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/Logic/Equiv/Set.lean
|
Equiv.subset_image'
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Sort u\nβ✝ : Sort v\nγ : Sort w\nα : Type u_1\nβ : Type u_2\ne : α ≃ β\ns : Set α\nt : Set β\n⊢ s ⊆ ↑e.symm '' t ↔ ↑e.symm.symm '' s ⊆ t",
"tactic": "rw [e.symm.subset_image]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Sort u\nβ✝ : Sort v\nγ : Sort w\nα : Type u_1\nβ : Type u_2\ne : α ≃ β\ns : Set α\nt : Set β\n⊢ ↑e.symm.symm '' s ⊆ t ↔ ↑e '' s ⊆ t",
"tactic": "rw [e.symm_symm]"
}
] |
[
74,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
11
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
csSup_le
|
[] |
[
447,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
446,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.mem_top
|
[] |
[
622,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.compl_image
|
[] |
[
530,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.aeval_X_left_apply
|
[] |
[
1463,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1462,
1
] |
Mathlib/Data/Nat/Digits.lean
|
Nat.dvd_ofDigits_sub_ofDigits
|
[
{
"state_after": "case nil\nn : ℕ\nα : Type u_1\ninst✝ : CommRing α\na b k : α\nh : k ∣ a - b\n⊢ k ∣ ofDigits a [] - ofDigits b []\n\ncase cons\nn : ℕ\nα : Type u_1\ninst✝ : CommRing α\na b k : α\nh : k ∣ a - b\nd : ℕ\nL : List ℕ\nih : k ∣ ofDigits a L - ofDigits b L\n⊢ k ∣ ofDigits a (d :: L) - ofDigits b (d :: L)",
"state_before": "n : ℕ\nα : Type u_1\ninst✝ : CommRing α\na b k : α\nh : k ∣ a - b\nL : List ℕ\n⊢ k ∣ ofDigits a L - ofDigits b L",
"tactic": "induction' L with d L ih"
},
{
"state_after": "case nil\nn : ℕ\nα : Type u_1\ninst✝ : CommRing α\na b k : α\nh : k ∣ a - b\n⊢ k ∣ 0 - 0",
"state_before": "case nil\nn : ℕ\nα : Type u_1\ninst✝ : CommRing α\na b k : α\nh : k ∣ a - b\n⊢ k ∣ ofDigits a [] - ofDigits b []",
"tactic": "change k ∣ 0 - 0"
},
{
"state_after": "no goals",
"state_before": "case nil\nn : ℕ\nα : Type u_1\ninst✝ : CommRing α\na b k : α\nh : k ∣ a - b\n⊢ k ∣ 0 - 0",
"tactic": "simp"
},
{
"state_after": "case cons\nn : ℕ\nα : Type u_1\ninst✝ : CommRing α\na b k : α\nh : k ∣ a - b\nd : ℕ\nL : List ℕ\nih : k ∣ ofDigits a L - ofDigits b L\n⊢ k ∣ a * ofDigits a L - b * ofDigits b L",
"state_before": "case cons\nn : ℕ\nα : Type u_1\ninst✝ : CommRing α\na b k : α\nh : k ∣ a - b\nd : ℕ\nL : List ℕ\nih : k ∣ ofDigits a L - ofDigits b L\n⊢ k ∣ ofDigits a (d :: L) - ofDigits b (d :: L)",
"tactic": "simp only [ofDigits, add_sub_add_left_eq_sub]"
},
{
"state_after": "no goals",
"state_before": "case cons\nn : ℕ\nα : Type u_1\ninst✝ : CommRing α\na b k : α\nh : k ∣ a - b\nd : ℕ\nL : List ℕ\nih : k ∣ ofDigits a L - ofDigits b L\n⊢ k ∣ a * ofDigits a L - b * ofDigits b L",
"tactic": "exact dvd_mul_sub_mul h ih"
}
] |
[
502,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
496,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
upperBounds_insert
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns✝ t : Set α\na✝ b a : α\ns : Set α\n⊢ upperBounds (insert a s) = Ici a ∩ upperBounds s",
"tactic": "rw [insert_eq, upperBounds_union, upperBounds_singleton]"
}
] |
[
973,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
971,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.dist_zero_of_empty
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.306222\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : IsEmpty α\n⊢ dist f g = 0",
"tactic": "rw [(ext isEmptyElim : f = g), dist_self]"
}
] |
[
250,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/NumberTheory/Padics/RingHoms.lean
|
PadicInt.lift_sub_val_mem_span
|
[
{
"state_after": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\n⊢ ↑(lift f_compat) r - ↑(ZMod.val (↑(f n) r)) ∈ Ideal.span {↑p ^ n}",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn : ℕ\n⊢ ↑(lift f_compat) r - ↑(ZMod.val (↑(f n) r)) ∈ Ideal.span {↑p ^ n}",
"tactic": "obtain ⟨k, hk⟩ :=\n limNthHom_spec f_compat r _\n (show (0 : ℝ) < (p : ℝ) ^ (-n : ℤ) from Nat.zpow_pos_of_pos hp_prime.1.pos _)"
},
{
"state_after": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k))‖ ≤ ↑p ^ (-↑n)\n⊢ ↑(lift f_compat) r - ↑(ZMod.val (↑(f n) r)) ∈ Ideal.span {↑p ^ n}",
"state_before": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\n⊢ ↑(lift f_compat) r - ↑(ZMod.val (↑(f n) r)) ∈ Ideal.span {↑p ^ n}",
"tactic": "have := le_of_lt (hk (max n k) (le_max_right _ _))"
},
{
"state_after": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑(lift f_compat) r - ↑(ZMod.val (↑(f n) r)) ∈ Ideal.span {↑p ^ n}",
"state_before": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k))‖ ≤ ↑p ^ (-↑n)\n⊢ ↑(lift f_compat) r - ↑(ZMod.val (↑(f n) r)) ∈ Ideal.span {↑p ^ n}",
"tactic": "rw [norm_le_pow_iff_mem_span_pow] at this"
},
{
"state_after": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ limNthHom f_compat r - ↑(ZMod.val (↑(f n) r)) ∈ Ideal.span {↑p ^ n}",
"state_before": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑(lift f_compat) r - ↑(ZMod.val (↑(f n) r)) ∈ Ideal.span {↑p ^ n}",
"tactic": "dsimp [lift]"
},
{
"state_after": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑(nthHom f r (max n k)) - ↑(ZMod.val (↑(f n) r)) + (limNthHom f_compat r - ↑(nthHom f r (max n k))) ∈\n Ideal.span {↑p ^ n}",
"state_before": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ limNthHom f_compat r - ↑(ZMod.val (↑(f n) r)) ∈ Ideal.span {↑p ^ n}",
"tactic": "rw [sub_eq_sub_add_sub (limNthHom f_compat r) _ ↑(nthHom f r (max n k))]"
},
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑(nthHom f r (max n k)) - ↑(ZMod.val (↑(f n) r)) ∈ Ideal.span {↑p ^ n}",
"state_before": "case intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑(nthHom f r (max n k)) - ↑(ZMod.val (↑(f n) r)) + (limNthHom f_compat r - ↑(nthHom f r (max n k))) ∈\n Ideal.span {↑p ^ n}",
"tactic": "apply Ideal.add_mem _ _ this"
},
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑p ^ n ∣ ↑(nthHom f r (max n k)) - ↑(ZMod.val (↑(f n) r))",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑(nthHom f r (max n k)) - ↑(ZMod.val (↑(f n) r)) ∈ Ideal.span {↑p ^ n}",
"tactic": "rw [Ideal.mem_span_singleton]"
},
{
"state_after": "case h.e'_3\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑p ^ n = ↑(Int.castRingHom ℤ_[p]) (↑p ^ n)\n\ncase h.e'_4\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑(nthHom f r (max n k)) - ↑(ZMod.val (↑(f n) r)) =\n ↑(Int.castRingHom ℤ_[p]) (nthHom (fun k2 => f k2) r (max n k) - nthHom (fun k2 => f k2) r n)",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑p ^ n ∣ ↑(nthHom f r (max n k)) - ↑(ZMod.val (↑(f n) r))",
"tactic": "convert\n (Int.castRingHom ℤ_[p]).map_dvd (pow_dvd_nthHom_sub f_compat r n (max n k) (le_max_left _ _))"
},
{
"state_after": "case h.e'_3\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑p ^ n = ↑(Int.castRingHom ℤ_[p]) ↑p ^ n",
"state_before": "case h.e'_3\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑p ^ n = ↑(Int.castRingHom ℤ_[p]) (↑p ^ n)",
"tactic": "rw [map_pow]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑p ^ n = ↑(Int.castRingHom ℤ_[p]) ↑p ^ n",
"tactic": "rfl"
},
{
"state_after": "case h.e'_4\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑(nthHom f r (max n k)) - ↑(ZMod.val (↑(f n) r)) =\n ↑(Int.castRingHom ℤ_[p]) (nthHom (fun k2 => f k2) r (max n k)) -\n ↑(Int.castRingHom ℤ_[p]) (nthHom (fun k2 => f k2) r n)",
"state_before": "case h.e'_4\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑(nthHom f r (max n k)) - ↑(ZMod.val (↑(f n) r)) =\n ↑(Int.castRingHom ℤ_[p]) (nthHom (fun k2 => f k2) r (max n k) - nthHom (fun k2 => f k2) r n)",
"tactic": "rw [map_sub]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\nn k : ℕ\nhk : ∀ (n_1 : ℕ), n_1 ≥ k → ‖limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r n_1)‖ < ↑p ^ (-↑n)\nthis : limNthHom f_compat r - ↑(nthHom (fun k2 => f k2) r (max n k)) ∈ Ideal.span {↑p ^ n}\n⊢ ↑(nthHom f r (max n k)) - ↑(ZMod.val (↑(f n) r)) =\n ↑(Int.castRingHom ℤ_[p]) (nthHom (fun k2 => f k2) r (max n k)) -\n ↑(Int.castRingHom ℤ_[p]) (nthHom (fun k2 => f k2) r n)",
"tactic": "rfl"
}
] |
[
635,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
nonpos_of_mul_nonpos_right
|
[] |
[
833,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
832,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.LeftInverse.surjective
|
[] |
[
362,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.injective
|
[] |
[
143,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
11
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.AEFinStronglyMeasurable.sub
|
[] |
[
1894,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1891,
11
] |
Mathlib/Order/WellFounded.lean
|
Function.not_lt_argminOn
|
[] |
[
212,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/SetTheory/Lists.lean
|
Lists'.subset_def
|
[
{
"state_after": "α : Type u_1\nl₁ l₂ : Lists' α true\nH : ∀ (a : Lists α), a ∈ toList l₁ → a ∈ l₂\n⊢ ofList (toList l₁) ⊆ l₂",
"state_before": "α : Type u_1\nl₁ l₂ : Lists' α true\nH : ∀ (a : Lists α), a ∈ toList l₁ → a ∈ l₂\n⊢ l₁ ⊆ l₂",
"tactic": "rw [← of_toList l₁]"
},
{
"state_after": "α : Type u_1\nl₁ l₂ : Lists' α true\n⊢ (∀ (a : Lists α), a ∈ toList l₁ → a ∈ l₂) → ofList (toList l₁) ⊆ l₂",
"state_before": "α : Type u_1\nl₁ l₂ : Lists' α true\nH : ∀ (a : Lists α), a ∈ toList l₁ → a ∈ l₂\n⊢ ofList (toList l₁) ⊆ l₂",
"tactic": "revert H"
},
{
"state_after": "case nil\nα : Type u_1\nl₁ l₂ : Lists' α true\nH : ∀ (a : Lists α), a ∈ [] → a ∈ l₂\n⊢ ofList [] ⊆ l₂\n\ncase cons\nα : Type u_1\nl₁ l₂ : Lists' α true\nh : Lists α\nt : List (Lists α)\nt_ih : (∀ (a : Lists α), a ∈ t → a ∈ l₂) → ofList t ⊆ l₂\nH : ∀ (a : Lists α), a ∈ h :: t → a ∈ l₂\n⊢ ofList (h :: t) ⊆ l₂",
"state_before": "α : Type u_1\nl₁ l₂ : Lists' α true\n⊢ (∀ (a : Lists α), a ∈ toList l₁ → a ∈ l₂) → ofList (toList l₁) ⊆ l₂",
"tactic": "induction' toList l₁ with h t t_ih <;> intro H"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nl₁ l₂ : Lists' α true\nH : ∀ (a : Lists α), a ∈ [] → a ∈ l₂\n⊢ ofList [] ⊆ l₂",
"tactic": "exact Subset.nil"
},
{
"state_after": "case cons\nα : Type u_1\nl₁ l₂ : Lists' α true\nh : Lists α\nt : List (Lists α)\nt_ih : (∀ (a : Lists α), a ∈ t → a ∈ l₂) → ofList t ⊆ l₂\nH : h ∈ l₂ ∧ ∀ (a : Lists α), a ∈ t → a ∈ l₂\n⊢ cons h (ofList t) ⊆ l₂",
"state_before": "case cons\nα : Type u_1\nl₁ l₂ : Lists' α true\nh : Lists α\nt : List (Lists α)\nt_ih : (∀ (a : Lists α), a ∈ t → a ∈ l₂) → ofList t ⊆ l₂\nH : ∀ (a : Lists α), a ∈ h :: t → a ∈ l₂\n⊢ ofList (h :: t) ⊆ l₂",
"tactic": "simp only [ofList, List.find?, List.mem_cons, forall_eq_or_imp] at *"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\nl₁ l₂ : Lists' α true\nh : Lists α\nt : List (Lists α)\nt_ih : (∀ (a : Lists α), a ∈ t → a ∈ l₂) → ofList t ⊆ l₂\nH : h ∈ l₂ ∧ ∀ (a : Lists α), a ∈ t → a ∈ l₂\n⊢ cons h (ofList t) ⊆ l₂",
"tactic": "exact cons_subset.2 ⟨H.1, t_ih H.2⟩"
}
] |
[
212,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/GroupTheory/Perm/Fin.lean
|
Equiv.Perm.decomposeFin_symm_apply_one
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\ne : Perm (Fin (n + 1))\np : Fin (n + 2)\n⊢ ↑(↑decomposeFin.symm (p, e)) 1 = ↑(swap 0 p) (Fin.succ (↑e 0))",
"tactic": "rw [← Fin.succ_zero_eq_one, Equiv.Perm.decomposeFin_symm_apply_succ e p 0]"
}
] |
[
63,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/Order/Heyting/Hom.lean
|
CoheytingHom.id_comp
|
[] |
[
480,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
479,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
EMetric.hausdorffEdist_comm
|
[
{
"state_after": "ι : Sort ?u.35598\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t u : Set α\nΦ : α → β\n⊢ ((⨆ (x : α) (_ : x ∈ s), infEdist x t) ⊔ ⨆ (y : α) (_ : y ∈ t), infEdist y s) =\n (⨆ (y : α) (_ : y ∈ t), infEdist y s) ⊔ ⨆ (x : α) (_ : x ∈ s), infEdist x t",
"state_before": "ι : Sort ?u.35598\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t u : Set α\nΦ : α → β\n⊢ hausdorffEdist s t = hausdorffEdist t s",
"tactic": "simp only [hausdorffEdist_def]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.35598\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t u : Set α\nΦ : α → β\n⊢ ((⨆ (x : α) (_ : x ∈ s), infEdist x t) ⊔ ⨆ (y : α) (_ : y ∈ t), infEdist y s) =\n (⨆ (y : α) (_ : y ∈ t), infEdist y s) ⊔ ⨆ (x : α) (_ : x ∈ s), infEdist x t",
"tactic": "apply sup_comm"
}
] |
[
277,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Std/Logic.lean
|
eq_iff_iff
|
[] |
[
53,
79
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
53,
9
] |
Mathlib/Data/Sum/Order.lean
|
OrderIso.sumDualDistrib_inr
|
[] |
[
622,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
1
] |
Mathlib/Data/Set/NAry.lean
|
Set.image2_distrib_subset_right
|
[
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_4\nα' : Type u_6\nβ : Type u_5\nβ' : Type u_7\nγ : Type u_3\nγ' : Type ?u.46897\nδ : Type u_2\nδ' : Type ?u.46903\nε : Type u_1\nε' : Type ?u.46909\nζ : Type ?u.46912\nζ' : Type ?u.46915\nν : Type ?u.46918\nf✝ f' : α → β → γ\ng✝ g'✝ : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na✝ a' : α\nb✝ b' : β\nc✝ c' : γ\nd d' : δ\nf : δ → γ → ε\ng : α → β → δ\nf₁ : α → γ → α'\nf₂ : β → γ → β'\ng' : α' → β' → ε\nh_distrib : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f₁ a c) (f₂ b c)\nc : γ\na : α\nb : β\nha : a ∈ s\nhb : b ∈ t\nhc : c ∈ u\n⊢ f (g a b) c ∈ image2 g' (image2 f₁ s u) (image2 f₂ t u)",
"state_before": "α : Type u_4\nα' : Type u_6\nβ : Type u_5\nβ' : Type u_7\nγ : Type u_3\nγ' : Type ?u.46897\nδ : Type u_2\nδ' : Type ?u.46903\nε : Type u_1\nε' : Type ?u.46909\nζ : Type ?u.46912\nζ' : Type ?u.46915\nν : Type ?u.46918\nf✝ f' : α → β → γ\ng✝ g'✝ : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nf : δ → γ → ε\ng : α → β → δ\nf₁ : α → γ → α'\nf₂ : β → γ → β'\ng' : α' → β' → ε\nh_distrib : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f₁ a c) (f₂ b c)\n⊢ image2 f (image2 g s t) u ⊆ image2 g' (image2 f₁ s u) (image2 f₂ t u)",
"tactic": "rintro _ ⟨_, c, ⟨a, b, ha, hb, rfl⟩, hc, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_4\nα' : Type u_6\nβ : Type u_5\nβ' : Type u_7\nγ : Type u_3\nγ' : Type ?u.46897\nδ : Type u_2\nδ' : Type ?u.46903\nε : Type u_1\nε' : Type ?u.46909\nζ : Type ?u.46912\nζ' : Type ?u.46915\nν : Type ?u.46918\nf✝ f' : α → β → γ\ng✝ g'✝ : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na✝ a' : α\nb✝ b' : β\nc✝ c' : γ\nd d' : δ\nf : δ → γ → ε\ng : α → β → δ\nf₁ : α → γ → α'\nf₂ : β → γ → β'\ng' : α' → β' → ε\nh_distrib : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f₁ a c) (f₂ b c)\nc : γ\na : α\nb : β\nha : a ∈ s\nhb : b ∈ t\nhc : c ∈ u\n⊢ g' (f₁ a c) (f₂ b c) ∈ image2 g' (image2 f₁ s u) (image2 f₂ t u)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_4\nα' : Type u_6\nβ : Type u_5\nβ' : Type u_7\nγ : Type u_3\nγ' : Type ?u.46897\nδ : Type u_2\nδ' : Type ?u.46903\nε : Type u_1\nε' : Type ?u.46909\nζ : Type ?u.46912\nζ' : Type ?u.46915\nν : Type ?u.46918\nf✝ f' : α → β → γ\ng✝ g'✝ : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na✝ a' : α\nb✝ b' : β\nc✝ c' : γ\nd d' : δ\nf : δ → γ → ε\ng : α → β → δ\nf₁ : α → γ → α'\nf₂ : β → γ → β'\ng' : α' → β' → ε\nh_distrib : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f₁ a c) (f₂ b c)\nc : γ\na : α\nb : β\nha : a ∈ s\nhb : b ∈ t\nhc : c ∈ u\n⊢ f (g a b) c ∈ image2 g' (image2 f₁ s u) (image2 f₂ t u)",
"tactic": "rw [h_distrib]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_4\nα' : Type u_6\nβ : Type u_5\nβ' : Type u_7\nγ : Type u_3\nγ' : Type ?u.46897\nδ : Type u_2\nδ' : Type ?u.46903\nε : Type u_1\nε' : Type ?u.46909\nζ : Type ?u.46912\nζ' : Type ?u.46915\nν : Type ?u.46918\nf✝ f' : α → β → γ\ng✝ g'✝ : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na✝ a' : α\nb✝ b' : β\nc✝ c' : γ\nd d' : δ\nf : δ → γ → ε\ng : α → β → δ\nf₁ : α → γ → α'\nf₂ : β → γ → β'\ng' : α' → β' → ε\nh_distrib : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f₁ a c) (f₂ b c)\nc : γ\na : α\nb : β\nha : a ∈ s\nhb : b ∈ t\nhc : c ∈ u\n⊢ g' (f₁ a c) (f₂ b c) ∈ image2 g' (image2 f₁ s u) (image2 f₂ t u)",
"tactic": "exact mem_image2_of_mem (mem_image2_of_mem ha hc) (mem_image2_of_mem hb hc)"
}
] |
[
394,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
1
] |
Mathlib/Data/Bool/Basic.lean
|
Bool.xor_iff_ne
|
[
{
"state_after": "no goals",
"state_before": "⊢ ∀ {x y : Bool}, xor x y = true ↔ x ≠ y",
"tactic": "decide"
}
] |
[
301,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.ofDual_max'
|
[
{
"state_after": "F : Type ?u.352825\nα : Type u_1\nβ : Type ?u.352831\nγ : Type ?u.352834\nι : Type ?u.352837\nκ : Type ?u.352840\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset αᵒᵈ\nhs : Finset.Nonempty s\n⊢ ↑(↑ofDual (max' s hs)) = ↑(min' (image (↑ofDual) s) (_ : Finset.Nonempty (image (↑ofDual) s)))",
"state_before": "F : Type ?u.352825\nα : Type u_1\nβ : Type ?u.352831\nγ : Type ?u.352834\nι : Type ?u.352837\nκ : Type ?u.352840\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset αᵒᵈ\nhs : Finset.Nonempty s\n⊢ ↑ofDual (max' s hs) = min' (image (↑ofDual) s) (_ : Finset.Nonempty (image (↑ofDual) s))",
"tactic": "rw [← WithTop.coe_eq_coe]"
},
{
"state_after": "F : Type ?u.352825\nα : Type u_1\nβ : Type ?u.352831\nγ : Type ?u.352834\nι : Type ?u.352837\nκ : Type ?u.352840\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset αᵒᵈ\nhs : Finset.Nonempty s\n⊢ inf s (WithTop.some ∘ fun x => ↑ofDual x) = inf s ((WithTop.some ∘ fun x => x) ∘ ↑ofDual)",
"state_before": "F : Type ?u.352825\nα : Type u_1\nβ : Type ?u.352831\nγ : Type ?u.352834\nι : Type ?u.352837\nκ : Type ?u.352840\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset αᵒᵈ\nhs : Finset.Nonempty s\n⊢ ↑(↑ofDual (max' s hs)) = ↑(min' (image (↑ofDual) s) (_ : Finset.Nonempty (image (↑ofDual) s)))",
"tactic": "simp only [max'_eq_sup', id_eq, ofDual_sup', Function.comp_apply, coe_inf', min'_eq_inf',\n inf_image]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.352825\nα : Type u_1\nβ : Type ?u.352831\nγ : Type ?u.352834\nι : Type ?u.352837\nκ : Type ?u.352840\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset αᵒᵈ\nhs : Finset.Nonempty s\n⊢ inf s (WithTop.some ∘ fun x => ↑ofDual x) = inf s ((WithTop.some ∘ fun x => x) ∘ ↑ofDual)",
"tactic": "rfl"
}
] |
[
1444,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1439,
1
] |
Mathlib/Data/Setoid/Partition.lean
|
IndexedPartition.proj_some_index
|
[] |
[
424,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
src/lean/Init/Data/Fin/Basic.lean
|
Fin.mlt
|
[] |
[
36,
22
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
32,
9
] |
Mathlib/Algebra/Order/Field/Power.lean
|
Nat.zpow_ne_zero_of_pos
|
[] |
[
56,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
|
generateFrom_eq_prod
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.808243\nβ : Type u_2\nβ' : Type ?u.808249\nγ : Type ?u.808252\nE : Type ?u.808255\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace α'\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace β'\ninst✝¹ : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝ : NormedAddCommGroup E\nC : Set (Set α)\nD : Set (Set β)\nhC : generateFrom C = inst✝⁵\nhD : generateFrom D = inst✝³\nh2C : IsCountablySpanning C\nh2D : IsCountablySpanning D\n⊢ generateFrom (image2 (fun x x_1 => x ×ˢ x_1) C D) = Prod.instMeasurableSpace",
"tactic": "rw [← hC, ← hD, generateFrom_prod_eq h2C h2D]"
}
] |
[
141,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Data/Finsupp/Order.lean
|
Finsupp.orderEmbeddingToFun_apply
|
[] |
[
70,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
isUpperSet_iInter₂
|
[] |
[
165,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.map_cons
|
[] |
[
1405,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1404,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDense.lean
|
MeasureTheory.SimpleFunc.edist_approxOn_mono
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.64039\nE : Type ?u.64042\nF : Type ?u.64045\n𝕜 : Type ?u.64048\ninst✝⁴ : MeasurableSpace α\ninst✝³ : PseudoEMetricSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : MeasurableSpace β\nf✝ f : β → α\nhf : Measurable f\ns : Set α\ny₀ : α\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nm n : ℕ\nh : m ≤ n\n⊢ edist (↑(nearestPt (fun k => Nat.rec y₀ (fun n n_ih => ↑(denseSeq (↑s) n)) k) n) (f x)) (f x) ≤\n edist (↑(nearestPt (fun k => Nat.rec y₀ (fun n n_ih => ↑(denseSeq (↑s) n)) k) m) (f x)) (f x)",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.64039\nE : Type ?u.64042\nF : Type ?u.64045\n𝕜 : Type ?u.64048\ninst✝⁴ : MeasurableSpace α\ninst✝³ : PseudoEMetricSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : MeasurableSpace β\nf✝ f : β → α\nhf : Measurable f\ns : Set α\ny₀ : α\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nm n : ℕ\nh : m ≤ n\n⊢ edist (↑(approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist (↑(approxOn f hf s y₀ h₀ m) x) (f x)",
"tactic": "dsimp only [approxOn, coe_comp, Function.comp]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.64039\nE : Type ?u.64042\nF : Type ?u.64045\n𝕜 : Type ?u.64048\ninst✝⁴ : MeasurableSpace α\ninst✝³ : PseudoEMetricSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : MeasurableSpace β\nf✝ f : β → α\nhf : Measurable f\ns : Set α\ny₀ : α\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nm n : ℕ\nh : m ≤ n\n⊢ edist (↑(nearestPt (fun k => Nat.rec y₀ (fun n n_ih => ↑(denseSeq (↑s) n)) k) n) (f x)) (f x) ≤\n edist (↑(nearestPt (fun k => Nat.rec y₀ (fun n n_ih => ↑(denseSeq (↑s) n)) k) m) (f x)) (f x)",
"tactic": "exact edist_nearestPt_le _ _ ((nearestPtInd_le _ _ _).trans h)"
}
] |
[
174,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Analysis/Complex/Isometry.lean
|
rotation_symm
|
[] |
[
59,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Topology/MetricSpace/Contracting.lean
|
ContractingWith.apriori_edist_iterate_efixedPoint_le'
|
[] |
[
221,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/FieldTheory/Subfield.lean
|
Subfield.toSubring_subtype_eq_subtype
|
[] |
[
423,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
|
ContinuousLinearEquiv.comp_right_hasFDerivAt_iff'
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_3\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.225072\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\niso : E ≃L[𝕜] F\nf : F → G\nx : E\nf' : E →L[𝕜] G\n⊢ HasFDerivAt (f ∘ ↑iso) f' x ↔ HasFDerivAt f (comp f' ↑(ContinuousLinearEquiv.symm iso)) (↑iso x)",
"tactic": "simp only [← hasFDerivWithinAt_univ, ← iso.comp_right_hasFDerivWithinAt_iff', preimage_univ]"
}
] |
[
234,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.prodMap_smul
|
[] |
[
368,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
366,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
Real.sinh_nonpos_iff
|
[
{
"state_after": "no goals",
"state_before": "x y z : ℝ\n⊢ sinh x ≤ 0 ↔ x ≤ 0",
"tactic": "simpa only [sinh_zero] using @sinh_le_sinh x 0"
}
] |
[
709,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
709,
1
] |
Mathlib/InformationTheory/Hamming.lean
|
hammingDist_comp_le_hammingDist
|
[] |
[
136,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.Monic.degree_mul_comm
|
[
{
"state_after": "case pos\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\nh : q = 0\n⊢ degree (p * q) = degree (q * p)\n\ncase neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\nh : ¬q = 0\n⊢ degree (p * q) = degree (q * p)",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\n⊢ degree (p * q) = degree (q * p)",
"tactic": "by_cases h : q = 0"
},
{
"state_after": "case neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\nh : ¬q = 0\n⊢ degree p + degree q = degree q + degree p\n\ncase neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\nh : ¬q = 0\n⊢ Polynomial.leadingCoeff p * Polynomial.leadingCoeff q ≠ 0",
"state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\nh : ¬q = 0\n⊢ degree (p * q) = degree (q * p)",
"tactic": "rw [degree_mul', hp.degree_mul]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\nh : q = 0\n⊢ degree (p * q) = degree (q * p)",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\nh : ¬q = 0\n⊢ degree p + degree q = degree q + degree p",
"tactic": "exact add_comm _ _"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q✝ r : R[X]\nhp : Monic p\nq : R[X]\nh : ¬q = 0\n⊢ Polynomial.leadingCoeff p * Polynomial.leadingCoeff q ≠ 0",
"tactic": "rwa [hp.leadingCoeff, one_mul, leadingCoeff_ne_zero]"
}
] |
[
184,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Topology/Instances/EReal.lean
|
EReal.tendsto_nhds_top_iff_real
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.12287\ninst✝ : TopologicalSpace α✝\nα : Type u_1\nm : α → EReal\nf : Filter α\n⊢ (∀ (i : ℝ), True → ∀ᶠ (x : α) in f, m x ∈ Ioi ↑i) ↔ ∀ (x : ℝ), ∀ᶠ (a : α) in f, ↑x < m a",
"tactic": "simp only [true_implies, mem_Ioi]"
}
] |
[
151,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/RingTheory/Polynomial/Opposites.lean
|
Polynomial.opRingEquiv_op_X
|
[] |
[
56,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Analysis/Calculus/Dslope.lean
|
ContinuousLinearMap.dslope_comp
|
[
{
"state_after": "case inl\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf✝ : 𝕜 → E\na b✝ : 𝕜\ns : Set 𝕜\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ng : 𝕜 → E\nb : 𝕜\nH : b = b → DifferentiableAt 𝕜 g b\n⊢ dslope (↑f ∘ g) b b = ↑f (dslope g b b)\n\ncase inr\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf✝ : 𝕜 → E\na✝ b✝ : 𝕜\ns : Set 𝕜\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ng : 𝕜 → E\na b : 𝕜\nH : a = b → DifferentiableAt 𝕜 g a\nhne : b ≠ a\n⊢ dslope (↑f ∘ g) a b = ↑f (dslope g a b)",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf✝ : 𝕜 → E\na✝ b✝ : 𝕜\ns : Set 𝕜\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ng : 𝕜 → E\na b : 𝕜\nH : a = b → DifferentiableAt 𝕜 g a\n⊢ dslope (↑f ∘ g) a b = ↑f (dslope g a b)",
"tactic": "rcases eq_or_ne b a with (rfl | hne)"
},
{
"state_after": "case inl\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf✝ : 𝕜 → E\na b✝ : 𝕜\ns : Set 𝕜\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ng : 𝕜 → E\nb : 𝕜\nH : b = b → DifferentiableAt 𝕜 g b\n⊢ deriv (↑f ∘ g) b = ↑f (deriv g b)",
"state_before": "case inl\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf✝ : 𝕜 → E\na b✝ : 𝕜\ns : Set 𝕜\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ng : 𝕜 → E\nb : 𝕜\nH : b = b → DifferentiableAt 𝕜 g b\n⊢ dslope (↑f ∘ g) b b = ↑f (dslope g b b)",
"tactic": "simp only [dslope_same]"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf✝ : 𝕜 → E\na b✝ : 𝕜\ns : Set 𝕜\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ng : 𝕜 → E\nb : 𝕜\nH : b = b → DifferentiableAt 𝕜 g b\n⊢ deriv (↑f ∘ g) b = ↑f (deriv g b)",
"tactic": "exact (f.hasFDerivAt.comp_hasDerivAt b (H rfl).hasDerivAt).deriv"
},
{
"state_after": "no goals",
"state_before": "case inr\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf✝ : 𝕜 → E\na✝ b✝ : 𝕜\ns : Set 𝕜\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ng : 𝕜 → E\na b : 𝕜\nH : a = b → DifferentiableAt 𝕜 g a\nhne : b ≠ a\n⊢ dslope (↑f ∘ g) a b = ↑f (dslope g a b)",
"tactic": "simpa only [dslope_of_ne _ hne] using f.toLinearMap.slope_comp g a b"
}
] |
[
55,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieSubmodule.bot_coeSubmodule
|
[] |
[
375,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.neg_mem_iff
|
[] |
[
656,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
655,
11
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.cthickening_subset_thickening'
|
[] |
[
1111,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1109,
1
] |
Mathlib/Order/BooleanAlgebra.lean
|
inf_compl_eq_bot'
|
[] |
[
558,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
557,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.ofLinearEquiv_symm
|
[] |
[
619,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
615,
1
] |
Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean
|
TrivSqZeroExt.exp_inr
|
[
{
"state_after": "case hx\n𝕜 : Type u_3\nR : Type u_1\nM : Type u_2\ninst✝¹⁷ : IsROrC 𝕜\ninst✝¹⁶ : NormedRing R\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : NormedAlgebra 𝕜 R\ninst✝¹³ : Module R M\ninst✝¹² : Module Rᵐᵒᵖ M\ninst✝¹¹ : SMulCommClass R Rᵐᵒᵖ M\ninst✝¹⁰ : Module 𝕜 M\ninst✝⁹ : IsScalarTower 𝕜 R M\ninst✝⁸ : IsScalarTower 𝕜 Rᵐᵒᵖ M\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : TopologicalRing R\ninst✝⁵ : TopologicalAddGroup M\ninst✝⁴ : ContinuousSMul R M\ninst✝³ : ContinuousSMul Rᵐᵒᵖ M\ninst✝² : CompleteSpace R\ninst✝¹ : T2Space R\ninst✝ : T2Space M\nm : M\n⊢ MulOpposite.op (fst (inr m)) • snd (inr m) = fst (inr m) • snd (inr m)",
"state_before": "𝕜 : Type u_3\nR : Type u_1\nM : Type u_2\ninst✝¹⁷ : IsROrC 𝕜\ninst✝¹⁶ : NormedRing R\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : NormedAlgebra 𝕜 R\ninst✝¹³ : Module R M\ninst✝¹² : Module Rᵐᵒᵖ M\ninst✝¹¹ : SMulCommClass R Rᵐᵒᵖ M\ninst✝¹⁰ : Module 𝕜 M\ninst✝⁹ : IsScalarTower 𝕜 R M\ninst✝⁸ : IsScalarTower 𝕜 Rᵐᵒᵖ M\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : TopologicalRing R\ninst✝⁵ : TopologicalAddGroup M\ninst✝⁴ : ContinuousSMul R M\ninst✝³ : ContinuousSMul Rᵐᵒᵖ M\ninst✝² : CompleteSpace R\ninst✝¹ : T2Space R\ninst✝ : T2Space M\nm : M\n⊢ exp 𝕜 (inr m) = 1 + inr m",
"tactic": "rw [exp_def_of_smul_comm, snd_inr, fst_inr, exp_zero, one_smul, inl_one]"
},
{
"state_after": "no goals",
"state_before": "case hx\n𝕜 : Type u_3\nR : Type u_1\nM : Type u_2\ninst✝¹⁷ : IsROrC 𝕜\ninst✝¹⁶ : NormedRing R\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : NormedAlgebra 𝕜 R\ninst✝¹³ : Module R M\ninst✝¹² : Module Rᵐᵒᵖ M\ninst✝¹¹ : SMulCommClass R Rᵐᵒᵖ M\ninst✝¹⁰ : Module 𝕜 M\ninst✝⁹ : IsScalarTower 𝕜 R M\ninst✝⁸ : IsScalarTower 𝕜 Rᵐᵒᵖ M\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : TopologicalRing R\ninst✝⁵ : TopologicalAddGroup M\ninst✝⁴ : ContinuousSMul R M\ninst✝³ : ContinuousSMul Rᵐᵒᵖ M\ninst✝² : CompleteSpace R\ninst✝¹ : T2Space R\ninst✝ : T2Space M\nm : M\n⊢ MulOpposite.op (fst (inr m)) • snd (inr m) = fst (inr m) • snd (inr m)",
"tactic": "rw [snd_inr, fst_inr, MulOpposite.op_zero, zero_smul, zero_smul]"
}
] |
[
128,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.leadingCoeff_X
|
[
{
"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.644630\n⊢ leadingCoeff X = 1",
"tactic": "simpa only [pow_one] using @leadingCoeff_X_pow R _ 1"
}
] |
[
822,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
821,
1
] |
Mathlib/SetTheory/Ordinal/NaturalOps.lean
|
NatOrdinal.toOrdinal_min
|
[] |
[
130,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Topology/UnitInterval.lean
|
unitInterval.coe_ne_one
|
[] |
[
84,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toNNReal_eq_zero_iff
|
[] |
[
268,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/Order/Atoms.lean
|
OrderEmbedding.isCoatom_of_map_top_of_image
|
[] |
[
708,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
706,
1
] |
Mathlib/Topology/UniformSpace/Equiv.lean
|
UniformEquiv.trans_apply
|
[] |
[
125,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.foldl_eq'
|
[
{
"state_after": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nf : β → α → β\nl₁ l₂ : List α\np : l₁ ~ l₂\nx y : α\nt₁ t₂ : List α\n_p : t₁ ~ t₂\nr :\n (∀ (x : α), x ∈ t₁ → ∀ (y : α), y ∈ t₁ → ∀ (z : β), f (f z x) y = f (f z y) x) →\n ∀ (b : β), foldl f b t₁ = foldl f b t₂\nH : ∀ (x_1 : α), x_1 ∈ y :: x :: t₁ → ∀ (y_1 : α), y_1 ∈ y :: x :: t₁ → ∀ (z : β), f (f z x_1) y_1 = f (f z y_1) x_1\nb : β\n⊢ foldl f (f (f b y) x) t₁ = foldl f (f (f b x) y) t₂",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nf : β → α → β\nl₁ l₂ : List α\np : l₁ ~ l₂\nx y : α\nt₁ t₂ : List α\n_p : t₁ ~ t₂\nr :\n (∀ (x : α), x ∈ t₁ → ∀ (y : α), y ∈ t₁ → ∀ (z : β), f (f z x) y = f (f z y) x) →\n ∀ (b : β), foldl f b t₁ = foldl f b t₂\nH : ∀ (x_1 : α), x_1 ∈ y :: x :: t₁ → ∀ (y_1 : α), y_1 ∈ y :: x :: t₁ → ∀ (z : β), f (f z x_1) y_1 = f (f z y_1) x_1\nb : β\n⊢ foldl f b (y :: x :: t₁) = foldl f b (x :: y :: t₂)",
"tactic": "simp only [foldl]"
},
{
"state_after": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nf : β → α → β\nl₁ l₂ : List α\np : l₁ ~ l₂\nx y : α\nt₁ t₂ : List α\n_p : t₁ ~ t₂\nr :\n (∀ (x : α), x ∈ t₁ → ∀ (y : α), y ∈ t₁ → ∀ (z : β), f (f z x) y = f (f z y) x) →\n ∀ (b : β), foldl f b t₁ = foldl f b t₂\nH : ∀ (x_1 : α), x_1 ∈ y :: x :: t₁ → ∀ (y_1 : α), y_1 ∈ y :: x :: t₁ → ∀ (z : β), f (f z x_1) y_1 = f (f z y_1) x_1\nb : β\n⊢ foldl f (f (f b y) x) t₁ = foldl f (f (f b y) x) t₂",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nf : β → α → β\nl₁ l₂ : List α\np : l₁ ~ l₂\nx y : α\nt₁ t₂ : List α\n_p : t₁ ~ t₂\nr :\n (∀ (x : α), x ∈ t₁ → ∀ (y : α), y ∈ t₁ → ∀ (z : β), f (f z x) y = f (f z y) x) →\n ∀ (b : β), foldl f b t₁ = foldl f b t₂\nH : ∀ (x_1 : α), x_1 ∈ y :: x :: t₁ → ∀ (y_1 : α), y_1 ∈ y :: x :: t₁ → ∀ (z : β), f (f z x_1) y_1 = f (f z y_1) x_1\nb : β\n⊢ foldl f (f (f b y) x) t₁ = foldl f (f (f b x) y) t₂",
"tactic": "rw [H x (.tail _ <| .head _) y (.head _)]"
},
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nf : β → α → β\nl₁ l₂ : List α\np : l₁ ~ l₂\nx y : α\nt₁ t₂ : List α\n_p : t₁ ~ t₂\nr :\n (∀ (x : α), x ∈ t₁ → ∀ (y : α), y ∈ t₁ → ∀ (z : β), f (f z x) y = f (f z y) x) →\n ∀ (b : β), foldl f b t₁ = foldl f b t₂\nH : ∀ (x_1 : α), x_1 ∈ y :: x :: t₁ → ∀ (y_1 : α), y_1 ∈ y :: x :: t₁ → ∀ (z : β), f (f z x_1) y_1 = f (f z y_1) x_1\nb : β\n⊢ foldl f (f (f b y) x) t₁ = foldl f (f (f b y) x) t₂",
"tactic": "exact r (fun x hx y hy => H _ (.tail _ <| .tail _ hx) _ (.tail _ <| .tail _ hy)) _"
}
] |
[
517,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
508,
1
] |
Mathlib/Analysis/LocallyConvex/Basic.lean
|
absorbs_zero_iff
|
[
{
"state_after": "𝕜 : Type u_1\n𝕝 : Type ?u.204287\nE : Type u_2\nι : Sort ?u.204293\nκ : ι → Sort ?u.204298\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\n⊢ Absorbs 𝕜 s 0 → 0 ∈ s",
"state_before": "𝕜 : Type u_1\n𝕝 : Type ?u.204287\nE : Type u_2\nι : Sort ?u.204293\nκ : ι → Sort ?u.204298\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\n⊢ Absorbs 𝕜 s 0 ↔ 0 ∈ s",
"tactic": "refine' ⟨_, fun h => ⟨1, zero_lt_one, fun a _ => zero_subset.2 <| zero_mem_smul_set h⟩⟩"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.204287\nE : Type u_2\nι : Sort ?u.204293\nκ : ι → Sort ?u.204298\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nr : ℝ\nhr : 0 < r\nh : ∀ (a : 𝕜), r ≤ ‖a‖ → 0 ⊆ a • s\n⊢ 0 ∈ s",
"state_before": "𝕜 : Type u_1\n𝕝 : Type ?u.204287\nE : Type u_2\nι : Sort ?u.204293\nκ : ι → Sort ?u.204298\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\n⊢ Absorbs 𝕜 s 0 → 0 ∈ s",
"tactic": "rintro ⟨r, hr, h⟩"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.204287\nE : Type u_2\nι : Sort ?u.204293\nκ : ι → Sort ?u.204298\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nr : ℝ\nhr : 0 < r\nh : ∀ (a : 𝕜), r ≤ ‖a‖ → 0 ⊆ a • s\na : 𝕜\nha : r < ‖a‖\n⊢ 0 ∈ s",
"state_before": "case intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.204287\nE : Type u_2\nι : Sort ?u.204293\nκ : ι → Sort ?u.204298\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nr : ℝ\nhr : 0 < r\nh : ∀ (a : 𝕜), r ≤ ‖a‖ → 0 ⊆ a • s\n⊢ 0 ∈ s",
"tactic": "obtain ⟨a, ha⟩ := NormedSpace.exists_lt_norm 𝕜 𝕜 r"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.204287\nE : Type u_2\nι : Sort ?u.204293\nκ : ι → Sort ?u.204298\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nr : ℝ\nhr : 0 < r\nh : ∀ (a : 𝕜), r ≤ ‖a‖ → 0 ⊆ a • s\na : 𝕜\nha : r < ‖a‖\nthis : 0 ⊆ a • s\n⊢ 0 ∈ s",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.204287\nE : Type u_2\nι : Sort ?u.204293\nκ : ι → Sort ?u.204298\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nr : ℝ\nhr : 0 < r\nh : ∀ (a : 𝕜), r ≤ ‖a‖ → 0 ⊆ a • s\na : 𝕜\nha : r < ‖a‖\n⊢ 0 ∈ s",
"tactic": "have := h _ ha.le"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.204287\nE : Type u_2\nι : Sort ?u.204293\nκ : ι → Sort ?u.204298\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nr : ℝ\nhr : 0 < r\nh : ∀ (a : 𝕜), r ≤ ‖a‖ → 0 ⊆ a • s\na : 𝕜\nha : r < ‖a‖\nthis : 0 ∈ a • s\n⊢ a ≠ 0",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.204287\nE : Type u_2\nι : Sort ?u.204293\nκ : ι → Sort ?u.204298\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nr : ℝ\nhr : 0 < r\nh : ∀ (a : 𝕜), r ≤ ‖a‖ → 0 ⊆ a • s\na : 𝕜\nha : r < ‖a‖\nthis : 0 ⊆ a • s\n⊢ 0 ∈ s",
"tactic": "rwa [zero_subset, zero_mem_smul_set_iff] at this"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.204287\nE : Type u_2\nι : Sort ?u.204293\nκ : ι → Sort ?u.204298\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nr : ℝ\nhr : 0 < r\nh : ∀ (a : 𝕜), r ≤ ‖a‖ → 0 ⊆ a • s\na : 𝕜\nha : r < ‖a‖\nthis : 0 ∈ a • s\n⊢ a ≠ 0",
"tactic": "exact norm_ne_zero_iff.1 (hr.trans ha).ne'"
}
] |
[
398,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
392,
1
] |
Mathlib/Data/Finset/Prod.lean
|
Finset.product_subset_product_right
|
[] |
[
102,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/RingTheory/Localization/Integral.lean
|
integralClosure.isFractionRing_of_finite_extension
|
[] |
[
379,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean
|
Asymptotics.IsEquivalent.neg
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v w : α → β\nl : Filter α\nhuv : u ~[l] v\n⊢ ((fun x => -u x) - fun x => -v x) =o[l] fun x => -v x",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v w : α → β\nl : Filter α\nhuv : u ~[l] v\n⊢ (fun x => -u x) ~[l] fun x => -v x",
"tactic": "rw [IsEquivalent]"
},
{
"state_after": "case h.e'_7.h\nα : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v w : α → β\nl : Filter α\nhuv : u ~[l] v\nx✝ : α\n⊢ ((fun x => -u x) - fun x => -v x) x✝ = -(u - v) x✝",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v w : α → β\nl : Filter α\nhuv : u ~[l] v\n⊢ ((fun x => -u x) - fun x => -v x) =o[l] fun x => -v x",
"tactic": "convert huv.isLittleO.neg_left.neg_right"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7.h\nα : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v w : α → β\nl : Filter α\nhuv : u ~[l] v\nx✝ : α\n⊢ ((fun x => -u x) - fun x => -v x) x✝ = -(u - v) x✝",
"tactic": "simp [neg_add_eq_sub]"
}
] |
[
186,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.card_roots'
|
[
{
"state_after": "case pos\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np✝ q p : R[X]\nhp0 : p = 0\n⊢ ↑Multiset.card (roots p) ≤ natDegree p\n\ncase neg\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np✝ q p : R[X]\nhp0 : ¬p = 0\n⊢ ↑Multiset.card (roots p) ≤ natDegree p",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np✝ q p : R[X]\n⊢ ↑Multiset.card (roots p) ≤ natDegree p",
"tactic": "by_cases hp0 : p = 0"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np✝ q p : R[X]\nhp0 : ¬p = 0\n⊢ ↑Multiset.card (roots p) ≤ natDegree p",
"tactic": "exact WithBot.coe_le_coe.1 (le_trans (card_roots hp0) (le_of_eq <| degree_eq_natDegree hp0))"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np✝ q p : R[X]\nhp0 : p = 0\n⊢ ↑Multiset.card (roots p) ≤ natDegree p",
"tactic": "simp [hp0]"
}
] |
[
547,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
544,
1
] |
Mathlib/Data/Nat/Prime.lean
|
Nat.coprime_primes
|
[] |
[
675,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
674,
1
] |
Mathlib/Data/List/Join.lean
|
List.sum_take_map_length_lt1
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.12788\nL : List (List α)\ni j : ℕ\nhi : i < length L\nhj : j < length (nthLe L i hi)\n⊢ sum (take i (map length L)) + j < sum (take (i + 1) (map length L))",
"tactic": "simp [hi, sum_take_succ, hj]"
}
] |
[
164,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/AlgebraicTopology/DoldKan/Projections.lean
|
AlgebraicTopology.DoldKan.P_f_naturality
|
[] |
[
201,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.BijOn.surjOn
|
[] |
[
916,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
915,
1
] |
Mathlib/Init/Data/List/Basic.lean
|
List.headI_nil
|
[] |
[
42,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
9
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.Equiv.ext
|
[] |
[
837,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
836,
1
] |
Mathlib/Algebra/CharP/Algebra.lean
|
algebraRat.charP_zero
|
[] |
[
80,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Data/Set/Intervals/OrdConnected.lean
|
Set.ordConnected_iInter
|
[] |
[
110,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/RepresentationTheory/Maschke.lean
|
LinearMap.equivariantProjection_apply
|
[
{
"state_after": "no goals",
"state_before": "k : Type u\ninst✝¹¹ : CommRing k\nG : Type u\ninst✝¹⁰ : Group G\nV : Type v\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module k V\ninst✝⁷ : Module (MonoidAlgebra k G) V\ninst✝⁶ : IsScalarTower k (MonoidAlgebra k G) V\nW : Type w\ninst✝⁵ : AddCommGroup W\ninst✝⁴ : Module k W\ninst✝³ : Module (MonoidAlgebra k G) W\ninst✝² : IsScalarTower k (MonoidAlgebra k G) W\nπ : W →ₗ[k] V\ni : V →ₗ[MonoidAlgebra k G] W\nh : ∀ (v : V), ↑π (↑i v) = v\ninst✝¹ : Fintype G\ninst✝ : Invertible ↑(Fintype.card G)\nv : W\n⊢ ↑(equivariantProjection G π) v = ⅟↑(Fintype.card G) • ∑ g : G, ↑(conjugate π g) v",
"tactic": "simp only [equivariantProjection, smul_apply, sumOfConjugatesEquivariant_apply]"
}
] |
[
131,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
MeasureTheory.FiniteMeasure.testAgainstNN_coe_eq
|
[] |
[
339,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/MeasureTheory/Function/UniformIntegrable.lean
|
MeasureTheory.tendsto_Lp_of_tendstoInMeasure
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nι : Type ?u.504674\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ℕ → α → β\ng : α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nhg : Memℒp g p\nhui : UnifIntegrable f p μ\nhfg : TendstoInMeasure μ f atTop g\nns : ℕ → ℕ\nhns : Tendsto ns atTop atTop\n⊢ ∃ ms, Tendsto (fun n => snorm (f (ns (ms n)) - g) p μ) atTop (𝓝 0)",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type ?u.504674\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ℕ → α → β\ng : α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nhg : Memℒp g p\nhui : UnifIntegrable f p μ\nhfg : TendstoInMeasure μ f atTop g\n⊢ Tendsto (fun n => snorm (f n - g) p μ) atTop (𝓝 0)",
"tactic": "refine' tendsto_of_subseq_tendsto fun ns hns => _"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type ?u.504674\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ℕ → α → β\ng : α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nhg : Memℒp g p\nhui : UnifIntegrable f p μ\nhfg : TendstoInMeasure μ f atTop g\nns : ℕ → ℕ\nhns : Tendsto ns atTop atTop\nms : ℕ → ℕ\nleft✝ : StrictMono ms\nhms' : ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns (ms i)) x) atTop (𝓝 (g x))\n⊢ ∃ ms, Tendsto (fun n => snorm (f (ns (ms n)) - g) p μ) atTop (𝓝 0)",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type ?u.504674\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ℕ → α → β\ng : α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nhg : Memℒp g p\nhui : UnifIntegrable f p μ\nhfg : TendstoInMeasure μ f atTop g\nns : ℕ → ℕ\nhns : Tendsto ns atTop atTop\n⊢ ∃ ms, Tendsto (fun n => snorm (f (ns (ms n)) - g) p μ) atTop (𝓝 0)",
"tactic": "obtain ⟨ms, _, hms'⟩ := TendstoInMeasure.exists_seq_tendsto_ae fun ε hε => (hfg ε hε).comp hns"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type ?u.504674\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ℕ → α → β\ng : α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ⊤\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nhg : Memℒp g p\nhui : UnifIntegrable f p μ\nhfg : TendstoInMeasure μ f atTop g\nns : ℕ → ℕ\nhns : Tendsto ns atTop atTop\nms : ℕ → ℕ\nleft✝ : StrictMono ms\nhms' : ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns (ms i)) x) atTop (𝓝 (g x))\n⊢ ∃ ms, Tendsto (fun n => snorm (f (ns (ms n)) - g) p μ) atTop (𝓝 0)",
"tactic": "exact ⟨ms,\n tendsto_Lp_of_tendsto_ae μ hp hp' (fun _ => hf _) hg (fun ε hε =>\n let ⟨δ, hδ, hδ'⟩ := hui hε\n ⟨δ, hδ, fun i s hs hμs => hδ' _ s hs hμs⟩)\n hms'⟩"
}
] |
[
620,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
611,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.set_smul_subset_set_smul_iff₀
|
[] |
[
1035,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1034,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
norm_fderiv_iteratedFDeriv
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\n⊢ ‖fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x‖ = ‖iteratedFDeriv 𝕜 (n + 1) f x‖",
"tactic": "rw [iteratedFDeriv_succ_eq_comp_left, comp_apply, LinearIsometryEquiv.norm_map]"
}
] |
[
1570,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1567,
1
] |
Mathlib/LinearAlgebra/Dfinsupp.lean
|
CompleteLattice.independent_of_dfinsupp_sumAddHom_injective'
|
[
{
"state_after": "ι : Type u_2\nR : Type ?u.632921\nS : Type ?u.632924\nM : ι → Type ?u.632929\nN : Type u_1\ndec_ι : DecidableEq ι\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → AddSubgroup N\nh : Function.Injective ↑(sumAddHom fun i => AddSubgroup.subtype (p i))\n⊢ Independent (↑AddSubgroup.toIntSubmodule ∘ p)",
"state_before": "ι : Type u_2\nR : Type ?u.632921\nS : Type ?u.632924\nM : ι → Type ?u.632929\nN : Type u_1\ndec_ι : DecidableEq ι\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → AddSubgroup N\nh : Function.Injective ↑(sumAddHom fun i => AddSubgroup.subtype (p i))\n⊢ Independent p",
"tactic": "rw [← independent_map_orderIso_iff (AddSubgroup.toIntSubmodule : AddSubgroup N ≃o _)]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nR : Type ?u.632921\nS : Type ?u.632924\nM : ι → Type ?u.632929\nN : Type u_1\ndec_ι : DecidableEq ι\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → AddSubgroup N\nh : Function.Injective ↑(sumAddHom fun i => AddSubgroup.subtype (p i))\n⊢ Independent (↑AddSubgroup.toIntSubmodule ∘ p)",
"tactic": "exact independent_of_dfinsupp_lsum_injective _ h"
}
] |
[
487,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Mathlib/Data/Multiset/Nodup.lean
|
Multiset.nodup_cons
|
[] |
[
45,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
exists_lt_mul_self
|
[] |
[
1940,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1937,
1
] |
Mathlib/Topology/Algebra/OpenSubgroup.lean
|
Subgroup.isOpen_of_openSubgroup
|
[] |
[
331,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/Data/Nat/Pairing.lean
|
Nat.pair_unpair
|
[
{
"state_after": "n : ℕ\n⊢ pair\n (if n - sqrt n * sqrt n < sqrt n then (n - sqrt n * sqrt n, sqrt n)\n else (sqrt n, n - sqrt n * sqrt n - sqrt n)).fst\n (if n - sqrt n * sqrt n < sqrt n then (n - sqrt n * sqrt n, sqrt n)\n else (sqrt n, n - sqrt n * sqrt n - sqrt n)).snd =\n n",
"state_before": "n : ℕ\n⊢ pair (unpair n).fst (unpair n).snd = n",
"tactic": "dsimp only [unpair]"
},
{
"state_after": "n : ℕ\ns : ℕ := sqrt n\n⊢ pair\n (if n - sqrt n * sqrt n < sqrt n then (n - sqrt n * sqrt n, sqrt n)\n else (sqrt n, n - sqrt n * sqrt n - sqrt n)).fst\n (if n - sqrt n * sqrt n < sqrt n then (n - sqrt n * sqrt n, sqrt n)\n else (sqrt n, n - sqrt n * sqrt n - sqrt n)).snd =\n n",
"state_before": "n : ℕ\n⊢ pair\n (if n - sqrt n * sqrt n < sqrt n then (n - sqrt n * sqrt n, sqrt n)\n else (sqrt n, n - sqrt n * sqrt n - sqrt n)).fst\n (if n - sqrt n * sqrt n < sqrt n then (n - sqrt n * sqrt n, sqrt n)\n else (sqrt n, n - sqrt n * sqrt n - sqrt n)).snd =\n n",
"tactic": "let s := sqrt n"
},
{
"state_after": "n : ℕ\ns : ℕ := sqrt n\nsm : s * s + (n - s * s) = n\n⊢ pair\n (if n - sqrt n * sqrt n < sqrt n then (n - sqrt n * sqrt n, sqrt n)\n else (sqrt n, n - sqrt n * sqrt n - sqrt n)).fst\n (if n - sqrt n * sqrt n < sqrt n then (n - sqrt n * sqrt n, sqrt n)\n else (sqrt n, n - sqrt n * sqrt n - sqrt n)).snd =\n n",
"state_before": "n : ℕ\ns : ℕ := sqrt n\n⊢ pair\n (if n - sqrt n * sqrt n < sqrt n then (n - sqrt n * sqrt n, sqrt n)\n else (sqrt n, n - sqrt n * sqrt n - sqrt n)).fst\n (if n - sqrt n * sqrt n < sqrt n then (n - sqrt n * sqrt n, sqrt n)\n else (sqrt n, n - sqrt n * sqrt n - sqrt n)).snd =\n n",
"tactic": "have sm : s * s + (n - s * s) = n := add_tsub_cancel_of_le (sqrt_le _)"
},
{
"state_after": "case inl\nn : ℕ\ns : ℕ := sqrt n\nsm : s * s + (n - s * s) = n\nh : n - sqrt n * sqrt n < sqrt n\n⊢ pair (n - sqrt n * sqrt n, sqrt n).fst (n - sqrt n * sqrt n, sqrt n).snd = n\n\ncase inr\nn : ℕ\ns : ℕ := sqrt n\nsm : s * s + (n - s * s) = n\nh : ¬n - sqrt n * sqrt n < sqrt n\n⊢ pair (sqrt n, n - sqrt n * sqrt n - sqrt n).fst (sqrt n, n - sqrt n * sqrt n - sqrt n).snd = n",
"state_before": "n : ℕ\ns : ℕ := sqrt n\nsm : s * s + (n - s * s) = n\n⊢ pair\n (if n - sqrt n * sqrt n < sqrt n then (n - sqrt n * sqrt n, sqrt n)\n else (sqrt n, n - sqrt n * sqrt n - sqrt n)).fst\n (if n - sqrt n * sqrt n < sqrt n then (n - sqrt n * sqrt n, sqrt n)\n else (sqrt n, n - sqrt n * sqrt n - sqrt n)).snd =\n n",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case inl\nn : ℕ\ns : ℕ := sqrt n\nsm : s * s + (n - s * s) = n\nh : n - sqrt n * sqrt n < sqrt n\n⊢ pair (n - sqrt n * sqrt n, sqrt n).fst (n - sqrt n * sqrt n, sqrt n).snd = n",
"tactic": "simp [pair, h, sm]"
},
{
"state_after": "case inr\nn : ℕ\ns : ℕ := sqrt n\nsm : s * s + (n - s * s) = n\nh : ¬n - sqrt n * sqrt n < sqrt n\nhl : n - s * s - s ≤ s\n⊢ pair (sqrt n, n - sqrt n * sqrt n - sqrt n).fst (sqrt n, n - sqrt n * sqrt n - sqrt n).snd = n",
"state_before": "case inr\nn : ℕ\ns : ℕ := sqrt n\nsm : s * s + (n - s * s) = n\nh : ¬n - sqrt n * sqrt n < sqrt n\n⊢ pair (sqrt n, n - sqrt n * sqrt n - sqrt n).fst (sqrt n, n - sqrt n * sqrt n - sqrt n).snd = n",
"tactic": "have hl : n - s * s - s ≤ s :=\n tsub_le_iff_left.mpr (tsub_le_iff_left.mpr <| by rw [← add_assoc] ; apply sqrt_le_add)"
},
{
"state_after": "no goals",
"state_before": "case inr\nn : ℕ\ns : ℕ := sqrt n\nsm : s * s + (n - s * s) = n\nh : ¬n - sqrt n * sqrt n < sqrt n\nhl : n - s * s - s ≤ s\n⊢ pair (sqrt n, n - sqrt n * sqrt n - sqrt n).fst (sqrt n, n - sqrt n * sqrt n - sqrt n).snd = n",
"tactic": "simp [pair, hl.not_lt, add_assoc, add_tsub_cancel_of_le (le_of_not_gt h), sm]"
},
{
"state_after": "n : ℕ\ns : ℕ := sqrt n\nsm : s * s + (n - s * s) = n\nh : ¬n - sqrt n * sqrt n < sqrt n\n⊢ n ≤ s * s + s + s",
"state_before": "n : ℕ\ns : ℕ := sqrt n\nsm : s * s + (n - s * s) = n\nh : ¬n - sqrt n * sqrt n < sqrt n\n⊢ n ≤ s * s + (s + s)",
"tactic": "rw [← add_assoc]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\ns : ℕ := sqrt n\nsm : s * s + (n - s * s) = n\nh : ¬n - sqrt n * sqrt n < sqrt n\n⊢ n ≤ s * s + s + s",
"tactic": "apply sqrt_le_add"
}
] |
[
60,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Data/PFunctor/Univariate/M.lean
|
PFunctor.Approx.agree_children
|
[
{
"state_after": "case intro\nF : PFunctor\nn : ℕ\na✝ : F.A\nx✝ : B F a✝ → CofixA F n\nx'✝ : B F a✝ → CofixA F (n + 1)\nhagree : ∀ (i : B F a✝), Agree (x✝ i) (x'✝ i)\ni : B F (head' (CofixA.intro a✝ x✝))\nj : B F (head' (CofixA.intro a✝ x'✝))\nh₀ : HEq i j\n⊢ Agree (children' (CofixA.intro a✝ x✝) i) (children' (CofixA.intro a✝ x'✝) j)",
"state_before": "F : PFunctor\nn : ℕ\nx : CofixA F (succ n)\ny : CofixA F (succ n + 1)\ni : B F (head' x)\nj : B F (head' y)\nh₀ : HEq i j\nh₁ : Agree x y\n⊢ Agree (children' x i) (children' y j)",
"tactic": "cases' h₁ with _ _ _ _ _ _ hagree"
},
{
"state_after": "case intro.refl\nF : PFunctor\nn : ℕ\na✝ : F.A\nx✝ : B F a✝ → CofixA F n\nx'✝ : B F a✝ → CofixA F (n + 1)\nhagree : ∀ (i : B F a✝), Agree (x✝ i) (x'✝ i)\ni : B F (head' (CofixA.intro a✝ x✝))\n⊢ Agree (children' (CofixA.intro a✝ x✝) i) (children' (CofixA.intro a✝ x'✝) i)",
"state_before": "case intro\nF : PFunctor\nn : ℕ\na✝ : F.A\nx✝ : B F a✝ → CofixA F n\nx'✝ : B F a✝ → CofixA F (n + 1)\nhagree : ∀ (i : B F a✝), Agree (x✝ i) (x'✝ i)\ni : B F (head' (CofixA.intro a✝ x✝))\nj : B F (head' (CofixA.intro a✝ x'✝))\nh₀ : HEq i j\n⊢ Agree (children' (CofixA.intro a✝ x✝) i) (children' (CofixA.intro a✝ x'✝) j)",
"tactic": "cases h₀"
},
{
"state_after": "no goals",
"state_before": "case intro.refl\nF : PFunctor\nn : ℕ\na✝ : F.A\nx✝ : B F a✝ → CofixA F n\nx'✝ : B F a✝ → CofixA F (n + 1)\nhagree : ∀ (i : B F a✝), Agree (x✝ i) (x'✝ i)\ni : B F (head' (CofixA.intro a✝ x✝))\n⊢ Agree (children' (CofixA.intro a✝ x✝) i) (children' (CofixA.intro a✝ x'✝) i)",
"tactic": "apply hagree"
}
] |
[
95,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Logic/Function/Iterate.lean
|
Function.Semiconj.iterate_right
|
[] |
[
109,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Topology/UniformSpace/Equicontinuity.lean
|
EquicontinuousAt.closure
|
[] |
[
392,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Std/Data/String/Lemmas.lean
|
String.posOfAux_eq
|
[] |
[
313,
69
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
313,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.mk_coeFn
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.308002\nδ : Type ?u.308005\ninst✝³ : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α →ₘ[μ] β\n⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = Quotient.mk'' (Quotient.out' f)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.308002\nδ : Type ?u.308005\ninst✝³ : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α →ₘ[μ] β\n⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = f",
"tactic": "conv_rhs => rw [← Quotient.out_eq' f]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.308002\nδ : Type ?u.308005\ninst✝³ : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α →ₘ[μ] β\ng : { f // AEStronglyMeasurable f μ } := Quotient.out' f\n⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = Quotient.mk'' g",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.308002\nδ : Type ?u.308005\ninst✝³ : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α →ₘ[μ] β\n⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = Quotient.mk'' (Quotient.out' f)",
"tactic": "set g : { f : α → β // AEStronglyMeasurable f μ } := Quotient.out' f"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.308002\nδ : Type ?u.308005\ninst✝³ : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α →ₘ[μ] β\ng : { f // AEStronglyMeasurable f μ } := Quotient.out' f\nthis : g = { val := ↑g, property := (_ : AEStronglyMeasurable (↑g) μ) }\n⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = Quotient.mk'' g",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.308002\nδ : Type ?u.308005\ninst✝³ : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α →ₘ[μ] β\ng : { f // AEStronglyMeasurable f μ } := Quotient.out' f\n⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = Quotient.mk'' g",
"tactic": "have : g = ⟨g.1, g.2⟩ := Subtype.eq rfl"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.308002\nδ : Type ?u.308005\ninst✝³ : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α →ₘ[μ] β\ng : { f // AEStronglyMeasurable f μ } := Quotient.out' f\nthis : g = { val := ↑g, property := (_ : AEStronglyMeasurable (↑g) μ) }\n⊢ ↑f =ᵐ[μ] ↑g",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.308002\nδ : Type ?u.308005\ninst✝³ : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α →ₘ[μ] β\ng : { f // AEStronglyMeasurable f μ } := Quotient.out' f\nthis : g = { val := ↑g, property := (_ : AEStronglyMeasurable (↑g) μ) }\n⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = Quotient.mk'' g",
"tactic": "rw [this, ← mk, mk_eq_mk]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.308002\nδ : Type ?u.308005\ninst✝³ : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α →ₘ[μ] β\ng : { f // AEStronglyMeasurable f μ } := Quotient.out' f\nthis : g = { val := ↑g, property := (_ : AEStronglyMeasurable (↑g) μ) }\n⊢ ↑f =ᵐ[μ] ↑g",
"tactic": "exact (AEStronglyMeasurable.ae_eq_mk _).symm"
}
] |
[
171,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Algebra/Order/Monoid/WithTop.lean
|
WithTop.zero_lt_coe
|
[] |
[
415,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/LinearAlgebra/Pi.lean
|
Submodule.iSup_map_single
|
[
{
"state_after": "case intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁴ : Semiring R\nφ : ι → Type u_1\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx : (i : ι) → φ i\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\n⊢ (⨆ (i : ι), map (single i) (p i)) = pi Set.univ p",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁴ : Semiring R\nφ : ι → Type u_1\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx : (i : ι) → φ i\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\n⊢ (⨆ (i : ι), map (single i) (p i)) = pi Set.univ p",
"tactic": "cases nonempty_fintype ι"
},
{
"state_after": "case intro.refine'_1\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁴ : Semiring R\nφ : ι → Type u_1\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx : (i : ι) → φ i\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\ni : ι\n⊢ map (single i) (p i) ≤ pi Set.univ p\n\ncase intro.refine'_2\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁴ : Semiring R\nφ : ι → Type u_1\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx : (i : ι) → φ i\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\n⊢ pi Set.univ p ≤ ⨆ (i : ι), map (single i) (p i)",
"state_before": "case intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁴ : Semiring R\nφ : ι → Type u_1\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx : (i : ι) → φ i\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\n⊢ (⨆ (i : ι), map (single i) (p i)) = pi Set.univ p",
"tactic": "refine' (iSup_le fun i => _).antisymm _"
},
{
"state_after": "case intro.refine'_1.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁴ : Semiring R\nφ : ι → Type u_1\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\ni : ι\nx : φ i\nhx : x ∈ p i\nj : ι\n⊢ ↑(single i) x j ∈ (fun i => ↑(p i)) j",
"state_before": "case intro.refine'_1\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁴ : Semiring R\nφ : ι → Type u_1\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx : (i : ι) → φ i\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\ni : ι\n⊢ map (single i) (p i) ≤ pi Set.univ p",
"tactic": "rintro _ ⟨x, hx : x ∈ p i, rfl⟩ j -"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_1.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁴ : Semiring R\nφ : ι → Type u_1\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\ni : ι\nx : φ i\nhx : x ∈ p i\nj : ι\n⊢ ↑(single i) x j ∈ (fun i => ↑(p i)) j",
"tactic": "rcases em (j = i) with (rfl | hj) <;> simp [*]"
},
{
"state_after": "case intro.refine'_2\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁴ : Semiring R\nφ : ι → Type u_1\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\nx : (i : ι) → φ i\nhx : x ∈ pi Set.univ p\n⊢ x ∈ ⨆ (i : ι), map (single i) (p i)",
"state_before": "case intro.refine'_2\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁴ : Semiring R\nφ : ι → Type u_1\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx : (i : ι) → φ i\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\n⊢ pi Set.univ p ≤ ⨆ (i : ι), map (single i) (p i)",
"tactic": "intro x hx"
},
{
"state_after": "case intro.refine'_2\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁴ : Semiring R\nφ : ι → Type u_1\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\nx : (i : ι) → φ i\nhx : x ∈ pi Set.univ p\n⊢ ∑ i : ι, Pi.single i (x i) ∈ ⨆ (i : ι), map (single i) (p i)",
"state_before": "case intro.refine'_2\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁴ : Semiring R\nφ : ι → Type u_1\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\nx : (i : ι) → φ i\nhx : x ∈ pi Set.univ p\n⊢ x ∈ ⨆ (i : ι), map (single i) (p i)",
"tactic": "rw [← Finset.univ_sum_single x]"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_2\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁴ : Semiring R\nφ : ι → Type u_1\ninst✝³ : (i : ι) → AddCommMonoid (φ i)\ninst✝² : (i : ι) → Module R (φ i)\nI : Set ι\np q : (i : ι) → Submodule R (φ i)\nx✝ : (i : ι) → φ i\ninst✝¹ : DecidableEq ι\ninst✝ : Finite ι\nval✝ : Fintype ι\nx : (i : ι) → φ i\nhx : x ∈ pi Set.univ p\n⊢ ∑ i : ι, Pi.single i (x i) ∈ ⨆ (i : ι), map (single i) (p i)",
"tactic": "exact sum_mem_iSup fun i => mem_map_of_mem (hx i trivial)"
}
] |
[
320,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
312,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometry.map_smulₛₗ
|
[] |
[
224,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
11
] |
Mathlib/Topology/LocallyConstant/Basic.lean
|
IsLocallyConstant.iff_isOpen_fiber
|
[] |
[
111,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Topology/Order.lean
|
discreteTopology_bot
|
[] |
[
276,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
275,
1
] |
Mathlib/Algebra/Module/LocalizedModule.lean
|
IsLocalizedModule.smul_inj
|
[] |
[
914,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
913,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.val_toFinset
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.455035\nγ : Type ?u.455038\ninst✝ : DecidableEq α\ns : Finset α\na✝ : α\n⊢ a✝ ∈ toFinset s.val ↔ a✝ ∈ s",
"state_before": "α : Type u_1\nβ : Type ?u.455035\nγ : Type ?u.455038\ninst✝ : DecidableEq α\ns : Finset α\n⊢ toFinset s.val = s",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.455035\nγ : Type ?u.455038\ninst✝ : DecidableEq α\ns : Finset α\na✝ : α\n⊢ a✝ ∈ toFinset s.val ↔ a✝ ∈ s",
"tactic": "rw [Multiset.mem_toFinset, ← mem_def]"
}
] |
[
3218,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3216,
1
] |
Mathlib/Data/Nat/Choose/Basic.lean
|
Nat.choose_eq_asc_factorial_div_factorial
|
[
{
"state_after": "n k : ℕ\n⊢ k ! * choose (n + k) k = k ! * (ascFactorial n k / k !)",
"state_before": "n k : ℕ\n⊢ choose (n + k) k = ascFactorial n k / k !",
"tactic": "apply mul_left_cancel₀ (factorial_ne_zero k)"
},
{
"state_after": "n k : ℕ\n⊢ ascFactorial n k = k ! * (ascFactorial n k / k !)",
"state_before": "n k : ℕ\n⊢ k ! * choose (n + k) k = k ! * (ascFactorial n k / k !)",
"tactic": "rw [← ascFactorial_eq_factorial_mul_choose]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\n⊢ ascFactorial n k = k ! * (ascFactorial n k / k !)",
"tactic": "exact (Nat.mul_div_cancel' <| factorial_dvd_ascFactorial _ _).symm"
}
] |
[
255,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
tendsto_one_div_add_atTop_nhds_0_nat
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.3724\nβ : Type ?u.3727\nι : Type ?u.3730\nthis : Tendsto (fun n => 1 / ↑(n + 1)) atTop (𝓝 0)\n⊢ Tendsto (fun n => 1 / (↑n + 1)) atTop (𝓝 0)",
"tactic": "simpa"
}
] |
[
56,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
mul_le_of_mul_le_of_nonneg_right
|
[] |
[
435,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/Topology/Order.lean
|
le_of_nhds_le_nhds
|
[
{
"state_after": "α : Type u_1\nt t₁ t₂ : TopologicalSpace α\ns✝ : Set α\nh : ∀ (x : α), 𝓝 x ≤ 𝓝 x\ns : Set α\n⊢ (∀ (a : α), a ∈ s → s ∈ 𝓝 a) → ∀ (a : α), a ∈ s → s ∈ 𝓝 a",
"state_before": "α : Type u_1\nt t₁ t₂ : TopologicalSpace α\ns✝ : Set α\nh : ∀ (x : α), 𝓝 x ≤ 𝓝 x\ns : Set α\n⊢ IsOpen s → IsOpen s",
"tactic": "rw [@isOpen_iff_mem_nhds _ t₁, @isOpen_iff_mem_nhds α t₂]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nt t₁ t₂ : TopologicalSpace α\ns✝ : Set α\nh : ∀ (x : α), 𝓝 x ≤ 𝓝 x\ns : Set α\n⊢ (∀ (a : α), a ∈ s → s ∈ 𝓝 a) → ∀ (a : α), a ∈ s → s ∈ 𝓝 a",
"tactic": "exact fun hs a ha => h _ (hs _ ha)"
}
] |
[
306,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioi_union_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.45790\ninst✝ : PartialOrder α\na b c x : α\n⊢ x ∈ Ioi a ∪ {a} ↔ x ∈ Ici a",
"tactic": "simp [eq_comm, le_iff_eq_or_lt]"
}
] |
[
848,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
847,
1
] |
Mathlib/Computability/NFA.lean
|
NFA.pumping_lemma
|
[
{
"state_after": "α : Type u\nσ σ' : Type v\nM : NFA α σ\ninst✝ : Fintype σ\nx : List α\nhx : x ∈ DFA.accepts (toDFA M)\nhlen : Fintype.card (Set σ) ≤ List.length x\n⊢ ∃ a b c,\n x = a ++ b ++ c ∧\n List.length a + List.length b ≤ Fintype.card (Set σ) ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ DFA.accepts (toDFA M)",
"state_before": "α : Type u\nσ σ' : Type v\nM : NFA α σ\ninst✝ : Fintype σ\nx : List α\nhx : x ∈ accepts M\nhlen : Fintype.card (Set σ) ≤ List.length x\n⊢ ∃ a b c,\n x = a ++ b ++ c ∧ List.length a + List.length b ≤ Fintype.card (Set σ) ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ accepts M",
"tactic": "rw [← toDFA_correct] at hx ⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nσ σ' : Type v\nM : NFA α σ\ninst✝ : Fintype σ\nx : List α\nhx : x ∈ DFA.accepts (toDFA M)\nhlen : Fintype.card (Set σ) ≤ List.length x\n⊢ ∃ a b c,\n x = a ++ b ++ c ∧\n List.length a + List.length b ≤ Fintype.card (Set σ) ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ DFA.accepts (toDFA M)",
"tactic": "exact M.toDFA.pumping_lemma hx hlen"
}
] |
[
135,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.zero_le
|
[
{
"state_after": "case mk\nα✝ β : Type u\na✝ : Cardinal\nα : Type u_1\n⊢ 0 ≤ Quot.mk Setoid.r α",
"state_before": "α β : Type u\n⊢ ∀ (a : Cardinal), 0 ≤ a",
"tactic": "rintro ⟨α⟩"
},
{
"state_after": "no goals",
"state_before": "case mk\nα✝ β : Type u\na✝ : Cardinal\nα : Type u_1\n⊢ 0 ≤ Quot.mk Setoid.r α",
"tactic": "exact ⟨Embedding.ofIsEmpty⟩"
}
] |
[
657,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
655,
11
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.radical_sup
|
[] |
[
923,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
921,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.int_cast_imI
|
[] |
[
1005,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1005,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.tail_interleave
|
[
{
"state_after": "α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\n⊢ tail\n (corec\n (fun x =>\n match x with\n | (s₁, snd) => head s₁)\n (fun x =>\n match x with\n | (s₁, s₂) => (s₂, tail s₁))\n (s₁, s₂)) =\n corec\n (fun x =>\n match x with\n | (s₁, snd) => head s₁)\n (fun x =>\n match x with\n | (s₁, s₂) => (s₂, tail s₁))\n (s₂, tail s₁)",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\n⊢ tail (s₁ ⋈ s₂) = s₂ ⋈ tail s₁",
"tactic": "unfold interleave corecOn"
},
{
"state_after": "α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\n⊢ tail\n ((match (s₁, s₂) with\n | (s₁, snd) => head s₁) ::\n corec\n (fun x =>\n match x with\n | (s₁, snd) => head s₁)\n (fun x =>\n match x with\n | (s₁, s₂) => (s₂, tail s₁))\n (match (s₁, s₂) with\n | (s₁, s₂) => (s₂, tail s₁))) =\n corec\n (fun x =>\n match x with\n | (s₁, snd) => head s₁)\n (fun x =>\n match x with\n | (s₁, s₂) => (s₂, tail s₁))\n (s₂, tail s₁)",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\n⊢ tail\n (corec\n (fun x =>\n match x with\n | (s₁, snd) => head s₁)\n (fun x =>\n match x with\n | (s₁, s₂) => (s₂, tail s₁))\n (s₁, s₂)) =\n corec\n (fun x =>\n match x with\n | (s₁, snd) => head s₁)\n (fun x =>\n match x with\n | (s₁, s₂) => (s₂, tail s₁))\n (s₂, tail s₁)",
"tactic": "rw [corec_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\ns₁ s₂ : Stream' α\n⊢ tail\n ((match (s₁, s₂) with\n | (s₁, snd) => head s₁) ::\n corec\n (fun x =>\n match x with\n | (s₁, snd) => head s₁)\n (fun x =>\n match x with\n | (s₁, s₂) => (s₂, tail s₁))\n (match (s₁, s₂) with\n | (s₁, s₂) => (s₂, tail s₁))) =\n corec\n (fun x =>\n match x with\n | (s₁, snd) => head s₁)\n (fun x =>\n match x with\n | (s₁, s₂) => (s₂, tail s₁))\n (s₂, tail s₁)",
"tactic": "rfl"
}
] |
[
424,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
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