file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.eq_of_mem_uIcc_of_mem_uIcc
|
[
{
"state_after": "ι : Type ?u.190771\nα : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\n⊢ b ⊓ c ≤ a ∧ a ≤ b ⊔ c → a ⊓ c ≤ b ∧ b ≤ a ⊔ c → a = b",
"state_before": "ι : Type ?u.190771\nα : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\n⊢ a ∈ [[b, c]] → b ∈ [[a, c]] → a = b",
"tactic": "simp_rw [mem_uIcc]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.190771\nα : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\n⊢ b ⊓ c ≤ a ∧ a ≤ b ⊔ c → a ⊓ c ≤ b ∧ b ≤ a ⊔ c → a = b",
"tactic": "exact Set.eq_of_mem_uIcc_of_mem_uIcc"
}
] |
[
968,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
966,
1
] |
Mathlib/Data/List/Forall2.lean
|
List.rel_bind
|
[] |
[
278,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/NumberTheory/PythagoreanTriples.lean
|
coprime_sq_sub_mul_of_even_odd
|
[
{
"state_after": "m n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\n⊢ False",
"state_before": "m n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\n⊢ Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1",
"tactic": "by_contra H"
},
{
"state_after": "case intro.intro.intro\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : p ∣ Int.natAbs (m ^ 2 - n ^ 2)\nhp2 : p ∣ Int.natAbs (2 * m * n)\n⊢ False",
"state_before": "m n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\n⊢ False",
"tactic": "obtain ⟨p, hp, hp1, hp2⟩ := Nat.Prime.not_coprime_iff_dvd.mp H"
},
{
"state_after": "case intro.intro.intro\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\n⊢ False",
"state_before": "case intro.intro.intro\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : p ∣ Int.natAbs (m ^ 2 - n ^ 2)\nhp2 : p ∣ Int.natAbs (2 * m * n)\n⊢ False",
"tactic": "rw [← Int.coe_nat_dvd_left] at hp1 hp2"
},
{
"state_after": "case intro.intro.intro\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\n⊢ False",
"state_before": "case intro.intro.intro\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\n⊢ False",
"tactic": "have hnp : ¬(p : ℤ) ∣ Int.gcd m n := by\n rw [h]\n norm_cast\n exact mt Nat.dvd_one.mp (Nat.Prime.ne_one hp)"
},
{
"state_after": "case intro.intro.intro.inl\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs (2 * m)\n⊢ False\n\ncase intro.intro.intro.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhpn : p ∣ Int.natAbs n\n⊢ False",
"state_before": "case intro.intro.intro\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\n⊢ False",
"tactic": "cases' Int.Prime.dvd_mul hp hp2 with hp2m hpn"
},
{
"state_after": "case intro.intro.intro.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd n m)\nhpn : p ∣ Int.natAbs n\n⊢ False",
"state_before": "case intro.intro.intro.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhpn : p ∣ Int.natAbs n\n⊢ False",
"tactic": "rw [Int.gcd_comm] at hnp"
},
{
"state_after": "case intro.intro.intro.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd n m)\nhpn : p ∣ Int.natAbs n\n⊢ ↑p ∣ m",
"state_before": "case intro.intro.intro.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd n m)\nhpn : p ∣ Int.natAbs n\n⊢ False",
"tactic": "apply mt (Int.dvd_gcd (Int.coe_nat_dvd_left.mpr hpn)) hnp"
},
{
"state_after": "case intro.intro.intro.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd n m)\nhpn : p ∣ Int.natAbs n\n⊢ ↑p ∣ m ∨ ↑p ∣ m",
"state_before": "case intro.intro.intro.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd n m)\nhpn : p ∣ Int.natAbs n\n⊢ ↑p ∣ m",
"tactic": "apply (or_self_iff _).mp"
},
{
"state_after": "case intro.intro.intro.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd n m)\nhpn : p ∣ Int.natAbs n\n⊢ ↑p ∣ m * m",
"state_before": "case intro.intro.intro.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd n m)\nhpn : p ∣ Int.natAbs n\n⊢ ↑p ∣ m ∨ ↑p ∣ m",
"tactic": "apply Int.Prime.dvd_mul' hp"
},
{
"state_after": "case intro.intro.intro.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd n m)\nhpn : p ∣ Int.natAbs n\n⊢ ↑p ∣ m ^ 2 - n ^ 2 + n * n",
"state_before": "case intro.intro.intro.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd n m)\nhpn : p ∣ Int.natAbs n\n⊢ ↑p ∣ m * m",
"tactic": "rw [(by ring : m * m = m ^ 2 - n ^ 2 + n * n)]"
},
{
"state_after": "case intro.intro.intro.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd n m)\nhpn : p ∣ Int.natAbs n\n⊢ ↑p ∣ n * n",
"state_before": "case intro.intro.intro.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd n m)\nhpn : p ∣ Int.natAbs n\n⊢ ↑p ∣ m ^ 2 - n ^ 2 + n * n",
"tactic": "apply dvd_add hp1"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd n m)\nhpn : p ∣ Int.natAbs n\n⊢ ↑p ∣ n * n",
"tactic": "exact (Int.coe_nat_dvd_left.mpr hpn).mul_right n"
},
{
"state_after": "m n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\n⊢ ¬↑p ∣ ↑1",
"state_before": "m n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\n⊢ ¬↑p ∣ ↑(Int.gcd m n)",
"tactic": "rw [h]"
},
{
"state_after": "m n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\n⊢ ¬p ∣ 1",
"state_before": "m n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\n⊢ ¬↑p ∣ ↑1",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "m n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\n⊢ ¬p ∣ 1",
"tactic": "exact mt Nat.dvd_one.mp (Nat.Prime.ne_one hp)"
},
{
"state_after": "case intro.intro.intro.inl\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\n⊢ False",
"state_before": "case intro.intro.intro.inl\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs (2 * m)\n⊢ False",
"tactic": "rw [Int.natAbs_mul] at hp2m"
},
{
"state_after": "case intro.intro.intro.inl.inl\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2✝ : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhp2 : p ∣ Int.natAbs 2\n⊢ False\n\ncase intro.intro.intro.inl.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhpm : p ∣ Int.natAbs m\n⊢ False",
"state_before": "case intro.intro.intro.inl\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\n⊢ False",
"tactic": "cases' (Nat.Prime.dvd_mul hp).mp hp2m with hp2 hpm"
},
{
"state_after": "case intro.intro.intro.inl.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhpm : p ∣ Int.natAbs m\n⊢ ↑p ∣ n",
"state_before": "case intro.intro.intro.inl.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhpm : p ∣ Int.natAbs m\n⊢ False",
"tactic": "apply mt (Int.dvd_gcd (Int.coe_nat_dvd_left.mpr hpm)) hnp"
},
{
"state_after": "case intro.intro.intro.inl.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhpm : p ∣ Int.natAbs m\n⊢ ↑p ∣ n ∨ ↑p ∣ n",
"state_before": "case intro.intro.intro.inl.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhpm : p ∣ Int.natAbs m\n⊢ ↑p ∣ n",
"tactic": "apply (or_self_iff _).mp"
},
{
"state_after": "case intro.intro.intro.inl.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhpm : p ∣ Int.natAbs m\n⊢ ↑p ∣ n * n",
"state_before": "case intro.intro.intro.inl.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhpm : p ∣ Int.natAbs m\n⊢ ↑p ∣ n ∨ ↑p ∣ n",
"tactic": "apply Int.Prime.dvd_mul' hp"
},
{
"state_after": "case intro.intro.intro.inl.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhpm : p ∣ Int.natAbs m\n⊢ ↑p ∣ -(m ^ 2 - n ^ 2) + m * m",
"state_before": "case intro.intro.intro.inl.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhpm : p ∣ Int.natAbs m\n⊢ ↑p ∣ n * n",
"tactic": "rw [(by ring : n * n = -(m ^ 2 - n ^ 2) + m * m)]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.inl.inr\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhpm : p ∣ Int.natAbs m\n⊢ ↑p ∣ -(m ^ 2 - n ^ 2) + m * m",
"tactic": "exact hp1.neg_right.add ((Int.coe_nat_dvd_left.2 hpm).mul_right _)"
},
{
"state_after": "case intro.intro.intro.inl.inl\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2✝ : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhp2 : p ∣ Int.natAbs 2\nhp2' : p = 2\n⊢ False",
"state_before": "case intro.intro.intro.inl.inl\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2✝ : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhp2 : p ∣ Int.natAbs 2\n⊢ False",
"tactic": "have hp2' : p = 2 := (Nat.le_of_dvd zero_lt_two hp2).antisymm hp.two_le"
},
{
"state_after": "case intro.intro.intro.inl.inl\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp2✝ : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhp2 : p ∣ Int.natAbs 2\nhp2' : p = 2\n⊢ ↑p ∣ m ^ 2 - n ^ 2 → False",
"state_before": "case intro.intro.intro.inl.inl\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2✝ : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhp2 : p ∣ Int.natAbs 2\nhp2' : p = 2\n⊢ False",
"tactic": "revert hp1"
},
{
"state_after": "case intro.intro.intro.inl.inl\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp2✝ : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhp2 : p ∣ Int.natAbs 2\nhp2' : p = 2\n⊢ ↑2 ∣ m ^ 2 - n ^ 2 → False",
"state_before": "case intro.intro.intro.inl.inl\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp2✝ : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhp2 : p ∣ Int.natAbs 2\nhp2' : p = 2\n⊢ ↑p ∣ m ^ 2 - n ^ 2 → False",
"tactic": "rw [hp2']"
},
{
"state_after": "case intro.intro.intro.inl.inl\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp2✝ : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhp2 : p ∣ Int.natAbs 2\nhp2' : p = 2\n⊢ ¬(m ^ 2 - n ^ 2) % ↑2 = 0",
"state_before": "case intro.intro.intro.inl.inl\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp2✝ : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhp2 : p ∣ Int.natAbs 2\nhp2' : p = 2\n⊢ ↑2 ∣ m ^ 2 - n ^ 2 → False",
"tactic": "apply mt Int.emod_eq_zero_of_dvd"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.inl.inl\nm n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp2✝ : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhp2 : p ∣ Int.natAbs 2\nhp2' : p = 2\n⊢ ¬(m ^ 2 - n ^ 2) % ↑2 = 0",
"tactic": "field_simp [sq, Int.sub_emod, Int.mul_emod, hm, hn]"
},
{
"state_after": "no goals",
"state_before": "m n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd m n)\nhp2m : p ∣ Int.natAbs 2 * Int.natAbs m\nhpm : p ∣ Int.natAbs m\n⊢ n * n = -(m ^ 2 - n ^ 2) + m * m",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "m n : ℤ\nh : Int.gcd m n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬Int.gcd (m ^ 2 - n ^ 2) (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(Int.gcd n m)\nhpn : p ∣ Int.natAbs n\n⊢ m * m = m ^ 2 - n ^ 2 + n * n",
"tactic": "ring"
}
] |
[
390,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
9
] |
Mathlib/Analysis/LocallyConvex/WeakDual.lean
|
LinearMap.toSeminorm_apply
|
[] |
[
68,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.smul_mem_smul_set_iff₀
|
[] |
[
1014,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1013,
1
] |
Mathlib/Order/Heyting/Regular.lean
|
Heyting.IsRegular.eq
|
[] |
[
50,
5
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
11
] |
Mathlib/Data/Set/Basic.lean
|
Membership.mem.out
|
[] |
[
268,
4
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/Analysis/Normed/Field/Basic.lean
|
List.norm_prod_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.113892\nγ : Type ?u.113895\nι : Type ?u.113898\ninst✝¹ : SeminormedRing α\ninst✝ : NormOneClass α\n⊢ ‖prod []‖ ≤ prod (map norm [])",
"tactic": "simp"
}
] |
[
329,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
div_eq_div_mul_div
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.69162\nβ : Type ?u.69165\nG : Type u_1\ninst✝ : CommGroup G\na✝ b✝ c✝ d a b c : G\n⊢ a / b = c / b * (a / c)",
"tactic": "simp [mul_left_comm c]"
}
] |
[
937,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
937,
1
] |
Mathlib/Algebra/QuaternionBasis.lean
|
QuaternionAlgebra.Basis.lift_one
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nA : Type u_1\nB : Type ?u.70086\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nc₁ c₂ : R\nq : Basis A c₁ c₂\n⊢ lift q 1 = 1",
"tactic": "simp [lift]"
}
] |
[
121,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.restrict_le_self
|
[] |
[
1574,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1571,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.lintegral_coe_le_coe_iff_integral_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.1088155\nF : Type ?u.1088158\n𝕜 : Type ?u.1088161\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1090852\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ≥0\nhfi : Integrable fun x => ↑(f x)\nb : ℝ≥0\n⊢ (∫⁻ (a : α), ↑(f a) ∂μ) ≤ ↑b ↔ (∫ (a : α), ↑(f a) ∂μ) ≤ ↑b",
"tactic": "rw [lintegral_coe_eq_integral f hfi, ENNReal.ofReal, ENNReal.coe_le_coe,\n Real.toNNReal_le_iff_le_coe]"
}
] |
[
1198,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1195,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
LieSubalgebra.coe_incl
|
[] |
[
281,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Topology/Hom/Open.lean
|
ContinuousOpenMap.id_comp
|
[] |
[
154,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Analysis/Convex/Topology.lean
|
Convex.combo_interior_closure_mem_interior
|
[] |
[
160,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
SymmetricRel.eq
|
[] |
[
236,
5
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/GroupTheory/Congruence.lean
|
Con.rel_mk
|
[] |
[
168,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Data/Set/Sigma.lean
|
Set.sigma_mono
|
[] |
[
75,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Topology/FiberBundle/Basic.lean
|
FiberBundleCore.mem_source_at
|
[
{
"state_after": "ι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.46750\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na : F\n⊢ { fst := b, snd := a }.fst ∈ (localTriv Z (indexAt Z b)).baseSet",
"state_before": "ι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.46750\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na : F\n⊢ { fst := b, snd := a } ∈ (localTrivAt Z b).toLocalHomeomorph.toLocalEquiv.source",
"tactic": "rw [localTrivAt, mem_localTriv_source]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.46750\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na : F\n⊢ { fst := b, snd := a }.fst ∈ (localTriv Z (indexAt Z b)).baseSet",
"tactic": "exact Z.mem_baseSet_at b"
}
] |
[
716,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
714,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.leadingCoeff_pow'
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.681610\n⊢ leadingCoeff p ^ Nat.zero ≠ 0 → leadingCoeff (p ^ Nat.zero) = leadingCoeff p ^ Nat.zero",
"tactic": "simp"
},
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.681610\nn : ℕ\nih : leadingCoeff p ^ n ≠ 0 → leadingCoeff (p ^ n) = leadingCoeff p ^ n\nh : leadingCoeff p ^ Nat.succ n ≠ 0\nh₁ : leadingCoeff p ^ n ≠ 0\n⊢ leadingCoeff (p ^ Nat.succ n) = leadingCoeff p ^ Nat.succ n",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.681610\nn : ℕ\nih : leadingCoeff p ^ n ≠ 0 → leadingCoeff (p ^ n) = leadingCoeff p ^ n\nh : leadingCoeff p ^ Nat.succ n ≠ 0\n⊢ leadingCoeff (p ^ Nat.succ n) = leadingCoeff p ^ Nat.succ n",
"tactic": "have h₁ : leadingCoeff p ^ n ≠ 0 := fun h₁ => h <| by rw [pow_succ, h₁, mul_zero]"
},
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.681610\nn : ℕ\nih : leadingCoeff p ^ n ≠ 0 → leadingCoeff (p ^ n) = leadingCoeff p ^ n\nh : leadingCoeff p ^ Nat.succ n ≠ 0\nh₁ : leadingCoeff p ^ n ≠ 0\nh₂ : leadingCoeff p * leadingCoeff (p ^ n) ≠ 0\n⊢ leadingCoeff (p ^ Nat.succ n) = leadingCoeff p ^ Nat.succ n",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.681610\nn : ℕ\nih : leadingCoeff p ^ n ≠ 0 → leadingCoeff (p ^ n) = leadingCoeff p ^ n\nh : leadingCoeff p ^ Nat.succ n ≠ 0\nh₁ : leadingCoeff p ^ n ≠ 0\n⊢ leadingCoeff (p ^ Nat.succ n) = leadingCoeff p ^ Nat.succ n",
"tactic": "have h₂ : leadingCoeff p * leadingCoeff (p ^ n) ≠ 0 := by rwa [pow_succ, ← ih h₁] at h"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.681610\nn : ℕ\nih : leadingCoeff p ^ n ≠ 0 → leadingCoeff (p ^ n) = leadingCoeff p ^ n\nh : leadingCoeff p ^ Nat.succ n ≠ 0\nh₁ : leadingCoeff p ^ n ≠ 0\nh₂ : leadingCoeff p * leadingCoeff (p ^ n) ≠ 0\n⊢ leadingCoeff (p ^ Nat.succ n) = leadingCoeff p ^ Nat.succ n",
"tactic": "rw [pow_succ, pow_succ, leadingCoeff_mul' h₂, ih h₁]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.681610\nn : ℕ\nih : leadingCoeff p ^ n ≠ 0 → leadingCoeff (p ^ n) = leadingCoeff p ^ n\nh : leadingCoeff p ^ Nat.succ n ≠ 0\nh₁ : leadingCoeff p ^ n = 0\n⊢ leadingCoeff p ^ Nat.succ n = 0",
"tactic": "rw [pow_succ, h₁, mul_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.681610\nn : ℕ\nih : leadingCoeff p ^ n ≠ 0 → leadingCoeff (p ^ n) = leadingCoeff p ^ n\nh : leadingCoeff p ^ Nat.succ n ≠ 0\nh₁ : leadingCoeff p ^ n ≠ 0\n⊢ leadingCoeff p * leadingCoeff (p ^ n) ≠ 0",
"tactic": "rwa [pow_succ, ← ih h₁] at h"
}
] |
[
975,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
971,
1
] |
Mathlib/Algebra/Order/Ring/Abs.lean
|
dvd_abs
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Ring α\ninst✝ : LinearOrder α\na✝ b✝ a b : α\n⊢ a ∣ abs b ↔ a ∣ b",
"tactic": "cases' abs_choice b with h h <;> simp only [h, dvd_neg]"
}
] |
[
134,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Algebra/Group/Prod.lean
|
MulHom.coe_prodMap
|
[] |
[
398,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
397,
1
] |
Mathlib/ModelTheory/Substructures.lean
|
FirstOrder.Language.Substructure.mem_map
|
[] |
[
454,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
Mathlib/Logic/Nonempty.lean
|
nonempty_plift
|
[] |
[
106,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.incl_range
|
[
{
"state_after": "case h\nV : Type ?u.489284\nW : Type ?u.489287\nV₁ : Type u_1\nV₂ : Type ?u.489293\nV₃ : Type ?u.489296\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\ns : AddSubgroup V₁\nx : V₁\n⊢ x ∈ range (incl s) ↔ x ∈ s",
"state_before": "V : Type ?u.489284\nW : Type ?u.489287\nV₁ : Type u_1\nV₂ : Type ?u.489293\nV₃ : Type ?u.489296\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\ns : AddSubgroup V₁\n⊢ range (incl s) = s",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nV : Type ?u.489284\nW : Type ?u.489287\nV₁ : Type u_1\nV₂ : Type ?u.489293\nV₃ : Type ?u.489296\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\ns : AddSubgroup V₁\nx : V₁\n⊢ x ∈ range (incl s) ↔ x ∈ s",
"tactic": "exact ⟨fun ⟨y, hy⟩ => by rw [← hy]; simp, fun hx => ⟨⟨x, hx⟩, by simp⟩⟩"
},
{
"state_after": "V : Type ?u.489284\nW : Type ?u.489287\nV₁ : Type u_1\nV₂ : Type ?u.489293\nV₃ : Type ?u.489296\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\ns : AddSubgroup V₁\nx : V₁\nx✝ : x ∈ range (incl s)\ny : { x // x ∈ s }\nhy : ↑(toAddMonoidHom (incl s)) y = x\n⊢ ↑(toAddMonoidHom (incl s)) y ∈ s",
"state_before": "V : Type ?u.489284\nW : Type ?u.489287\nV₁ : Type u_1\nV₂ : Type ?u.489293\nV₃ : Type ?u.489296\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\ns : AddSubgroup V₁\nx : V₁\nx✝ : x ∈ range (incl s)\ny : { x // x ∈ s }\nhy : ↑(toAddMonoidHom (incl s)) y = x\n⊢ x ∈ s",
"tactic": "rw [← hy]"
},
{
"state_after": "no goals",
"state_before": "V : Type ?u.489284\nW : Type ?u.489287\nV₁ : Type u_1\nV₂ : Type ?u.489293\nV₃ : Type ?u.489296\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\ns : AddSubgroup V₁\nx : V₁\nx✝ : x ∈ range (incl s)\ny : { x // x ∈ s }\nhy : ↑(toAddMonoidHom (incl s)) y = x\n⊢ ↑(toAddMonoidHom (incl s)) y ∈ s",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "V : Type ?u.489284\nW : Type ?u.489287\nV₁ : Type u_1\nV₂ : Type ?u.489293\nV₃ : Type ?u.489296\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\ns : AddSubgroup V₁\nx : V₁\nhx : x ∈ s\n⊢ ↑(toAddMonoidHom (incl s)) { val := x, property := hx } = x",
"tactic": "simp"
}
] |
[
809,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
807,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.AbsolutelyContinuous.rfl
|
[] |
[
2384,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2384,
11
] |
Mathlib/MeasureTheory/Integral/MeanInequalities.lean
|
NNReal.lintegral_mul_le_Lp_mul_Lq
|
[
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nf g : α → ℝ≥0\nhf : AEMeasurable f\nhg : AEMeasurable g\n⊢ (∫⁻ (a : α), ↑(f a) * ↑(g a) ∂μ) ≤ (∫⁻ (a : α), ↑(f a) ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), ↑(g a) ^ q ∂μ) ^ (1 / q)",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nf g : α → ℝ≥0\nhf : AEMeasurable f\nhg : AEMeasurable g\n⊢ (∫⁻ (a : α), ↑((f * g) a) ∂μ) ≤ (∫⁻ (a : α), ↑(f a) ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), ↑(g a) ^ q ∂μ) ^ (1 / q)",
"tactic": "simp_rw [Pi.mul_apply, ENNReal.coe_mul]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nf g : α → ℝ≥0\nhf : AEMeasurable f\nhg : AEMeasurable g\n⊢ (∫⁻ (a : α), ↑(f a) * ↑(g a) ∂μ) ≤ (∫⁻ (a : α), ↑(f a) ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), ↑(g a) ^ q ∂μ) ^ (1 / q)",
"tactic": "exact ENNReal.lintegral_mul_le_Lp_mul_Lq μ hpq hf.coe_nnreal_ennreal hg.coe_nnreal_ennreal"
}
] |
[
407,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
402,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.yn_zero
|
[] |
[
138,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.tan_pi_div_four
|
[
{
"state_after": "⊢ sqrt 2 / 2 / (sqrt 2 / 2) = 1",
"state_before": "⊢ tan (π / 4) = 1",
"tactic": "rw [tan_eq_sin_div_cos, cos_pi_div_four, sin_pi_div_four]"
},
{
"state_after": "h : sqrt 2 / 2 > 0\n⊢ sqrt 2 / 2 / (sqrt 2 / 2) = 1",
"state_before": "⊢ sqrt 2 / 2 / (sqrt 2 / 2) = 1",
"tactic": "have h : sqrt 2 / 2 > 0 := by cancel_denoms"
},
{
"state_after": "no goals",
"state_before": "h : sqrt 2 / 2 > 0\n⊢ sqrt 2 / 2 / (sqrt 2 / 2) = 1",
"tactic": "exact div_self (ne_of_gt h)"
},
{
"state_after": "no goals",
"state_before": "⊢ sqrt 2 / 2 > 0",
"tactic": "cancel_denoms"
}
] |
[
932,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
929,
1
] |
Mathlib/Data/List/Basic.lean
|
List.take_left
|
[] |
[
1943,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1941,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Icc_subset_Ioc_iff
|
[] |
[
570,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
568,
1
] |
Mathlib/Analysis/Normed/Field/Basic.lean
|
List.norm_prod
|
[] |
[
557,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
556,
11
] |
Mathlib/Data/List/Basic.lean
|
List.reduceOption_get?_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.369711\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nl : List (Option α)\nx : α\n⊢ (∃ i, get? l i = some (some x)) ↔ ∃ i, get? (reduceOption l) i = some x",
"tactic": "rw [← mem_iff_get?, ← mem_iff_get?, reduceOption_mem_iff]"
}
] |
[
3502,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3500,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.coe_zero
|
[] |
[
264,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
Ideal.adic_module_basis
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\ni : ℕ\n⊢ ↑(I ^ i • ⊤) = ↑((fun i => toAddSubgroup (I ^ i • ⊤)) i)",
"tactic": "simp"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\ni : ℕ\na : R\na_in : a ∈ I ^ i\n⊢ a ∈ (fun x => x • m) ⁻¹' ↑(I ^ i • ⊤)",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\ni : ℕ\na : R\na_in : a ∈ ↑(I ^ i • ⊤)\n⊢ a ∈ (fun x => x • m) ⁻¹' ↑(I ^ i • ⊤)",
"tactic": "replace a_in : a ∈ I ^ i := by simpa [(I ^ i).mul_top] using a_in"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\ni : ℕ\na : R\na_in : a ∈ I ^ i\n⊢ a ∈ (fun x => x • m) ⁻¹' ↑(I ^ i • ⊤)",
"tactic": "exact smul_mem_smul a_in mem_top"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\ni : ℕ\na : R\na_in : a ∈ ↑(I ^ i • ⊤)\n⊢ a ∈ I ^ i",
"tactic": "simpa [(I ^ i).mul_top] using a_in"
}
] |
[
130,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.edgeSet_singletonSubgraph
|
[] |
[
877,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
876,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.dvd_add_right
|
[] |
[
730,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
729,
11
] |
Mathlib/Data/Polynomial/Derivative.lean
|
Polynomial.iterate_derivative_X_pow_eq_smul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : CommSemiring R\nn k : ℕ\n⊢ (↑derivative^[k]) (X ^ n) = ↑(Nat.descFactorial n k) • X ^ (n - k)",
"tactic": "rw [iterate_derivative_X_pow_eq_C_mul n k, smul_eq_C_mul]"
}
] |
[
520,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
518,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
Submodule.dualRestrict_ker_eq_dualAnnihilator
|
[] |
[
816,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
813,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean
|
Complex.cos_surjective
|
[
{
"state_after": "x : ℂ\n⊢ ∃ a, cos a = x",
"state_before": "⊢ Function.Surjective cos",
"tactic": "intro x"
},
{
"state_after": "case intro.intro\nx w : ℂ\nw₀ : w ≠ 0\nhw : 1 * w * w + -2 * x * w + 1 = 0\n⊢ ∃ a, cos a = x",
"state_before": "x : ℂ\n⊢ ∃ a, cos a = x",
"tactic": "obtain ⟨w, w₀, hw⟩ : ∃ (w : _) (_ : w ≠ 0), 1 * w * w + -2 * x * w + 1 = 0 := by\n rcases exists_quadratic_eq_zero one_ne_zero\n ⟨_, (cpow_nat_inv_pow _ two_ne_zero).symm.trans <| pow_two _⟩ with\n ⟨w, hw⟩\n refine' ⟨w, _, hw⟩\n rintro rfl\n simp only [zero_add, one_ne_zero, MulZeroClass.mul_zero] at hw"
},
{
"state_after": "case intro.intro\nx w : ℂ\nw₀ : w ≠ 0\nhw : 1 * w * w + -2 * x * w + 1 = 0\n⊢ exp (log w / I * I) ^ 2 - 2 * x * exp (log w / I * I) + 1 = 0",
"state_before": "case intro.intro\nx w : ℂ\nw₀ : w ≠ 0\nhw : 1 * w * w + -2 * x * w + 1 = 0\n⊢ ∃ a, cos a = x",
"tactic": "refine' ⟨log w / I, cos_eq_iff_quadratic.2 _⟩"
},
{
"state_after": "case intro.intro\nx w : ℂ\nw₀ : w ≠ 0\nhw : 1 * w * w + -2 * x * w + 1 = 0\n⊢ w ^ 2 - 2 * x * w + 1 = 0",
"state_before": "case intro.intro\nx w : ℂ\nw₀ : w ≠ 0\nhw : 1 * w * w + -2 * x * w + 1 = 0\n⊢ exp (log w / I * I) ^ 2 - 2 * x * exp (log w / I * I) + 1 = 0",
"tactic": "rw [div_mul_cancel _ I_ne_zero, exp_log w₀]"
},
{
"state_after": "case h.e'_2\nx w : ℂ\nw₀ : w ≠ 0\nhw : 1 * w * w + -2 * x * w + 1 = 0\n⊢ w ^ 2 - 2 * x * w + 1 = 1 * w * w + -2 * x * w + 1",
"state_before": "case intro.intro\nx w : ℂ\nw₀ : w ≠ 0\nhw : 1 * w * w + -2 * x * w + 1 = 0\n⊢ w ^ 2 - 2 * x * w + 1 = 0",
"tactic": "convert hw using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nx w : ℂ\nw₀ : w ≠ 0\nhw : 1 * w * w + -2 * x * w + 1 = 0\n⊢ w ^ 2 - 2 * x * w + 1 = 1 * w * w + -2 * x * w + 1",
"tactic": "ring"
},
{
"state_after": "case intro\nx w : ℂ\nhw : 1 * w * w + ?m.62077 * w + ?m.62078 = 0\n⊢ ∃ w x_1, 1 * w * w + -2 * x * w + 1 = 0",
"state_before": "x : ℂ\n⊢ ∃ w x_1, 1 * w * w + -2 * x * w + 1 = 0",
"tactic": "rcases exists_quadratic_eq_zero one_ne_zero\n ⟨_, (cpow_nat_inv_pow _ two_ne_zero).symm.trans <| pow_two _⟩ with\n ⟨w, hw⟩"
},
{
"state_after": "case intro\nx w : ℂ\nhw : 1 * w * w + -2 * x * w + 1 = 0\n⊢ w ≠ 0",
"state_before": "case intro\nx w : ℂ\nhw : 1 * w * w + ?m.62077 * w + ?m.62078 = 0\n⊢ ∃ w x_1, 1 * w * w + -2 * x * w + 1 = 0",
"tactic": "refine' ⟨w, _, hw⟩"
},
{
"state_after": "case intro\nx : ℂ\nhw : 1 * 0 * 0 + -2 * x * 0 + 1 = 0\n⊢ False",
"state_before": "case intro\nx w : ℂ\nhw : 1 * w * w + -2 * x * w + 1 = 0\n⊢ w ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case intro\nx : ℂ\nhw : 1 * 0 * 0 + -2 * x * 0 + 1 = 0\n⊢ False",
"tactic": "simp only [zero_add, one_ne_zero, MulZeroClass.mul_zero] at hw"
}
] |
[
187,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Data/Semiquot.lean
|
Semiquot.mem_pure_self
|
[] |
[
168,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.le_of_mem_factorization
|
[] |
[
149,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.memℒp_of_memℒp_trim
|
[] |
[
1006,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1004,
1
] |
Mathlib/Topology/Algebra/Algebra.lean
|
Subalgebra.isClosed_topologicalClosure
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u\ninst✝³ : TopologicalSpace A\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : TopologicalSemiring A\ns : Subalgebra R A\n⊢ IsClosed ↑(topologicalClosure s)",
"tactic": "convert @isClosed_closure A _ s"
}
] |
[
121,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.leadingCoeff_monic_mul
|
[
{
"state_after": "case inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.691467\np : R[X]\nhp : Monic p\n⊢ leadingCoeff (p * 0) = leadingCoeff 0\n\ncase inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.691467\np q : R[X]\nhp : Monic p\nH : q ≠ 0\n⊢ leadingCoeff (p * q) = leadingCoeff q",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.691467\np q : R[X]\nhp : Monic p\n⊢ leadingCoeff (p * q) = leadingCoeff q",
"tactic": "rcases eq_or_ne q 0 with (rfl | H)"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.691467\np : R[X]\nhp : Monic p\n⊢ leadingCoeff (p * 0) = leadingCoeff 0",
"tactic": "simp"
},
{
"state_after": "case inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.691467\np q : R[X]\nhp : Monic p\nH : q ≠ 0\n⊢ leadingCoeff p * leadingCoeff q ≠ 0",
"state_before": "case inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.691467\np q : R[X]\nhp : Monic p\nH : q ≠ 0\n⊢ leadingCoeff (p * q) = leadingCoeff q",
"tactic": "rw [leadingCoeff_mul', hp.leadingCoeff, one_mul]"
},
{
"state_after": "no goals",
"state_before": "case inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.691467\np q : R[X]\nhp : Monic p\nH : q ≠ 0\n⊢ leadingCoeff p * leadingCoeff q ≠ 0",
"tactic": "rwa [hp.leadingCoeff, one_mul, Ne.def, leadingCoeff_eq_zero]"
}
] |
[
1007,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1002,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.coe_ofRepr
|
[] |
[
139,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/GroupTheory/QuotientGroup.lean
|
QuotientGroup.quotientMapSubgroupOfOfLe_mk
|
[] |
[
469,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
Mathlib/Geometry/Manifold/ChartedSpace.lean
|
StructureGroupoid.compatible
|
[] |
[
872,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
869,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean
|
AddSubmonoid.closure_mul_closure
|
[
{
"state_after": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\n⊢ closure S * closure T ≤ closure (S * T)\n\ncase a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\n⊢ closure (S * T) ≤ closure S * closure T",
"state_before": "α : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\n⊢ closure S * closure T = closure (S * T)",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na : R\nha : a ∈ closure S\nb : R\nhb : b ∈ closure T\n⊢ a * b ∈ closure (S * T)",
"state_before": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\n⊢ closure S * closure T ≤ closure (S * T)",
"tactic": "refine mul_le.2 fun a ha b hb => ?_"
},
{
"state_after": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na : R\nha : a ∈ closure S\nb : R\nhb : b ∈ closure T\n⊢ a ∈ comap (AddMonoidHom.mulRight b) (closure (S * T))",
"state_before": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na : R\nha : a ∈ closure S\nb : R\nhb : b ∈ closure T\n⊢ a * b ∈ closure (S * T)",
"tactic": "rw [← AddMonoidHom.mulRight_apply, ← AddSubmonoid.mem_comap]"
},
{
"state_after": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na : R\nha : a ∈ closure S\nb : R\nhb : b ∈ closure T\na' : R\nha' : a' ∈ S\n⊢ a' ∈ ↑(comap (AddMonoidHom.mulRight b) (closure (S * T)))",
"state_before": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na : R\nha : a ∈ closure S\nb : R\nhb : b ∈ closure T\n⊢ a ∈ comap (AddMonoidHom.mulRight b) (closure (S * T))",
"tactic": "refine (closure_le.2 fun a' ha' => ?_) ha"
},
{
"state_after": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na : R\nha : a ∈ closure S\nb : R\nhb : b ∈ closure T\na' : R\nha' : a' ∈ S\n⊢ b ∈ comap (AddMonoidHom.mulLeft a') (closure (S * T))",
"state_before": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na : R\nha : a ∈ closure S\nb : R\nhb : b ∈ closure T\na' : R\nha' : a' ∈ S\n⊢ a' ∈ ↑(comap (AddMonoidHom.mulRight b) (closure (S * T)))",
"tactic": "change b ∈ (closure (S * T)).comap (AddMonoidHom.mulLeft a')"
},
{
"state_after": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na : R\nha : a ∈ closure S\nb : R\nhb : b ∈ closure T\na' : R\nha' : a' ∈ S\nb' : R\nhb' : b' ∈ T\n⊢ b' ∈ ↑(comap (AddMonoidHom.mulLeft a') (closure (S * T)))",
"state_before": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na : R\nha : a ∈ closure S\nb : R\nhb : b ∈ closure T\na' : R\nha' : a' ∈ S\n⊢ b ∈ comap (AddMonoidHom.mulLeft a') (closure (S * T))",
"tactic": "refine (closure_le.2 fun b' hb' => ?_) hb"
},
{
"state_after": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na : R\nha : a ∈ closure S\nb : R\nhb : b ∈ closure T\na' : R\nha' : a' ∈ S\nb' : R\nhb' : b' ∈ T\n⊢ a' * b' ∈ closure (S * T)",
"state_before": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na : R\nha : a ∈ closure S\nb : R\nhb : b ∈ closure T\na' : R\nha' : a' ∈ S\nb' : R\nhb' : b' ∈ T\n⊢ b' ∈ ↑(comap (AddMonoidHom.mulLeft a') (closure (S * T)))",
"tactic": "change a' * b' ∈ closure (S * T)"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na : R\nha : a ∈ closure S\nb : R\nhb : b ∈ closure T\na' : R\nha' : a' ∈ S\nb' : R\nhb' : b' ∈ T\n⊢ a' * b' ∈ closure (S * T)",
"tactic": "exact subset_closure (Set.mul_mem_mul ha' hb')"
},
{
"state_after": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\n⊢ S * T ⊆ ↑(closure S * closure T)",
"state_before": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\n⊢ closure (S * T) ≤ closure S * closure T",
"tactic": "rw [closure_le]"
},
{
"state_after": "case a.intro.intro.intro.intro\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na b : R\nha : a ∈ S\nhb : b ∈ T\n⊢ (fun x x_1 => x * x_1) a b ∈ ↑(closure S * closure T)",
"state_before": "case a\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\n⊢ S * T ⊆ ↑(closure S * closure T)",
"tactic": "rintro _ ⟨a, b, ha, hb, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case a.intro.intro.intro.intro\nα : Type ?u.244690\nG : Type ?u.244693\nM : Type ?u.244696\nR : Type u_1\nA : Type ?u.244702\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : NonUnitalNonAssocSemiring R\nS T : Set R\na b : R\nha : a ∈ S\nhb : b ∈ T\n⊢ (fun x x_1 => x * x_1) a b ∈ ↑(closure S * closure T)",
"tactic": "exact mul_mem_mul (subset_closure ha) (subset_closure hb)"
}
] |
[
565,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
554,
1
] |
Mathlib/Combinatorics/Hall/Finite.lean
|
HallMarriageTheorem.hall_hard_inductive_step_B
|
[
{
"state_after": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = n + 1\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = n + 1\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"tactic": "haveI := Classical.decEq ι"
},
{
"state_after": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = n + 1\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"tactic": "rw [Nat.add_one] at hn"
},
{
"state_after": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"tactic": "have card_ι'_le : Fintype.card s ≤ n := by\n apply Nat.le_of_lt_succ\n calc\n Fintype.card s = s.card := Fintype.card_coe _\n _ < Fintype.card ι := (card_lt_iff_ne_univ _).mpr hns\n _ = n.succ := hn"
},
{
"state_after": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"tactic": "let t' : s → Finset α := fun x' => t x'"
},
{
"state_after": "case intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"tactic": "rcases ih t' card_ι'_le (hall_cond_of_restrict ht) with ⟨f', hf', hsf'⟩"
},
{
"state_after": "case intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"state_before": "case intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"tactic": "set ι'' := (s : Set ι)ᶜ"
},
{
"state_after": "case intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"state_before": "case intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"tactic": "let t'' : ι'' → Finset α := fun a'' => t a'' \\ s.biUnion t"
},
{
"state_after": "case intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"state_before": "case intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"tactic": "have card_ι''_le : Fintype.card ι'' ≤ n := by\n simp_rw [← Nat.lt_succ_iff, ← hn, ← Finset.coe_compl, coe_sort_coe]\n rwa [Fintype.card_coe, card_compl_lt_iff_nonempty]"
},
{
"state_after": "case intro.intro.intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"state_before": "case intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"tactic": "rcases ih t'' card_ι''_le (hall_cond_of_compl hus ht) with ⟨f'', hf'', hsf''⟩"
},
{
"state_after": "case intro.intro.intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"state_before": "case intro.intro.intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"tactic": "have f'_mem_biUnion : ∀ (x') (hx' : x' ∈ s), f' ⟨x', hx'⟩ ∈ s.biUnion t := by\n intro x' hx'\n rw [mem_biUnion]\n exact ⟨x', hx', hsf' _⟩"
},
{
"state_after": "case intro.intro.intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"state_before": "case intro.intro.intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"tactic": "have f''_not_mem_biUnion : ∀ (x'') (hx'' : ¬x'' ∈ s), ¬f'' ⟨x'', hx''⟩ ∈ s.biUnion t := by\n intro x'' hx''\n have h := hsf'' ⟨x'', hx''⟩\n rw [mem_sdiff] at h\n exact h.2"
},
{
"state_after": "case intro.intro.intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nim_disj :\n ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s),\n f' { val := x', property := hx' } ≠ f'' { val := x'', property := hx'' }\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"state_before": "case intro.intro.intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"tactic": "have im_disj :\n ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s), f' ⟨x', hx'⟩ ≠ f'' ⟨x'', hx''⟩ := by\n intro x x' hx' hx'' h\n apply f''_not_mem_biUnion x' hx''\n rw [← h]\n apply f'_mem_biUnion x"
},
{
"state_after": "case intro.intro.intro.intro.refine'_1\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nim_disj :\n ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s),\n f' { val := x', property := hx' } ≠ f'' { val := x'', property := hx'' }\n⊢ Function.Injective fun x => if h : x ∈ s then f' { val := x, property := h } else f'' { val := x, property := h }\n\ncase intro.intro.intro.intro.refine'_2\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nim_disj :\n ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s),\n f' { val := x', property := hx' } ≠ f'' { val := x'', property := hx'' }\n⊢ ∀ (x : ι), (fun x => if h : x ∈ s then f' { val := x, property := h } else f'' { val := x, property := h }) x ∈ t x",
"state_before": "case intro.intro.intro.intro\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nim_disj :\n ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s),\n f' { val := x', property := hx' } ≠ f'' { val := x'', property := hx'' }\n⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x",
"tactic": "refine' ⟨fun x => if h : x ∈ s then f' ⟨x, h⟩ else f'' ⟨x, h⟩, _, _⟩"
},
{
"state_after": "case a\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\n⊢ Fintype.card { x // x ∈ s } < Nat.succ n",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\n⊢ Fintype.card { x // x ∈ s } ≤ n",
"tactic": "apply Nat.le_of_lt_succ"
},
{
"state_after": "no goals",
"state_before": "case a\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\n⊢ Fintype.card { x // x ∈ s } < Nat.succ n",
"tactic": "calc\n Fintype.card s = s.card := Fintype.card_coe _\n _ < Fintype.card ι := (card_lt_iff_ne_univ _).mpr hns\n _ = n.succ := hn"
},
{
"state_after": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\n⊢ Fintype.card { x // x ∈ sᶜ } < Fintype.card ι",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\n⊢ Fintype.card ↑ι'' ≤ n",
"tactic": "simp_rw [← Nat.lt_succ_iff, ← hn, ← Finset.coe_compl, coe_sort_coe]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\n⊢ Fintype.card { x // x ∈ sᶜ } < Fintype.card ι",
"tactic": "rwa [Fintype.card_coe, card_compl_lt_iff_nonempty]"
},
{
"state_after": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nx' : ι\nhx' : x' ∈ s\n⊢ f' { val := x', property := hx' } ∈ Finset.biUnion s t",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\n⊢ ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t",
"tactic": "intro x' hx'"
},
{
"state_after": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nx' : ι\nhx' : x' ∈ s\n⊢ ∃ a, a ∈ s ∧ f' { val := x', property := hx' } ∈ t a",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nx' : ι\nhx' : x' ∈ s\n⊢ f' { val := x', property := hx' } ∈ Finset.biUnion s t",
"tactic": "rw [mem_biUnion]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nx' : ι\nhx' : x' ∈ s\n⊢ ∃ a, a ∈ s ∧ f' { val := x', property := hx' } ∈ t a",
"tactic": "exact ⟨x', hx', hsf' _⟩"
},
{
"state_after": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nx'' : ι\nhx'' : ¬x'' ∈ s\n⊢ ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\n⊢ ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t",
"tactic": "intro x'' hx''"
},
{
"state_after": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nx'' : ι\nhx'' : ¬x'' ∈ s\nh : f'' { val := x'', property := hx'' } ∈ t'' { val := x'', property := hx'' }\n⊢ ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nx'' : ι\nhx'' : ¬x'' ∈ s\n⊢ ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t",
"tactic": "have h := hsf'' ⟨x'', hx''⟩"
},
{
"state_after": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nx'' : ι\nhx'' : ¬x'' ∈ s\nh :\n f'' { val := x'', property := hx'' } ∈ t ↑{ val := x'', property := hx'' } ∧\n ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\n⊢ ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nx'' : ι\nhx'' : ¬x'' ∈ s\nh : f'' { val := x'', property := hx'' } ∈ t'' { val := x'', property := hx'' }\n⊢ ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t",
"tactic": "rw [mem_sdiff] at h"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nx'' : ι\nhx'' : ¬x'' ∈ s\nh :\n f'' { val := x'', property := hx'' } ∈ t ↑{ val := x'', property := hx'' } ∧\n ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\n⊢ ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t",
"tactic": "exact h.2"
},
{
"state_after": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nx x' : ι\nhx' : x ∈ s\nhx'' : ¬x' ∈ s\nh : f' { val := x, property := hx' } = f'' { val := x', property := hx'' }\n⊢ False",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\n⊢ ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s),\n f' { val := x', property := hx' } ≠ f'' { val := x'', property := hx'' }",
"tactic": "intro x x' hx' hx'' h"
},
{
"state_after": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nx x' : ι\nhx' : x ∈ s\nhx'' : ¬x' ∈ s\nh : f' { val := x, property := hx' } = f'' { val := x', property := hx'' }\n⊢ f'' { val := x', property := hx'' } ∈ Finset.biUnion s t",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nx x' : ι\nhx' : x ∈ s\nhx'' : ¬x' ∈ s\nh : f' { val := x, property := hx' } = f'' { val := x', property := hx'' }\n⊢ False",
"tactic": "apply f''_not_mem_biUnion x' hx''"
},
{
"state_after": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nx x' : ι\nhx' : x ∈ s\nhx'' : ¬x' ∈ s\nh : f' { val := x, property := hx' } = f'' { val := x', property := hx'' }\n⊢ f' { val := x, property := hx' } ∈ Finset.biUnion s t",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nx x' : ι\nhx' : x ∈ s\nhx'' : ¬x' ∈ s\nh : f' { val := x, property := hx' } = f'' { val := x', property := hx'' }\n⊢ f'' { val := x', property := hx'' } ∈ Finset.biUnion s t",
"tactic": "rw [← h]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nx x' : ι\nhx' : x ∈ s\nhx'' : ¬x' ∈ s\nh : f' { val := x, property := hx' } = f'' { val := x', property := hx'' }\n⊢ f' { val := x, property := hx' } ∈ Finset.biUnion s t",
"tactic": "apply f'_mem_biUnion x"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_1\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nim_disj :\n ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s),\n f' { val := x', property := hx' } ≠ f'' { val := x'', property := hx'' }\n⊢ Function.Injective fun x => if h : x ∈ s then f' { val := x, property := h } else f'' { val := x, property := h }",
"tactic": "refine' hf'.dite _ hf'' (@fun x x' => im_disj x x' _ _)"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nim_disj :\n ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s),\n f' { val := x', property := hx' } ≠ f'' { val := x'', property := hx'' }\nx : ι\n⊢ (fun x => if h : x ∈ s then f' { val := x, property := h } else f'' { val := x, property := h }) x ∈ t x",
"state_before": "case intro.intro.intro.intro.refine'_2\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nim_disj :\n ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s),\n f' { val := x', property := hx' } ≠ f'' { val := x'', property := hx'' }\n⊢ ∀ (x : ι), (fun x => if h : x ∈ s then f' { val := x, property := h } else f'' { val := x, property := h }) x ∈ t x",
"tactic": "intro x"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nim_disj :\n ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s),\n f' { val := x', property := hx' } ≠ f'' { val := x'', property := hx'' }\nx : ι\n⊢ (if h : x ∈ s then f' { val := x, property := h } else f'' { val := x, property := h }) ∈ t x",
"state_before": "case intro.intro.intro.intro.refine'_2\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nim_disj :\n ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s),\n f' { val := x', property := hx' } ≠ f'' { val := x'', property := hx'' }\nx : ι\n⊢ (fun x => if h : x ∈ s then f' { val := x, property := h } else f'' { val := x, property := h }) x ∈ t x",
"tactic": "simp only [of_eq_true]"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2.inl\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nim_disj :\n ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s),\n f' { val := x', property := hx' } ≠ f'' { val := x'', property := hx'' }\nx : ι\nh : x ∈ s\n⊢ f' { val := x, property := h } ∈ t x\n\ncase intro.intro.intro.intro.refine'_2.inr\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nim_disj :\n ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s),\n f' { val := x', property := hx' } ≠ f'' { val := x'', property := hx'' }\nx : ι\nh : ¬x ∈ s\n⊢ f'' { val := x, property := h } ∈ t x",
"state_before": "case intro.intro.intro.intro.refine'_2\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nim_disj :\n ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s),\n f' { val := x', property := hx' } ≠ f'' { val := x'', property := hx'' }\nx : ι\n⊢ (if h : x ∈ s then f' { val := x, property := h } else f'' { val := x, property := h }) ∈ t x",
"tactic": "split_ifs with h <;> simp"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_2.inl\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nim_disj :\n ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s),\n f' { val := x', property := hx' } ≠ f'' { val := x'', property := hx'' }\nx : ι\nh : x ∈ s\n⊢ f' { val := x, property := h } ∈ t x",
"tactic": "exact hsf' ⟨x, h⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_2.inr\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = Nat.succ n\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), card s' ≤ card (Finset.biUnion s' t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x\ns : Finset ι\nhs : Finset.Nonempty s\nhns : s ≠ univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq ι\ncard_ι'_le : Fintype.card { x // x ∈ s } ≤ n\nt' : { x // x ∈ s } → Finset α := fun x' => t ↑x'\nf' : { x // x ∈ s } → α\nhf' : Function.Injective f'\nhsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x\nι'' : Set ι := ↑sᶜ\nt'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ Finset.biUnion s t\ncard_ι''_le : Fintype.card ↑ι'' ≤ n\nf'' : ↑ι'' → α\nhf'' : Function.Injective f''\nhsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x\nf'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' { val := x', property := hx' } ∈ Finset.biUnion s t\nf''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : ¬x'' ∈ s), ¬f'' { val := x'', property := hx'' } ∈ Finset.biUnion s t\nim_disj :\n ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : ¬x'' ∈ s),\n f' { val := x', property := hx' } ≠ f'' { val := x'', property := hx'' }\nx : ι\nh : ¬x ∈ s\n⊢ f'' { val := x, property := h } ∈ t x",
"tactic": "exact sdiff_subset _ _ (hsf'' ⟨x, h⟩)"
}
] |
[
219,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Analysis/SpecialFunctions/Polynomials.lean
|
Polynomial.abs_tendsto_atTop
|
[
{
"state_after": "case inl\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : 0 < degree P\nhP : 0 ≤ leadingCoeff P\n⊢ Tendsto (fun x => abs (eval x P)) atTop atTop\n\ncase inr\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : 0 < degree P\nhP : leadingCoeff P ≤ 0\n⊢ Tendsto (fun x => abs (eval x P)) atTop atTop",
"state_before": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : 0 < degree P\n⊢ Tendsto (fun x => abs (eval x P)) atTop atTop",
"tactic": "cases' le_total 0 P.leadingCoeff with hP hP"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : 0 < degree P\nhP : 0 ≤ leadingCoeff P\n⊢ Tendsto (fun x => abs (eval x P)) atTop atTop",
"tactic": "exact tendsto_abs_atTop_atTop.comp (P.tendsto_atTop_of_leadingCoeff_nonneg hdeg hP)"
},
{
"state_after": "no goals",
"state_before": "case inr\n𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : 0 < degree P\nhP : leadingCoeff P ≤ 0\n⊢ Tendsto (fun x => abs (eval x P)) atTop atTop",
"tactic": "exact tendsto_abs_atBot_atTop.comp (P.tendsto_atBot_of_leadingCoeff_nonpos hdeg hP)"
}
] |
[
91,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
ContinuousOn.zpow
|
[] |
[
549,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
548,
1
] |
Mathlib/RingTheory/ClassGroup.lean
|
toPrincipalIdeal_eq_iff
|
[
{
"state_after": "R : Type u_2\nK : Type u_1\nL : Type ?u.116215\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : DecidableEq L\ninst✝⁵ : Algebra R K\ninst✝⁴ : IsFractionRing R K\ninst✝³ : Algebra K L\ninst✝² : FiniteDimensional K L\ninst✝¹ : Algebra R L\ninst✝ : IsScalarTower R K L\nI : (FractionalIdeal R⁰ K)ˣ\nx : Kˣ\n⊢ ↑{\n toOneHom :=\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : Kˣ),\n OneHom.toFun\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) }\n y) }\n x =\n I ↔\n spanSingleton R⁰ ↑x = ↑I",
"state_before": "R : Type u_2\nK : Type u_1\nL : Type ?u.116215\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : DecidableEq L\ninst✝⁵ : Algebra R K\ninst✝⁴ : IsFractionRing R K\ninst✝³ : Algebra K L\ninst✝² : FiniteDimensional K L\ninst✝¹ : Algebra R L\ninst✝ : IsScalarTower R K L\nI : (FractionalIdeal R⁰ K)ˣ\nx : Kˣ\n⊢ ↑(toPrincipalIdeal R K) x = I ↔ spanSingleton R⁰ ↑x = ↑I",
"tactic": "simp only [toPrincipalIdeal]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nK : Type u_1\nL : Type ?u.116215\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : DecidableEq L\ninst✝⁵ : Algebra R K\ninst✝⁴ : IsFractionRing R K\ninst✝³ : Algebra K L\ninst✝² : FiniteDimensional K L\ninst✝¹ : Algebra R L\ninst✝ : IsScalarTower R K L\nI : (FractionalIdeal R⁰ K)ˣ\nx : Kˣ\n⊢ ↑{\n toOneHom :=\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n ∀ (x y : Kˣ),\n OneHom.toFun\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) },\n map_one' :=\n (_ :\n (fun x =>\n { val := spanSingleton R⁰ ↑x, inv := spanSingleton R⁰ (↑x)⁻¹,\n val_inv := (_ : spanSingleton R⁰ ↑x * spanSingleton R⁰ (↑x)⁻¹ = 1),\n inv_val := (_ : spanSingleton R⁰ (↑x)⁻¹ * spanSingleton R⁰ ↑x = 1) })\n 1 =\n 1) }\n y) }\n x =\n I ↔\n spanSingleton R⁰ ↑x = ↑I",
"tactic": "exact Units.ext_iff"
}
] |
[
76,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
derivWithin_sin
|
[] |
[
843,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
841,
1
] |
Mathlib/CategoryTheory/Sites/Canonical.lean
|
CategoryTheory.Sheaf.sheaf_for_finestTopology
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nP : Cᵒᵖ ⥤ Type v\nX✝ Y : C\nS✝ : Sieve X✝\nR : Presieve X✝\nJ J₂ : GrothendieckTopology C\nPs : Set (Cᵒᵖ ⥤ Type v)\nh : P ∈ Ps\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves (finestTopology Ps) X\n⊢ Presieve.IsSheafFor P S.arrows",
"tactic": "simpa using hS _ ⟨⟨_, _, ⟨_, h, rfl⟩, rfl⟩, rfl⟩ _ (𝟙 _)"
}
] |
[
197,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.headD_cons
|
[] |
[
27,
63
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
27,
14
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
linearIndependent_sum
|
[
{
"state_after": "ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\n⊢ LinearIndependent R v ↔\n LinearIndependent R (v ∘ Sum.inl) ∧\n LinearIndependent R (v ∘ Sum.inr) ∧ Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))",
"state_before": "ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\n⊢ LinearIndependent R v ↔\n LinearIndependent R (v ∘ Sum.inl) ∧\n LinearIndependent R (v ∘ Sum.inr) ∧ Disjoint (span R (range (v ∘ Sum.inl))) (span R (range (v ∘ Sum.inr)))",
"tactic": "rw [range_comp v, range_comp v]"
},
{
"state_after": "case refine'_1\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\n⊢ LinearIndependent R v →\n LinearIndependent R (v ∘ Sum.inl) ∧\n LinearIndependent R (v ∘ Sum.inr) ∧ Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\n\ncase refine'_2\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\n⊢ LinearIndependent R (v ∘ Sum.inl) ∧\n LinearIndependent R (v ∘ Sum.inr) ∧ Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr)) →\n LinearIndependent R v",
"state_before": "ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\n⊢ LinearIndependent R v ↔\n LinearIndependent R (v ∘ Sum.inl) ∧\n LinearIndependent R (v ∘ Sum.inr) ∧ Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))",
"tactic": "refine' ⟨_, _⟩"
},
{
"state_after": "case refine'_2.intro.intro\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : LinearIndependent R (v ∘ Sum.inl)\nhr : LinearIndependent R (v ∘ Sum.inr)\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\n⊢ LinearIndependent R v",
"state_before": "case refine'_2\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\n⊢ LinearIndependent R (v ∘ Sum.inl) ∧\n LinearIndependent R (v ∘ Sum.inr) ∧ Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr)) →\n LinearIndependent R v",
"tactic": "rintro ⟨hl, hr, hlr⟩"
},
{
"state_after": "case refine'_2.intro.intro\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\n⊢ ∀ (s : Finset (ι ⊕ ι')) (g : ι ⊕ ι' → R), ∑ i in s, g i • v i = 0 → ∀ (i : ι ⊕ ι'), i ∈ s → g i = 0",
"state_before": "case refine'_2.intro.intro\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : LinearIndependent R (v ∘ Sum.inl)\nhr : LinearIndependent R (v ∘ Sum.inr)\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\n⊢ LinearIndependent R v",
"tactic": "rw [linearIndependent_iff'] at *"
},
{
"state_after": "case refine'_2.intro.intro\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni : ι ⊕ ι'\nhi : i ∈ s\n⊢ g i = 0",
"state_before": "case refine'_2.intro.intro\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\n⊢ ∀ (s : Finset (ι ⊕ ι')) (g : ι ⊕ ι' → R), ∑ i in s, g i • v i = 0 → ∀ (i : ι ⊕ ι'), i ∈ s → g i = 0",
"tactic": "intro s g hg i hi"
},
{
"state_after": "case refine'_1\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nh : LinearIndependent R v\n⊢ LinearIndependent R (v ∘ Sum.inl) ∧\n LinearIndependent R (v ∘ Sum.inr) ∧ Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))",
"state_before": "case refine'_1\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\n⊢ LinearIndependent R v →\n LinearIndependent R (v ∘ Sum.inl) ∧\n LinearIndependent R (v ∘ Sum.inr) ∧ Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))",
"tactic": "intro h"
},
{
"state_after": "case refine'_1\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nh : LinearIndependent R v\n⊢ Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))",
"state_before": "case refine'_1\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nh : LinearIndependent R v\n⊢ LinearIndependent R (v ∘ Sum.inl) ∧\n LinearIndependent R (v ∘ Sum.inr) ∧ Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))",
"tactic": "refine' ⟨h.comp _ Sum.inl_injective, h.comp _ Sum.inr_injective, _⟩"
},
{
"state_after": "case refine'_1\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nh : LinearIndependent R v\n⊢ Disjoint (range Sum.inl) (range Sum.inr)",
"state_before": "case refine'_1\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nh : LinearIndependent R v\n⊢ Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))",
"tactic": "refine' h.disjoint_span_image _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nh : LinearIndependent R v\n⊢ Disjoint (range Sum.inl) (range Sum.inr)",
"tactic": "exact IsCompl.disjoint isCompl_range_inl_range_inr"
},
{
"state_after": "ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni : ι ⊕ ι'\nhi : i ∈ s\n⊢ ∑ x in Finset.filter (fun a => a ∈ range Sum.inl ∨ a ∈ range Sum.inr) s, g x • v x = 0\n\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni : ι ⊕ ι'\nhi : i ∈ s\n⊢ Disjoint (Finset.filter (fun x => x ∈ range Sum.inl) s) (Finset.filter (fun x => x ∈ range Sum.inr) s)",
"state_before": "ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni : ι ⊕ ι'\nhi : i ∈ s\n⊢ ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) +\n ∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i) =\n 0",
"tactic": "rw [Finset.sum_preimage' (g := fun x => g x • v x),\n Finset.sum_preimage' (g := fun x => g x • v x), ← Finset.sum_union, ← Finset.filter_or]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni : ι ⊕ ι'\nhi : i ∈ s\n⊢ ∑ x in Finset.filter (fun a => a ∈ range Sum.inl ∨ a ∈ range Sum.inr) s, g x • v x = 0",
"tactic": "simpa only [← mem_union, range_inl_union_range_inr, mem_univ, Finset.filter_True]"
},
{
"state_after": "ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝¹ y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni : ι ⊕ ι'\nhi : i ∈ s\nx : ι ⊕ ι'\nx✝ : x ∈ s\nhx : x ∈ range Sum.inl\n⊢ Disjoint (range Sum.inl) (range Sum.inr)",
"state_before": "ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni : ι ⊕ ι'\nhi : i ∈ s\n⊢ Disjoint (Finset.filter (fun x => x ∈ range Sum.inl) s) (Finset.filter (fun x => x ∈ range Sum.inr) s)",
"tactic": "refine Finset.disjoint_filter.2 fun x _ hx =>\n disjoint_left.1 ?_ hx"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝¹ y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni : ι ⊕ ι'\nhi : i ∈ s\nx : ι ⊕ ι'\nx✝ : x ∈ s\nhx : x ∈ range Sum.inl\n⊢ Disjoint (range Sum.inl) (range Sum.inr)",
"tactic": "exact IsCompl.disjoint isCompl_range_inl_range_inr"
},
{
"state_after": "case refine'_2.intro.intro\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni : ι ⊕ ι'\nhi : i ∈ s\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) =\n -∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i)\n⊢ g i = 0",
"state_before": "case refine'_2.intro.intro\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni : ι ⊕ ι'\nhi : i ∈ s\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) +\n ∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i) =\n 0\n⊢ g i = 0",
"tactic": "rw [← eq_neg_iff_add_eq_zero] at this"
},
{
"state_after": "case refine'_2.intro.intro\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : ∀ (x : M), x ∈ span R (v '' range Sum.inl) → ∀ (y : M), y ∈ span R (v '' range Sum.inr) → x = y → x = 0\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni : ι ⊕ ι'\nhi : i ∈ s\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) =\n -∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i)\n⊢ g i = 0",
"state_before": "case refine'_2.intro.intro\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni : ι ⊕ ι'\nhi : i ∈ s\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) =\n -∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i)\n⊢ g i = 0",
"tactic": "rw [disjoint_def'] at hlr"
},
{
"state_after": "case refine'_2.intro.intro.inl\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : ∀ (x : M), x ∈ span R (v '' range Sum.inl) → ∀ (y : M), y ∈ span R (v '' range Sum.inr) → x = y → x = 0\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) =\n -∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i)\nA : ∑ c in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl c) = 0\ni : ι\nhi : Sum.inl i ∈ s\n⊢ g (Sum.inl i) = 0\n\ncase refine'_2.intro.intro.inr\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : ∀ (x : M), x ∈ span R (v '' range Sum.inl) → ∀ (y : M), y ∈ span R (v '' range Sum.inr) → x = y → x = 0\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) =\n -∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i)\nA : ∑ c in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl c) = 0\ni : ι'\nhi : Sum.inr i ∈ s\n⊢ g (Sum.inr i) = 0",
"state_before": "case refine'_2.intro.intro\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : ∀ (x : M), x ∈ span R (v '' range Sum.inl) → ∀ (y : M), y ∈ span R (v '' range Sum.inr) → x = y → x = 0\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni : ι ⊕ ι'\nhi : i ∈ s\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) =\n -∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i)\nA : ∑ c in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl c) = 0\n⊢ g i = 0",
"tactic": "cases' i with i i"
},
{
"state_after": "case refine_1\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : ∀ (x : M), x ∈ span R (v '' range Sum.inl) → ∀ (y : M), y ∈ span R (v '' range Sum.inr) → x = y → x = 0\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni✝ : ι ⊕ ι'\nhi : i✝ ∈ s\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) =\n -∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i)\ni : ι\nx✝ : i ∈ Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s))\n⊢ (fun x => g x • v x) (Sum.inl i) ∈ span R (v '' range Sum.inl)\n\ncase refine_2\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : ∀ (x : M), x ∈ span R (v '' range Sum.inl) → ∀ (y : M), y ∈ span R (v '' range Sum.inr) → x = y → x = 0\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni✝ : ι ⊕ ι'\nhi : i✝ ∈ s\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) =\n -∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i)\ni : ι'\nx✝ : i ∈ Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s))\n⊢ (fun x => g x • v x) (Sum.inr i) ∈ span R (v '' range Sum.inr)",
"state_before": "ι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : ∀ (x : M), x ∈ span R (v '' range Sum.inl) → ∀ (y : M), y ∈ span R (v '' range Sum.inr) → x = y → x = 0\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni : ι ⊕ ι'\nhi : i ∈ s\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) =\n -∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i)\n⊢ ?m.371363",
"tactic": "refine hlr _ (sum_mem fun i _ => ?_) _ (neg_mem <| sum_mem fun i _ => ?_) this"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : ∀ (x : M), x ∈ span R (v '' range Sum.inl) → ∀ (y : M), y ∈ span R (v '' range Sum.inr) → x = y → x = 0\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni✝ : ι ⊕ ι'\nhi : i✝ ∈ s\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) =\n -∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i)\ni : ι\nx✝ : i ∈ Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s))\n⊢ (fun x => g x • v x) (Sum.inl i) ∈ span R (v '' range Sum.inl)",
"tactic": "exact smul_mem _ _ (subset_span ⟨Sum.inl i, mem_range_self _, rfl⟩)"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : ∀ (x : M), x ∈ span R (v '' range Sum.inl) → ∀ (y : M), y ∈ span R (v '' range Sum.inr) → x = y → x = 0\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\ni✝ : ι ⊕ ι'\nhi : i✝ ∈ s\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) =\n -∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i)\ni : ι'\nx✝ : i ∈ Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s))\n⊢ (fun x => g x • v x) (Sum.inr i) ∈ span R (v '' range Sum.inr)",
"tactic": "exact smul_mem _ _ (subset_span ⟨Sum.inr i, mem_range_self _, rfl⟩)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.inl\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : ∀ (x : M), x ∈ span R (v '' range Sum.inl) → ∀ (y : M), y ∈ span R (v '' range Sum.inr) → x = y → x = 0\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) =\n -∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i)\nA : ∑ c in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl c) = 0\ni : ι\nhi : Sum.inl i ∈ s\n⊢ g (Sum.inl i) = 0",
"tactic": "exact hl _ _ A i (Finset.mem_preimage.2 hi)"
},
{
"state_after": "case refine'_2.intro.intro.inr\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : ∀ (x : M), x ∈ span R (v '' range Sum.inl) → ∀ (y : M), y ∈ span R (v '' range Sum.inr) → x = y → x = 0\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) =\n -∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i)\nA : ∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i) = 0\ni : ι'\nhi : Sum.inr i ∈ s\n⊢ g (Sum.inr i) = 0",
"state_before": "case refine'_2.intro.intro.inr\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : ∀ (x : M), x ∈ span R (v '' range Sum.inl) → ∀ (y : M), y ∈ span R (v '' range Sum.inr) → x = y → x = 0\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) =\n -∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i)\nA : ∑ c in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl c) = 0\ni : ι'\nhi : Sum.inr i ∈ s\n⊢ g (Sum.inr i) = 0",
"tactic": "rw [this, neg_eq_zero] at A"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.inr\nι : Type u'\nι' : Type u_1\nR : Type u_2\nK : Type ?u.352990\nM : Type u_3\nM' : Type ?u.352996\nM'' : Type ?u.352999\nV : Type u\nV' : Type ?u.353004\nv✝ : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nv : ι ⊕ ι' → M\nhl : ∀ (s : Finset ι) (g : ι → R), ∑ i in s, g i • (v ∘ Sum.inl) i = 0 → ∀ (i : ι), i ∈ s → g i = 0\nhr : ∀ (s : Finset ι') (g : ι' → R), ∑ i in s, g i • (v ∘ Sum.inr) i = 0 → ∀ (i : ι'), i ∈ s → g i = 0\nhlr : ∀ (x : M), x ∈ span R (v '' range Sum.inl) → ∀ (y : M), y ∈ span R (v '' range Sum.inr) → x = y → x = 0\ns : Finset (ι ⊕ ι')\ng : ι ⊕ ι' → R\nhg : ∑ i in s, g i • v i = 0\nthis :\n ∑ i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inl i) =\n -∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i)\nA : ∑ i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr ⁻¹' ↑s)), (fun x => g x • v x) (Sum.inr i) = 0\ni : ι'\nhi : Sum.inr i ∈ s\n⊢ g (Sum.inr i) = 0",
"tactic": "exact hr _ _ A i (Finset.mem_preimage.2 hi)"
}
] |
[
679,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
642,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.iUnion_Ico_coe_nat
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.136095\nβ : Type ?u.136098\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ (⋃ (n : ℕ), Ico a ↑n) = Ici a \\ {⊤}",
"tactic": "simp only [← Ici_inter_Iio, ← inter_iUnion, iUnion_Iio_coe_nat, diff_eq]"
}
] |
[
889,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
888,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieIdeal.comap_incl_self
|
[
{
"state_after": "case h\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nm✝ : { x // x ∈ ↑I }\n⊢ m✝ ∈ comap (incl I) I ↔ m✝ ∈ ⊤",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n⊢ comap (incl I) I = ⊤",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nm✝ : { x // x ∈ ↑I }\n⊢ m✝ ∈ comap (incl I) I ↔ m✝ ∈ ⊤",
"tactic": "simp"
}
] |
[
1125,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1125,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.measure_union'
|
[] |
[
133,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/Analysis/Convex/Segment.lean
|
insert_endpoints_openSegment
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.40657\nG : Type ?u.40660\nι : Type ?u.40663\nπ : ι → Type ?u.40668\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nx✝ y✝ z x y : E\n⊢ [x-[𝕜]y] ⊆ insert x (insert y (openSegment 𝕜 x y))",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.40657\nG : Type ?u.40660\nι : Type ?u.40663\nπ : ι → Type ?u.40668\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nx✝ y✝ z x y : E\n⊢ insert x (insert y (openSegment 𝕜 x y)) = [x-[𝕜]y]",
"tactic": "simp only [subset_antisymm_iff, insert_subset, left_mem_segment, right_mem_segment,\n openSegment_subset_segment, true_and_iff]"
},
{
"state_after": "case intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.40657\nG : Type ?u.40660\nι : Type ?u.40663\nπ : ι → Type ?u.40668\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nx✝ y✝ z x y : E\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • x + b • y ∈ insert x (insert y (openSegment 𝕜 x y))",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.40657\nG : Type ?u.40660\nι : Type ?u.40663\nπ : ι → Type ?u.40668\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nx✝ y✝ z x y : E\n⊢ [x-[𝕜]y] ⊆ insert x (insert y (openSegment 𝕜 x y))",
"tactic": "rintro z ⟨a, b, ha, hb, hab, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.refine'_1\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.40657\nG : Type ?u.40660\nι : Type ?u.40663\nπ : ι → Type ?u.40668\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nx✝ y✝ z x y : E\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ b = 0 → a • x + b • y = x\n\ncase intro.intro.intro.intro.intro.refine'_2\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.40657\nG : Type ?u.40660\nι : Type ?u.40663\nπ : ι → Type ?u.40668\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nx✝ y✝ z x y : E\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhb' : 0 < b\n⊢ a = 0 → a • x + b • y = y\n\ncase intro.intro.intro.intro.intro.refine'_3\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.40657\nG : Type ?u.40660\nι : Type ?u.40663\nπ : ι → Type ?u.40668\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nx✝ y✝ z x y : E\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhb' : 0 < b\nha' : 0 < a\n⊢ a • x + b • y ∈ openSegment 𝕜 x y",
"state_before": "case intro.intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.40657\nG : Type ?u.40660\nι : Type ?u.40663\nπ : ι → Type ?u.40668\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nx✝ y✝ z x y : E\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • x + b • y ∈ insert x (insert y (openSegment 𝕜 x y))",
"tactic": "refine' hb.eq_or_gt.imp _ fun hb' => ha.eq_or_gt.imp _ fun ha' => _"
},
{
"state_after": "case intro.intro.intro.intro.intro.refine'_1\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.40657\nG : Type ?u.40660\nι : Type ?u.40663\nπ : ι → Type ?u.40668\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nx✝ y✝ z x y : E\na : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ 0\nhab : a + 0 = 1\n⊢ a • x + 0 • y = x",
"state_before": "case intro.intro.intro.intro.intro.refine'_1\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.40657\nG : Type ?u.40660\nι : Type ?u.40663\nπ : ι → Type ?u.40668\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nx✝ y✝ z x y : E\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ b = 0 → a • x + b • y = x",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.refine'_1\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.40657\nG : Type ?u.40660\nι : Type ?u.40663\nπ : ι → Type ?u.40668\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nx✝ y✝ z x y : E\na : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ 0\nhab : a + 0 = 1\n⊢ a • x + 0 • y = x",
"tactic": "rw [← add_zero a, hab, one_smul, zero_smul, add_zero]"
},
{
"state_after": "case intro.intro.intro.intro.intro.refine'_2\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.40657\nG : Type ?u.40660\nι : Type ?u.40663\nπ : ι → Type ?u.40668\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nx✝ y✝ z x y : E\nb : 𝕜\nhb : 0 ≤ b\nhb' : 0 < b\nha : 0 ≤ 0\nhab : 0 + b = 1\n⊢ 0 • x + b • y = y",
"state_before": "case intro.intro.intro.intro.intro.refine'_2\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.40657\nG : Type ?u.40660\nι : Type ?u.40663\nπ : ι → Type ?u.40668\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nx✝ y✝ z x y : E\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhb' : 0 < b\n⊢ a = 0 → a • x + b • y = y",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.refine'_2\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.40657\nG : Type ?u.40660\nι : Type ?u.40663\nπ : ι → Type ?u.40668\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nx✝ y✝ z x y : E\nb : 𝕜\nhb : 0 ≤ b\nhb' : 0 < b\nha : 0 ≤ 0\nhab : 0 + b = 1\n⊢ 0 • x + b • y = y",
"tactic": "rw [← zero_add b, hab, one_smul, zero_smul, zero_add]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.refine'_3\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.40657\nG : Type ?u.40660\nι : Type ?u.40663\nπ : ι → Type ?u.40668\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nx✝ y✝ z x y : E\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhb' : 0 < b\nha' : 0 < a\n⊢ a • x + b • y ∈ openSegment 𝕜 x y",
"tactic": "exact ⟨a, b, ha', hb', hab, rfl⟩"
}
] |
[
150,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
Sbtw.ne_right
|
[] |
[
251,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.subset_div
|
[] |
[
703,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
701,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.typein_lt_type
|
[] |
[
442,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
441,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.not_terminates_empty
|
[] |
[
405,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/Logic/Basic.lean
|
BAll.imp_left
|
[] |
[
1055,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1054,
1
] |
Mathlib/Topology/Maps.lean
|
inducing_iff_nhds
|
[] |
[
92,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Topology/Algebra/Module/Multilinear.lean
|
ContinuousMultilinearMap.coe_pi
|
[] |
[
264,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Topology/Homotopy/Basic.lean
|
ContinuousMap.HomotopyWith.ext
|
[] |
[
437,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/Topology/IsLocallyHomeomorph.lean
|
isLocallyHomeomorph_iff_isLocallyHomeomorphOn_univ
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.3146\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\ng : Y → Z\nf : X → Y\ns : Set X\nt : Set Y\n⊢ IsLocallyHomeomorph f ↔ IsLocallyHomeomorphOn f Set.univ",
"tactic": "simp only [IsLocallyHomeomorph, IsLocallyHomeomorphOn, Set.mem_univ, forall_true_left]"
}
] |
[
95,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/LinearAlgebra/Matrix/ZPow.lean
|
Matrix.pow_inv_comm'
|
[
{
"state_after": "case zero\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ m : ℕ\n⊢ A⁻¹ ^ m ⬝ A ^ Nat.zero = A ^ Nat.zero ⬝ A⁻¹ ^ m\n\ncase succ\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\nm : ℕ\n⊢ A⁻¹ ^ m ⬝ A ^ Nat.succ n = A ^ Nat.succ n ⬝ A⁻¹ ^ m",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm n : ℕ\n⊢ A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m",
"tactic": "induction' n with n IH generalizing m"
},
{
"state_after": "case succ.zero\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\n⊢ A⁻¹ ^ Nat.zero ⬝ A ^ Nat.succ n = A ^ Nat.succ n ⬝ A⁻¹ ^ Nat.zero\n\ncase succ.succ\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\nm : ℕ\n⊢ A⁻¹ ^ Nat.succ m ⬝ A ^ Nat.succ n = A ^ Nat.succ n ⬝ A⁻¹ ^ Nat.succ m",
"state_before": "case succ\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\nm : ℕ\n⊢ A⁻¹ ^ m ⬝ A ^ Nat.succ n = A ^ Nat.succ n ⬝ A⁻¹ ^ m",
"tactic": "cases' m with m m"
},
{
"state_after": "case succ.succ.inl.intro\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\nm : ℕ\nh : A⁻¹ ⬝ A = 1\nh' : A ⬝ A⁻¹ = 1\n⊢ A⁻¹ ^ Nat.succ m ⬝ A ^ Nat.succ n = A ^ Nat.succ n ⬝ A⁻¹ ^ Nat.succ m\n\ncase succ.succ.inr\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\nm : ℕ\nh : A⁻¹ = 0\n⊢ A⁻¹ ^ Nat.succ m ⬝ A ^ Nat.succ n = A ^ Nat.succ n ⬝ A⁻¹ ^ Nat.succ m",
"state_before": "case succ.succ\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\nm : ℕ\n⊢ A⁻¹ ^ Nat.succ m ⬝ A ^ Nat.succ n = A ^ Nat.succ n ⬝ A⁻¹ ^ Nat.succ m",
"tactic": "rcases nonsing_inv_cancel_or_zero A with (⟨h, h'⟩ | h)"
},
{
"state_after": "no goals",
"state_before": "case zero\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ m : ℕ\n⊢ A⁻¹ ^ m ⬝ A ^ Nat.zero = A ^ Nat.zero ⬝ A⁻¹ ^ m",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ.zero\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\n⊢ A⁻¹ ^ Nat.zero ⬝ A ^ Nat.succ n = A ^ Nat.succ n ⬝ A⁻¹ ^ Nat.zero",
"tactic": "simp"
},
{
"state_after": "case succ.succ.inl.intro\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\nm : ℕ\nh : A⁻¹ ⬝ A = 1\nh' : A ⬝ A⁻¹ = 1\n⊢ A⁻¹ ^ (m + 1) ⬝ A ^ (n + 1) = A ^ (n + 1) ⬝ A⁻¹ ^ (m + 1)",
"state_before": "case succ.succ.inl.intro\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\nm : ℕ\nh : A⁻¹ ⬝ A = 1\nh' : A ⬝ A⁻¹ = 1\n⊢ A⁻¹ ^ Nat.succ m ⬝ A ^ Nat.succ n = A ^ Nat.succ n ⬝ A⁻¹ ^ Nat.succ m",
"tactic": "simp only [Nat.succ_eq_add_one]"
},
{
"state_after": "no goals",
"state_before": "case succ.succ.inl.intro\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\nm : ℕ\nh : A⁻¹ ⬝ A = 1\nh' : A ⬝ A⁻¹ = 1\n⊢ A⁻¹ ^ (m + 1) ⬝ A ^ (n + 1) = A ^ (n + 1) ⬝ A⁻¹ ^ (m + 1)",
"tactic": "calc\n A⁻¹ ^ (m + 1) ⬝ A ^ (n + 1) = A⁻¹ ^ m ⬝ (A⁻¹ ⬝ A) ⬝ A ^ n := by\n simp only [pow_succ' A⁻¹, pow_succ A, mul_eq_mul, Matrix.mul_assoc]\n _ = A ^ n ⬝ A⁻¹ ^ m := by simp only [h, Matrix.mul_one, Matrix.one_mul, IH m]\n _ = A ^ n ⬝ (A ⬝ A⁻¹) ⬝ A⁻¹ ^ m := by simp only [h', Matrix.mul_one, Matrix.one_mul]\n _ = A ^ (n + 1) ⬝ A⁻¹ ^ (m + 1) := by\n simp only [pow_succ' A, pow_succ A⁻¹, mul_eq_mul, Matrix.mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\nm : ℕ\nh : A⁻¹ ⬝ A = 1\nh' : A ⬝ A⁻¹ = 1\n⊢ A⁻¹ ^ (m + 1) ⬝ A ^ (n + 1) = A⁻¹ ^ m ⬝ (A⁻¹ ⬝ A) ⬝ A ^ n",
"tactic": "simp only [pow_succ' A⁻¹, pow_succ A, mul_eq_mul, Matrix.mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\nm : ℕ\nh : A⁻¹ ⬝ A = 1\nh' : A ⬝ A⁻¹ = 1\n⊢ A⁻¹ ^ m ⬝ (A⁻¹ ⬝ A) ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m",
"tactic": "simp only [h, Matrix.mul_one, Matrix.one_mul, IH m]"
},
{
"state_after": "no goals",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\nm : ℕ\nh : A⁻¹ ⬝ A = 1\nh' : A ⬝ A⁻¹ = 1\n⊢ A ^ n ⬝ A⁻¹ ^ m = A ^ n ⬝ (A ⬝ A⁻¹) ⬝ A⁻¹ ^ m",
"tactic": "simp only [h', Matrix.mul_one, Matrix.one_mul]"
},
{
"state_after": "no goals",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\nm : ℕ\nh : A⁻¹ ⬝ A = 1\nh' : A ⬝ A⁻¹ = 1\n⊢ A ^ n ⬝ (A ⬝ A⁻¹) ⬝ A⁻¹ ^ m = A ^ (n + 1) ⬝ A⁻¹ ^ (m + 1)",
"tactic": "simp only [pow_succ' A, pow_succ A⁻¹, mul_eq_mul, Matrix.mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "case succ.succ.inr\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm✝ n : ℕ\nIH : ∀ (m : ℕ), A⁻¹ ^ m ⬝ A ^ n = A ^ n ⬝ A⁻¹ ^ m\nm : ℕ\nh : A⁻¹ = 0\n⊢ A⁻¹ ^ Nat.succ m ⬝ A ^ Nat.succ n = A ^ Nat.succ n ⬝ A⁻¹ ^ Nat.succ m",
"tactic": "simp [h]"
}
] |
[
72,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.sum_centroidWeightsIndicator
|
[] |
[
911,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
909,
1
] |
Mathlib/MeasureTheory/Group/Measure.lean
|
MeasureTheory.measure_lt_top_of_isCompact_of_isMulLeftInvariant
|
[
{
"state_after": "𝕜 : Type ?u.535820\nG : Type u_1\nH : Type ?u.535826\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : MeasurableSpace H\ninst✝⁴ : TopologicalSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : IsMulLeftInvariant μ\nU : Set G\nhU : IsOpen U\nh'U : Set.Nonempty (interior U)\nh : ↑↑μ U ≠ ⊤\nK : Set G\nhK : IsCompact K\n⊢ ↑↑μ K < ⊤",
"state_before": "𝕜 : Type ?u.535820\nG : Type u_1\nH : Type ?u.535826\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : MeasurableSpace H\ninst✝⁴ : TopologicalSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : IsMulLeftInvariant μ\nU : Set G\nhU : IsOpen U\nh'U : Set.Nonempty U\nh : ↑↑μ U ≠ ⊤\nK : Set G\nhK : IsCompact K\n⊢ ↑↑μ K < ⊤",
"tactic": "rw [← hU.interior_eq] at h'U"
},
{
"state_after": "case intro\n𝕜 : Type ?u.535820\nG : Type u_1\nH : Type ?u.535826\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : MeasurableSpace H\ninst✝⁴ : TopologicalSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : IsMulLeftInvariant μ\nU : Set G\nhU : IsOpen U\nh'U : Set.Nonempty (interior U)\nh : ↑↑μ U ≠ ⊤\nK : Set G\nhK : IsCompact K\nt : Finset G\nhKt : K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' U\n⊢ ↑↑μ K < ⊤",
"state_before": "𝕜 : Type ?u.535820\nG : Type u_1\nH : Type ?u.535826\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : MeasurableSpace H\ninst✝⁴ : TopologicalSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : IsMulLeftInvariant μ\nU : Set G\nhU : IsOpen U\nh'U : Set.Nonempty (interior U)\nh : ↑↑μ U ≠ ⊤\nK : Set G\nhK : IsCompact K\n⊢ ↑↑μ K < ⊤",
"tactic": "obtain ⟨t, hKt⟩ : ∃ t : Finset G, K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h : G => g * h) ⁻¹' U :=\n compact_covered_by_mul_left_translates hK h'U"
},
{
"state_after": "no goals",
"state_before": "case intro\n𝕜 : Type ?u.535820\nG : Type u_1\nH : Type ?u.535826\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : MeasurableSpace H\ninst✝⁴ : TopologicalSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : IsMulLeftInvariant μ\nU : Set G\nhU : IsOpen U\nh'U : Set.Nonempty (interior U)\nh : ↑↑μ U ≠ ⊤\nK : Set G\nhK : IsCompact K\nt : Finset G\nhKt : K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' U\n⊢ ↑↑μ K < ⊤",
"tactic": "calc\n μ K ≤ μ (⋃ (g : G) (_ : g ∈ t), (fun h : G => g * h) ⁻¹' U) := measure_mono hKt\n _ ≤ ∑ g in t, μ ((fun h : G => g * h) ⁻¹' U) := (measure_biUnion_finset_le _ _)\n _ = Finset.card t * μ U := by simp only [measure_preimage_mul, Finset.sum_const, nsmul_eq_mul]\n _ < ∞ := ENNReal.mul_lt_top (ENNReal.nat_ne_top _) h"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.535820\nG : Type u_1\nH : Type ?u.535826\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : MeasurableSpace H\ninst✝⁴ : TopologicalSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : IsMulLeftInvariant μ\nU : Set G\nhU : IsOpen U\nh'U : Set.Nonempty (interior U)\nh : ↑↑μ U ≠ ⊤\nK : Set G\nhK : IsCompact K\nt : Finset G\nhKt : K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' U\n⊢ ∑ g in t, ↑↑μ ((fun h => g * h) ⁻¹' U) = ↑(Finset.card t) * ↑↑μ U",
"tactic": "simp only [measure_preimage_mul, Finset.sum_const, nsmul_eq_mul]"
}
] |
[
591,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
582,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.prod.diag_map_fst_snd_comp
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nX✝ Y✝ : C\ninst✝ : HasLimitsOfShape (Discrete WalkingPair) C\nX X' Y Y' : C\ng : X ⟶ Y\ng' : X' ⟶ Y'\n⊢ diag (X ⨯ X') ≫ map (fst ≫ g) (snd ≫ g') = map g g'",
"tactic": "simp"
}
] |
[
815,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
813,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.cons_self_tail
|
[
{
"state_after": "case h\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\n⊢ cons (q 0) (tail q) j = q j",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\n⊢ cons (q 0) (tail q) = q",
"tactic": "ext j"
},
{
"state_after": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : j = 0\n⊢ cons (q 0) (tail q) j = q j\n\ncase neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\n⊢ cons (q 0) (tail q) j = q j",
"state_before": "case h\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\n⊢ cons (q 0) (tail q) j = q j",
"tactic": "by_cases h : j = 0"
},
{
"state_after": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : j = 0\n⊢ cons (q 0) (tail q) 0 = q 0",
"state_before": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : j = 0\n⊢ cons (q 0) (tail q) j = q j",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : j = 0\n⊢ cons (q 0) (tail q) 0 = q 0",
"tactic": "simp"
},
{
"state_after": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\nj' : Fin n := pred j h\n⊢ cons (q 0) (tail q) j = q j",
"state_before": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\n⊢ cons (q 0) (tail q) j = q j",
"tactic": "let j' := pred j h"
},
{
"state_after": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\nj' : Fin n := pred j h\nthis : succ j' = j\n⊢ cons (q 0) (tail q) j = q j",
"state_before": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\nj' : Fin n := pred j h\n⊢ cons (q 0) (tail q) j = q j",
"tactic": "have : j'.succ = j := succ_pred j h"
},
{
"state_after": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\nj' : Fin n := pred j h\nthis : succ j' = j\n⊢ cons (q 0) (tail q) (succ j') = q (succ j')",
"state_before": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\nj' : Fin n := pred j h\nthis : succ j' = j\n⊢ cons (q 0) (tail q) j = q j",
"tactic": "rw [← this]"
},
{
"state_after": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\nj' : Fin n := pred j h\nthis : succ j' = j\n⊢ cons (q 0) (fun i => q (succ i)) (succ j') = q (succ j')",
"state_before": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\nj' : Fin n := pred j h\nthis : succ j' = j\n⊢ cons (q 0) (tail q) (succ j') = q (succ j')",
"tactic": "unfold tail"
},
{
"state_after": "no goals",
"state_before": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\nj' : Fin n := pred j h\nthis : succ j' = j\n⊢ cons (q 0) (fun i => q (succ i)) (succ j') = q (succ j')",
"tactic": "rw [cons_succ]"
}
] |
[
145,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Analysis/Calculus/IteratedDeriv.lean
|
iteratedDerivWithin_eq_equiv_comp
|
[
{
"state_after": "case h\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.5253\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx✝ x : 𝕜\n⊢ iteratedDerivWithin n f s x =\n (↑(LinearIsometryEquiv.symm (ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F)) ∘ iteratedFDerivWithin 𝕜 n f s) x",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.5253\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\n⊢ iteratedDerivWithin n f s =\n ↑(LinearIsometryEquiv.symm (ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F)) ∘ iteratedFDerivWithin 𝕜 n f s",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.5253\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx✝ x : 𝕜\n⊢ iteratedDerivWithin n f s x =\n (↑(LinearIsometryEquiv.symm (ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F)) ∘ iteratedFDerivWithin 𝕜 n f s) x",
"tactic": "rfl"
}
] |
[
91,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.IsSpanning.card_verts
|
[
{
"state_after": "ι : Sort ?u.199392\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\ninst✝¹ : Fintype V\nG' : Subgraph G\ninst✝ : Fintype ↑G'.verts\nh : IsSpanning G'\n⊢ Finset.card Finset.univ = Fintype.card V",
"state_before": "ι : Sort ?u.199392\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\ninst✝¹ : Fintype V\nG' : Subgraph G\ninst✝ : Fintype ↑G'.verts\nh : IsSpanning G'\n⊢ Finset.card (Set.toFinset G'.verts) = Fintype.card V",
"tactic": "simp only [isSpanning_iff.1 h, Set.toFinset_univ]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.199392\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\ninst✝¹ : Fintype V\nG' : Subgraph G\ninst✝ : Fintype ↑G'.verts\nh : IsSpanning G'\n⊢ Finset.card Finset.univ = Fintype.card V",
"tactic": "congr"
}
] |
[
794,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
791,
1
] |
Mathlib/Order/Max.lean
|
IsTop.prod_mk
|
[] |
[
410,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/Topology/Algebra/Affine.lean
|
AffineMap.lineMap_continuous
|
[] |
[
56,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/AlgebraicTopology/SimplicialObject.lean
|
CategoryTheory.CosimplicialObject.δ_comp_δ
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nX : CosimplicialObject C\nn : ℕ\ni j : Fin (n + 2)\nH : i ≤ j\n⊢ X.map (SimplexCategory.δ i) ≫ X.map (SimplexCategory.δ (Fin.succ j)) =\n X.map (SimplexCategory.δ j) ≫ X.map (SimplexCategory.δ (↑Fin.castSucc i))",
"state_before": "C : Type u\ninst✝ : Category C\nX : CosimplicialObject C\nn : ℕ\ni j : Fin (n + 2)\nH : i ≤ j\n⊢ δ X i ≫ δ X (Fin.succ j) = δ X j ≫ δ X (↑Fin.castSucc i)",
"tactic": "dsimp [δ]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX : CosimplicialObject C\nn : ℕ\ni j : Fin (n + 2)\nH : i ≤ j\n⊢ X.map (SimplexCategory.δ i) ≫ X.map (SimplexCategory.δ (Fin.succ j)) =\n X.map (SimplexCategory.δ j) ≫ X.map (SimplexCategory.δ (↑Fin.castSucc i))",
"tactic": "simp only [← X.map_comp, SimplexCategory.δ_comp_δ H]"
}
] |
[
485,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
482,
1
] |
Mathlib/Topology/Maps.lean
|
OpenEmbedding.tendsto_nhds_iff'
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.238827\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α → β\nhf : OpenEmbedding f\ng : β → γ\nl : Filter γ\na : α\n⊢ map g (𝓝 (f a)) ≤ l ↔ Tendsto g (𝓝 (f a)) l",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.238827\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α → β\nhf : OpenEmbedding f\ng : β → γ\nl : Filter γ\na : α\n⊢ Tendsto (g ∘ f) (𝓝 a) l ↔ Tendsto g (𝓝 (f a)) l",
"tactic": "rw [Tendsto, ← map_map, hf.map_nhds_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.238827\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α → β\nhf : OpenEmbedding f\ng : β → γ\nl : Filter γ\na : α\n⊢ map g (𝓝 (f a)) ≤ l ↔ Tendsto g (𝓝 (f a)) l",
"tactic": "rfl"
}
] |
[
575,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
573,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.intDegree_mul
|
[
{
"state_after": "K : Type u\ninst✝ : Field K\nx y : RatFunc K\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ ↑(natDegree (num (x * y))) + (↑(natDegree (denom x)) + ↑(natDegree (denom y))) =\n ↑(natDegree (num x)) + ↑(natDegree (num y)) + ↑(natDegree (denom (x * y)))",
"state_before": "K : Type u\ninst✝ : Field K\nx y : RatFunc K\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ intDegree (x * y) = intDegree x + intDegree y",
"tactic": "simp only [intDegree, add_sub, sub_add, sub_sub_eq_add_sub, sub_sub, sub_eq_sub_iff_add_eq_add]"
},
{
"state_after": "K : Type u\ninst✝ : Field K\nx y : RatFunc K\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ natDegree (num (x * y)) + (natDegree (denom x) + natDegree (denom y)) =\n natDegree (num x) + natDegree (num y) + natDegree (denom (x * y))",
"state_before": "K : Type u\ninst✝ : Field K\nx y : RatFunc K\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ ↑(natDegree (num (x * y))) + (↑(natDegree (denom x)) + ↑(natDegree (denom y))) =\n ↑(natDegree (num x)) + ↑(natDegree (num y)) + ↑(natDegree (denom (x * y)))",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝ : Field K\nx y : RatFunc K\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ natDegree (num (x * y)) + (natDegree (denom x) + natDegree (denom y)) =\n natDegree (num x) + natDegree (num y) + natDegree (denom (x * y))",
"tactic": "rw [← Polynomial.natDegree_mul x.denom_ne_zero y.denom_ne_zero, ←\n Polynomial.natDegree_mul (RatFunc.num_ne_zero (mul_ne_zero hx hy))\n (mul_ne_zero x.denom_ne_zero y.denom_ne_zero),\n ← Polynomial.natDegree_mul (RatFunc.num_ne_zero hx) (RatFunc.num_ne_zero hy), ←\n Polynomial.natDegree_mul (mul_ne_zero (RatFunc.num_ne_zero hx) (RatFunc.num_ne_zero hy))\n (x * y).denom_ne_zero,\n RatFunc.num_denom_mul]"
}
] |
[
1610,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1600,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
|
Asymptotics.IsLittleO.rpow
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nr c✝ : ℝ\nl : Filter α\nf g : α → ℝ\nhr : 0 < r\nhg : 0 ≤ᶠ[l] g\nh : f =o[l] g\nc : ℝ\nhc : 0 < c\n⊢ (c ^ r⁻¹) ^ r = c",
"tactic": "rw [← rpow_mul hc.le, inv_mul_cancel hr.ne', Real.rpow_one]"
}
] |
[
240,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/Order/Lattice.lean
|
Monotone.max
|
[] |
[
1087,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1084,
11
] |
Mathlib/Algebra/BigOperators/Finprod.lean
|
finprod_mem_congr
|
[] |
[
582,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/Data/List/MinMax.lean
|
List.not_lt_minimum_of_mem'
|
[] |
[
338,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.IsFundamentalSequence.succ
|
[
{
"state_after": "case refine'_1\nα : Type ?u.69314\nr : α → α → Prop\na o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal\n⊢ 1 ≤ ord (cof (succ o))\n\ncase refine'_2\nα : Type ?u.69314\nr : α → α → Prop\na o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal\ni j : Ordinal\nhi : i < 1\nhj : j < 1\nh : i < j\n⊢ (fun x x => o) i hi < (fun x x => o) j hj",
"state_before": "α : Type ?u.69314\nr : α → α → Prop\na o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal\n⊢ IsFundamentalSequence (succ o) 1 fun x x => o",
"tactic": "refine' ⟨_, @fun i j hi hj h => _, blsub_const Ordinal.one_ne_zero o⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type ?u.69314\nr : α → α → Prop\na o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal\n⊢ 1 ≤ ord (cof (succ o))",
"tactic": "rw [cof_succ, ord_one]"
},
{
"state_after": "case refine'_2\nα : Type ?u.69314\nr : α → α → Prop\na o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal\ni j : Ordinal\nhi✝ : i < 1\nhi : i = 0\nhj✝ : j < 1\nhj : j = 0\nh : i < j\n⊢ (fun x x => o) i hi✝ < (fun x x => o) j hj✝",
"state_before": "case refine'_2\nα : Type ?u.69314\nr : α → α → Prop\na o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal\ni j : Ordinal\nhi : i < 1\nhj : j < 1\nh : i < j\n⊢ (fun x x => o) i hi < (fun x x => o) j hj",
"tactic": "rw [lt_one_iff_zero] at hi hj"
},
{
"state_after": "case refine'_2\nα : Type ?u.69314\nr : α → α → Prop\na o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal\ni j : Ordinal\nhi✝ : i < 1\nhi : i = 0\nhj✝ : j < 1\nhj : j = 0\nh : 0 < 0\n⊢ (fun x x => o) i hi✝ < (fun x x => o) j hj✝",
"state_before": "case refine'_2\nα : Type ?u.69314\nr : α → α → Prop\na o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal\ni j : Ordinal\nhi✝ : i < 1\nhi : i = 0\nhj✝ : j < 1\nhj : j = 0\nh : i < j\n⊢ (fun x x => o) i hi✝ < (fun x x => o) j hj✝",
"tactic": "rw [hi, hj] at h"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type ?u.69314\nr : α → α → Prop\na o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal\ni j : Ordinal\nhi✝ : i < 1\nhi : i = 0\nhj✝ : j < 1\nhj : j = 0\nh : 0 < 0\n⊢ (fun x x => o) i hi✝ < (fun x x => o) j hj✝",
"tactic": "exact h.false.elim"
}
] |
[
591,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
586,
11
] |
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
VitaliFamily.ae_eventually_measure_pos
|
[
{
"state_after": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\n⊢ ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\n⊢ ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a",
"tactic": "set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs"
},
{
"state_after": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\n⊢ ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\n⊢ ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a",
"tactic": "simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs"
},
{
"state_after": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\n⊢ ↑↑μ s = 0",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\n⊢ ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a",
"tactic": "change μ s = 0"
},
{
"state_after": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\n⊢ ↑↑μ s = 0",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\n⊢ ↑↑μ s = 0",
"tactic": "let f : α → Set (Set α) := fun _ => {a | μ a = 0}"
},
{
"state_after": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nh : FineSubfamilyOn v f s\n⊢ ↑↑μ s = 0",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\n⊢ ↑↑μ s = 0",
"tactic": "have h : v.FineSubfamilyOn f s := by\n intro x hx ε εpos\n rw [hs] at hx\n simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx\n rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩\n exact ⟨a, ⟨a_sets, μa⟩, ax⟩"
},
{
"state_after": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nh : FineSubfamilyOn v f s\n⊢ ↑↑μ s ≤ 0",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nh : FineSubfamilyOn v f s\n⊢ ↑↑μ s = 0",
"tactic": "refine' le_antisymm _ bot_le"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nh : FineSubfamilyOn v f s\n⊢ ↑↑μ s ≤ 0",
"tactic": "calc\n μ s ≤ ∑' x : h.index, μ (h.covering x) := h.measure_le_tsum\n _ = ∑' x : h.index, 0 := by congr; ext1 x; exact h.covering_mem x.2\n _ = 0 := by simp only [tsum_zero, add_zero]"
},
{
"state_after": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nx : α\nhx : x ∈ s\nε : ℝ\nεpos : ε > 0\n⊢ ∃ a, a ∈ setsAt v x ∩ f x ∧ a ⊆ closedBall x ε",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\n⊢ FineSubfamilyOn v f s",
"tactic": "intro x hx ε εpos"
},
{
"state_after": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nx : α\nhx : x ∈ {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nε : ℝ\nεpos : ε > 0\n⊢ ∃ a, a ∈ setsAt v x ∩ f x ∧ a ⊆ closedBall x ε",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nx : α\nhx : x ∈ s\nε : ℝ\nεpos : ε > 0\n⊢ ∃ a, a ∈ setsAt v x ∩ f x ∧ a ⊆ closedBall x ε",
"tactic": "rw [hs] at hx"
},
{
"state_after": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nx : α\nε : ℝ\nεpos : ε > 0\nhx : ∀ (ε : ℝ), 0 < ε → ∃ a, a ∈ setsAt v x ∧ a ⊆ closedBall x ε ∧ ↑↑μ a = 0\n⊢ ∃ a, a ∈ setsAt v x ∩ f x ∧ a ⊆ closedBall x ε",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nx : α\nhx : x ∈ {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nε : ℝ\nεpos : ε > 0\n⊢ ∃ a, a ∈ setsAt v x ∩ f x ∧ a ⊆ closedBall x ε",
"tactic": "simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nx : α\nε : ℝ\nεpos : ε > 0\nhx : ∀ (ε : ℝ), 0 < ε → ∃ a, a ∈ setsAt v x ∧ a ⊆ closedBall x ε ∧ ↑↑μ a = 0\na : Set α\na_sets : a ∈ setsAt v x\nax : a ⊆ closedBall x ε\nμa : ↑↑μ a = 0\n⊢ ∃ a, a ∈ setsAt v x ∩ f x ∧ a ⊆ closedBall x ε",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nx : α\nε : ℝ\nεpos : ε > 0\nhx : ∀ (ε : ℝ), 0 < ε → ∃ a, a ∈ setsAt v x ∧ a ⊆ closedBall x ε ∧ ↑↑μ a = 0\n⊢ ∃ a, a ∈ setsAt v x ∩ f x ∧ a ⊆ closedBall x ε",
"tactic": "rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nx : α\nε : ℝ\nεpos : ε > 0\nhx : ∀ (ε : ℝ), 0 < ε → ∃ a, a ∈ setsAt v x ∧ a ⊆ closedBall x ε ∧ ↑↑μ a = 0\na : Set α\na_sets : a ∈ setsAt v x\nax : a ⊆ closedBall x ε\nμa : ↑↑μ a = 0\n⊢ ∃ a, a ∈ setsAt v x ∩ f x ∧ a ⊆ closedBall x ε",
"tactic": "exact ⟨a, ⟨a_sets, μa⟩, ax⟩"
},
{
"state_after": "case e_f\nα : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nh : FineSubfamilyOn v f s\n⊢ (fun x => ↑↑μ (FineSubfamilyOn.covering h ↑x)) = fun x => 0",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nh : FineSubfamilyOn v f s\n⊢ (∑' (x : ↑(FineSubfamilyOn.index h)), ↑↑μ (FineSubfamilyOn.covering h ↑x)) = ∑' (x : ↑(FineSubfamilyOn.index h)), 0",
"tactic": "congr"
},
{
"state_after": "case e_f.h\nα : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nh : FineSubfamilyOn v f s\nx : ↑(FineSubfamilyOn.index h)\n⊢ ↑↑μ (FineSubfamilyOn.covering h ↑x) = 0",
"state_before": "case e_f\nα : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nh : FineSubfamilyOn v f s\n⊢ (fun x => ↑↑μ (FineSubfamilyOn.covering h ↑x)) = fun x => 0",
"tactic": "ext1 x"
},
{
"state_after": "no goals",
"state_before": "case e_f.h\nα : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nh : FineSubfamilyOn v f s\nx : ↑(FineSubfamilyOn.index h)\n⊢ ↑↑μ (FineSubfamilyOn.covering h ↑x) = 0",
"tactic": "exact h.covering_mem x.2"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.472138\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}\nf : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}\nh : FineSubfamilyOn v f s\n⊢ (∑' (x : ↑(FineSubfamilyOn.index h)), 0) = 0",
"tactic": "simp only [tsum_zero, add_zero]"
}
] |
[
119,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Analysis/Convex/Complex.lean
|
convex_halfspace_re_gt
|
[] |
[
31,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
30,
1
] |
Std/Logic.lean
|
and_imp
|
[] |
[
314,
57
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
313,
9
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.exists_ball_inter_eq_singleton_of_mem_discrete
|
[] |
[
1122,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1120,
1
] |
Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
|
Matrix.Nondegenerate.toBilin'
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.2760443\nM✝ : Type ?u.2760446\ninst✝¹⁹ : Semiring R\ninst✝¹⁸ : AddCommMonoid M✝\ninst✝¹⁷ : Module R M✝\nR₁ : Type ?u.2760482\nM₁ : Type ?u.2760485\ninst✝¹⁶ : Ring R₁\ninst✝¹⁵ : AddCommGroup M₁\ninst✝¹⁴ : Module R₁ M₁\nR₂ : Type ?u.2761094\nM₂ : Type ?u.2761097\ninst✝¹³ : CommSemiring R₂\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2761287\ninst✝¹⁰ : CommRing R₃\ninst✝⁹ : AddCommGroup M₃\ninst✝⁸ : Module R₃ M₃\nV : Type ?u.2761875\nK : Type ?u.2761878\ninst✝⁷ : Field K\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module K V\nB : BilinForm R M✝\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nA : Type ?u.2763092\ninst✝⁴ : CommRing A\ninst✝³ : IsDomain A\ninst✝² : Module A M₃\nB₃ : BilinForm A M₃\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nM : Matrix ι ι R₃\nh : Matrix.Nondegenerate M\nx : ι → R₃\nhx : ∀ (n : ι → R₃), bilin (↑Matrix.toBilin' M) x n = 0\ny : ι → R₃\n⊢ x ⬝ᵥ mulVec M y = 0",
"tactic": "simpa only [toBilin'_apply'] using hx y"
}
] |
[
554,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
552,
1
] |
Mathlib/Data/List/BigOperators/Basic.lean
|
List.length_pos_of_one_lt_prod
|
[] |
[
200,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
PSet.mem_wf_aux
|
[
{
"state_after": "case mk.intro\nα : Type u\nA : α → PSet\nβ : Type u\nB : β → PSet\nH : Equiv (mk α A) (mk β B)\nγ : Type u\nC : γ → PSet\nb : Type (mk β B)\nhc : Equiv (mk γ C) (Func (mk β B) b)\n⊢ Acc (fun x x_1 => x ∈ x_1) (mk γ C)",
"state_before": "α : Type u\nA : α → PSet\nβ : Type u\nB : β → PSet\nH : Equiv (mk α A) (mk β B)\n⊢ ∀ (y : PSet), y ∈ mk β B → Acc (fun x x_1 => x ∈ x_1) y",
"tactic": "rintro ⟨γ, C⟩ ⟨b, hc⟩"
},
{
"state_after": "case mk.intro.intro\nα : Type u\nA : α → PSet\nβ : Type u\nB : β → PSet\nH : Equiv (mk α A) (mk β B)\nγ : Type u\nC : γ → PSet\nb : Type (mk β B)\nhc : Equiv (mk γ C) (Func (mk β B) b)\na : Type (mk α A)\nha : Equiv (Func (mk α A) a) (Func (mk β B) b)\n⊢ Acc (fun x x_1 => x ∈ x_1) (mk γ C)",
"state_before": "case mk.intro\nα : Type u\nA : α → PSet\nβ : Type u\nB : β → PSet\nH : Equiv (mk α A) (mk β B)\nγ : Type u\nC : γ → PSet\nb : Type (mk β B)\nhc : Equiv (mk γ C) (Func (mk β B) b)\n⊢ Acc (fun x x_1 => x ∈ x_1) (mk γ C)",
"tactic": "cases' H.exists_right b with a ha"
},
{
"state_after": "case mk.intro.intro\nα : Type u\nA : α → PSet\nβ : Type u\nB : β → PSet\nH✝ : Equiv (mk α A) (mk β B)\nγ : Type u\nC : γ → PSet\nb : Type (mk β B)\nhc : Equiv (mk γ C) (Func (mk β B) b)\na : Type (mk α A)\nha : Equiv (Func (mk α A) a) (Func (mk β B) b)\nH : Equiv (Func (mk α A) a) (mk γ C)\n⊢ Acc (fun x x_1 => x ∈ x_1) (mk γ C)",
"state_before": "case mk.intro.intro\nα : Type u\nA : α → PSet\nβ : Type u\nB : β → PSet\nH : Equiv (mk α A) (mk β B)\nγ : Type u\nC : γ → PSet\nb : Type (mk β B)\nhc : Equiv (mk γ C) (Func (mk β B) b)\na : Type (mk α A)\nha : Equiv (Func (mk α A) a) (Func (mk β B) b)\n⊢ Acc (fun x x_1 => x ∈ x_1) (mk γ C)",
"tactic": "have H := ha.trans hc.symm"
},
{
"state_after": "case mk.intro.intro\nα : Type u\nA : α → PSet\nβ : Type u\nB : β → PSet\nH✝ : Equiv (mk α A) (mk β B)\nγ : Type u\nC : γ → PSet\nb : Type (mk β B)\nhc : Equiv (mk γ C) (Func (mk β B) b)\na : Type (mk α A)\nha : Equiv (Func (mk α A) a) (Func (mk β B) b)\nH : Equiv (A a) (mk γ C)\n⊢ Acc (fun x x_1 => x ∈ x_1) (mk γ C)",
"state_before": "case mk.intro.intro\nα : Type u\nA : α → PSet\nβ : Type u\nB : β → PSet\nH✝ : Equiv (mk α A) (mk β B)\nγ : Type u\nC : γ → PSet\nb : Type (mk β B)\nhc : Equiv (mk γ C) (Func (mk β B) b)\na : Type (mk α A)\nha : Equiv (Func (mk α A) a) (Func (mk β B) b)\nH : Equiv (Func (mk α A) a) (mk γ C)\n⊢ Acc (fun x x_1 => x ∈ x_1) (mk γ C)",
"tactic": "rw [mk_func] at H"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro\nα : Type u\nA : α → PSet\nβ : Type u\nB : β → PSet\nH✝ : Equiv (mk α A) (mk β B)\nγ : Type u\nC : γ → PSet\nb : Type (mk β B)\nhc : Equiv (mk γ C) (Func (mk β B) b)\na : Type (mk α A)\nha : Equiv (Func (mk α A) a) (Func (mk β B) b)\nH : Equiv (A a) (mk γ C)\n⊢ Acc (fun x x_1 => x ∈ x_1) (mk γ C)",
"tactic": "exact mem_wf_aux H"
}
] |
[
321,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
9
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.cos_pos_of_mem_Ioo
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x ∈ Ioo (-(π / 2)) (π / 2)\n⊢ 0 < x + π / 2",
"tactic": "linarith [hx.1]"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x ∈ Ioo (-(π / 2)) (π / 2)\n⊢ x + π / 2 < π",
"tactic": "linarith [hx.2]"
}
] |
[
465,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
464,
1
] |
Mathlib/Order/CompleteLattice.lean
|
monotone_sInf_of_monotone
|
[] |
[
1814,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1813,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.set_eventuallyEq_iff_inf_principal
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.209553\nι : Sort x\ns t : Set α\nl : Filter α\n⊢ s =ᶠ[l] t ↔ l ⊓ 𝓟 s = l ⊓ 𝓟 t",
"tactic": "simp only [eventuallyLE_antisymm_iff, le_antisymm_iff, set_eventuallyLE_iff_inf_principal_le]"
}
] |
[
1749,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1747,
1
] |
Mathlib/Topology/Homotopy/Equiv.lean
|
ContinuousMap.HomotopyEquiv.symm_trans
|
[] |
[
162,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Topology/Separation.lean
|
IsCompact.inter
|
[] |
[
1297,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1295,
1
] |
Mathlib/Order/BoundedOrder.lean
|
Monotone.ball
|
[] |
[
588,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
587,
1
] |
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