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 |
|---|---|---|---|---|---|---|
Std/Data/Array/Init/Lemmas.lean
|
Array.toListAppend_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\narr : Array α\nl : List α\n⊢ toListAppend arr l = arr.data ++ l",
"tactic": "simp [toListAppend, foldr_eq_foldr_data]"
}
] |
[
93,
43
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
92,
9
] |
Mathlib/Data/Rat/Cast.lean
|
Rat.cast_mul_of_ne_zero
|
[
{
"state_after": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\n⊢ ↑(mk' n₁ d₁ * mk' n₂ d₂) = ↑(mk' n₁ d₁) * ↑(mk' n₂ d₂)",
"state_before": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\n⊢ ↑(mk' n₁ d₁ * mk' n₂ d₂) = ↑(mk' n₁ d₁) * ↑(mk' n₂ d₂)",
"tactic": "have d₁0' : (d₁ : ℤ) ≠ 0 :=\n Int.coe_nat_ne_zero.2 fun e => by rw [e] at d₁0 ; exact d₁0 Nat.cast_zero"
},
{
"state_after": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\nd₂0' : ↑d₂ ≠ 0\n⊢ ↑(mk' n₁ d₁ * mk' n₂ d₂) = ↑(mk' n₁ d₁) * ↑(mk' n₂ d₂)",
"state_before": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\n⊢ ↑(mk' n₁ d₁ * mk' n₂ d₂) = ↑(mk' n₁ d₁) * ↑(mk' n₂ d₂)",
"tactic": "have d₂0' : (d₂ : ℤ) ≠ 0 :=\n Int.coe_nat_ne_zero.2 fun e => by rw [e] at d₂0 ; exact d₂0 Nat.cast_zero"
},
{
"state_after": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\nd₂0' : ↑d₂ ≠ 0\n⊢ ↑(n₁ * n₂ /. (↑d₁ * ↑d₂)) = ↑(n₁ /. ↑d₁) * ↑(n₂ /. ↑d₂)",
"state_before": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\nd₂0' : ↑d₂ ≠ 0\n⊢ ↑(mk' n₁ d₁ * mk' n₂ d₂) = ↑(mk' n₁ d₁) * ↑(mk' n₂ d₂)",
"tactic": "rw [num_den', num_den', mul_def' d₁0' d₂0']"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\nd₂0' : ↑d₂ ≠ 0\n⊢ ↑n₁ * (↑n₂ * (↑d₂)⁻¹ * (↑d₁)⁻¹) = ↑n₁ * ((↑d₁)⁻¹ * (↑n₂ * (↑d₂)⁻¹))",
"tactic": "rw [(d₁.commute_cast (_ : α)).inv_right₀.eq]"
},
{
"state_after": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑0 ≠ 0\nd₂0 : ↑d₂ ≠ 0\ne : d₁ = 0\n⊢ False",
"state_before": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\ne : d₁ = 0\n⊢ False",
"tactic": "rw [e] at d₁0"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑0 ≠ 0\nd₂0 : ↑d₂ ≠ 0\ne : d₁ = 0\n⊢ False",
"tactic": "exact d₁0 Nat.cast_zero"
},
{
"state_after": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑0 ≠ 0\nd₁0' : ↑d₁ ≠ 0\ne : d₂ = 0\n⊢ False",
"state_before": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\ne : d₂ = 0\n⊢ False",
"tactic": "rw [e] at d₂0"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑0 ≠ 0\nd₁0' : ↑d₁ ≠ 0\ne : d₂ = 0\n⊢ False",
"tactic": "exact d₂0 Nat.cast_zero"
},
{
"state_after": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\nd₂0' : ↑d₂ ≠ 0\nthis : ↑n₁ * (↑n₂ * (↑d₂)⁻¹ * (↑d₁)⁻¹) = ↑n₁ * ((↑d₁)⁻¹ * (↑n₂ * (↑d₂)⁻¹))\n⊢ ↑(n₁ * n₂) / ↑(↑d₁ * ↑d₂) = ↑n₁ / ↑↑d₁ * (↑n₂ / ↑↑d₂)\n\ncase b0\nF : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\nd₂0' : ↑d₂ ≠ 0\nthis : ↑n₁ * (↑n₂ * (↑d₂)⁻¹ * (↑d₁)⁻¹) = ↑n₁ * ((↑d₁)⁻¹ * (↑n₂ * (↑d₂)⁻¹))\n⊢ ↑↑d₂ ≠ 0\n\ncase b0\nF : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\nd₂0' : ↑d₂ ≠ 0\nthis : ↑n₁ * (↑n₂ * (↑d₂)⁻¹ * (↑d₁)⁻¹) = ↑n₁ * ((↑d₁)⁻¹ * (↑n₂ * (↑d₂)⁻¹))\n⊢ ↑↑d₁ ≠ 0\n\ncase b0\nF : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\nd₂0' : ↑d₂ ≠ 0\nthis : ↑n₁ * (↑n₂ * (↑d₂)⁻¹ * (↑d₁)⁻¹) = ↑n₁ * ((↑d₁)⁻¹ * (↑n₂ * (↑d₂)⁻¹))\n⊢ ↑(↑d₁ * ↑d₂) ≠ 0",
"state_before": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\nd₂0' : ↑d₂ ≠ 0\nthis : ↑n₁ * (↑n₂ * (↑d₂)⁻¹ * (↑d₁)⁻¹) = ↑n₁ * ((↑d₁)⁻¹ * (↑n₂ * (↑d₂)⁻¹))\n⊢ ↑(n₁ * n₂ /. (↑d₁ * ↑d₂)) = ↑(n₁ /. ↑d₁) * ↑(n₂ /. ↑d₂)",
"tactic": "rw [cast_mk_of_ne_zero, cast_mk_of_ne_zero, cast_mk_of_ne_zero]"
},
{
"state_after": "no goals",
"state_before": "case b0\nF : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\nd₂0' : ↑d₂ ≠ 0\nthis : ↑n₁ * (↑n₂ * (↑d₂)⁻¹ * (↑d₁)⁻¹) = ↑n₁ * ((↑d₁)⁻¹ * (↑n₂ * (↑d₂)⁻¹))\n⊢ ↑↑d₂ ≠ 0\n\ncase b0\nF : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\nd₂0' : ↑d₂ ≠ 0\nthis : ↑n₁ * (↑n₂ * (↑d₂)⁻¹ * (↑d₁)⁻¹) = ↑n₁ * ((↑d₁)⁻¹ * (↑n₂ * (↑d₂)⁻¹))\n⊢ ↑↑d₁ ≠ 0\n\ncase b0\nF : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\nd₂0' : ↑d₂ ≠ 0\nthis : ↑n₁ * (↑n₂ * (↑d₂)⁻¹ * (↑d₁)⁻¹) = ↑n₁ * ((↑d₁)⁻¹ * (↑n₂ * (↑d₂)⁻¹))\n⊢ ↑(↑d₁ * ↑d₂) ≠ 0",
"tactic": "all_goals simp [d₁0, d₂0]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\nd₂0' : ↑d₂ ≠ 0\nthis : ↑n₁ * (↑n₂ * (↑d₂)⁻¹ * (↑d₁)⁻¹) = ↑n₁ * ((↑d₁)⁻¹ * (↑n₂ * (↑d₂)⁻¹))\n⊢ ↑(n₁ * n₂) / ↑(↑d₁ * ↑d₂) = ↑n₁ / ↑↑d₁ * (↑n₂ / ↑↑d₂)",
"tactic": "simpa [division_def, mul_inv_rev, d₁0, d₂0, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "case b0\nF : Type ?u.23959\nι : Type ?u.23962\nα : Type u_1\nβ : Type ?u.23968\ninst✝ : DivisionRing α\nn₁ : ℤ\nd₁ : ℕ\nh₁ : d₁ ≠ 0\nc₁ : Nat.coprime (Int.natAbs n₁) d₁\nn₂ : ℤ\nd₂ : ℕ\nh₂ : d₂ ≠ 0\nc₂ : Nat.coprime (Int.natAbs n₂) d₂\nd₁0 : ↑d₁ ≠ 0\nd₂0 : ↑d₂ ≠ 0\nd₁0' : ↑d₁ ≠ 0\nd₂0' : ↑d₂ ≠ 0\nthis : ↑n₁ * (↑n₂ * (↑d₂)⁻¹ * (↑d₁)⁻¹) = ↑n₁ * ((↑d₁)⁻¹ * (↑n₂ * (↑d₂)⁻¹))\n⊢ ↑(↑d₁ * ↑d₂) ≠ 0",
"tactic": "simp [d₁0, d₂0]"
}
] |
[
150,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean
|
ContDiffBump.nonneg'
|
[] |
[
396,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.row_mul_col_apply
|
[] |
[
2729,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2727,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Real.lean
|
cauchySeq_of_summable_dist
|
[] |
[
72,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.bddAbove_range
|
[
{
"state_after": "case intro\nα β ι : Type u\nf : ι → Cardinal\ni : ι\n⊢ f i ≤ ?m.80685 f",
"state_before": "α β ι : Type u\nf : ι → Cardinal\n⊢ ?m.80685 f ∈ upperBounds (range f)",
"tactic": "rintro a ⟨i, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα β ι : Type u\nf : ι → Cardinal\ni : ι\n⊢ f i ≤ ?m.80685 f",
"tactic": "exact le_sum.{v,u} f i"
}
] |
[
943,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
939,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.vsub_subset_iff
|
[] |
[
677,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
676,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.one_add_omega
|
[
{
"state_after": "α : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ 1 + ω ≤ ω",
"state_before": "α : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ 1 + ω = ω",
"tactic": "refine' le_antisymm _ (le_add_left _ _)"
},
{
"state_after": "α : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ type (Sum.Lex EmptyRelation fun x x_1 => x < x_1) ≤ type fun x x_1 => x < x_1",
"state_before": "α : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ 1 + ω ≤ ω",
"tactic": "rw [omega, ← lift_one.{_, 0}, ← lift_add, lift_le, ← type_unit, ← type_sum_lex]"
},
{
"state_after": "case refine'_1\nα : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ Unit ⊕ ℕ → ℕ\n\ncase refine'_2\nα : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ ∀ (a b : Unit ⊕ ℕ), Sum.Lex EmptyRelation (fun x x_1 => x < x_1) a b → ?refine'_1 a < ?refine'_1 b",
"state_before": "α : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ type (Sum.Lex EmptyRelation fun x x_1 => x < x_1) ≤ type fun x x_1 => x < x_1",
"tactic": "refine' ⟨RelEmbedding.collapse (RelEmbedding.ofMonotone _ _)⟩"
},
{
"state_after": "case refine'_1.inl\nα : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ Unit → ℕ\n\ncase refine'_1.inr\nα : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ ℕ → ℕ",
"state_before": "case refine'_1\nα : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ Unit ⊕ ℕ → ℕ",
"tactic": "apply Sum.rec"
},
{
"state_after": "case refine'_1.inr\nα : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ ℕ → ℕ",
"state_before": "case refine'_1.inl\nα : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ Unit → ℕ\n\ncase refine'_1.inr\nα : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ ℕ → ℕ",
"tactic": "exact fun _ => 0"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.inr\nα : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ ℕ → ℕ",
"tactic": "exact Nat.succ"
},
{
"state_after": "case refine'_2\nα : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Unit ⊕ ℕ\n⊢ Sum.Lex EmptyRelation (fun x x_1 => x < x_1) a b → Sum.rec (fun x => 0) Nat.succ a < Sum.rec (fun x => 0) Nat.succ b",
"state_before": "case refine'_2\nα : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ ∀ (a b : Unit ⊕ ℕ),\n Sum.Lex EmptyRelation (fun x x_1 => x < x_1) a b → Sum.rec (fun x => 0) Nat.succ a < Sum.rec (fun x => 0) Nat.succ b",
"tactic": "intro a b"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type ?u.134018\nβ : Type ?u.134021\nγ : Type ?u.134024\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Unit ⊕ ℕ\n⊢ Sum.Lex EmptyRelation (fun x x_1 => x < x_1) a b → Sum.rec (fun x => 0) Nat.succ a < Sum.rec (fun x => 0) Nat.succ b",
"tactic": "cases a <;> cases b <;> intro H <;> cases' H with _ _ H _ _ H <;>\n [exact H.elim; exact Nat.succ_pos _; exact Nat.succ_lt_succ H]"
}
] |
[
627,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
618,
1
] |
Mathlib/RingTheory/FinitePresentation.lean
|
Algebra.FinitePresentation.polynomial
|
[] |
[
124,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
LinearMap.SeparatingLeft.toMatrix₂'
|
[] |
[
725,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
723,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
LinearMap.dualMap_bijective_iff
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝⁵ : Field K\nV₁ : Type v'\nV₂ : Type v''\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K V₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V₂\nf : V₁ →ₗ[K] V₂\n⊢ Function.Bijective ↑(dualMap f) ↔ Function.Bijective ↑f",
"tactic": "simp_rw [Function.Bijective, dualMap_surjective_iff, dualMap_injective_iff, and_comm]"
}
] |
[
1515,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1513,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_add_le_of_degree_le
|
[] |
[
643,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
641,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.image_neg_Ioi
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ Neg.neg '' Ioi a = Iio (-a)",
"tactic": "simp"
}
] |
[
297,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/Order/Lattice.lean
|
Monotone.of_map_sup
|
[] |
[
1114,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1112,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
MeasureTheory.IntegrableOn.mono
|
[] |
[
127,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
|
CategoryTheory.Limits.IsZero.iff_isSplitEpi_eq_zero
|
[
{
"state_after": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitEpi f\n⊢ 𝟙 Y = 0 ↔ f = 0",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitEpi f\n⊢ IsZero Y ↔ f = 0",
"tactic": "rw [iff_id_eq_zero]"
},
{
"state_after": "case mp\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitEpi f\n⊢ 𝟙 Y = 0 → f = 0\n\ncase mpr\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitEpi f\n⊢ f = 0 → 𝟙 Y = 0",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitEpi f\n⊢ 𝟙 Y = 0 ↔ f = 0",
"tactic": "constructor"
},
{
"state_after": "case mp\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitEpi f\nh : 𝟙 Y = 0\n⊢ f = 0",
"state_before": "case mp\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitEpi f\n⊢ 𝟙 Y = 0 → f = 0",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mp\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitEpi f\nh : 𝟙 Y = 0\n⊢ f = 0",
"tactic": "rw [← Category.comp_id f, h, comp_zero]"
},
{
"state_after": "case mpr\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitEpi f\nh : f = 0\n⊢ 𝟙 Y = 0",
"state_before": "case mpr\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitEpi f\n⊢ f = 0 → 𝟙 Y = 0",
"tactic": "intro h"
},
{
"state_after": "case mpr\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitEpi f\nh : f = 0\n⊢ section_ f ≫ f = 0",
"state_before": "case mpr\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitEpi f\nh : f = 0\n⊢ 𝟙 Y = 0",
"tactic": "rw [← IsSplitEpi.id f]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitEpi f\nh : f = 0\n⊢ section_ f ≫ f = 0",
"tactic": "simp [h]"
}
] |
[
232,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.sup_limZero
|
[
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nf g : CauSeq α abs\nhf : LimZero f\nhg : LimZero g\nε : α\nε0 : ε > 0\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → abs (↑f j) < ε ∧ abs (↑g j) < ε\nj : ℕ\nij : j ≥ i\nH₁ : abs (↑f j) < ε\nH₂ : abs (↑g j) < ε\n⊢ abs (↑(f ⊔ g) j) < ε",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nf g : CauSeq α abs\nhf : LimZero f\nhg : LimZero g\nε : α\nε0 : ε > 0\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → abs (↑f j) < ε ∧ abs (↑g j) < ε\nj : ℕ\nij : j ≥ i\n⊢ abs (↑(f ⊔ g) j) < ε",
"tactic": "let ⟨H₁, H₂⟩ := H _ ij"
},
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nf g : CauSeq α abs\nhf : LimZero f\nhg : LimZero g\nε : α\nε0 : ε > 0\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → abs (↑f j) < ε ∧ abs (↑g j) < ε\nj : ℕ\nij : j ≥ i\nH₁ : -ε < ↑f j ∧ ↑f j < ε\nH₂ : -ε < ↑g j ∧ ↑g j < ε\n⊢ -ε < ↑(f ⊔ g) j ∧ ↑(f ⊔ g) j < ε",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nf g : CauSeq α abs\nhf : LimZero f\nhg : LimZero g\nε : α\nε0 : ε > 0\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → abs (↑f j) < ε ∧ abs (↑g j) < ε\nj : ℕ\nij : j ≥ i\nH₁ : abs (↑f j) < ε\nH₂ : abs (↑g j) < ε\n⊢ abs (↑(f ⊔ g) j) < ε",
"tactic": "rw [abs_lt] at H₁ H₂⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nf g : CauSeq α abs\nhf : LimZero f\nhg : LimZero g\nε : α\nε0 : ε > 0\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → abs (↑f j) < ε ∧ abs (↑g j) < ε\nj : ℕ\nij : j ≥ i\nH₁ : -ε < ↑f j ∧ ↑f j < ε\nH₂ : -ε < ↑g j ∧ ↑g j < ε\n⊢ -ε < ↑(f ⊔ g) j ∧ ↑(f ⊔ g) j < ε",
"tactic": "exact ⟨lt_sup_iff.mpr (Or.inl H₁.1), sup_lt_iff.mpr ⟨H₁.2, H₂.2⟩⟩"
}
] |
[
853,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
848,
1
] |
Mathlib/Order/Partition/Finpartition.lean
|
Finpartition.card_parts_le_card
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP✝ P : Finpartition s\n⊢ card P.parts ≤ card ⊥.parts",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP✝ P : Finpartition s\n⊢ card P.parts ≤ card s",
"tactic": "rw [← card_bot s]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP✝ P : Finpartition s\n⊢ card P.parts ≤ card ⊥.parts",
"tactic": "exact card_mono bot_le"
}
] |
[
518,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
516,
1
] |
Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
|
ModuleCat.Free.associativity
|
[
{
"state_after": "R : Type u\ninst✝ : CommRing R\nX Y Z : Type u\n⊢ ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom =\n (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom",
"state_before": "R : Type u\ninst✝ : CommRing R\nX Y Z : Type u\n⊢ ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom =\n (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom",
"tactic": "intros"
},
{
"state_after": "case H\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\n⊢ LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom) =\n LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)",
"state_before": "R : Type u\ninst✝ : CommRing R\nX Y Z : Type u\n⊢ ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom =\n (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom",
"tactic": "apply TensorProduct.ext"
},
{
"state_after": "case H.H\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\n⊢ LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)) =\n LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom))",
"state_before": "case H\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\n⊢ LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom) =\n LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)",
"tactic": "apply TensorProduct.ext"
},
{
"state_after": "case H.H.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\n⊢ ∀ (a : X),\n LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle a) =\n LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle a)",
"state_before": "case H.H\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\n⊢ LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)) =\n LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom))",
"tactic": "apply Finsupp.lhom_ext'"
},
{
"state_after": "case H.H.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\n⊢ LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x) =\n LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x)",
"state_before": "case H.H.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\n⊢ ∀ (a : X),\n LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle a) =\n LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle a)",
"tactic": "intro x"
},
{
"state_after": "case H.H.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\n⊢ ↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1 =\n ↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1",
"state_before": "case H.H.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\n⊢ LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x) =\n LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x)",
"tactic": "apply LinearMap.ext_ring"
},
{
"state_after": "case H.H.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\n⊢ ∀ (a : Y),\n LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle a) =\n LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle a)",
"state_before": "case H.H.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\n⊢ ↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1 =\n ↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1",
"tactic": "apply Finsupp.lhom_ext'"
},
{
"state_after": "case H.H.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\n⊢ LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y) =\n LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y)",
"state_before": "case H.H.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\n⊢ ∀ (a : Y),\n LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle a) =\n LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle a)",
"tactic": "intro y"
},
{
"state_after": "case H.H.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\n⊢ ↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1 =\n ↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1",
"state_before": "case H.H.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\n⊢ LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y) =\n LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y)",
"tactic": "apply LinearMap.ext_ring"
},
{
"state_after": "case H.H.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\n⊢ ∀ (a : Z),\n LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle a) =\n LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle a)",
"state_before": "case H.H.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\n⊢ ↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1 =\n ↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1",
"tactic": "apply Finsupp.lhom_ext'"
},
{
"state_after": "case H.H.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\nz : Z\n⊢ LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z) =\n LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z)",
"state_before": "case H.H.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\n⊢ ∀ (a : Z),\n LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle a) =\n LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle a)",
"tactic": "intro z"
},
{
"state_after": "case H.H.h.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\nz : Z\n⊢ ↑(LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z))\n 1 =\n ↑(LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z))\n 1",
"state_before": "case H.H.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\nz : Z\n⊢ LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z) =\n LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z)",
"tactic": "apply LinearMap.ext_ring"
},
{
"state_after": "case H.H.h.h.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\nz : Z\n⊢ ∀ (a : X ⊗ Y ⊗ Z),\n ↑(↑(LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂\n (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z))\n 1)\n a =\n ↑(↑(LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂\n (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z))\n 1)\n a",
"state_before": "case H.H.h.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\nz : Z\n⊢ ↑(LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z))\n 1 =\n ↑(LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z))\n 1",
"tactic": "apply Finsupp.ext"
},
{
"state_after": "case H.H.h.h.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\nz : Z\na : X ⊗ Y ⊗ Z\n⊢ ↑(↑(LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z))\n 1)\n a =\n ↑(↑(LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z))\n 1)\n a",
"state_before": "case H.H.h.h.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\nz : Z\n⊢ ∀ (a : X ⊗ Y ⊗ Z),\n ↑(↑(LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂\n (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z))\n 1)\n a =\n ↑(↑(LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂\n (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z))\n 1)\n a",
"tactic": "intro a"
},
{
"state_after": "case H.H.h.h.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\nz : Z\na : X ⊗ Y ⊗ Z\n⊢ ↑(Finsupp.mapDomain (α_ X Y Z).hom\n (↑(finsuppTensorFinsupp' R (X ⊗ Y) Z)\n (↑(finsuppTensorFinsupp' R X Y) (Finsupp.single x 1 ⊗ₜ[R] Finsupp.single y 1) ⊗ₜ[R] Finsupp.single z 1)))\n a =\n ↑(↑(finsuppTensorFinsupp' R X (Y ⊗ Z))\n (Finsupp.single x 1 ⊗ₜ[R] ↑(finsuppTensorFinsupp' R Y Z) (Finsupp.single y 1 ⊗ₜ[R] Finsupp.single z 1)))\n a",
"state_before": "case H.H.h.h.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\nz : Z\na : X ⊗ Y ⊗ Z\n⊢ ↑(↑(LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫\n (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z))\n 1)\n a =\n ↑(↑(LinearMap.comp\n (↑(LinearMap.comp\n (↑(LinearMap.comp\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))\n (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))\n ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫\n (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)))\n (Finsupp.lsingle x))\n 1)\n (Finsupp.lsingle y))\n 1)\n (Finsupp.lsingle z))\n 1)\n a",
"tactic": "change Finsupp.mapDomain (α_ X Y Z).hom (finsuppTensorFinsupp' R (X ⊗ Y) Z\n (finsuppTensorFinsupp' R X Y\n (Finsupp.single x 1 ⊗ₜ[R] Finsupp.single y 1) ⊗ₜ[R] Finsupp.single z 1)) a =\n finsuppTensorFinsupp' R X (Y ⊗ Z)\n (Finsupp.single x 1 ⊗ₜ[R]\n finsuppTensorFinsupp' R Y Z (Finsupp.single y 1 ⊗ₜ[R] Finsupp.single z 1)) a"
},
{
"state_after": "no goals",
"state_before": "case H.H.h.h.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nx : X\ny : Y\nz : Z\na : X ⊗ Y ⊗ Z\n⊢ ↑(Finsupp.mapDomain (α_ X Y Z).hom\n (↑(finsuppTensorFinsupp' R (X ⊗ Y) Z)\n (↑(finsuppTensorFinsupp' R X Y) (Finsupp.single x 1 ⊗ₜ[R] Finsupp.single y 1) ⊗ₜ[R] Finsupp.single z 1)))\n a =\n ↑(↑(finsuppTensorFinsupp' R X (Y ⊗ Z))\n (Finsupp.single x 1 ⊗ₜ[R] ↑(finsuppTensorFinsupp' R Y Z) (Finsupp.single y 1 ⊗ₜ[R] Finsupp.single z 1)))\n a",
"tactic": "simp_rw [finsuppTensorFinsupp'_single_tmul_single, Finsupp.mapDomain_single, mul_one,\n CategoryTheory.associator_hom_apply]"
}
] |
[
182,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.eq_pow_of_pow_eq_one
|
[
{
"state_after": "case intro\nM : Type ?u.3458904\nN : Type ?u.3458907\nG : Type ?u.3458910\nR : Type u_1\nS : Type ?u.3458916\nF : Type ?u.3458919\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nk : ℕ\nξ : R\nhξ : ξ ^ k = 1\nh0 : 0 < k\nζ : Rˣ\nh : IsPrimitiveRoot (↑ζ) k\n⊢ ∃ i, i < k ∧ ↑ζ ^ i = ξ",
"state_before": "M : Type ?u.3458904\nN : Type ?u.3458907\nG : Type ?u.3458910\nR : Type u_1\nS : Type ?u.3458916\nF : Type ?u.3458919\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nk : ℕ\nζ ξ : R\nh : IsPrimitiveRoot ζ k\nhξ : ξ ^ k = 1\nh0 : 0 < k\n⊢ ∃ i, i < k ∧ ζ ^ i = ξ",
"tactic": "lift ζ to Rˣ using h.isUnit h0"
},
{
"state_after": "case intro.intro\nM : Type ?u.3458904\nN : Type ?u.3458907\nG : Type ?u.3458910\nR : Type u_1\nS : Type ?u.3458916\nF : Type ?u.3458919\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nk : ℕ\nh0 : 0 < k\nζ : Rˣ\nh : IsPrimitiveRoot (↑ζ) k\nξ : Rˣ\nhξ : ↑ξ ^ k = 1\n⊢ ∃ i, i < k ∧ ↑ζ ^ i = ↑ξ",
"state_before": "case intro\nM : Type ?u.3458904\nN : Type ?u.3458907\nG : Type ?u.3458910\nR : Type u_1\nS : Type ?u.3458916\nF : Type ?u.3458919\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nk : ℕ\nξ : R\nhξ : ξ ^ k = 1\nh0 : 0 < k\nζ : Rˣ\nh : IsPrimitiveRoot (↑ζ) k\n⊢ ∃ i, i < k ∧ ↑ζ ^ i = ξ",
"tactic": "lift ξ to Rˣ using isUnit_ofPowEqOne hξ h0.ne'"
},
{
"state_after": "case intro.intro.intro\nM : Type ?u.3458904\nN : Type ?u.3458907\nG : Type ?u.3458910\nR : Type u_1\nS : Type ?u.3458916\nF : Type ?u.3458919\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nζ ξ : Rˣ\nk : ℕ+\nh : IsPrimitiveRoot ↑ζ ↑k\nhξ : ↑ξ ^ ↑k = 1\n⊢ ∃ i, i < ↑k ∧ ↑ζ ^ i = ↑ξ",
"state_before": "case intro.intro\nM : Type ?u.3458904\nN : Type ?u.3458907\nG : Type ?u.3458910\nR : Type u_1\nS : Type ?u.3458916\nF : Type ?u.3458919\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nk : ℕ\nh0 : 0 < k\nζ : Rˣ\nh : IsPrimitiveRoot (↑ζ) k\nξ : Rˣ\nhξ : ↑ξ ^ k = 1\n⊢ ∃ i, i < k ∧ ↑ζ ^ i = ↑ξ",
"tactic": "lift k to ℕ+ using h0"
},
{
"state_after": "case intro.intro.intro\nM : Type ?u.3458904\nN : Type ?u.3458907\nG : Type ?u.3458910\nR : Type u_1\nS : Type ?u.3458916\nF : Type ?u.3458919\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nζ ξ : Rˣ\nk : ℕ+\nh : IsPrimitiveRoot ↑ζ ↑k\nhξ : ↑ξ ^ ↑k = 1\n⊢ ∃ i, i < ↑k ∧ ζ ^ i = ξ",
"state_before": "case intro.intro.intro\nM : Type ?u.3458904\nN : Type ?u.3458907\nG : Type ?u.3458910\nR : Type u_1\nS : Type ?u.3458916\nF : Type ?u.3458919\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nζ ξ : Rˣ\nk : ℕ+\nh : IsPrimitiveRoot ↑ζ ↑k\nhξ : ↑ξ ^ ↑k = 1\n⊢ ∃ i, i < ↑k ∧ ↑ζ ^ i = ↑ξ",
"tactic": "simp only [← Units.val_pow_eq_pow_val, ← Units.ext_iff]"
},
{
"state_after": "case intro.intro.intro\nM : Type ?u.3458904\nN : Type ?u.3458907\nG : Type ?u.3458910\nR : Type u_1\nS : Type ?u.3458916\nF : Type ?u.3458919\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nζ ξ : Rˣ\nk : ℕ+\nh : IsPrimitiveRoot ζ ↑k\nhξ : ↑ξ ^ ↑k = 1\n⊢ ∃ i, i < ↑k ∧ ζ ^ i = ξ",
"state_before": "case intro.intro.intro\nM : Type ?u.3458904\nN : Type ?u.3458907\nG : Type ?u.3458910\nR : Type u_1\nS : Type ?u.3458916\nF : Type ?u.3458919\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nζ ξ : Rˣ\nk : ℕ+\nh : IsPrimitiveRoot ↑ζ ↑k\nhξ : ↑ξ ^ ↑k = 1\n⊢ ∃ i, i < ↑k ∧ ζ ^ i = ξ",
"tactic": "rw [coe_units_iff] at h"
},
{
"state_after": "case intro.intro.intro\nM : Type ?u.3458904\nN : Type ?u.3458907\nG : Type ?u.3458910\nR : Type u_1\nS : Type ?u.3458916\nF : Type ?u.3458919\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nζ ξ : Rˣ\nk : ℕ+\nh : IsPrimitiveRoot ζ ↑k\nhξ : ↑ξ ^ ↑k = 1\n⊢ ξ ∈ rootsOfUnity k R",
"state_before": "case intro.intro.intro\nM : Type ?u.3458904\nN : Type ?u.3458907\nG : Type ?u.3458910\nR : Type u_1\nS : Type ?u.3458916\nF : Type ?u.3458919\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nζ ξ : Rˣ\nk : ℕ+\nh : IsPrimitiveRoot ζ ↑k\nhξ : ↑ξ ^ ↑k = 1\n⊢ ∃ i, i < ↑k ∧ ζ ^ i = ξ",
"tactic": "apply h.eq_pow_of_mem_rootsOfUnity"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nM : Type ?u.3458904\nN : Type ?u.3458907\nG : Type ?u.3458910\nR : Type u_1\nS : Type ?u.3458916\nF : Type ?u.3458919\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nζ ξ : Rˣ\nk : ℕ+\nh : IsPrimitiveRoot ζ ↑k\nhξ : ↑ξ ^ ↑k = 1\n⊢ ξ ∈ rootsOfUnity k R",
"tactic": "rw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, hξ, Units.val_one]"
}
] |
[
778,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
770,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.sInf_apply
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nm : Set (OuterMeasure α)\ns : Set α\nh : Set.Nonempty m\n⊢ ↑(sInf m) s = ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨅ (μ : OuterMeasure α) (_ : μ ∈ m), ↑μ (t n)",
"tactic": "simp_rw [sInf_eq_boundedBy_sInfGen, boundedBy_apply, iSup_sInfGen_nonempty h]"
}
] |
[
1175,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1172,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.tan_add_pi
|
[] |
[
805,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
804,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.WalkingPair.swap_apply_left
|
[] |
[
66,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean
|
PadicInt.valuation_p
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ valuation ↑p = 1",
"tactic": "simp [valuation]"
}
] |
[
394,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
394,
1
] |
Mathlib/Data/Nat/Bitwise.lean
|
Nat.land'_comm
|
[] |
[
181,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Data/Real/NNReal.lean
|
Set.OrdConnected.preimage_real_toNNReal
|
[] |
[
1042,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1041,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.infs_assoc
|
[] |
[
374,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.frequently_bot
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.169823\nι : Sort x\np : α → Prop\n⊢ ¬∃ᶠ (x : α) in ⊥, p x",
"tactic": "simp"
}
] |
[
1386,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1386,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
HasDerivAt.continuousOn
|
[] |
[
728,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
727,
11
] |
Mathlib/Data/Finset/Slice.lean
|
Finset.subset_powersetLen_univ_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nι : Sort ?u.3644\nκ : ι → Sort ?u.3649\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\ns : Finset α\nr : ℕ\nA : Finset α\n⊢ A ∈ 𝒜 → A ∈ powersetLen r univ ↔ A ∈ ↑𝒜 → card A = r",
"tactic": "rw [mem_powerset_len_univ_iff, mem_coe]"
}
] |
[
110,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/Analysis/Calculus/LHopital.lean
|
deriv.lhopital_zero_atBot_on_Iio
|
[
{
"state_after": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf✝ : DifferentiableOn ℝ f (Iio a)\nhg' : ∀ (x : ℝ), x ∈ Iio a → deriv g x ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l\nhdf : ∀ (x : ℝ), x ∈ Iio a → DifferentiableAt ℝ f x\n⊢ Tendsto (fun x => f x / g x) atBot l",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : DifferentiableOn ℝ f (Iio a)\nhg' : ∀ (x : ℝ), x ∈ Iio a → deriv g x ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l\n⊢ Tendsto (fun x => f x / g x) atBot l",
"tactic": "have hdf : ∀ x ∈ Iio a, DifferentiableAt ℝ f x := fun x hx =>\n (hdf x hx).differentiableAt (Iio_mem_nhds hx)"
},
{
"state_after": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf✝ : DifferentiableOn ℝ f (Iio a)\nhg' : ∀ (x : ℝ), x ∈ Iio a → deriv g x ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l\nhdf : ∀ (x : ℝ), x ∈ Iio a → DifferentiableAt ℝ f x\nhdg : ∀ (x : ℝ), x ∈ Iio a → DifferentiableAt ℝ g x\n⊢ Tendsto (fun x => f x / g x) atBot l",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf✝ : DifferentiableOn ℝ f (Iio a)\nhg' : ∀ (x : ℝ), x ∈ Iio a → deriv g x ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l\nhdf : ∀ (x : ℝ), x ∈ Iio a → DifferentiableAt ℝ f x\n⊢ Tendsto (fun x => f x / g x) atBot l",
"tactic": "have hdg : ∀ x ∈ Iio a, DifferentiableAt ℝ g x := fun x hx =>\n by_contradiction fun h => hg' x hx (deriv_zero_of_not_differentiableAt h)"
},
{
"state_after": "no goals",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf✝ : DifferentiableOn ℝ f (Iio a)\nhg' : ∀ (x : ℝ), x ∈ Iio a → deriv g x ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l\nhdf : ∀ (x : ℝ), x ∈ Iio a → DifferentiableAt ℝ f x\nhdg : ∀ (x : ℝ), x ∈ Iio a → DifferentiableAt ℝ g x\n⊢ Tendsto (fun x => f x / g x) atBot l",
"tactic": "exact HasDerivAt.lhopital_zero_atBot_on_Iio (fun x hx => (hdf x hx).hasDerivAt)\n (fun x hx => (hdg x hx).hasDerivAt) hg' hfbot hgbot hdiv"
}
] |
[
269,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/GroupTheory/Subsemigroup/Basic.lean
|
Subsemigroup.closure_induction
|
[] |
[
359,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Mathlib/Algebra/Star/Order.lean
|
star_mul_self_nonneg
|
[] |
[
157,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean
|
Real.logb_neg_of_base_lt_one
|
[] |
[
302,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.inter_self
|
[] |
[
912,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
911,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
Submodule.toConvexCone_bot
|
[] |
[
533,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
532,
1
] |
Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
|
Matrix.SpecialLinearGroup.det_ne_zero
|
[
{
"state_after": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nA B : SpecialLinearGroup n R\ninst✝ : Nontrivial R\ng : SpecialLinearGroup n R\n⊢ 1 ≠ 0",
"state_before": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nA B : SpecialLinearGroup n R\ninst✝ : Nontrivial R\ng : SpecialLinearGroup n R\n⊢ det ↑g ≠ 0",
"tactic": "rw [g.det_coe]"
},
{
"state_after": "no goals",
"state_before": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nA B : SpecialLinearGroup n R\ninst✝ : Nontrivial R\ng : SpecialLinearGroup n R\n⊢ 1 ≠ 0",
"tactic": "norm_num"
}
] |
[
166,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Order/JordanHolder.lean
|
CompositionSeries.bot_eraseTop
|
[] |
[
411,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
inf_eq_bot_of_coprime
|
[
{
"state_after": "G✝ : Type u\nA : Type v\nx✝ y : G✝\na b : A\nn m : ℕ\ninst✝⁵ : Group G✝\ninst✝⁴ : AddGroup A\ninst✝³ : Fintype G✝\nG : Type u_1\ninst✝² : Group G\nH K : Subgroup G\ninst✝¹ : Fintype { x // x ∈ H }\ninst✝ : Fintype { x // x ∈ K }\nh : coprime (Fintype.card { x // x ∈ H }) (Fintype.card { x // x ∈ K })\nx : G\nhx : x ∈ H ⊓ K\n⊢ x = 1",
"state_before": "G✝ : Type u\nA : Type v\nx y : G✝\na b : A\nn m : ℕ\ninst✝⁵ : Group G✝\ninst✝⁴ : AddGroup A\ninst✝³ : Fintype G✝\nG : Type u_1\ninst✝² : Group G\nH K : Subgroup G\ninst✝¹ : Fintype { x // x ∈ H }\ninst✝ : Fintype { x // x ∈ K }\nh : coprime (Fintype.card { x // x ∈ H }) (Fintype.card { x // x ∈ K })\n⊢ H ⊓ K = ⊥",
"tactic": "refine' (H ⊓ K).eq_bot_iff_forall.mpr fun x hx => _"
},
{
"state_after": "G✝ : Type u\nA : Type v\nx✝ y : G✝\na b : A\nn m : ℕ\ninst✝⁵ : Group G✝\ninst✝⁴ : AddGroup A\ninst✝³ : Fintype G✝\nG : Type u_1\ninst✝² : Group G\nH K : Subgroup G\ninst✝¹ : Fintype { x // x ∈ H }\ninst✝ : Fintype { x // x ∈ K }\nh : coprime (Fintype.card { x // x ∈ H }) (Fintype.card { x // x ∈ K })\nx : G\nhx : x ∈ H ⊓ K\n⊢ orderOf x ∣ Fintype.card { x // x ∈ H } ∧ orderOf x ∣ Fintype.card { x // x ∈ K }",
"state_before": "G✝ : Type u\nA : Type v\nx✝ y : G✝\na b : A\nn m : ℕ\ninst✝⁵ : Group G✝\ninst✝⁴ : AddGroup A\ninst✝³ : Fintype G✝\nG : Type u_1\ninst✝² : Group G\nH K : Subgroup G\ninst✝¹ : Fintype { x // x ∈ H }\ninst✝ : Fintype { x // x ∈ K }\nh : coprime (Fintype.card { x // x ∈ H }) (Fintype.card { x // x ∈ K })\nx : G\nhx : x ∈ H ⊓ K\n⊢ x = 1",
"tactic": "rw [← orderOf_eq_one_iff, ← Nat.dvd_one, ← h.gcd_eq_one, Nat.dvd_gcd_iff]"
}
] |
[
1004,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
998,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.finite_subsets
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nα : Type u\na : Set α\nh : Set.Finite a\ns : Set α\n⊢ s ∈ Finset.map Finset.coeEmb.toEmbedding (Finset.powerset (Finite.toFinset h)) ↔ s ∈ {b | b ⊆ a}",
"tactic": "simpa [← @exists_finite_iff_finset α fun t => t ⊆ a ∧ t = s, Finite.subset_toFinset, ←\n and_assoc, Finset.coeEmb] using h.subset"
}
] |
[
1000,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
997,
1
] |
Mathlib/Topology/Instances/AddCircle.lean
|
AddCircle.coe_image_Ioc_eq
|
[
{
"state_after": "𝕜 : Type u_1\nB : Type ?u.105763\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\n⊢ (range fun x => ↑↑x) = univ",
"state_before": "𝕜 : Type u_1\nB : Type ?u.105763\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\n⊢ QuotientAddGroup.mk '' Ioc a (a + p) = univ",
"tactic": "rw [image_eq_range]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nB : Type ?u.105763\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\np : 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\n⊢ (range fun x => ↑↑x) = univ",
"tactic": "exact (equivIoc p a).symm.range_eq_univ"
}
] |
[
308,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/CategoryTheory/Functor/EpiMono.lean
|
CategoryTheory.Functor.mono_map_iff_mono
|
[
{
"state_after": "case mp\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\nhF₁ : PreservesMonomorphisms F\nhF₂ : ReflectsMonomorphisms F\n⊢ Mono (F.map f) → Mono f\n\ncase mpr\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\nhF₁ : PreservesMonomorphisms F\nhF₂ : ReflectsMonomorphisms F\n⊢ Mono f → Mono (F.map f)",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\nhF₁ : PreservesMonomorphisms F\nhF₂ : ReflectsMonomorphisms F\n⊢ Mono (F.map f) ↔ Mono f",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case mp\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\nhF₁ : PreservesMonomorphisms F\nhF₂ : ReflectsMonomorphisms F\n⊢ Mono (F.map f) → Mono f",
"tactic": "exact F.mono_of_mono_map"
},
{
"state_after": "case mpr\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\nhF₁ : PreservesMonomorphisms F\nhF₂ : ReflectsMonomorphisms F\nh : Mono f\n⊢ Mono (F.map f)",
"state_before": "case mpr\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\nhF₁ : PreservesMonomorphisms F\nhF₂ : ReflectsMonomorphisms F\n⊢ Mono f → Mono (F.map f)",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mpr\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\nhF₁ : PreservesMonomorphisms F\nhF₂ : ReflectsMonomorphisms F\nh : Mono f\n⊢ Mono (F.map f)",
"tactic": "exact F.map_mono f"
}
] |
[
293,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean
|
MulChar.inv_apply_eq_inv
|
[] |
[
339,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/FieldTheory/Finiteness.lean
|
IsNoetherian.finite_basis_index
|
[] |
[
71,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/CategoryTheory/EssentiallySmall.lean
|
CategoryTheory.essentiallySmall_iff_of_thin
|
[
{
"state_after": "no goals",
"state_before": "C✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : Quiver.IsThin C\n⊢ EssentiallySmall C ↔ Small (Skeleton C)",
"tactic": "simp [essentiallySmall_iff, CategoryTheory.locallySmall_of_thin]"
}
] |
[
238,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/Algebra/Group/WithOne/Basic.lean
|
WithOne.map_map
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Mul α\ninst✝¹ : Mul β\ninst✝ : Mul γ\nf : α →ₙ* β\ng : β →ₙ* γ\nx : WithOne α\n⊢ ↑(map g) (↑(map f) x) = ↑(map (MulHom.comp g f)) x",
"tactic": "induction x using WithOne.cases_on <;> rfl"
}
] |
[
124,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.neg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\n⊢ (multiplicity a (-b)).Dom ↔ (multiplicity a b).Dom",
"tactic": "simp only [multiplicity, PartENat.find, dvd_neg]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nh₁ : (multiplicity a (-b)).Dom\nh₂ : (multiplicity a b).Dom\n⊢ multiplicity a (-b) = ↑(Part.get (multiplicity a b) h₂)",
"state_before": "α : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nh₁ : (multiplicity a (-b)).Dom\nh₂ : (multiplicity a b).Dom\n⊢ ↑(Part.get (multiplicity a (-b)) h₁) = ↑(Part.get (multiplicity a b) h₂)",
"tactic": "rw [PartENat.natCast_get]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nh₁ : (multiplicity a (-b)).Dom\nh₂ : (multiplicity a b).Dom\n⊢ multiplicity a (-b) = ↑(Part.get (multiplicity a b) h₂)",
"tactic": "exact Eq.symm\n (unique (pow_multiplicity_dvd _).neg_right\n (mt dvd_neg.1 (is_greatest' _ (lt_succ_self _))))"
}
] |
[
428,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
422,
11
] |
Mathlib/Order/Circular.lean
|
btw_refl_right
|
[] |
[
311,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
310,
1
] |
Mathlib/RingTheory/Ideal/LocalRing.lean
|
LocalRing.ResidueField.lift_comp_residue
|
[] |
[
397,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
395,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.Nonempty.of_vsub_right
|
[] |
[
645,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
644,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.card_congr
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nh₁ : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nh₂ : ∀ (a b : α) (ha : a ∈ s) (hb : b ∈ s), f a ha = f b hb → a = b\nh₃ : ∀ (b : β), b ∈ t → ∃ a ha, f a ha = b\n⊢ card s = card t",
"tactic": "classical calc\n s.card = s.attach.card := card_attach.symm\n _ = (s.attach.image fun a : { a // a ∈ s } => f a.1 a.2).card :=\n Eq.symm ((card_image_of_injective _) fun a b h => Subtype.eq <| h₂ _ _ _ _ h)\n _ = t.card :=\n congr_arg card\n (Finset.ext fun b =>\n ⟨fun h =>\n let ⟨a, _, ha₂⟩ := mem_image.1 h\n ha₂ ▸ h₁ _ _,\n fun h =>\n let ⟨a, ha₁, ha₂⟩ := h₃ b h\n mem_image.2 ⟨⟨a, ha₁⟩, by simp [ha₂]⟩⟩)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nh₁ : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nh₂ : ∀ (a b : α) (ha : a ∈ s) (hb : b ∈ s), f a ha = f b hb → a = b\nh₃ : ∀ (b : β), b ∈ t → ∃ a ha, f a ha = b\n⊢ card s = card t",
"tactic": "calc\ns.card = s.attach.card := card_attach.symm\n_ = (s.attach.image fun a : { a // a ∈ s } => f a.1 a.2).card :=\nEq.symm ((card_image_of_injective _) fun a b h => Subtype.eq <| h₂ _ _ _ _ h)\n_ = t.card :=\ncongr_arg card\n(Finset.ext fun b =>\n⟨fun h =>\n let ⟨a, _, ha₂⟩ := mem_image.1 h\n ha₂ ▸ h₁ _ _,\n fun h =>\n let ⟨a, ha₁, ha₂⟩ := h₃ b h\n mem_image.2 ⟨⟨a, ha₁⟩, by simp [ha₂]⟩⟩)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : (a : α) → a ∈ s → β\nh₁ : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nh₂ : ∀ (a b : α) (ha : a ∈ s) (hb : b ∈ s), f a ha = f b hb → a = b\nh₃ : ∀ (b : β), b ∈ t → ∃ a ha, f a ha = b\nb : β\nh : b ∈ t\na : α\nha₁ : a ∈ s\nha₂ : f a ha₁ = b\n⊢ { val := a, property := ha₁ } ∈ attach s ∧\n f ↑{ val := a, property := ha₁ } (_ : ↑{ val := a, property := ha₁ } ∈ s) = b",
"tactic": "simp [ha₂]"
}
] |
[
327,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
312,
1
] |
Mathlib/Topology/Algebra/Module/Multilinear.lean
|
ContinuousMultilinearMap.sum_apply
|
[] |
[
226,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.limit.isoLimitCone_hom_π
|
[
{
"state_after": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u\ninst✝¹ : Category C\nF✝ F : J ⥤ C\ninst✝ : HasLimit F\nt : LimitCone F\nj : J\n⊢ IsLimit.lift t.isLimit (cone F) ≫ t.cone.π.app j = π F j",
"state_before": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u\ninst✝¹ : Category C\nF✝ F : J ⥤ C\ninst✝ : HasLimit F\nt : LimitCone F\nj : J\n⊢ (isoLimitCone t).hom ≫ t.cone.π.app j = π F j",
"tactic": "dsimp [limit.isoLimitCone, IsLimit.conePointUniqueUpToIso]"
},
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u\ninst✝¹ : Category C\nF✝ F : J ⥤ C\ninst✝ : HasLimit F\nt : LimitCone F\nj : J\n⊢ IsLimit.lift t.isLimit (cone F) ≫ t.cone.π.app j = π F j",
"tactic": "aesop_cat"
}
] |
[
257,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Data/List/Basic.lean
|
List.head?_append
|
[
{
"state_after": "case nil\nι : Type ?u.42760\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ t : List α\nx : α\nh : x ∈ head? []\n⊢ x ∈ head? ([] ++ t)\n\ncase cons\nι : Type ?u.42760\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ t : List α\nx head✝ : α\ntail✝ : List α\nh : x ∈ head? (head✝ :: tail✝)\n⊢ x ∈ head? (head✝ :: tail✝ ++ t)",
"state_before": "ι : Type ?u.42760\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ s t : List α\nx : α\nh : x ∈ head? s\n⊢ x ∈ head? (s ++ t)",
"tactic": "cases s"
},
{
"state_after": "case cons\nι : Type ?u.42760\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ t : List α\nx head✝ : α\ntail✝ : List α\nh : x ∈ head? (head✝ :: tail✝)\n⊢ x ∈ head? (head✝ :: tail✝ ++ t)",
"state_before": "case nil\nι : Type ?u.42760\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ t : List α\nx : α\nh : x ∈ head? []\n⊢ x ∈ head? ([] ++ t)\n\ncase cons\nι : Type ?u.42760\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ t : List α\nx head✝ : α\ntail✝ : List α\nh : x ∈ head? (head✝ :: tail✝)\n⊢ x ∈ head? (head✝ :: tail✝ ++ t)",
"tactic": "contradiction"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type ?u.42760\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ t : List α\nx head✝ : α\ntail✝ : List α\nh : x ∈ head? (head✝ :: tail✝)\n⊢ x ∈ head? (head✝ :: tail✝ ++ t)",
"tactic": "exact h"
}
] |
[
898,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
897,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.rpow_one
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ x ^ 1 = x",
"tactic": "simp [rpow_def]"
}
] |
[
129,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/GroupTheory/Coset.lean
|
QuotientGroup.rightRel_r_eq_rightCosetEquivalence
|
[
{
"state_after": "case h.h.a\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx✝¹ x✝ : α\n⊢ Setoid.r x✝¹ x✝ ↔ RightCosetEquivalence (↑s) x✝¹ x✝",
"state_before": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\n⊢ Setoid.r = RightCosetEquivalence ↑s",
"tactic": "ext"
},
{
"state_after": "case h.h.a\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx✝¹ x✝ : α\n⊢ (fun x y => y * x⁻¹ ∈ s) x✝¹ x✝ ↔ RightCosetEquivalence (↑s) x✝¹ x✝",
"state_before": "case h.h.a\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx✝¹ x✝ : α\n⊢ Setoid.r x✝¹ x✝ ↔ RightCosetEquivalence (↑s) x✝¹ x✝",
"tactic": "rw [rightRel_eq]"
},
{
"state_after": "no goals",
"state_before": "case h.h.a\nα : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx✝¹ x✝ : α\n⊢ (fun x y => y * x⁻¹ ∈ s) x✝¹ x✝ ↔ RightCosetEquivalence (↑s) x✝¹ x✝",
"tactic": "exact (rightCoset_eq_iff s).symm"
}
] |
[
399,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
395,
1
] |
Mathlib/Algebra/Order/SMul.lean
|
smul_nonpos_of_nonneg_of_nonpos
|
[] |
[
111,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Init/Data/Nat/Bitwise.lean
|
Nat.ldiff'_bit
|
[] |
[
483,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
482,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderEmbedding.isWellOrder
|
[] |
[
682,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
681,
11
] |
Mathlib/CategoryTheory/Sites/Sieves.lean
|
CategoryTheory.Sieve.pullback_top
|
[] |
[
469,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
468,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiag'_sub
|
[] |
[
906,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
904,
1
] |
Mathlib/Order/Filter/Cofinite.lean
|
Set.Finite.eventually_cofinite_nmem
|
[] |
[
81,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.nfp_lt_ord
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.35734\nr : α → α → Prop\nf : Ordinal → Ordinal\nc : Ordinal\nhc : ℵ₀ < cof c\nhf : ∀ (i : Ordinal), i < c → f i < c\na : Ordinal\n⊢ Cardinal.lift (#Unit) < cof c",
"tactic": "simpa using Cardinal.one_lt_aleph0.trans hc"
}
] |
[
411,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
409,
1
] |
Mathlib/Algebra/GradedMonoid.lean
|
GradedMonoid.GMonoid.gnpowRec_zero
|
[] |
[
162,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_neg_atTop_iff
|
[] |
[
859,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
858,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.inv_map_iInf
|
[] |
[
516,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
himp_le_himp
|
[] |
[
423,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
422,
1
] |
Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean
|
IsConformalMap.injective
|
[
{
"state_after": "case intro.intro.intro\nR : Type u_1\nM : Type ?u.238550\nN : Type u_3\nG : Type ?u.238556\nM' : Type u_2\ninst✝⁸ : NormedField R\ninst✝⁷ : SeminormedAddCommGroup M\ninst✝⁶ : SeminormedAddCommGroup N\ninst✝⁵ : SeminormedAddCommGroup G\ninst✝⁴ : NormedSpace R M\ninst✝³ : NormedSpace R N\ninst✝² : NormedSpace R G\ninst✝¹ : NormedAddCommGroup M'\ninst✝ : NormedSpace R M'\nf : M →L[R] N\ng : N →L[R] G\nc✝ c : R\nhc : c ≠ 0\nli : M' →ₗᵢ[R] N\n⊢ Injective ↑(c • toContinuousLinearMap li)",
"state_before": "R : Type u_1\nM : Type ?u.238550\nN : Type u_3\nG : Type ?u.238556\nM' : Type u_2\ninst✝⁸ : NormedField R\ninst✝⁷ : SeminormedAddCommGroup M\ninst✝⁶ : SeminormedAddCommGroup N\ninst✝⁵ : SeminormedAddCommGroup G\ninst✝⁴ : NormedSpace R M\ninst✝³ : NormedSpace R N\ninst✝² : NormedSpace R G\ninst✝¹ : NormedAddCommGroup M'\ninst✝ : NormedSpace R M'\nf✝ : M →L[R] N\ng : N →L[R] G\nc : R\nf : M' →L[R] N\nh : IsConformalMap f\n⊢ Injective ↑f",
"tactic": "rcases h with ⟨c, hc, li, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nR : Type u_1\nM : Type ?u.238550\nN : Type u_3\nG : Type ?u.238556\nM' : Type u_2\ninst✝⁸ : NormedField R\ninst✝⁷ : SeminormedAddCommGroup M\ninst✝⁶ : SeminormedAddCommGroup N\ninst✝⁵ : SeminormedAddCommGroup G\ninst✝⁴ : NormedSpace R M\ninst✝³ : NormedSpace R N\ninst✝² : NormedSpace R G\ninst✝¹ : NormedAddCommGroup M'\ninst✝ : NormedSpace R M'\nf : M →L[R] N\ng : N →L[R] G\nc✝ c : R\nhc : c ≠ 0\nli : M' →ₗᵢ[R] N\n⊢ Injective ↑(c • toContinuousLinearMap li)",
"tactic": "exact (smul_right_injective _ hc).comp li.injective"
}
] |
[
97,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
11
] |
Mathlib/RingTheory/IsTensorProduct.lean
|
IsBaseChange.equiv_tmul
|
[] |
[
255,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/NumberTheory/Padics/Hensel.lean
|
newton_seq_succ_dist
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nhnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (↑Polynomial.derivative F)‖ ^ 2\nhnsol : Polynomial.eval a F ≠ 0\nn : ℕ\n⊢ ‖Polynomial.eval (newton_seq n) F‖ / ‖Polynomial.eval (newton_seq n) (↑Polynomial.derivative F)‖ =\n ‖Polynomial.eval (newton_seq n) F‖ / ‖Polynomial.eval a (↑Polynomial.derivative F)‖",
"tactic": "rw [newton_seq_deriv_norm]"
}
] |
[
276,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
9
] |
Mathlib/GroupTheory/PGroup.lean
|
IsPGroup.comap_of_injective
|
[] |
[
307,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.two_zsmul_eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ 2 • θ = 0 ↔ θ = 0 ∨ θ = ↑π",
"tactic": "simp_rw [two_zsmul, ← two_nsmul, two_nsmul_eq_zero_iff]"
}
] |
[
207,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Data/Int/Order/Lemmas.lean
|
Int.natAbs_lt_iff_mul_self_lt
|
[
{
"state_after": "a✝ b✝ : ℤ\nn : ℕ\na b : ℤ\n⊢ natAbs a < natAbs b ↔ ↑(natAbs a) < ↑(natAbs b)",
"state_before": "a✝ b✝ : ℤ\nn : ℕ\na b : ℤ\n⊢ natAbs a < natAbs b ↔ a * a < b * b",
"tactic": "rw [← abs_lt_iff_mul_self_lt, abs_eq_natAbs, abs_eq_natAbs]"
},
{
"state_after": "no goals",
"state_before": "a✝ b✝ : ℤ\nn : ℕ\na b : ℤ\n⊢ natAbs a < natAbs b ↔ ↑(natAbs a) < ↑(natAbs b)",
"tactic": "exact Int.ofNat_lt.symm"
}
] |
[
42,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.castLT_mk
|
[] |
[
1008,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1007,
1
] |
Mathlib/Data/PFunctor/Univariate/Basic.lean
|
PFunctor.fst_map
|
[
{
"state_after": "case mk\nP : PFunctor\nα✝ β✝ α β : Type u\nf : α → β\nfst✝ : P.A\nsnd✝ : B P fst✝ → α\n⊢ (f <$> { fst := fst✝, snd := snd✝ }).fst = { fst := fst✝, snd := snd✝ }.fst",
"state_before": "P : PFunctor\nα✝ β✝ α β : Type u\nx : Obj P α\nf : α → β\n⊢ (f <$> x).fst = x.fst",
"tactic": "cases x"
},
{
"state_after": "no goals",
"state_before": "case mk\nP : PFunctor\nα✝ β✝ α β : Type u\nf : α → β\nfst✝ : P.A\nsnd✝ : B P fst✝ → α\n⊢ (f <$> { fst := fst✝, snd := snd✝ }).fst = { fst := fst✝, snd := snd✝ }.fst",
"tactic": "rfl"
}
] |
[
143,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Analysis/Normed/Group/HomCompletion.lean
|
NormedAddGroupHom.completion_comp
|
[
{
"state_after": "case H\nG : Type u_1\ninst✝² : SeminormedAddCommGroup G\nH : Type u_2\ninst✝¹ : SeminormedAddCommGroup H\nK : Type u_3\ninst✝ : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\ng : NormedAddGroupHom H K\nx : Completion G\n⊢ ↑(NormedAddGroupHom.comp (completion g) (completion f)) x = ↑(completion (NormedAddGroupHom.comp g f)) x",
"state_before": "G : Type u_1\ninst✝² : SeminormedAddCommGroup G\nH : Type u_2\ninst✝¹ : SeminormedAddCommGroup H\nK : Type u_3\ninst✝ : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\ng : NormedAddGroupHom H K\n⊢ NormedAddGroupHom.comp (completion g) (completion f) = completion (NormedAddGroupHom.comp g f)",
"tactic": "ext x"
},
{
"state_after": "case H\nG : Type u_1\ninst✝² : SeminormedAddCommGroup G\nH : Type u_2\ninst✝¹ : SeminormedAddCommGroup H\nK : Type u_3\ninst✝ : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\ng : NormedAddGroupHom H K\nx : Completion G\n⊢ Completion.map (↑g ∘ ↑f) x = Completion.map (↑(NormedAddGroupHom.comp g f)) x",
"state_before": "case H\nG : Type u_1\ninst✝² : SeminormedAddCommGroup G\nH : Type u_2\ninst✝¹ : SeminormedAddCommGroup H\nK : Type u_3\ninst✝ : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\ng : NormedAddGroupHom H K\nx : Completion G\n⊢ ↑(NormedAddGroupHom.comp (completion g) (completion f)) x = ↑(completion (NormedAddGroupHom.comp g f)) x",
"tactic": "rw [NormedAddGroupHom.coe_comp, NormedAddGroupHom.completion_def,\n NormedAddGroupHom.completion_coe_to_fun, NormedAddGroupHom.completion_coe_to_fun,\n Completion.map_comp g.uniformContinuous f.uniformContinuous]"
},
{
"state_after": "no goals",
"state_before": "case H\nG : Type u_1\ninst✝² : SeminormedAddCommGroup G\nH : Type u_2\ninst✝¹ : SeminormedAddCommGroup H\nK : Type u_3\ninst✝ : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\ng : NormedAddGroupHom H K\nx : Completion G\n⊢ Completion.map (↑g ∘ ↑f) x = Completion.map (↑(NormedAddGroupHom.comp g f)) x",
"tactic": "rfl"
}
] |
[
116,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
lt_of_tsub_lt_tsub_right_of_le
|
[
{
"state_after": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nh : c ≤ b\nh2 : a - c < b - c\n⊢ a ≠ b",
"state_before": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nh : c ≤ b\nh2 : a - c < b - c\n⊢ a < b",
"tactic": "refine' ((tsub_le_tsub_iff_right h).mp h2.le).lt_of_ne _"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na c d : α\nh : c ≤ a\nh2 : a - c < a - c\n⊢ False",
"state_before": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nh : c ≤ b\nh2 : a - c < b - c\n⊢ a ≠ b",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na c d : α\nh : c ≤ a\nh2 : a - c < a - c\n⊢ False",
"tactic": "exact h2.false"
}
] |
[
62,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/MeasureTheory/Lattice.lean
|
AEMeasurable.sup_const
|
[] |
[
133,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/GroupTheory/Perm/Sign.lean
|
Equiv.Perm.sign_extendDomain
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝⁴ : DecidableEq α\ninst✝³ : Fintype α\ninst✝² : DecidableEq β\ninst✝¹ : Fintype β\ne : Perm α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\n⊢ ↑sign (extendDomain e f) = ↑sign e",
"tactic": "simp only [Equiv.Perm.extendDomain, sign_subtypeCongr, sign_permCongr, sign_refl, mul_one]"
}
] |
[
794,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
792,
1
] |
Mathlib/Analysis/Convex/Extreme.lean
|
mem_extremePoints_iff_extreme_singleton
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.5166\nι : Type ?u.5169\nπ : ι → Type ?u.5174\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx : E\n⊢ x ∈ extremePoints 𝕜 A → IsExtreme 𝕜 A {x}",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.5166\nι : Type ?u.5169\nπ : ι → Type ?u.5174\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx : E\n⊢ x ∈ extremePoints 𝕜 A ↔ IsExtreme 𝕜 A {x}",
"tactic": "refine' ⟨_, fun hx ↦ ⟨singleton_subset_iff.1 hx.1, fun x₁ hx₁ x₂ hx₂ ↦ hx.2 hx₁ hx₂ rfl⟩⟩"
},
{
"state_after": "case intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.5166\nι : Type ?u.5169\nπ : ι → Type ?u.5174\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx : E\nhxA : x ∈ A\nhAx : ∀ ⦃x₁ : E⦄, x₁ ∈ A → ∀ ⦃x₂ : E⦄, x₂ ∈ A → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x\n⊢ IsExtreme 𝕜 A {x}",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.5166\nι : Type ?u.5169\nπ : ι → Type ?u.5174\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx : E\n⊢ x ∈ extremePoints 𝕜 A → IsExtreme 𝕜 A {x}",
"tactic": "rintro ⟨hxA, hAx⟩"
},
{
"state_after": "case intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.5166\nι : Type ?u.5169\nπ : ι → Type ?u.5174\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx : E\nhxA : x ∈ A\nhAx : ∀ ⦃x₁ : E⦄, x₁ ∈ A → ∀ ⦃x₂ : E⦄, x₂ ∈ A → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x\n⊢ ∀ ⦃x₁ : E⦄, x₁ ∈ A → ∀ ⦃x₂ : E⦄, x₂ ∈ A → ∀ ⦃x_1 : E⦄, x_1 ∈ {x} → x_1 ∈ openSegment 𝕜 x₁ x₂ → x₁ ∈ {x} ∧ x₂ ∈ {x}",
"state_before": "case intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.5166\nι : Type ?u.5169\nπ : ι → Type ?u.5174\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx : E\nhxA : x ∈ A\nhAx : ∀ ⦃x₁ : E⦄, x₁ ∈ A → ∀ ⦃x₂ : E⦄, x₂ ∈ A → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x\n⊢ IsExtreme 𝕜 A {x}",
"tactic": "use singleton_subset_iff.2 hxA"
},
{
"state_after": "case intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.5166\nι : Type ?u.5169\nπ : ι → Type ?u.5174\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx₁ : E\nhx₁A : x₁ ∈ A\nx₂ : E\nhx₂A : x₂ ∈ A\ny : E\nhxA : y ∈ A\nhAx : ∀ ⦃x₁ : E⦄, x₁ ∈ A → ∀ ⦃x₂ : E⦄, x₂ ∈ A → y ∈ openSegment 𝕜 x₁ x₂ → x₁ = y ∧ x₂ = y\n⊢ y ∈ openSegment 𝕜 x₁ x₂ → x₁ ∈ {y} ∧ x₂ ∈ {y}",
"state_before": "case intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.5166\nι : Type ?u.5169\nπ : ι → Type ?u.5174\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx : E\nhxA : x ∈ A\nhAx : ∀ ⦃x₁ : E⦄, x₁ ∈ A → ∀ ⦃x₂ : E⦄, x₂ ∈ A → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x\n⊢ ∀ ⦃x₁ : E⦄, x₁ ∈ A → ∀ ⦃x₂ : E⦄, x₂ ∈ A → ∀ ⦃x_1 : E⦄, x_1 ∈ {x} → x_1 ∈ openSegment 𝕜 x₁ x₂ → x₁ ∈ {x} ∧ x₂ ∈ {x}",
"tactic": "rintro x₁ hx₁A x₂ hx₂A y (rfl : y = x)"
},
{
"state_after": "no goals",
"state_before": "case intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.5166\nι : Type ?u.5169\nπ : ι → Type ?u.5174\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx₁ : E\nhx₁A : x₁ ∈ A\nx₂ : E\nhx₂A : x₂ ∈ A\ny : E\nhxA : y ∈ A\nhAx : ∀ ⦃x₁ : E⦄, x₁ ∈ A → ∀ ⦃x₂ : E⦄, x₂ ∈ A → y ∈ openSegment 𝕜 x₁ x₂ → x₁ = y ∧ x₂ = y\n⊢ y ∈ openSegment 𝕜 x₁ x₂ → x₁ ∈ {y} ∧ x₂ ∈ {y}",
"tactic": "exact hAx hx₁A hx₂A"
}
] |
[
148,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/MeasureTheory/Function/LpOrder.lean
|
MeasureTheory.Memℒp.sup
|
[] |
[
73,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Data/Polynomial/IntegralNormalization.lean
|
Polynomial.integralNormalization_support
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\nf : R[X]\na✝ : ℕ\n⊢ a✝ ∈ support (integralNormalization f) → a✝ ∈ support f",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\nf : R[X]\n⊢ support (integralNormalization f) ⊆ support f",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\nf : R[X]\na✝ : ℕ\n⊢ a✝ ∈ support (integralNormalization f) → a✝ ∈ support f",
"tactic": "simp (config := { contextual := true }) [integralNormalization, coeff_monomial, mem_support_iff]"
}
] |
[
62,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
nnnorm_prod_le
|
[
{
"state_after": "𝓕 : Type ?u.1102940\n𝕜 : Type ?u.1102943\nα : Type ?u.1102946\nι : Type u_1\nκ : Type ?u.1102952\nE : Type u_2\nF : Type ?u.1102958\nG : Type ?u.1102961\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ns : Finset ι\nf : ι → E\n⊢ ‖∏ a in s, f a‖ ≤ ∑ x in s, ‖f x‖",
"state_before": "𝓕 : Type ?u.1102940\n𝕜 : Type ?u.1102943\nα : Type ?u.1102946\nι : Type u_1\nκ : Type ?u.1102952\nE : Type u_2\nF : Type ?u.1102958\nG : Type ?u.1102961\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ns : Finset ι\nf : ι → E\n⊢ ↑‖∏ a in s, f a‖₊ ≤ ↑(∑ a in s, ‖f a‖₊)",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.1102940\n𝕜 : Type ?u.1102943\nα : Type ?u.1102946\nι : Type u_1\nκ : Type ?u.1102952\nE : Type u_2\nF : Type ?u.1102958\nG : Type ?u.1102961\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ns : Finset ι\nf : ι → E\n⊢ ‖∏ a in s, f a‖ ≤ ∑ x in s, ‖f x‖",
"tactic": "exact norm_prod_le _ _"
}
] |
[
1690,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1687,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Icc_subset_Ioo
|
[] |
[
472,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
self_eq_mul_right
|
[] |
[
190,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Data/Sym/Sym2.lean
|
Sym2.map_pair_eq
|
[] |
[
266,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
Matrix.toLinearMapₛₗ₂'_symm
|
[] |
[
241,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Mathlib/Data/Rat/Order.lean
|
Rat.num_pos_iff_pos
|
[
{
"state_after": "no goals",
"state_before": "a✝ b c a : ℚ\n⊢ a.num ≤ 0 ↔ a ≤ 0",
"tactic": "simpa [(by cases a; rfl : (-a).num = -a.num)] using @num_nonneg_iff_zero_le (-a)"
},
{
"state_after": "case mk'\na b c : ℚ\nnum✝ : ℤ\nden✝ : ℕ\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\n⊢ (-mk' num✝ den✝).num = -(mk' num✝ den✝).num",
"state_before": "a✝ b c a : ℚ\n⊢ (-a).num = -a.num",
"tactic": "cases a"
},
{
"state_after": "no goals",
"state_before": "case mk'\na b c : ℚ\nnum✝ : ℤ\nden✝ : ℕ\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs num✝) den✝\n⊢ (-mk' num✝ den✝).num = -(mk' num✝ den✝).num",
"tactic": "rfl"
}
] |
[
285,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/GroupTheory/Coset.lean
|
QuotientGroup.induction_on
|
[] |
[
479,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
478,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
eq_one_div_of_mul_eq_one_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.21993\nG : Type ?u.21996\ninst✝ : DivisionMonoid α\na b c : α\nh : b * a = 1\n⊢ b = 1 / a",
"tactic": "rw [eq_inv_of_mul_eq_one_left h, one_div]"
}
] |
[
379,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.mapRange_sub'
|
[] |
[
1267,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1264,
1
] |
Mathlib/Algebra/Group/WithOne/Defs.lean
|
WithZero.coe_pow
|
[] |
[
268,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
dist_self_div_left
|
[
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.640919\n𝕜 : Type ?u.640922\nα : Type ?u.640925\nι : Type ?u.640928\nκ : Type ?u.640931\nE : Type u_1\nF : Type ?u.640937\nG : Type ?u.640940\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na✝ a₁ a₂ b✝ b₁ b₂ : E\nr r₁ r₂ : ℝ\na b : E\n⊢ dist (a / b) a = ‖b‖",
"tactic": "rw [dist_comm, dist_self_div_right]"
}
] |
[
1423,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1422,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.eq_ones_iff_le_length
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\nc✝ c : Composition n\n⊢ c = ones n ↔ n ≤ length c",
"tactic": "simp [eq_ones_iff_length, le_antisymm_iff, c.length_le]"
}
] |
[
566,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
565,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.coe_inf
|
[] |
[
970,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
969,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.im_sq_le_normSq
|
[] |
[
668,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
667,
1
] |
Mathlib/Algebra/Module/Injective.lean
|
Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_wd
|
[
{
"state_after": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr r' : R\neq1 : r • y = r' • y\n⊢ ↑(extendIdealTo i f h y) (r - r') = 0",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr r' : R\neq1 : r • y = r' • y\n⊢ ↑(extendIdealTo i f h y) r = ↑(extendIdealTo i f h y) r'",
"tactic": "rw [← sub_eq_zero, ← map_sub]"
},
{
"state_after": "case convert_2\nR : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr r' : R\neq1 : r • y = r' • y\n⊢ (r - r') • y = 0",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr r' : R\neq1 : r • y = r' • y\n⊢ ↑(extendIdealTo i f h y) (r - r') = 0",
"tactic": "convert ExtensionOfMaxAdjoin.extendIdealTo_wd' i f h (r - r') _"
},
{
"state_after": "no goals",
"state_before": "case convert_2\nR : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\nh : Baer R Q\ny : N\nr r' : R\neq1 : r • y = r' • y\n⊢ (r - r') • y = 0",
"tactic": "rw [sub_smul, sub_eq_zero, eq1]"
}
] |
[
349,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
AffineIndependent.affineIndependent_of_not_mem_span
|
[
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\n⊢ ∀ (i : ι), i ∈ s → w i = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\n⊢ AffineIndependent k p",
"tactic": "intro s w hw hs"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\n⊢ ∀ (i : ι), i ∈ s → w i = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\n⊢ ∀ (i : ι), i ∈ s → w i = 0",
"tactic": "let p' : { y // y ≠ i } → P := fun x => p x"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\n⊢ ∀ (i : ι), i ∈ s → w i = 0\n\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬(i ∈ s ∧ w i ≠ 0)\n⊢ ∀ (i : ι), i ∈ s → w i = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\n⊢ ∀ (i : ι), i ∈ s → w i = 0",
"tactic": "by_cases his : i ∈ s ∧ w i ≠ 0"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\n⊢ p i ∈ affineSpan k (p '' {x | x ≠ i})",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\n⊢ ∀ (i : ι), i ∈ s → w i = 0",
"tactic": "refine' False.elim (hi _)"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\n⊢ p i ∈ affineSpan k (p '' {x | x ≠ i})",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\n⊢ p i ∈ affineSpan k (p '' {x | x ≠ i})",
"tactic": "let wm : ι → k := -(w i)⁻¹ • w"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\n⊢ p i ∈ affineSpan k (p '' {x | x ≠ i})",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\n⊢ p i ∈ affineSpan k (p '' {x | x ≠ i})",
"tactic": "have hms : s.weightedVSub p wm = (0 : V) := by simp [hs]"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwm : ∑ i in s, wm i = 0\n⊢ p i ∈ affineSpan k (p '' {x | x ≠ i})",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\n⊢ p i ∈ affineSpan k (p '' {x | x ≠ i})",
"tactic": "have hwm : (∑ i in s, wm i) = 0 := by simp [← Finset.mul_sum, hw]"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwm : ∑ i in s, wm i = 0\nhwmi : wm i = -1\n⊢ p i ∈ affineSpan k (p '' {x | x ≠ i})",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwm : ∑ i in s, wm i = 0\n⊢ p i ∈ affineSpan k (p '' {x | x ≠ i})",
"tactic": "have hwmi : wm i = -1 := by simp [his.2]"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwm : ∑ i in s, wm i = 0\nhwmi : wm i = -1\nw' : { y // y ≠ i } → k := fun x => wm ↑x\n⊢ p i ∈ affineSpan k (p '' {x | x ≠ i})",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwm : ∑ i in s, wm i = 0\nhwmi : wm i = -1\n⊢ p i ∈ affineSpan k (p '' {x | x ≠ i})",
"tactic": "let w' : { y // y ≠ i } → k := fun x => wm x"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwm : ∑ i in s, wm i = 0\nhwmi : wm i = -1\nw' : { y // y ≠ i } → k := fun x => wm ↑x\nhw' : ∑ x in s', w' x = 1\n⊢ (↑(Finset.affineCombination k (Finset.subtype (fun x => x ≠ i) s) fun i_1 => p ↑i_1) fun i_1 => wm ↑i_1) ∈\n affineSpan k (Set.range (p ∘ Subtype.val))",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwm : ∑ i in s, wm i = 0\nhwmi : wm i = -1\nw' : { y // y ≠ i } → k := fun x => wm ↑x\nhw' : ∑ x in s', w' x = 1\n⊢ p i ∈ affineSpan k (p '' {x | x ≠ i})",
"tactic": "rw [← s.affineCombination_eq_of_weightedVSub_eq_zero_of_eq_neg_one hms his.1 hwmi, ←\n (Subtype.range_coe : _ = { x | x ≠ i }), ← Set.range_comp, ←\n s.affineCombination_subtype_eq_filter]"
},
{
"state_after": "no goals",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwm : ∑ i in s, wm i = 0\nhwmi : wm i = -1\nw' : { y // y ≠ i } → k := fun x => wm ↑x\nhw' : ∑ x in s', w' x = 1\n⊢ (↑(Finset.affineCombination k (Finset.subtype (fun x => x ≠ i) s) fun i_1 => p ↑i_1) fun i_1 => wm ↑i_1) ∈\n affineSpan k (Set.range (p ∘ Subtype.val))",
"tactic": "exact affineCombination_mem_affineSpan hw' p'"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\n⊢ ↑(Finset.weightedVSub s p) wm = 0",
"tactic": "simp [hs]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\n⊢ ∑ i in s, wm i = 0",
"tactic": "simp [← Finset.mul_sum, hw]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwm : ∑ i in s, wm i = 0\n⊢ wm i = -1",
"tactic": "simp [his.2]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwm : ∑ i in s, wm i = 0\nhwmi : wm i = -1\nw' : { y // y ≠ i } → k := fun x => wm ↑x\n⊢ ∑ x in Finset.filter (fun x => x ≠ i) s, (-(w i)⁻¹ • w) x = 1",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwm : ∑ i in s, wm i = 0\nhwmi : wm i = -1\nw' : { y // y ≠ i } → k := fun x => wm ↑x\n⊢ ∑ x in s', w' x = 1",
"tactic": "simp_rw [Finset.sum_subtype_eq_sum_filter]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwmi : wm i = -1\nw' : { y // y ≠ i } → k := fun x => wm ↑x\nhwm :\n ∑ x in Finset.filter (fun x => x ≠ i) s, (-(w i)⁻¹ • w) x +\n ∑ x in Finset.filter (fun x => x = i) s, (-(w i)⁻¹ • w) x =\n 0\n⊢ ∑ x in Finset.filter (fun x => x ≠ i) s, (-(w i)⁻¹ • w) x = 1",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwm : ∑ x in Finset.filter (fun x => x ≠ i) s, wm x + ∑ x in Finset.filter (fun x => ¬x ≠ i) s, wm x = 0\nhwmi : wm i = -1\nw' : { y // y ≠ i } → k := fun x => wm ↑x\n⊢ ∑ x in Finset.filter (fun x => x ≠ i) s, (-(w i)⁻¹ • w) x = 1",
"tactic": "simp_rw [Classical.not_not] at hwm"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwmi : wm i = -1\nw' : { y // y ≠ i } → k := fun x => wm ↑x\nhwm : ∑ x in Finset.filter (fun x => x ≠ i) s, (-(w i)⁻¹ • w) x + ∑ x in if i ∈ s then {i} else ∅, (-(w i)⁻¹ • w) x = 0\n⊢ ∑ x in Finset.filter (fun x => x ≠ i) s, (-(w i)⁻¹ • w) x = 1",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwmi : wm i = -1\nw' : { y // y ≠ i } → k := fun x => wm ↑x\nhwm :\n ∑ x in Finset.filter (fun x => x ≠ i) s, (-(w i)⁻¹ • w) x +\n ∑ x in Finset.filter (fun x => x = i) s, (-(w i)⁻¹ • w) x =\n 0\n⊢ ∑ x in Finset.filter (fun x => x ≠ i) s, (-(w i)⁻¹ • w) x = 1",
"tactic": "erw [Finset.filter_eq'] at hwm"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwmi : wm i = -1\nw' : { y // y ≠ i } → k := fun x => wm ↑x\nhwm : ∑ x in Finset.filter (fun x => x ≠ i) s, (-(w i)⁻¹ • w) x = 1\n⊢ ∑ x in Finset.filter (fun x => x ≠ i) s, (-(w i)⁻¹ • w) x = 1",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwmi : wm i = -1\nw' : { y // y ≠ i } → k := fun x => wm ↑x\nhwm : ∑ x in Finset.filter (fun x => x ≠ i) s, (-(w i)⁻¹ • w) x + ∑ x in if i ∈ s then {i} else ∅, (-(w i)⁻¹ • w) x = 0\n⊢ ∑ x in Finset.filter (fun x => x ≠ i) s, (-(w i)⁻¹ • w) x = 1",
"tactic": "simp_rw [if_pos his.1, Finset.sum_singleton, hwmi, ← sub_eq_add_neg, sub_eq_zero] at hwm"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : i ∈ s ∧ w i ≠ 0\nwm : ι → k := -(w i)⁻¹ • w\nhms : ↑(Finset.weightedVSub s p) wm = 0\nhwmi : wm i = -1\nw' : { y // y ≠ i } → k := fun x => wm ↑x\nhwm : ∑ x in Finset.filter (fun x => x ≠ i) s, (-(w i)⁻¹ • w) x = 1\n⊢ ∑ x in Finset.filter (fun x => x ≠ i) s, (-(w i)⁻¹ • w) x = 1",
"tactic": "exact hwm"
},
{
"state_after": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\n⊢ ∀ (i : ι), i ∈ s → w i = 0",
"state_before": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬(i ∈ s ∧ w i ≠ 0)\n⊢ ∀ (i : ι), i ∈ s → w i = 0",
"tactic": "rw [not_and_or, Classical.not_not] at his"
},
{
"state_after": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\n⊢ ∀ (i : ι), i ∈ s → w i = 0",
"state_before": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\n⊢ ∀ (i : ι), i ∈ s → w i = 0",
"tactic": "let w' : { y // y ≠ i } → k := fun x => w x"
},
{
"state_after": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\n⊢ ∀ (i : ι), i ∈ s → w i = 0",
"state_before": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\n⊢ ∀ (i : ι), i ∈ s → w i = 0",
"tactic": "have hw' : (∑ x in s', w' x) = 0 := by\n simp_rw [Finset.sum_subtype_eq_sum_filter]\n rw [Finset.sum_filter_of_ne, hw]\n rintro x hxs hwx rfl\n exact hwx (his.neg_resolve_left hxs)"
},
{
"state_after": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\nhs' : ↑(Finset.weightedVSub s' p') w' = 0\n⊢ ∀ (i : ι), i ∈ s → w i = 0",
"state_before": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\n⊢ ∀ (i : ι), i ∈ s → w i = 0",
"tactic": "have hs' : s'.weightedVSub p' w' = (0 : V) := by\n simp_rw [Finset.weightedVSub_subtype_eq_filter]\n rw [Finset.weightedVSub_filter_of_ne, hs]\n rintro x hxs hwx rfl\n exact hwx (his.neg_resolve_left hxs)"
},
{
"state_after": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\nhs' : ↑(Finset.weightedVSub s' p') w' = 0\nj : ι\nhj : j ∈ s\n⊢ w j = 0",
"state_before": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\nhs' : ↑(Finset.weightedVSub s' p') w' = 0\n⊢ ∀ (i : ι), i ∈ s → w i = 0",
"tactic": "intro j hj"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\nhs' : ↑(Finset.weightedVSub s' p') w' = 0\nj : ι\nhj : j ∈ s\nhji : j = i\n⊢ w j = 0\n\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\nhs' : ↑(Finset.weightedVSub s' p') w' = 0\nj : ι\nhj : j ∈ s\nhji : ¬j = i\n⊢ w j = 0",
"state_before": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\nhs' : ↑(Finset.weightedVSub s' p') w' = 0\nj : ι\nhj : j ∈ s\n⊢ w j = 0",
"tactic": "by_cases hji : j = i"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\n⊢ ∑ x in Finset.filter (fun x => x ≠ i) s, w x = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\n⊢ ∑ x in s', w' x = 0",
"tactic": "simp_rw [Finset.sum_subtype_eq_sum_filter]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\n⊢ ∀ (x : ι), x ∈ s → w x ≠ 0 → x ≠ i",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\n⊢ ∑ x in Finset.filter (fun x => x ≠ i) s, w x = 0",
"tactic": "rw [Finset.sum_filter_of_ne, hw]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\nx : ι\nhxs : x ∈ s\nhwx : w x ≠ 0\nha : AffineIndependent k fun x_1 => p ↑x_1\nhi : ¬p x ∈ affineSpan k (p '' {x_1 | x_1 ≠ x})\ns' : Finset { y // y ≠ x } := Finset.subtype (fun x_1 => x_1 ≠ x) s\np' : { y // y ≠ x } → P := fun x_1 => p ↑x_1\nhis : ¬x ∈ s ∨ w x = 0\nw' : { y // y ≠ x } → k := fun x_1 => w ↑x_1\n⊢ False",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\n⊢ ∀ (x : ι), x ∈ s → w x ≠ 0 → x ≠ i",
"tactic": "rintro x hxs hwx rfl"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\nx : ι\nhxs : x ∈ s\nhwx : w x ≠ 0\nha : AffineIndependent k fun x_1 => p ↑x_1\nhi : ¬p x ∈ affineSpan k (p '' {x_1 | x_1 ≠ x})\ns' : Finset { y // y ≠ x } := Finset.subtype (fun x_1 => x_1 ≠ x) s\np' : { y // y ≠ x } → P := fun x_1 => p ↑x_1\nhis : ¬x ∈ s ∨ w x = 0\nw' : { y // y ≠ x } → k := fun x_1 => w ↑x_1\n⊢ False",
"tactic": "exact hwx (his.neg_resolve_left hxs)"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\n⊢ ↑(Finset.weightedVSub (Finset.filter (fun x => x ≠ i) s) p) w = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\n⊢ ↑(Finset.weightedVSub s' p') w' = 0",
"tactic": "simp_rw [Finset.weightedVSub_subtype_eq_filter]"
},
{
"state_after": "case h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\n⊢ ∀ (i_1 : ι), i_1 ∈ s → w i_1 ≠ 0 → i_1 ≠ i",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\n⊢ ↑(Finset.weightedVSub (Finset.filter (fun x => x ≠ i) s) p) w = 0",
"tactic": "rw [Finset.weightedVSub_filter_of_ne, hs]"
},
{
"state_after": "case h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\nx : ι\nhxs : x ∈ s\nhwx : w x ≠ 0\nha : AffineIndependent k fun x_1 => p ↑x_1\nhi : ¬p x ∈ affineSpan k (p '' {x_1 | x_1 ≠ x})\ns' : Finset { y // y ≠ x } := Finset.subtype (fun x_1 => x_1 ≠ x) s\np' : { y // y ≠ x } → P := fun x_1 => p ↑x_1\nhis : ¬x ∈ s ∨ w x = 0\nw' : { y // y ≠ x } → k := fun x_1 => w ↑x_1\nhw' : ∑ x in s', w' x = 0\n⊢ False",
"state_before": "case h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\n⊢ ∀ (i_1 : ι), i_1 ∈ s → w i_1 ≠ 0 → i_1 ≠ i",
"tactic": "rintro x hxs hwx rfl"
},
{
"state_after": "no goals",
"state_before": "case h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\nx : ι\nhxs : x ∈ s\nhwx : w x ≠ 0\nha : AffineIndependent k fun x_1 => p ↑x_1\nhi : ¬p x ∈ affineSpan k (p '' {x_1 | x_1 ≠ x})\ns' : Finset { y // y ≠ x } := Finset.subtype (fun x_1 => x_1 ≠ x) s\np' : { y // y ≠ x } → P := fun x_1 => p ↑x_1\nhis : ¬x ∈ s ∨ w x = 0\nw' : { y // y ≠ x } → k := fun x_1 => w ↑x_1\nhw' : ∑ x in s', w' x = 0\n⊢ False",
"tactic": "exact hwx (his.neg_resolve_left hxs)"
},
{
"state_after": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\nhs' : ↑(Finset.weightedVSub s' p') w' = 0\nj : ι\nhj : i ∈ s\nhji : j = i\n⊢ w j = 0",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\nhs' : ↑(Finset.weightedVSub s' p') w' = 0\nj : ι\nhj : j ∈ s\nhji : j = i\n⊢ w j = 0",
"tactic": "rw [hji] at hj"
},
{
"state_after": "no goals",
"state_before": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\nhs' : ↑(Finset.weightedVSub s' p') w' = 0\nj : ι\nhj : i ∈ s\nhji : j = i\n⊢ w j = 0",
"tactic": "exact hji.symm ▸ his.neg_resolve_left hj"
},
{
"state_after": "no goals",
"state_before": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x => p ↑x\nhi : ¬p i ∈ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 0\nhs : ↑(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y ≠ i } := Finset.subtype (fun x => x ≠ i) s\np' : { y // y ≠ i } → P := fun x => p ↑x\nhis : ¬i ∈ s ∨ w i = 0\nw' : { y // y ≠ i } → k := fun x => w ↑x\nhw' : ∑ x in s', w' x = 0\nhs' : ↑(Finset.weightedVSub s' p') w' = 0\nj : ι\nhj : j ∈ s\nhji : ¬j = i\n⊢ w j = 0",
"tactic": "exact ha s' w' hw' hs' ⟨j, hji⟩ (Finset.mem_subtype.2 hj)"
}
] |
[
675,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
633,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.zero_inv
|
[
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type ?u.1763981\nk : Type u_2\ninst✝ : Field k\n⊢ 0⁻¹ = 0",
"tactic": "rw [inv_eq_zero, constantCoeff_zero]"
}
] |
[
965,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
964,
1
] |
Mathlib/Order/OrdContinuous.lean
|
LeftOrdContinuous.map_ciSup
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : ConditionallyCompleteLattice α\ninst✝¹ : ConditionallyCompleteLattice β\ninst✝ : Nonempty ι\nf : α → β\nhf : LeftOrdContinuous f\ng : ι → α\nhg : BddAbove (range g)\n⊢ sSup (range (f ∘ g)) = sSup (range fun i => f (g i))",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : ConditionallyCompleteLattice α\ninst✝¹ : ConditionallyCompleteLattice β\ninst✝ : Nonempty ι\nf : α → β\nhf : LeftOrdContinuous f\ng : ι → α\nhg : BddAbove (range g)\n⊢ f (⨆ (i : ι), g i) = ⨆ (i : ι), f (g i)",
"tactic": "simp only [iSup, hf.map_csSup (range_nonempty _) hg, ← range_comp]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : ConditionallyCompleteLattice α\ninst✝¹ : ConditionallyCompleteLattice β\ninst✝ : Nonempty ι\nf : α → β\nhf : LeftOrdContinuous f\ng : ι → α\nhg : BddAbove (range g)\n⊢ sSup (range (f ∘ g)) = sSup (range fun i => f (g i))",
"tactic": "rfl"
}
] |
[
157,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
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