : CommSemiring R\nR₁ : Type ?u.234236\nR₂ : Type ?u.234239\ns : ι → Type u_2\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type ?u.234429\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_4\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type ?u.234697\ninst✝ : AddCommMonoid F\nφ₁ φ₂ : (⨂[R] (i : ι), s i) →ₗ[R] E\nH : LinearMap.compMultilinearMap φ₁ (tprod R) = LinearMap.compMultilinearMap φ₂ (tprod R)\nz : ⨂[R] (i : ι), s i\nr : R\nf : (i : ι) → s i\n⊢ r • ↑φ₁ (↑(tprod R) f) = r • ↑φ₂ (↑(tprod R) f)",
"state_before": "ι : Type u_3\nι₂ : Type ?u.234224\nι₃ : Type ?u.234227\nR : Type u_1\ninst✝⁷ : CommSemiring R\nR₁ : Type ?u.234236\nR₂ : Type ?u.234239\ns : ι → Type u_2\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type ?u.234429\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_4\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type ?u.234697\ninst✝ : AddCommMonoid F\nφ₁ φ₂ : (⨂[R] (i : ι), s i) →ₗ[R] E\nH : LinearMap.compMultilinearMap φ₁ (tprod R) = LinearMap.compMultilinearMap φ₂ (tprod R)\nz : ⨂[R] (i : ι), s i\nr : R\nf : (i : ι) → s i\n⊢ ↑φ₁ (tprodCoeff R r f) = ↑φ₂ (tprodCoeff R r f)",
"tactic": "rw [tprodCoeff_eq_smul_tprod, φ₁.map_smul, φ₂.map_smul]"
},
{
"state_after": "case h\nι : Type u_3\nι₂ : Type ?u.234224\nι₃ : Type ?u.234227\nR : Type u_1\ninst✝⁷ : CommSemiring R\nR₁ : Type ?u.234236\nR₂ : Type ?u.234239\ns : ι → Type u_2\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type ?u.234429\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_4\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type ?u.234697\ninst✝ : AddCommMonoid F\nφ₁ φ₂ : (⨂[R] (i : ι), s i) →ₗ[R] E\nH : LinearMap.compMultilinearMap φ₁ (tprod R) = LinearMap.compMultilinearMap φ₂ (tprod R)\nz : ⨂[R] (i : ι), s i\nr : R\nf : (i : ι) → s i\n⊢ ↑φ₁ (↑(tprod R) f) = ↑φ₂ (↑(tprod R) f)",
"state_before": "ι : Type u_3\nι₂ : Type ?u.234224\nι₃ : Type ?u.234227\nR : Type u_1\ninst✝⁷ : CommSemiring R\nR₁ : Type ?u.234236\nR₂ : Type ?u.234239\ns : ι → Type u_2\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type ?u.234429\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_4\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type ?u.234697\ninst✝ : AddCommMonoid F\nφ₁ φ₂ : (⨂[R] (i : ι), s i) →ₗ[R] E\nH : LinearMap.compMultilinearMap φ₁ (tprod R) = LinearMap.compMultilinearMap φ₂ (tprod R)\nz : ⨂[R] (i : ι), s i\nr : R\nf : (i : ι) → s i\n⊢ r • ↑φ₁ (↑(tprod R) f) = r • ↑φ₂ (↑(tprod R) f)",
"tactic": "apply _root_.congr_arg"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u_3\nι₂ : Type ?u.234224\nι₃ : Type ?u.234227\nR : Type u_1\ninst✝⁷ : CommSemiring R\nR₁ : Type ?u.234236\nR₂ : Type ?u.234239\ns : ι → Type u_2\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type ?u.234429\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_4\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type ?u.234697\ninst✝ : AddCommMonoid F\nφ₁ φ₂ : (⨂[R] (i : ι), s i) →ₗ[R] E\nH : LinearMap.compMultilinearMap φ₁ (tprod R) = LinearMap.compMultilinearMap φ₂ (tprod R)\nz : ⨂[R] (i : ι), s i\nr : R\nf : (i : ι) → s i\n⊢ ↑φ₁ (↑(tprod R) f) = ↑φ₂ (↑(tprod R) f)",
"tactic": "exact MultilinearMap.congr_fun H f"
}
] |
[
348,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/Analysis/LocallyConvex/ContinuousOfBounded.lean
|
LinearMap.clmOfExistsBoundedImage_coe
|
[] |
[
81,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
|
ContinuousWithinAt.rpow_const
|
[] |
[
313,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
8
] |
Mathlib/Order/Atoms.lean
|
isAtomic_of_isCoatomic_of_complementedLattice_of_isModular
|
[] |
[
876,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
874,
1
] |
Mathlib/Data/PNat/Xgcd.lean
|
PNat.XgcdType.finish_isReduced
|
[
{
"state_after": "u : XgcdType\n⊢ (finish u).ap = (finish u).bp",
"state_before": "u : XgcdType\n⊢ IsReduced (finish u)",
"tactic": "dsimp [IsReduced]"
},
{
"state_after": "no goals",
"state_before": "u : XgcdType\n⊢ (finish u).ap = (finish u).bp",
"tactic": "rfl"
}
] |
[
280,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
278,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
squeeze_one_norm'
|
[] |
[
1058,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1055,
1
] |
Mathlib/Topology/Instances/Matrix.lean
|
HasSum.matrix_transpose
|
[] |
[
296,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieHom.coe_mk
|
[] |
[
406,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/Algebra/Order/Hom/Monoid.lean
|
OrderMonoidWithZeroHom.toMonoidWithZeroHom_injective
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.124488\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.124497\nδ : Type ?u.124500\ninst✝⁷ : Preorder α\ninst✝⁶ : Preorder β\ninst✝⁵ : Preorder γ\ninst✝⁴ : Preorder δ\ninst✝³ : MulZeroOneClass α\ninst✝² : MulZeroOneClass β\ninst✝¹ : MulZeroOneClass γ\ninst✝ : MulZeroOneClass δ\nf✝ g✝ f g : α →*₀o β\nh : f.toMonoidWithZeroHom = g.toMonoidWithZeroHom\n⊢ ∀ (a : α), ↑f a = ↑g a",
"tactic": "convert FunLike.ext_iff.1 h using 0"
}
] |
[
641,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
640,
1
] |
Mathlib/GroupTheory/EckmannHilton.lean
|
EckmannHilton.mul_assoc
|
[
{
"state_after": "no goals",
"state_before": "X : Type u\nm₁ m₂ : X → X → X\ne₁ e₂ : X\nh₁ : IsUnital m₁ e₁\nh₂ : IsUnital m₂ e₂\ndistrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)\na b c : X\n⊢ m₂ (m₂ a b) c = m₂ a (m₂ b c)",
"tactic": "simpa [mul h₁ h₂ distrib, h₂.left_id, h₂.right_id] using distrib a b e₂ c"
}
] |
[
89,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Order/Filter/Prod.lean
|
Filter.map_const_principal_coprod_map_id_principal
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.65740\nβ✝ : Type ?u.65743\nγ : Type ?u.65746\nδ : Type ?u.65749\nι✝ : Sort ?u.65752\nf : Filter α✝\ng : Filter β✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\na : α\nb : β\ni : ι\n⊢ Filter.coprod (map (fun x => b) (𝓟 {a})) (map id (𝓟 {i})) = 𝓟 ({b} ×ˢ univ ∪ univ ×ˢ {i})",
"tactic": "simp only [map_principal, Filter.coprod, comap_principal, sup_principal, image_singleton,\n image_id, prod_univ, univ_prod, id]"
}
] |
[
542,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
538,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.tendsto_atBot
|
[] |
[
547,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
544,
11
] |
Mathlib/Combinatorics/Additive/Etransform.lean
|
Finset.mulEtransformRight.fst_mul_snd_subset
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Group α\ne : α\nx : Finset α × Finset α\n⊢ op e • x.fst * e⁻¹ • x.snd ⊆ x.fst * x.snd",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Group α\ne : α\nx : Finset α × Finset α\n⊢ (mulEtransformRight e x).fst * (mulEtransformRight e x).snd ⊆ x.fst * x.snd",
"tactic": "refine' union_mul_inter_subset_union.trans (union_subset Subset.rfl _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Group α\ne : α\nx : Finset α × Finset α\n⊢ op e • x.fst * e⁻¹ • x.snd ⊆ x.fst * x.snd",
"tactic": "rw [op_smul_finset_mul_eq_mul_smul_finset, smul_inv_smul]"
}
] |
[
156,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Ordered.lean
|
lineMap_le_map_iff_slope_le_slope
|
[] |
[
288,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/Algebra/Module/Basic.lean
|
inv_int_cast_smul_comm
|
[] |
[
538,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.mul_dvd_mul_left
|
[] |
[
687,
37
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
686,
11
] |
Mathlib/Analysis/NormedSpace/Basic.lean
|
interior_sphere'
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.309886\nβ : Type ?u.309889\nγ : Type ?u.309892\nι : Type ?u.309895\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.309988\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\n⊢ interior (sphere x r) = ∅",
"tactic": "rw [← frontier_closedBall' x, interior_frontier isClosed_ball]"
}
] |
[
397,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/Order/Hom/CompleteLattice.lean
|
CompleteLatticeHom.setPreimage_comp
|
[] |
[
940,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
938,
1
] |
Mathlib/Data/Nat/Prime.lean
|
Nat.minFac_eq_one_iff
|
[
{
"state_after": "case mp\nn : ℕ\n⊢ minFac n = 1 → n = 1\n\ncase mpr\nn : ℕ\n⊢ n = 1 → minFac n = 1",
"state_before": "n : ℕ\n⊢ minFac n = 1 ↔ n = 1",
"tactic": "constructor"
},
{
"state_after": "case mp\nn : ℕ\nh : minFac n = 1\n⊢ n = 1",
"state_before": "case mp\nn : ℕ\n⊢ minFac n = 1 → n = 1",
"tactic": "intro h"
},
{
"state_after": "case mp\nn : ℕ\nh : minFac n = 1\nhn : ¬n = 1\n⊢ False",
"state_before": "case mp\nn : ℕ\nh : minFac n = 1\n⊢ n = 1",
"tactic": "by_contra hn"
},
{
"state_after": "case mp\nn : ℕ\nh : minFac n = 1\nhn : ¬n = 1\nthis : Prime (minFac n)\n⊢ False",
"state_before": "case mp\nn : ℕ\nh : minFac n = 1\nhn : ¬n = 1\n⊢ False",
"tactic": "have := minFac_prime hn"
},
{
"state_after": "case mp\nn : ℕ\nh : minFac n = 1\nhn : ¬n = 1\nthis : Prime 1\n⊢ False",
"state_before": "case mp\nn : ℕ\nh : minFac n = 1\nhn : ¬n = 1\nthis : Prime (minFac n)\n⊢ False",
"tactic": "rw [h] at this"
},
{
"state_after": "no goals",
"state_before": "case mp\nn : ℕ\nh : minFac n = 1\nhn : ¬n = 1\nthis : Prime 1\n⊢ False",
"tactic": "exact not_prime_one this"
},
{
"state_after": "case mpr\n\n⊢ minFac 1 = 1",
"state_before": "case mpr\nn : ℕ\n⊢ n = 1 → minFac n = 1",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case mpr\n\n⊢ minFac 1 = 1",
"tactic": "rfl"
}
] |
[
429,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/Analysis/Convex/StrictConvexSpace.lean
|
StrictConvexSpace.ofPairwiseSphereNormNeTwo
|
[] |
[
141,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Data/Nat/Dist.lean
|
Nat.dist_tri_left'
|
[
{
"state_after": "n m : ℕ\n⊢ n ≤ dist m n + m",
"state_before": "n m : ℕ\n⊢ n ≤ dist n m + m",
"tactic": "rw [dist_comm]"
},
{
"state_after": "no goals",
"state_before": "n m : ℕ\n⊢ n ≤ dist m n + m",
"tactic": "apply dist_tri_left"
}
] |
[
64,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.bsup_le_iff
|
[
{
"state_after": "α : Type ?u.309348\nβ : Type ?u.309351\nγ : Type ?u.309354\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\na : Ordinal\nh : ∀ (i : (Quotient.out o).α), familyOfBFamily o f i ≤ a\ni : Ordinal\nhi : i < o\n⊢ familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1)) ≤ a",
"state_before": "α : Type ?u.309348\nβ : Type ?u.309351\nγ : Type ?u.309354\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\na : Ordinal\nh : ∀ (i : (Quotient.out o).α), familyOfBFamily o f i ≤ a\ni : Ordinal\nhi : i < o\n⊢ f i hi ≤ a",
"tactic": "rw [← familyOfBFamily_enum o f]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.309348\nβ : Type ?u.309351\nγ : Type ?u.309354\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\na : Ordinal\nh : ∀ (i : (Quotient.out o).α), familyOfBFamily o f i ≤ a\ni : Ordinal\nhi : i < o\n⊢ familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1)) ≤ a",
"tactic": "exact h _"
}
] |
[
1492,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1488,
1
] |
Mathlib/Data/Vector/Zip.lean
|
Vector.prod_mul_prod_eq_prod_zipWith
|
[] |
[
54,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
intervalIntegral.intervalIntegrable_zpow
|
[] |
[
63,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/Algebra/Ring/BooleanRing.lean
|
RingHom.asBoolAlg_id
|
[] |
[
371,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
370,
1
] |
Mathlib/Algebra/Hom/Aut.lean
|
MulAut.one_apply
|
[] |
[
98,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Analysis/Convex/Gauge.lean
|
interior_subset_gauge_lt_one
|
[
{
"state_after": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\n⊢ x ∈ {x | gauge s x < 1}",
"state_before": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\n⊢ interior s ⊆ {x | gauge s x < 1}",
"tactic": "intro x hx"
},
{
"state_after": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\n⊢ x ∈ {x | gauge s x < 1}",
"state_before": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\n⊢ x ∈ {x | gauge s x < 1}",
"tactic": "let f : ℝ → E := fun t => t • x"
},
{
"state_after": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\n⊢ x ∈ {x | gauge s x < 1}",
"state_before": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\n⊢ x ∈ {x | gauge s x < 1}",
"tactic": "have hf : Continuous f := by continuity"
},
{
"state_after": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\n⊢ x ∈ {x | gauge s x < 1}",
"state_before": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\n⊢ x ∈ {x | gauge s x < 1}",
"tactic": "let s' := f ⁻¹' interior s"
},
{
"state_after": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\n⊢ x ∈ {x | gauge s x < 1}",
"state_before": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\n⊢ x ∈ {x | gauge s x < 1}",
"tactic": "have hs' : IsOpen s' := hf.isOpen_preimage _ isOpen_interior"
},
{
"state_after": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\n⊢ x ∈ {x | gauge s x < 1}",
"state_before": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\n⊢ x ∈ {x | gauge s x < 1}",
"tactic": "have one_mem : (1 : ℝ) ∈ s' := by simpa only [Set.mem_preimage, one_smul]"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Metric.closedBall 1 ε ⊆ s'\n⊢ x ∈ {x | gauge s x < 1}",
"state_before": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\n⊢ x ∈ {x | gauge s x < 1}",
"tactic": "obtain ⟨ε, hε₀, hε⟩ := (Metric.nhds_basis_closedBall.1 _).1 (isOpen_iff_mem_nhds.1 hs' 1 one_mem)"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Icc (1 - ε) (1 + ε) ⊆ s'\n⊢ x ∈ {x | gauge s x < 1}",
"state_before": "case intro.intro\n𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Metric.closedBall 1 ε ⊆ s'\n⊢ x ∈ {x | gauge s x < 1}",
"tactic": "rw [Real.closedBall_eq_Icc] at hε"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Icc (1 - ε) (1 + ε) ⊆ s'\nhε₁ : 0 < 1 + ε\n⊢ x ∈ {x | gauge s x < 1}",
"state_before": "case intro.intro\n𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Icc (1 - ε) (1 + ε) ⊆ s'\n⊢ x ∈ {x | gauge s x < 1}",
"tactic": "have hε₁ : 0 < 1 + ε := hε₀.trans (lt_one_add ε)"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Icc (1 - ε) (1 + ε) ⊆ s'\nhε₁ : 0 < 1 + ε\nthis : (1 + ε)⁻¹ < 1\n⊢ x ∈ {x | gauge s x < 1}",
"state_before": "case intro.intro\n𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Icc (1 - ε) (1 + ε) ⊆ s'\nhε₁ : 0 < 1 + ε\n⊢ x ∈ {x | gauge s x < 1}",
"tactic": "have : (1 + ε)⁻¹ < 1 := by\n rw [inv_lt_one_iff]\n right\n linarith"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Icc (1 - ε) (1 + ε) ⊆ s'\nhε₁ : 0 < 1 + ε\nthis : (1 + ε)⁻¹ < 1\n⊢ x ∈ (1 + ε)⁻¹ • s",
"state_before": "case intro.intro\n𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Icc (1 - ε) (1 + ε) ⊆ s'\nhε₁ : 0 < 1 + ε\nthis : (1 + ε)⁻¹ < 1\n⊢ x ∈ {x | gauge s x < 1}",
"tactic": "refine' (gauge_le_of_mem (inv_nonneg.2 hε₁.le) _).trans_lt this"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Icc (1 - ε) (1 + ε) ⊆ s'\nhε₁ : 0 < 1 + ε\nthis : (1 + ε)⁻¹ < 1\n⊢ (1 + ε) • x ∈ s",
"state_before": "case intro.intro\n𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Icc (1 - ε) (1 + ε) ⊆ s'\nhε₁ : 0 < 1 + ε\nthis : (1 + ε)⁻¹ < 1\n⊢ x ∈ (1 + ε)⁻¹ • s",
"tactic": "rw [mem_inv_smul_set_iff₀ hε₁.ne']"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\n𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Icc (1 - ε) (1 + ε) ⊆ s'\nhε₁ : 0 < 1 + ε\nthis : (1 + ε)⁻¹ < 1\n⊢ (1 + ε) • x ∈ s",
"tactic": "exact\n interior_subset\n (hε ⟨(sub_le_self _ hε₀.le).trans ((le_add_iff_nonneg_right _).2 hε₀.le), le_rfl⟩)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\n⊢ Continuous f",
"tactic": "continuity"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\n⊢ 1 ∈ s'",
"tactic": "simpa only [Set.mem_preimage, one_smul]"
},
{
"state_after": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Icc (1 - ε) (1 + ε) ⊆ s'\nhε₁ : 0 < 1 + ε\n⊢ 1 + ε ≤ 0 ∨ 1 < 1 + ε",
"state_before": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Icc (1 - ε) (1 + ε) ⊆ s'\nhε₁ : 0 < 1 + ε\n⊢ (1 + ε)⁻¹ < 1",
"tactic": "rw [inv_lt_one_iff]"
},
{
"state_after": "case h\n𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Icc (1 - ε) (1 + ε) ⊆ s'\nhε₁ : 0 < 1 + ε\n⊢ 1 < 1 + ε",
"state_before": "𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Icc (1 - ε) (1 + ε) ⊆ s'\nhε₁ : 0 < 1 + ε\n⊢ 1 + ε ≤ 0 ∨ 1 < 1 + ε",
"tactic": "right"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type ?u.178960\nE : Type u_1\nF : Type ?u.178966\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns✝ t : Set E\na : ℝ\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx : E\nhx : x ∈ interior s\nf : ℝ → E := fun t => t • x\nhf : Continuous f\ns' : Set ℝ := f ⁻¹' interior s\nhs' : IsOpen s'\none_mem : 1 ∈ s'\nε : ℝ\nhε₀ : 0 < ε\nhε : Icc (1 - ε) (1 + ε) ⊆ s'\nhε₁ : 0 < 1 + ε\n⊢ 1 < 1 + ε",
"tactic": "linarith"
}
] |
[
351,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigOWith.sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.223355\nE : Type ?u.223358\nF : Type u_3\nG : Type ?u.223364\nE' : Type u_2\nF' : Type ?u.223370\nG' : Type ?u.223373\nE'' : Type ?u.223376\nF'' : Type ?u.223379\nG'' : Type ?u.223382\nR : Type ?u.223385\nR' : Type ?u.223388\n𝕜 : Type ?u.223391\n𝕜' : Type ?u.223394\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf₁ f₂ : α → E'\ng₁ g₂ : α → F'\nh₁ : IsBigOWith c₁ l f₁ g\nh₂ : IsBigOWith c₂ l f₂ g\n⊢ IsBigOWith (c₁ + c₂) l (fun x => f₁ x - f₂ x) g",
"tactic": "simpa only [sub_eq_add_neg] using h₁.add h₂.neg_left"
}
] |
[
1106,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1104,
1
] |
Mathlib/Order/CompleteLattice.lean
|
le_iInf
|
[] |
[
821,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
820,
1
] |
Mathlib/MeasureTheory/Group/FundamentalDomain.lean
|
MeasureTheory.IsFundamentalDomain.measure_fundamentalInterior
|
[] |
[
677,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
676,
1
] |
Mathlib/Algebra/Hom/Equiv/Basic.lean
|
MulHom.toMulEquiv_apply
|
[] |
[
745,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
742,
1
] |
Mathlib/Algebra/Ring/Commute.lean
|
Commute.bit1_left
|
[] |
[
68,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Std/Data/List/Lemmas.lean
|
List.erase_sublist
|
[] |
[
1075,
41
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1074,
1
] |
Std/Data/List/Lemmas.lean
|
List.set_comm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\na b : α\nx✝² x✝¹ : Nat\nx✝ : x✝² ≠ x✝¹\n⊢ set (set [] x✝² a) x✝¹ b = set (set [] x✝¹ b) x✝² a",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\na b : α\nn : Nat\nhead✝ : α\ntail✝ : List α\nx✝ : n + 1 ≠ 0\n⊢ set (set (head✝ :: tail✝) (n + 1) a) 0 b = set (set (head✝ :: tail✝) 0 b) (n + 1) a",
"tactic": "simp [set]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\na b : α\nm : Nat\nhead✝ : α\ntail✝ : List α\nx✝ : 0 ≠ m + 1\n⊢ set (set (head✝ :: tail✝) 0 a) (m + 1) b = set (set (head✝ :: tail✝) (m + 1) b) 0 a",
"tactic": "simp [set]"
}
] |
[
828,
69
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
822,
1
] |
Mathlib/Algebra/Lie/Solvable.lean
|
LieIdeal.derivedSeries_eq_derivedSeriesOfIdeal_map
|
[
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nk : ℕ\n⊢ derivedSeriesOfIdeal R L k I ≤ I",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nk : ℕ\n⊢ map (incl I) (derivedSeries R { x // x ∈ ↑I } k) = derivedSeriesOfIdeal R L k I",
"tactic": "rw [derivedSeries_eq_derivedSeriesOfIdeal_comap, map_comap_incl, inf_eq_right]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nk : ℕ\n⊢ derivedSeriesOfIdeal R L k I ≤ I",
"tactic": "apply derivedSeriesOfIdeal_le_self"
}
] |
[
166,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/LinearAlgebra/StdBasis.lean
|
LinearMap.stdBasis_eq_single
|
[] |
[
163,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Linear.lean
|
ContinuousLinearMap.hasStrictFDerivAt
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.13611\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.13706\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx✝¹ : E\ns t : Set E\nL L₁ L₂ : Filter E\nx✝ : E\nx : E × E\n⊢ 0 = ↑e x.fst - ↑e x.snd - ↑e (x.fst - x.snd)",
"tactic": "simp only [e.map_sub, sub_self]"
}
] |
[
66,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
11
] |
Mathlib/Data/Int/GCD.lean
|
Int.coe_nat_gcd
|
[] |
[
185,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
11
] |
Mathlib/Data/Rat/Cast.lean
|
MonoidWithZeroHom.ext_rat'
|
[
{
"state_after": "F : Type u_2\nι : Type ?u.82971\nα : Type ?u.82974\nβ : Type ?u.82977\nM₀ : Type u_1\ninst✝¹ : MonoidWithZero M₀\ninst✝ : MonoidWithZeroHomClass F ℚ M₀\nf g : F\nh : ∀ (m : ℤ), ↑f ↑m = ↑g ↑m\nr : ℚ\n⊢ ↑f ↑↑r.den = ↑g ↑↑r.den",
"state_before": "F : Type u_2\nι : Type ?u.82971\nα : Type ?u.82974\nβ : Type ?u.82977\nM₀ : Type u_1\ninst✝¹ : MonoidWithZero M₀\ninst✝ : MonoidWithZeroHomClass F ℚ M₀\nf g : F\nh : ∀ (m : ℤ), ↑f ↑m = ↑g ↑m\nr : ℚ\n⊢ ↑f r = ↑g r",
"tactic": "rw [← r.num_div_den, div_eq_mul_inv, map_mul, map_mul, h, ← Int.cast_ofNat,\n eq_on_inv₀ f g]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_2\nι : Type ?u.82971\nα : Type ?u.82974\nβ : Type ?u.82977\nM₀ : Type u_1\ninst✝¹ : MonoidWithZero M₀\ninst✝ : MonoidWithZeroHomClass F ℚ M₀\nf g : F\nh : ∀ (m : ℤ), ↑f ↑m = ↑g ↑m\nr : ℚ\n⊢ ↑f ↑↑r.den = ↑g ↑↑r.den",
"tactic": "apply h"
}
] |
[
457,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
453,
1
] |
Mathlib/Topology/MetricSpace/Antilipschitz.lean
|
AntilipschitzWith.closedEmbedding
|
[] |
[
200,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.semiconjBy_iff_semiconj
|
[] |
[
274,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.map_sup_comap_of_surjective
|
[] |
[
911,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
909,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.mk_one
|
[] |
[
830,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
829,
1
] |
Mathlib/RingTheory/DedekindDomain/Dvr.lean
|
IsLocalization.AtPrime.isDedekindDomain
|
[] |
[
117,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Order/Filter/Extr.lean
|
IsMaxFilter.sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ninst✝ : OrderedAddCommGroup β\nf g : α → β\na : α\ns : Set α\nl : Filter α\nhf : IsMaxFilter f l a\nhg : IsMinFilter g l a\n⊢ IsMaxFilter (fun x => f x - g x) l a",
"tactic": "simpa only [sub_eq_add_neg] using hf.add hg.neg"
}
] |
[
511,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
510,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
MulLECancellable.Injective
|
[] |
[
1604,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1602,
11
] |
Mathlib/Analysis/Analytic/Basic.lean
|
HasFPowerSeriesOnBall.eventually_hasSum
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.445837\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\n⊢ ∀ᶠ (y : E) in 𝓝 0, HasSum (fun n => ↑(p n) fun x => y) (f (x + y))",
"tactic": "filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos]using fun _ => hf.hasSum"
}
] |
[
486,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.mapRange_finset_sum
|
[] |
[
244,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
invOf_exp_of_mem_ball
|
[
{
"state_after": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.124210\ninst✝⁷ : NontriviallyNormedField 𝕂\ninst✝⁶ : NormedRing 𝔸\ninst✝⁵ : NormedRing 𝔹\ninst✝⁴ : NormedAlgebra 𝕂 𝔸\ninst✝³ : NormedAlgebra 𝕂 𝔹\ninst✝² : CompleteSpace 𝔸\ninst✝¹ : CharZero 𝕂\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\ninst✝ : Invertible (exp 𝕂 x)\nthis : Invertible (exp 𝕂 x) := invertibleExpOfMemBall hx\n⊢ ⅟(exp 𝕂 x) = exp 𝕂 (-x)",
"state_before": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.124210\ninst✝⁷ : NontriviallyNormedField 𝕂\ninst✝⁶ : NormedRing 𝔸\ninst✝⁵ : NormedRing 𝔹\ninst✝⁴ : NormedAlgebra 𝕂 𝔸\ninst✝³ : NormedAlgebra 𝕂 𝔹\ninst✝² : CompleteSpace 𝔸\ninst✝¹ : CharZero 𝕂\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\ninst✝ : Invertible (exp 𝕂 x)\n⊢ ⅟(exp 𝕂 x) = exp 𝕂 (-x)",
"tactic": "letI := invertibleExpOfMemBall hx"
},
{
"state_after": "no goals",
"state_before": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type ?u.124210\ninst✝⁷ : NontriviallyNormedField 𝕂\ninst✝⁶ : NormedRing 𝔸\ninst✝⁵ : NormedRing 𝔹\ninst✝⁴ : NormedAlgebra 𝕂 𝔸\ninst✝³ : NormedAlgebra 𝕂 𝔹\ninst✝² : CompleteSpace 𝔸\ninst✝¹ : CharZero 𝕂\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\ninst✝ : Invertible (exp 𝕂 x)\nthis : Invertible (exp 𝕂 x) := invertibleExpOfMemBall hx\n⊢ ⅟(exp 𝕂 x) = exp 𝕂 (-x)",
"tactic": "convert(rfl : ⅟ (exp 𝕂 x) = _)"
}
] |
[
309,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.fst_apply
|
[] |
[
79,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/FieldTheory/Minpoly/Basic.lean
|
minpoly.monic
|
[
{
"state_after": "A : Type u_1\nB : Type u_2\nB' : Type ?u.2178\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\nhx : IsIntegral A x\n⊢ Monic\n (if hx : IsIntegral A x then\n WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx\n else 0)",
"state_before": "A : Type u_1\nB : Type u_2\nB' : Type ?u.2178\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\nhx : IsIntegral A x\n⊢ Monic (minpoly A x)",
"tactic": "delta minpoly"
},
{
"state_after": "A : Type u_1\nB : Type u_2\nB' : Type ?u.2178\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\nhx : IsIntegral A x\n⊢ Monic\n (WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx)",
"state_before": "A : Type u_1\nB : Type u_2\nB' : Type ?u.2178\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\nhx : IsIntegral A x\n⊢ Monic\n (if hx : IsIntegral A x then\n WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx\n else 0)",
"tactic": "rw [dif_pos hx]"
},
{
"state_after": "no goals",
"state_before": "A : Type u_1\nB : Type u_2\nB' : Type ?u.2178\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\nhx : IsIntegral A x\n⊢ Monic\n (WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx)",
"tactic": "exact (degree_lt_wf.min_mem _ hx).1"
}
] |
[
58,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Data/Option/Basic.lean
|
Option.pmap_map
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.7286\np : α → Prop\nf : (a : α) → p a → β\nx✝ : Option α\ng : γ → α\nx : Option γ\nH : ∀ (a : α), a ∈ Option.map g x → p a\n⊢ pmap f (Option.map g x) H = pmap (fun a h => f (g a) h) x (_ : ∀ (a : γ), a ∈ x → p (g a))",
"tactic": "cases x <;> simp only [map_none', map_some', pmap]"
}
] |
[
185,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/FieldTheory/Separable.lean
|
isSeparable_tower_top_of_isSeparable
|
[] |
[
544,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
542,
1
] |
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