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/Vector.lean
|
Vector.cons_head_tail
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nφ : Type w\nn : ℕ\nh : List.length [] = succ n\n⊢ cons (head { val := [], property := h }) (tail { val := [], property := h }) = { val := [], property := h }",
"tactic": "contradiction"
}
] |
[
84,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
PosNum.cast_sub'
|
[
{
"state_after": "α : Type u_1\ninst✝ : AddGroupWithOne α\na : PosNum\n⊢ ↑(↑a - 1) = ↑a - ↑1",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na : PosNum\n⊢ ↑(sub' a 1) = ↑a - ↑1",
"tactic": "rw [sub'_one, Num.cast_toZNum, ← Num.cast_to_nat, pred'_to_nat, ← Nat.sub_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na : PosNum\n⊢ ↑(↑a - 1) = ↑a - ↑1",
"tactic": "simp [PosNum.cast_pos]"
},
{
"state_after": "α : Type u_1\ninst✝ : AddGroupWithOne α\nb : PosNum\n⊢ ↑(↑b - 1) = ↑b - ↑1",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\nb : PosNum\n⊢ ↑(sub' 1 b) = ↑1 - ↑b",
"tactic": "rw [one_sub', Num.cast_toZNumNeg, ← neg_sub, neg_inj, ← Num.cast_to_nat, pred'_to_nat,\n ← Nat.sub_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\nb : PosNum\n⊢ ↑(↑b - 1) = ↑b - ↑1",
"tactic": "simp [PosNum.cast_pos]"
},
{
"state_after": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ _root_.bit0 (↑a - ↑b) = ↑(bit0 a) - ↑(bit0 b)",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ ↑(sub' (bit0 a) (bit0 b)) = ↑(bit0 a) - ↑(bit0 b)",
"tactic": "rw [sub', ZNum.cast_bit0, cast_sub' a b]"
},
{
"state_after": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\nthis : ↑(↑a + -↑b + (↑a + -↑b)) = ↑a + ↑a + (-↑b + -↑b)\n⊢ _root_.bit0 (↑a - ↑b) = ↑(bit0 a) - ↑(bit0 b)",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ _root_.bit0 (↑a - ↑b) = ↑(bit0 a) - ↑(bit0 b)",
"tactic": "have : ((a + -b + (a + -b) : ℤ) : α) = a + a + (-b + -b) := by simp [add_left_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\nthis : ↑(↑a + -↑b + (↑a + -↑b)) = ↑a + ↑a + (-↑b + -↑b)\n⊢ _root_.bit0 (↑a - ↑b) = ↑(bit0 a) - ↑(bit0 b)",
"tactic": "simpa [_root_.bit0, sub_eq_add_neg] using this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ ↑(↑a + -↑b + (↑a + -↑b)) = ↑a + ↑a + (-↑b + -↑b)",
"tactic": "simp [add_left_comm]"
},
{
"state_after": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ _root_.bit0 (↑a - ↑b) - 1 = ↑(bit0 a) - ↑(bit1 b)",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ ↑(sub' (bit0 a) (bit1 b)) = ↑(bit0 a) - ↑(bit1 b)",
"tactic": "rw [sub', ZNum.cast_bitm1, cast_sub' a b]"
},
{
"state_after": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\nthis : ↑(-↑b + (↑a + (-↑b + -1))) = ↑(↑a + -1 + (-↑b + -↑b))\n⊢ _root_.bit0 (↑a - ↑b) - 1 = ↑(bit0 a) - ↑(bit1 b)",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ _root_.bit0 (↑a - ↑b) - 1 = ↑(bit0 a) - ↑(bit1 b)",
"tactic": "have : ((-b + (a + (-b + -1)) : ℤ) : α) = (a + -1 + (-b + -b) : ℤ) := by\n simp [add_comm, add_left_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\nthis : ↑(-↑b + (↑a + (-↑b + -1))) = ↑(↑a + -1 + (-↑b + -↑b))\n⊢ _root_.bit0 (↑a - ↑b) - 1 = ↑(bit0 a) - ↑(bit1 b)",
"tactic": "simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ ↑(-↑b + (↑a + (-↑b + -1))) = ↑(↑a + -1 + (-↑b + -↑b))",
"tactic": "simp [add_comm, add_left_comm]"
},
{
"state_after": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ _root_.bit1 (↑a - ↑b) = ↑(bit1 a) - ↑(bit0 b)",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ ↑(sub' (bit1 a) (bit0 b)) = ↑(bit1 a) - ↑(bit0 b)",
"tactic": "rw [sub', ZNum.cast_bit1, cast_sub' a b]"
},
{
"state_after": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\nthis : ↑(-↑b + (↑a + (-↑b + 1))) = ↑(↑a + 1 + (-↑b + -↑b))\n⊢ _root_.bit1 (↑a - ↑b) = ↑(bit1 a) - ↑(bit0 b)",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ _root_.bit1 (↑a - ↑b) = ↑(bit1 a) - ↑(bit0 b)",
"tactic": "have : ((-b + (a + (-b + 1)) : ℤ) : α) = (a + 1 + (-b + -b) : ℤ) := by\n simp [add_comm, add_left_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\nthis : ↑(-↑b + (↑a + (-↑b + 1))) = ↑(↑a + 1 + (-↑b + -↑b))\n⊢ _root_.bit1 (↑a - ↑b) = ↑(bit1 a) - ↑(bit0 b)",
"tactic": "simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ ↑(-↑b + (↑a + (-↑b + 1))) = ↑(↑a + 1 + (-↑b + -↑b))",
"tactic": "simp [add_comm, add_left_comm]"
},
{
"state_after": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ _root_.bit0 (↑a - ↑b) = ↑(bit1 a) - ↑(bit1 b)",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ ↑(sub' (bit1 a) (bit1 b)) = ↑(bit1 a) - ↑(bit1 b)",
"tactic": "rw [sub', ZNum.cast_bit0, cast_sub' a b]"
},
{
"state_after": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\nthis : ↑(-↑b + (↑a + -↑b)) = ↑a + (-↑b + -↑b)\n⊢ _root_.bit0 (↑a - ↑b) = ↑(bit1 a) - ↑(bit1 b)",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ _root_.bit0 (↑a - ↑b) = ↑(bit1 a) - ↑(bit1 b)",
"tactic": "have : ((-b + (a + -b) : ℤ) : α) = a + (-b + -b) := by simp [add_left_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\nthis : ↑(-↑b + (↑a + -↑b)) = ↑a + (-↑b + -↑b)\n⊢ _root_.bit0 (↑a - ↑b) = ↑(bit1 a) - ↑(bit1 b)",
"tactic": "simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\na b : PosNum\n⊢ ↑(-↑b + (↑a + -↑b)) = ↑a + (-↑b + -↑b)",
"tactic": "simp [add_left_comm]"
}
] |
[
1213,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1188,
1
] |
Mathlib/Algebra/Group/Defs.lean
|
inv_mul_cancel_right
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝ : Group G\na✝ b✝ c a b : G\n⊢ a * b⁻¹ * b = a",
"tactic": "rw [mul_assoc, mul_left_inv, mul_one]"
}
] |
[
1125,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1124,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.infiniteNeg_mul_infiniteNeg
|
[] |
[
869,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
867,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean
|
HomogeneousLocalization.NumDenSameDeg.deg_zero
|
[] |
[
147,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion_ite
|
[] |
[
633,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
631,
1
] |
Mathlib/LinearAlgebra/SesquilinearForm.lean
|
LinearMap.IsRefl.ker_eq_bot_iff_ker_flip_eq_bot
|
[
{
"state_after": "R : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.124036\nR₃ : Type ?u.124039\nM : Type ?u.124042\nM₁ : Type u_3\nM₂ : Type ?u.124048\nMₗ₁ : Type ?u.124051\nMₗ₁' : Type ?u.124054\nMₗ₂ : Type ?u.124057\nMₗ₂' : Type ?u.124060\nK : Type ?u.124063\nK₁ : Type ?u.124066\nK₂ : Type ?u.124069\nV : Type ?u.124072\nV₁ : Type ?u.124075\nV₂ : Type ?u.124078\nn : Type ?u.124081\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\nH✝ H : IsRefl B\nh : ker (flip B) = ⊥\n⊢ ker B = ⊥",
"state_before": "R : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.124036\nR₃ : Type ?u.124039\nM : Type ?u.124042\nM₁ : Type u_3\nM₂ : Type ?u.124048\nMₗ₁ : Type ?u.124051\nMₗ₁' : Type ?u.124054\nMₗ₂ : Type ?u.124057\nMₗ₂' : Type ?u.124060\nK : Type ?u.124063\nK₁ : Type ?u.124066\nK₂ : Type ?u.124069\nV : Type ?u.124072\nV₁ : Type ?u.124075\nV₂ : Type ?u.124078\nn : Type ?u.124081\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\nH✝ H : IsRefl B\n⊢ ker B = ⊥ ↔ ker (flip B) = ⊥",
"tactic": "refine' ⟨ker_flip_eq_bot H, fun h ↦ _⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.124036\nR₃ : Type ?u.124039\nM : Type ?u.124042\nM₁ : Type u_3\nM₂ : Type ?u.124048\nMₗ₁ : Type ?u.124051\nMₗ₁' : Type ?u.124054\nMₗ₂ : Type ?u.124057\nMₗ₂' : Type ?u.124060\nK : Type ?u.124063\nK₁ : Type ?u.124066\nK₂ : Type ?u.124069\nV : Type ?u.124072\nV₁ : Type ?u.124075\nV₂ : Type ?u.124078\nn : Type ?u.124081\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\nH✝ H : IsRefl B\nh : ker (flip B) = ⊥\n⊢ ker B = ⊥",
"tactic": "exact (congr_arg _ B.flip_flip.symm).trans (ker_flip_eq_bot (flip_isRefl_iff.mpr H) h)"
}
] |
[
205,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.head_zip
|
[] |
[
213,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Order/Lattice.lean
|
right_eq_sup
|
[] |
[
195,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Data/Nat/Choose/Central.lean
|
Nat.succ_mul_centralBinom_succ
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ choose (2 * n + 2) (n + 1) * (n + 1) = choose (2 * n + 1) n * (2 * n + 2)",
"tactic": "rw [choose_succ_right_eq, choose_mul_succ_eq]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ choose (2 * n + 1) n * (2 * n + 2) = 2 * (choose (2 * n + 1) n * (n + 1))",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ 2 * (choose (2 * n + 1) n * (n + 1)) = 2 * (choose (2 * n + 1) n * (2 * n + 1 - n))",
"tactic": "rw [two_mul n, add_assoc,\n Nat.add_sub_cancel_left]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ 2 * (choose (2 * n + 1) n * (2 * n + 1 - n)) = 2 * (choose (2 * n) n * (2 * n + 1))",
"tactic": "rw [choose_mul_succ_eq]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ 2 * (choose (2 * n) n * (2 * n + 1)) = 2 * (2 * n + 1) * choose (2 * n) n",
"tactic": "rw [mul_assoc, mul_comm (2 * n + 1)]"
}
] |
[
83,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Algebra/Module/LinearMap.lean
|
image_smul_setₛₗ
|
[
{
"state_after": "case h₁\nR : Type u_2\nR₁ : Type ?u.144905\nR₂ : Type ?u.144908\nR₃ : Type ?u.144911\nk : Type ?u.144914\nS : Type u_3\nS₃ : Type ?u.144920\nT : Type ?u.144923\nM : Type u_4\nM₁ : Type ?u.144929\nM₂ : Type ?u.144932\nM₃ : Type u_5\nN₁ : Type ?u.144938\nN₂ : Type ?u.144941\nN₃ : Type ?u.144944\nι : Type ?u.144947\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid M₃\ninst✝⁶ : AddCommMonoid N₁\ninst✝⁵ : AddCommMonoid N₂\ninst✝⁴ : AddCommMonoid N₃\ninst✝³ : Module R M\ninst✝² : Module R M₂\ninst✝¹ : Module S M₃\nσ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf g : M →ₛₗ[σ] M₃\nF : Type u_1\nh : F\ninst✝ : SemilinearMapClass F σ M M₃\nc : R\ns : Set M\n⊢ ↑h '' (c • s) ⊆ ↑σ c • ↑h '' s\n\ncase h₂\nR : Type u_2\nR₁ : Type ?u.144905\nR₂ : Type ?u.144908\nR₃ : Type ?u.144911\nk : Type ?u.144914\nS : Type u_3\nS₃ : Type ?u.144920\nT : Type ?u.144923\nM : Type u_4\nM₁ : Type ?u.144929\nM₂ : Type ?u.144932\nM₃ : Type u_5\nN₁ : Type ?u.144938\nN₂ : Type ?u.144941\nN₃ : Type ?u.144944\nι : Type ?u.144947\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid M₃\ninst✝⁶ : AddCommMonoid N₁\ninst✝⁵ : AddCommMonoid N₂\ninst✝⁴ : AddCommMonoid N₃\ninst✝³ : Module R M\ninst✝² : Module R M₂\ninst✝¹ : Module S M₃\nσ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf g : M →ₛₗ[σ] M₃\nF : Type u_1\nh : F\ninst✝ : SemilinearMapClass F σ M M₃\nc : R\ns : Set M\n⊢ ↑σ c • ↑h '' s ⊆ ↑h '' (c • s)",
"state_before": "R : Type u_2\nR₁ : Type ?u.144905\nR₂ : Type ?u.144908\nR₃ : Type ?u.144911\nk : Type ?u.144914\nS : Type u_3\nS₃ : Type ?u.144920\nT : Type ?u.144923\nM : Type u_4\nM₁ : Type ?u.144929\nM₂ : Type ?u.144932\nM₃ : Type u_5\nN₁ : Type ?u.144938\nN₂ : Type ?u.144941\nN₃ : Type ?u.144944\nι : Type ?u.144947\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid M₃\ninst✝⁶ : AddCommMonoid N₁\ninst✝⁵ : AddCommMonoid N₂\ninst✝⁴ : AddCommMonoid N₃\ninst✝³ : Module R M\ninst✝² : Module R M₂\ninst✝¹ : Module S M₃\nσ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf g : M →ₛₗ[σ] M₃\nF : Type u_1\nh : F\ninst✝ : SemilinearMapClass F σ M M₃\nc : R\ns : Set M\n⊢ ↑h '' (c • s) = ↑σ c • ↑h '' s",
"tactic": "apply Set.Subset.antisymm"
},
{
"state_after": "case h₁.intro.intro.intro.intro\nR : Type u_2\nR₁ : Type ?u.144905\nR₂ : Type ?u.144908\nR₃ : Type ?u.144911\nk : Type ?u.144914\nS : Type u_3\nS₃ : Type ?u.144920\nT : Type ?u.144923\nM : Type u_4\nM₁ : Type ?u.144929\nM₂ : Type ?u.144932\nM₃ : Type u_5\nN₁ : Type ?u.144938\nN₂ : Type ?u.144941\nN₃ : Type ?u.144944\nι : Type ?u.144947\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid M₃\ninst✝⁶ : AddCommMonoid N₁\ninst✝⁵ : AddCommMonoid N₂\ninst✝⁴ : AddCommMonoid N₃\ninst✝³ : Module R M\ninst✝² : Module R M₂\ninst✝¹ : Module S M₃\nσ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf g : M →ₛₗ[σ] M₃\nF : Type u_1\nh : F\ninst✝ : SemilinearMapClass F σ M M₃\nc : R\ns : Set M\nz : M\nzs : z ∈ s\n⊢ ↑h ((fun x => c • x) z) ∈ ↑σ c • ↑h '' s",
"state_before": "case h₁\nR : Type u_2\nR₁ : Type ?u.144905\nR₂ : Type ?u.144908\nR₃ : Type ?u.144911\nk : Type ?u.144914\nS : Type u_3\nS₃ : Type ?u.144920\nT : Type ?u.144923\nM : Type u_4\nM₁ : Type ?u.144929\nM₂ : Type ?u.144932\nM₃ : Type u_5\nN₁ : Type ?u.144938\nN₂ : Type ?u.144941\nN₃ : Type ?u.144944\nι : Type ?u.144947\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid M₃\ninst✝⁶ : AddCommMonoid N₁\ninst✝⁵ : AddCommMonoid N₂\ninst✝⁴ : AddCommMonoid N₃\ninst✝³ : Module R M\ninst✝² : Module R M₂\ninst✝¹ : Module S M₃\nσ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf g : M →ₛₗ[σ] M₃\nF : Type u_1\nh : F\ninst✝ : SemilinearMapClass F σ M M₃\nc : R\ns : Set M\n⊢ ↑h '' (c • s) ⊆ ↑σ c • ↑h '' s",
"tactic": "rintro x ⟨y, ⟨z, zs, rfl⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h₁.intro.intro.intro.intro\nR : Type u_2\nR₁ : Type ?u.144905\nR₂ : Type ?u.144908\nR₃ : Type ?u.144911\nk : Type ?u.144914\nS : Type u_3\nS₃ : Type ?u.144920\nT : Type ?u.144923\nM : Type u_4\nM₁ : Type ?u.144929\nM₂ : Type ?u.144932\nM₃ : Type u_5\nN₁ : Type ?u.144938\nN₂ : Type ?u.144941\nN₃ : Type ?u.144944\nι : Type ?u.144947\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid M₃\ninst✝⁶ : AddCommMonoid N₁\ninst✝⁵ : AddCommMonoid N₂\ninst✝⁴ : AddCommMonoid N₃\ninst✝³ : Module R M\ninst✝² : Module R M₂\ninst✝¹ : Module S M₃\nσ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf g : M →ₛₗ[σ] M₃\nF : Type u_1\nh : F\ninst✝ : SemilinearMapClass F σ M M₃\nc : R\ns : Set M\nz : M\nzs : z ∈ s\n⊢ ↑h ((fun x => c • x) z) ∈ ↑σ c • ↑h '' s",
"tactic": "exact ⟨h z, Set.mem_image_of_mem _ zs, (map_smulₛₗ _ _ _).symm⟩"
},
{
"state_after": "case h₂.intro.intro.intro.intro\nR : Type u_2\nR₁ : Type ?u.144905\nR₂ : Type ?u.144908\nR₃ : Type ?u.144911\nk : Type ?u.144914\nS : Type u_3\nS₃ : Type ?u.144920\nT : Type ?u.144923\nM : Type u_4\nM₁ : Type ?u.144929\nM₂ : Type ?u.144932\nM₃ : Type u_5\nN₁ : Type ?u.144938\nN₂ : Type ?u.144941\nN₃ : Type ?u.144944\nι : Type ?u.144947\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid M₃\ninst✝⁶ : AddCommMonoid N₁\ninst✝⁵ : AddCommMonoid N₂\ninst✝⁴ : AddCommMonoid N₃\ninst✝³ : Module R M\ninst✝² : Module R M₂\ninst✝¹ : Module S M₃\nσ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf g : M →ₛₗ[σ] M₃\nF : Type u_1\nh : F\ninst✝ : SemilinearMapClass F σ M M₃\nc : R\ns : Set M\nz : M\nhz : z ∈ s\n⊢ (fun x => ↑σ c • x) (↑h z) ∈ ↑h '' (c • s)",
"state_before": "case h₂\nR : Type u_2\nR₁ : Type ?u.144905\nR₂ : Type ?u.144908\nR₃ : Type ?u.144911\nk : Type ?u.144914\nS : Type u_3\nS₃ : Type ?u.144920\nT : Type ?u.144923\nM : Type u_4\nM₁ : Type ?u.144929\nM₂ : Type ?u.144932\nM₃ : Type u_5\nN₁ : Type ?u.144938\nN₂ : Type ?u.144941\nN₃ : Type ?u.144944\nι : Type ?u.144947\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid M₃\ninst✝⁶ : AddCommMonoid N₁\ninst✝⁵ : AddCommMonoid N₂\ninst✝⁴ : AddCommMonoid N₃\ninst✝³ : Module R M\ninst✝² : Module R M₂\ninst✝¹ : Module S M₃\nσ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf g : M →ₛₗ[σ] M₃\nF : Type u_1\nh : F\ninst✝ : SemilinearMapClass F σ M M₃\nc : R\ns : Set M\n⊢ ↑σ c • ↑h '' s ⊆ ↑h '' (c • s)",
"tactic": "rintro x ⟨y, ⟨z, hz, rfl⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h₂.intro.intro.intro.intro\nR : Type u_2\nR₁ : Type ?u.144905\nR₂ : Type ?u.144908\nR₃ : Type ?u.144911\nk : Type ?u.144914\nS : Type u_3\nS₃ : Type ?u.144920\nT : Type ?u.144923\nM : Type u_4\nM₁ : Type ?u.144929\nM₂ : Type ?u.144932\nM₃ : Type u_5\nN₁ : Type ?u.144938\nN₂ : Type ?u.144941\nN₃ : Type ?u.144944\nι : Type ?u.144947\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid M₃\ninst✝⁶ : AddCommMonoid N₁\ninst✝⁵ : AddCommMonoid N₂\ninst✝⁴ : AddCommMonoid N₃\ninst✝³ : Module R M\ninst✝² : Module R M₂\ninst✝¹ : Module S M₃\nσ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf g : M →ₛₗ[σ] M₃\nF : Type u_1\nh : F\ninst✝ : SemilinearMapClass F σ M M₃\nc : R\ns : Set M\nz : M\nhz : z ∈ s\n⊢ (fun x => ↑σ c • x) (↑h z) ∈ ↑h '' (c • s)",
"tactic": "exact (Set.mem_image _ _ _).2 ⟨c • z, Set.smul_mem_smul_set hz, map_smulₛₗ _ _ _⟩"
}
] |
[
367,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.piecewise_mem_Icc
|
[] |
[
2585,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2583,
1
] |
Mathlib/Data/Nat/Cast/Basic.lean
|
toDual_natCast
|
[] |
[
354,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Tactic/NormNum/GCD.lean
|
Tactic.NormNum.nat_gcd_helper_dvd_left
|
[] |
[
31,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
30,
1
] |
Mathlib/Analysis/Complex/CauchyIntegral.lean
|
Complex.two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable
|
[
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = f w",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = f w",
"tactic": "have hR : 0 < R := dist_nonneg.trans_lt hw"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\n⊢ w ∈ closure (ball c R \\ s)",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = f w",
"tactic": "suffices w ∈ closure (ball c R \\ s) by\n lift R to ℝ≥0 using hR.le\n have A : ContinuousAt (fun w => (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) w := by\n have := hasFPowerSeriesOn_cauchy_integral\n ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR\n refine' this.continuousOn.continuousAt (EMetric.isOpen_ball.mem_nhds _)\n rwa [Metric.emetric_ball_nnreal]\n have B : ContinuousAt f w := hc.continuousAt (closedBall_mem_nhds_of_mem hw)\n refine' tendsto_nhds_unique_of_frequently_eq A B ((mem_closure_iff_frequently.1 this).mono _)\n intro z hz\n rw [circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux hs hz hc hd,\n inv_smul_smul₀]\n simp [Real.pi_ne_zero, I_ne_zero]"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\n⊢ Set.Nonempty (t ∩ (ball c R \\ s))",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\n⊢ w ∈ closure (ball c R \\ s)",
"tactic": "refine' mem_closure_iff_nhds.2 fun t ht => _"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\ng : ℝ → ℂ := fun x => w + ↑ofReal x\n⊢ Set.Nonempty (t ∩ (ball c R \\ s))",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\n⊢ Set.Nonempty (t ∩ (ball c R \\ s))",
"tactic": "set g : ℝ → ℂ := fun x => w + ofReal x"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\ng : ℝ → ℂ := fun x => w + ↑ofReal x\nthis : Tendsto g (𝓝 0) (𝓝 w)\n⊢ Set.Nonempty (t ∩ (ball c R \\ s))",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\ng : ℝ → ℂ := fun x => w + ↑ofReal x\n⊢ Set.Nonempty (t ∩ (ball c R \\ s))",
"tactic": "have : Tendsto g (𝓝 0) (𝓝 w) :=\n (continuous_const.add continuous_ofReal).tendsto' 0 w (add_zero _)"
},
{
"state_after": "case intro.intro.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\ng : ℝ → ℂ := fun x => w + ↑ofReal x\nthis : Tendsto g (𝓝 0) (𝓝 w)\nl u : ℝ\nhlu₀ : 0 ∈ Ioo l u\nhlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)\n⊢ Set.Nonempty (t ∩ (ball c R \\ s))",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\ng : ℝ → ℂ := fun x => w + ↑ofReal x\nthis : Tendsto g (𝓝 0) (𝓝 w)\n⊢ Set.Nonempty (t ∩ (ball c R \\ s))",
"tactic": "rcases mem_nhds_iff_exists_Ioo_subset.1 (this <| inter_mem ht <| isOpen_ball.mem_nhds hw) with\n ⟨l, u, hlu₀, hlu_sub⟩"
},
{
"state_after": "case intro.intro.intro.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\ng : ℝ → ℂ := fun x => w + ↑ofReal x\nthis : Tendsto g (𝓝 0) (𝓝 w)\nl u : ℝ\nhlu₀ : 0 ∈ Ioo l u\nhlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)\nx : ℝ\nhx : x ∈ Ioo l u \\ g ⁻¹' s\n⊢ Set.Nonempty (t ∩ (ball c R \\ s))",
"state_before": "case intro.intro.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\ng : ℝ → ℂ := fun x => w + ↑ofReal x\nthis : Tendsto g (𝓝 0) (𝓝 w)\nl u : ℝ\nhlu₀ : 0 ∈ Ioo l u\nhlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)\n⊢ Set.Nonempty (t ∩ (ball c R \\ s))",
"tactic": "obtain ⟨x, hx⟩ : (Ioo l u \\ g ⁻¹' s).Nonempty := by\n refine' nonempty_diff.2 fun hsub => _\n have : (Ioo l u).Countable :=\n (hs.preimage ((add_right_injective w).comp ofReal_injective)).mono hsub\n rw [← Cardinal.le_aleph0_iff_set_countable, Cardinal.mk_Ioo_real (hlu₀.1.trans hlu₀.2)] at this\n exact this.not_lt Cardinal.aleph0_lt_continuum"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\ng : ℝ → ℂ := fun x => w + ↑ofReal x\nthis : Tendsto g (𝓝 0) (𝓝 w)\nl u : ℝ\nhlu₀ : 0 ∈ Ioo l u\nhlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)\nx : ℝ\nhx : x ∈ Ioo l u \\ g ⁻¹' s\n⊢ Set.Nonempty (t ∩ (ball c R \\ s))",
"tactic": "exact ⟨g x, (hlu_sub hx.1).1, (hlu_sub hx.1).2, hx.2⟩"
},
{
"state_after": "case intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis : w ∈ closure (ball c ↑R \\ s)\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) = f w",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nthis : w ∈ closure (ball c R \\ s)\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = f w",
"tactic": "lift R to ℝ≥0 using hR.le"
},
{
"state_after": "case intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis : w ∈ closure (ball c ↑R \\ s)\nA : ContinuousAt (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) = f w",
"state_before": "case intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis : w ∈ closure (ball c ↑R \\ s)\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) = f w",
"tactic": "have A : ContinuousAt (fun w => (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) w := by\n have := hasFPowerSeriesOn_cauchy_integral\n ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR\n refine' this.continuousOn.continuousAt (EMetric.isOpen_ball.mem_nhds _)\n rwa [Metric.emetric_ball_nnreal]"
},
{
"state_after": "case intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis : w ∈ closure (ball c ↑R \\ s)\nA : ContinuousAt (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w\nB : ContinuousAt f w\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) = f w",
"state_before": "case intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis : w ∈ closure (ball c ↑R \\ s)\nA : ContinuousAt (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) = f w",
"tactic": "have B : ContinuousAt f w := hc.continuousAt (closedBall_mem_nhds_of_mem hw)"
},
{
"state_after": "case intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis : w ∈ closure (ball c ↑R \\ s)\nA : ContinuousAt (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w\nB : ContinuousAt f w\n⊢ ∀ (x : ℂ), x ∈ ball c ↑R \\ s → ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - x)⁻¹ • f z) = f x",
"state_before": "case intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis : w ∈ closure (ball c ↑R \\ s)\nA : ContinuousAt (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w\nB : ContinuousAt f w\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) = f w",
"tactic": "refine' tendsto_nhds_unique_of_frequently_eq A B ((mem_closure_iff_frequently.1 this).mono _)"
},
{
"state_after": "case intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis : w ∈ closure (ball c ↑R \\ s)\nA : ContinuousAt (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w\nB : ContinuousAt f w\nz : ℂ\nhz : z ∈ ball c ↑R \\ s\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z_1 : ℂ) in C(c, ↑R), (z_1 - z)⁻¹ • f z_1) = f z",
"state_before": "case intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis : w ∈ closure (ball c ↑R \\ s)\nA : ContinuousAt (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w\nB : ContinuousAt f w\n⊢ ∀ (x : ℂ), x ∈ ball c ↑R \\ s → ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - x)⁻¹ • f z) = f x",
"tactic": "intro z hz"
},
{
"state_after": "case intro.hc\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis : w ∈ closure (ball c ↑R \\ s)\nA : ContinuousAt (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w\nB : ContinuousAt f w\nz : ℂ\nhz : z ∈ ball c ↑R \\ s\n⊢ 2 * ↑π * I ≠ 0",
"state_before": "case intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis : w ∈ closure (ball c ↑R \\ s)\nA : ContinuousAt (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w\nB : ContinuousAt f w\nz : ℂ\nhz : z ∈ ball c ↑R \\ s\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z_1 : ℂ) in C(c, ↑R), (z_1 - z)⁻¹ • f z_1) = f z",
"tactic": "rw [circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux hs hz hc hd,\n inv_smul_smul₀]"
},
{
"state_after": "no goals",
"state_before": "case intro.hc\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis : w ∈ closure (ball c ↑R \\ s)\nA : ContinuousAt (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w\nB : ContinuousAt f w\nz : ℂ\nhz : z ∈ ball c ↑R \\ s\n⊢ 2 * ↑π * I ≠ 0",
"tactic": "simp [Real.pi_ne_zero, I_ne_zero]"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis✝ : w ∈ closure (ball c ↑R \\ s)\nthis :\n HasFPowerSeriesOnBall (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) (cauchyPowerSeries f c ↑R) c\n ↑R\n⊢ ContinuousAt (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis : w ∈ closure (ball c ↑R \\ s)\n⊢ ContinuousAt (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w",
"tactic": "have := hasFPowerSeriesOn_cauchy_integral\n ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis✝ : w ∈ closure (ball c ↑R \\ s)\nthis :\n HasFPowerSeriesOnBall (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) (cauchyPowerSeries f c ↑R) c\n ↑R\n⊢ w ∈ EMetric.ball c ↑R",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis✝ : w ∈ closure (ball c ↑R \\ s)\nthis :\n HasFPowerSeriesOnBall (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) (cauchyPowerSeries f c ↑R) c\n ↑R\n⊢ ContinuousAt (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w",
"tactic": "refine' this.continuousOn.continuousAt (EMetric.isOpen_ball.mem_nhds _)"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nR : ℝ≥0\nhw : w ∈ ball c ↑R\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ (x : ℂ), x ∈ ball c ↑R \\ s → DifferentiableAt ℂ f x\nhR : 0 < ↑R\nthis✝ : w ∈ closure (ball c ↑R \\ s)\nthis :\n HasFPowerSeriesOnBall (fun w => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) (cauchyPowerSeries f c ↑R) c\n ↑R\n⊢ w ∈ EMetric.ball c ↑R",
"tactic": "rwa [Metric.emetric_ball_nnreal]"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\ng : ℝ → ℂ := fun x => w + ↑ofReal x\nthis : Tendsto g (𝓝 0) (𝓝 w)\nl u : ℝ\nhlu₀ : 0 ∈ Ioo l u\nhlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)\nhsub : Ioo l u ⊆ g ⁻¹' s\n⊢ False",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\ng : ℝ → ℂ := fun x => w + ↑ofReal x\nthis : Tendsto g (𝓝 0) (𝓝 w)\nl u : ℝ\nhlu₀ : 0 ∈ Ioo l u\nhlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)\n⊢ Set.Nonempty (Ioo l u \\ g ⁻¹' s)",
"tactic": "refine' nonempty_diff.2 fun hsub => _"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\ng : ℝ → ℂ := fun x => w + ↑ofReal x\nthis✝ : Tendsto g (𝓝 0) (𝓝 w)\nl u : ℝ\nhlu₀ : 0 ∈ Ioo l u\nhlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)\nhsub : Ioo l u ⊆ g ⁻¹' s\nthis : Set.Countable (Ioo l u)\n⊢ False",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\ng : ℝ → ℂ := fun x => w + ↑ofReal x\nthis : Tendsto g (𝓝 0) (𝓝 w)\nl u : ℝ\nhlu₀ : 0 ∈ Ioo l u\nhlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)\nhsub : Ioo l u ⊆ g ⁻¹' s\n⊢ False",
"tactic": "have : (Ioo l u).Countable :=\n (hs.preimage ((add_right_injective w).comp ofReal_injective)).mono hsub"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\ng : ℝ → ℂ := fun x => w + ↑ofReal x\nthis✝ : Tendsto g (𝓝 0) (𝓝 w)\nl u : ℝ\nhlu₀ : 0 ∈ Ioo l u\nhlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)\nhsub : Ioo l u ⊆ g ⁻¹' s\nthis : Cardinal.continuum ≤ Cardinal.aleph0\n⊢ False",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\ng : ℝ → ℂ := fun x => w + ↑ofReal x\nthis✝ : Tendsto g (𝓝 0) (𝓝 w)\nl u : ℝ\nhlu₀ : 0 ∈ Ioo l u\nhlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)\nhsub : Ioo l u ⊆ g ⁻¹' s\nthis : Set.Countable (Ioo l u)\n⊢ False",
"tactic": "rw [← Cardinal.le_aleph0_iff_set_countable, Cardinal.mk_Ioo_real (hlu₀.1.trans hlu₀.2)] at this"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : Set.Countable s\nhw : w ∈ ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ (x : ℂ), x ∈ ball c R \\ s → DifferentiableAt ℂ f x\nhR : 0 < R\nt : Set ℂ\nht : t ∈ 𝓝 w\ng : ℝ → ℂ := fun x => w + ↑ofReal x\nthis✝ : Tendsto g (𝓝 0) (𝓝 w)\nl u : ℝ\nhlu₀ : 0 ∈ Ioo l u\nhlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)\nhsub : Ioo l u ⊆ g ⁻¹' s\nthis : Cardinal.continuum ≤ Cardinal.aleph0\n⊢ False",
"tactic": "exact this.not_lt Cardinal.aleph0_lt_continuum"
}
] |
[
494,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
463,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
DifferentiableOn.hasFDerivAt
|
[] |
[
526,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
524,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Submodule.smul_assoc
|
[] |
[
221,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
11
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.AbsTheory.mul_self_abs
|
[] |
[
908,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
907,
9
] |
Mathlib/Order/MinMax.lean
|
max_eq_right_iff
|
[] |
[
152,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.intervalIntegral_eq_integral_uIoc
|
[
{
"state_after": "case inl\nι : Type ?u.10052960\n𝕜 : Type ?u.10052963\nE : Type u_1\nF : Type ?u.10052969\nA : Type ?u.10052972\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ : ℝ\nf✝ g : ℝ → E\nμ✝ : MeasureTheory.Measure ℝ\nf : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nh : a ≤ b\n⊢ (∫ (x : ℝ) in a..b, f x ∂μ) = 1 • ∫ (x : ℝ) in Ι a b, f x ∂μ\n\ncase inr\nι : Type ?u.10052960\n𝕜 : Type ?u.10052963\nE : Type u_1\nF : Type ?u.10052969\nA : Type ?u.10052972\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ : ℝ\nf✝ g : ℝ → E\nμ✝ : MeasureTheory.Measure ℝ\nf : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nh : ¬a ≤ b\n⊢ (∫ (x : ℝ) in a..b, f x ∂μ) = -1 • ∫ (x : ℝ) in Ι a b, f x ∂μ",
"state_before": "ι : Type ?u.10052960\n𝕜 : Type ?u.10052963\nE : Type u_1\nF : Type ?u.10052969\nA : Type ?u.10052972\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ : ℝ\nf✝ g : ℝ → E\nμ✝ : MeasureTheory.Measure ℝ\nf : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\n⊢ (∫ (x : ℝ) in a..b, f x ∂μ) = (if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, f x ∂μ",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type ?u.10052960\n𝕜 : Type ?u.10052963\nE : Type u_1\nF : Type ?u.10052969\nA : Type ?u.10052972\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ : ℝ\nf✝ g : ℝ → E\nμ✝ : MeasureTheory.Measure ℝ\nf : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nh : a ≤ b\n⊢ (∫ (x : ℝ) in a..b, f x ∂μ) = 1 • ∫ (x : ℝ) in Ι a b, f x ∂μ",
"tactic": "simp only [integral_of_le h, uIoc_of_le h, one_smul]"
},
{
"state_after": "no goals",
"state_before": "case inr\nι : Type ?u.10052960\n𝕜 : Type ?u.10052963\nE : Type u_1\nF : Type ?u.10052969\nA : Type ?u.10052972\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ : ℝ\nf✝ g : ℝ → E\nμ✝ : MeasureTheory.Measure ℝ\nf : ℝ → E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nh : ¬a ≤ b\n⊢ (∫ (x : ℝ) in a..b, f x ∂μ) = -1 • ∫ (x : ℝ) in Ι a b, f x ∂μ",
"tactic": "simp only [integral_of_ge (not_le.1 h).le, uIoc_of_lt (not_le.1 h), neg_one_smul]"
}
] |
[
488,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
UpperSet.mem_mk
|
[] |
[
441,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
440,
1
] |
Mathlib/Data/PFun.lean
|
PFun.comp_apply
|
[] |
[
583,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
582,
1
] |
Mathlib/Analysis/InnerProductSpace/Positive.lean
|
ContinuousLinearMap.IsPositive.adjoint_conj
|
[
{
"state_after": "case h.e'_7.h.e'_24.h.e'_24\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] E\nhT : IsPositive T\nS : F →L[𝕜] E\n⊢ S = ↑adjoint (↑adjoint S)",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] E\nhT : IsPositive T\nS : F →L[𝕜] E\n⊢ IsPositive (comp (↑adjoint S) (comp T S))",
"tactic": "convert hT.conj_adjoint (S†)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7.h.e'_24.h.e'_24\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] E\nhT : IsPositive T\nS : F →L[𝕜] E\n⊢ S = ↑adjoint (↑adjoint S)",
"tactic": "rw [adjoint_adjoint]"
}
] |
[
104,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
SupBotHom.symm_dual_id
|
[] |
[
1463,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1462,
1
] |
Mathlib/GroupTheory/Submonoid/Basic.lean
|
Submonoid.closure_union
|
[] |
[
530,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
lcm_dvd_lcm_mul_right
|
[] |
[
853,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
852,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.card_functions_sum
|
[
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\ni : ℕ\n⊢ (#Functions (Language.sum L L') i) = lift (#Functions L i) + lift (#Functions L' i)",
"tactic": "simp [Language.sum]"
}
] |
[
256,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.prod_mono_left
|
[] |
[
1066,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1065,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.stronglyMeasurable_const
|
[] |
[
162,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
InitialSeg.ordinal_type_le
|
[] |
[
367,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPolynomial.coe_eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ : MvPolynomial σ R\n⊢ ↑φ = 0 ↔ φ = 0",
"tactic": "rw [← coe_zero, coe_inj]"
}
] |
[
1150,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1150,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.eq_symm_apply
|
[] |
[
411,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/Order/Hom/Basic.lean
|
codisjoint_map_orderIso_iff
|
[] |
[
1233,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1230,
1
] |
Mathlib/Algebra/Order/UpperLower.lean
|
IsUpperSet.mul_right
|
[
{
"state_after": "α : Type u_1\ninst✝ : OrderedCommGroup α\ns t : Set α\na : α\nhs : IsUpperSet s\n⊢ IsUpperSet (t * s)",
"state_before": "α : Type u_1\ninst✝ : OrderedCommGroup α\ns t : Set α\na : α\nhs : IsUpperSet s\n⊢ IsUpperSet (s * t)",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedCommGroup α\ns t : Set α\na : α\nhs : IsUpperSet s\n⊢ IsUpperSet (t * s)",
"tactic": "exact hs.mul_left"
}
] |
[
77,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.lt_of_lt_pred
|
[] |
[
258,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Order/Antichain.lean
|
IsAntichain.insert
|
[] |
[
110,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
11
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.mk'_eq_iff_eq_mul
|
[] |
[
291,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/Order/Monotone/Union.lean
|
StrictAntiOn.union
|
[] |
[
69,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
11
] |
Mathlib/Data/Sign.lean
|
SignType.coe_one
|
[] |
[
265,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
ringChar.of_eq
|
[] |
[
226,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Topology/Homeomorph.lean
|
Homeomorph.quotientMap
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.58426\nδ : Type ?u.58429\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\n⊢ QuotientMap (↑h ∘ ↑(Homeomorph.symm h))",
"tactic": "simp only [self_comp_symm, QuotientMap.id]"
}
] |
[
238,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
11
] |
Mathlib/Order/Hom/Lattice.lean
|
LatticeHom.ext
|
[] |
[
1059,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1058,
1
] |
Mathlib/Order/RelIso/Basic.lean
|
RelEmbedding.isStrictTotalOrder
|
[] |
[
388,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
11
] |
Mathlib/Order/Heyting/Basic.lean
|
inf_himp
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.28319\nα : Type u_1\nβ : Type ?u.28325\ninst✝ : GeneralizedHeytingAlgebra α\na✝ b✝ c d a b : α\n⊢ a ⊓ (a ⇨ b) ≤ b",
"tactic": "rw [inf_comm, ← le_himp_iff]"
}
] |
[
356,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
Mathlib/Algebra/Polynomial/GroupRingAction.lean
|
prodXSubSmul.coeff
|
[
{
"state_after": "no goals",
"state_before": "M : Type ?u.109284\ninst✝⁴ : Monoid M\nG : Type u_2\ninst✝³ : Group G\ninst✝² : Fintype G\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : MulSemiringAction G R\nx : R\ng : G\nn : ℕ\n⊢ g • Polynomial.coeff (prodXSubSmul G R x) n = Polynomial.coeff (prodXSubSmul G R x) n",
"tactic": "rw [← Polynomial.coeff_smul, prodXSubSmul.smul]"
}
] |
[
123,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
TrivSqZeroExt.snd_sum
|
[] |
[
304,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/LinearAlgebra/Finrank.lean
|
FiniteDimensional.finrank_self
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝⁵ : Ring K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : StrongRankCondition K\n⊢ Module.rank K K = ↑1",
"tactic": "simp"
}
] |
[
144,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.mem_sups
|
[] |
[
63,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/GroupTheory/PGroup.lean
|
IsPGroup.orderOf_coprime
|
[] |
[
103,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/GroupTheory/Subsemigroup/Basic.lean
|
Subsemigroup.mk_le_mk
|
[] |
[
132,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Algebra/Regular/SMul.lean
|
IsSMulRegular.smul
|
[] |
[
72,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Algebra/Order/Rearrangement.lean
|
MonovaryOn.sum_smul_comp_perm_lt_sum_smul_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : LinearOrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\nhfg : MonovaryOn f g ↑s\nhσ : {x | ↑σ x ≠ x} ⊆ ↑s\n⊢ ∑ i in s, f i • g (↑σ i) < ∑ i in s, f i • g i ↔ ¬MonovaryOn f (g ∘ ↑σ) ↑s",
"tactic": "simp [← hfg.sum_smul_comp_perm_eq_sum_smul_iff hσ, lt_iff_le_and_ne,\n hfg.sum_smul_comp_perm_le_sum_smul hσ]"
}
] |
[
149,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_node3R
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : Ordnode α\nx : α\nm : Ordnode α\ny : α\nr : Ordnode α\n⊢ dual (node3R l x m y r) = node3L (dual r) y (dual m) x (dual l)",
"tactic": "simp [node3L, node3R, dual_node', add_comm]"
}
] |
[
322,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Std/Data/String/Lemmas.lean
|
String.Iterator.Valid.prev
|
[] |
[
644,
35
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
640,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Pretrivialization.coe_coe
|
[] |
[
113,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
YoungDiagram.coe_bot
|
[
{
"state_after": "case refine'_1\n\n⊢ ↑⊥.cells ⊆ ∅\n\ncase refine'_2\n\n⊢ ∅ ⊆ ↑⊥.cells",
"state_before": "⊢ ↑⊥.cells = ∅",
"tactic": "refine' Set.eq_of_subset_of_subset _ _"
},
{
"state_after": "case refine'_1\nx : ℕ × ℕ\nh : x ∈ ↑⊥.cells\n⊢ x ∈ ∅\n\ncase refine'_2\n\n⊢ ∅ ⊆ ↑⊥.cells",
"state_before": "case refine'_1\n\n⊢ ↑⊥.cells ⊆ ∅\n\ncase refine'_2\n\n⊢ ∅ ⊆ ↑⊥.cells",
"tactic": "intros x h"
},
{
"state_after": "case refine'_2\n\n⊢ ∅ ⊆ ↑⊥.cells",
"state_before": "case refine'_1\nx : ℕ × ℕ\nh : x ∈ ↑⊥.cells\n⊢ x ∈ ∅\n\ncase refine'_2\n\n⊢ ∅ ⊆ ↑⊥.cells",
"tactic": "simp [mem_mk, Finset.coe_empty, Set.mem_empty_iff_false] at h"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\n\n⊢ ∅ ⊆ ↑⊥.cells",
"tactic": "simp only [cells_bot, Finset.coe_empty, Set.empty_subset]"
}
] |
[
182,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pullbackAssoc_inv_fst_snd
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁷ : Category C\nD : Type u₂\ninst✝⁶ : Category D\nW X Y Z X₁ X₂ X₃ Y₁ Y₂ : C\nf₁ : X₁ ⟶ Y₁\nf₂ : X₂ ⟶ Y₁\nf₃ : X₂ ⟶ Y₂\nf₄ : X₃ ⟶ Y₂\ninst✝⁵ : HasPullback f₁ f₂\ninst✝⁴ : HasPullback f₃ f₄\ninst✝³ : HasPullback (pullback.snd ≫ sorryAx (sorryAx C true ⟶ Y₂) true) f₄\ninst✝² : HasPullback f₁ (pullback.fst ≫ f₂)\ninst✝¹ : HasPullback (pullback.snd ≫ f₃) f₄\ninst✝ : HasPullback f₁ (pullback.fst ≫ f₂)\n⊢ (pullbackAssoc f₁ f₂ f₃ f₄).inv ≫ pullback.fst ≫ pullback.snd = pullback.snd ≫ pullback.fst",
"tactic": "rw [Iso.inv_comp_eq, pullbackAssoc_hom_snd_fst]"
}
] |
[
2441,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2438,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1014853\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\nA : ContinuousOn (fun y => y - x) (EMetric.ball x r)\n⊢ TendstoLocallyUniformlyOn (fun n y => FormalMultilinearSeries.partialSum p n (y - x)) f atTop (EMetric.ball x r)",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1014853\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\n⊢ TendstoLocallyUniformlyOn (fun n y => FormalMultilinearSeries.partialSum p n (y - x)) f atTop (EMetric.ball x r)",
"tactic": "have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=\n (continuous_id.sub continuous_const).continuousOn"
},
{
"state_after": "case h.e'_7\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1014853\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\nA : ContinuousOn (fun y => y - x) (EMetric.ball x r)\n⊢ f = (fun y => f (x + y)) ∘ fun y => y - x\n\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1014853\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\nA : ContinuousOn (fun y => y - x) (EMetric.ball x r)\n⊢ MapsTo (fun y => y - x) (EMetric.ball x r) (EMetric.ball 0 r)",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1014853\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\nA : ContinuousOn (fun y => y - x) (EMetric.ball x r)\n⊢ TendstoLocallyUniformlyOn (fun n y => FormalMultilinearSeries.partialSum p n (y - x)) f atTop (EMetric.ball x r)",
"tactic": "convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1"
},
{
"state_after": "case h.e'_7.h\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1014853\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\nA : ContinuousOn (fun y => y - x) (EMetric.ball x r)\nz : E\n⊢ f z = ((fun y => f (x + y)) ∘ fun y => y - x) z",
"state_before": "case h.e'_7\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1014853\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\nA : ContinuousOn (fun y => y - x) (EMetric.ball x r)\n⊢ f = (fun y => f (x + y)) ∘ fun y => y - x",
"tactic": "ext z"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7.h\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1014853\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\nA : ContinuousOn (fun y => y - x) (EMetric.ball x r)\nz : E\n⊢ f z = ((fun y => f (x + y)) ∘ fun y => y - x) z",
"tactic": "simp"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1014853\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\nA : ContinuousOn (fun y => y - x) (EMetric.ball x r)\nz : E\n⊢ z ∈ EMetric.ball x r → (fun y => y - x) z ∈ EMetric.ball 0 r",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1014853\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\nA : ContinuousOn (fun y => y - x) (EMetric.ball x r)\n⊢ MapsTo (fun y => y - x) (EMetric.ball x r) (EMetric.ball 0 r)",
"tactic": "intro z"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1014853\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\nA : ContinuousOn (fun y => y - x) (EMetric.ball x r)\nz : E\n⊢ z ∈ EMetric.ball x r → (fun y => y - x) z ∈ EMetric.ball 0 r",
"tactic": "simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]"
}
] |
[
864,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
855,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
one_le_div_iff
|
[
{
"state_after": "case inl\nι : Type ?u.164600\nα : Type u_1\nβ : Type ?u.164606\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nhb : b < 0\n⊢ 1 ≤ a / b ↔ 0 < b ∧ b ≤ a ∨ b < 0 ∧ a ≤ b\n\ncase inr.inl\nι : Type ?u.164600\nα : Type u_1\nβ : Type ?u.164606\ninst✝ : LinearOrderedField α\na c d : α\nn : ℤ\n⊢ 1 ≤ a / 0 ↔ 0 < 0 ∧ 0 ≤ a ∨ 0 < 0 ∧ a ≤ 0\n\ncase inr.inr\nι : Type ?u.164600\nα : Type u_1\nβ : Type ?u.164606\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nhb : 0 < b\n⊢ 1 ≤ a / b ↔ 0 < b ∧ b ≤ a ∨ b < 0 ∧ a ≤ b",
"state_before": "ι : Type ?u.164600\nα : Type u_1\nβ : Type ?u.164606\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\n⊢ 1 ≤ a / b ↔ 0 < b ∧ b ≤ a ∨ b < 0 ∧ a ≤ b",
"tactic": "rcases lt_trichotomy b 0 with (hb | rfl | hb)"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type ?u.164600\nα : Type u_1\nβ : Type ?u.164606\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nhb : b < 0\n⊢ 1 ≤ a / b ↔ 0 < b ∧ b ≤ a ∨ b < 0 ∧ a ≤ b",
"tactic": "simp [hb, hb.not_lt, one_le_div_of_neg]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nι : Type ?u.164600\nα : Type u_1\nβ : Type ?u.164606\ninst✝ : LinearOrderedField α\na c d : α\nn : ℤ\n⊢ 1 ≤ a / 0 ↔ 0 < 0 ∧ 0 ≤ a ∨ 0 < 0 ∧ a ≤ 0",
"tactic": "simp [lt_irrefl, zero_lt_one.not_le, zero_lt_one]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nι : Type ?u.164600\nα : Type u_1\nβ : Type ?u.164606\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nhb : 0 < b\n⊢ 1 ≤ a / b ↔ 0 < b ∧ b ≤ a ∨ b < 0 ∧ a ≤ b",
"tactic": "simp [hb, hb.not_lt, one_le_div]"
}
] |
[
846,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
842,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.ofReal_neg
|
[] |
[
226,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Data/Nat/Pow.lean
|
Nat.pow_lt_pow_of_lt_left
|
[] |
[
31,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
30,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.comap_inf
|
[] |
[
2184,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2184,
9
] |
Mathlib/Data/Set/Intervals/Disjoint.lean
|
IsGLB.biUnion_Ici_eq_Ioi
|
[
{
"state_after": "case refine'_1\nι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\na_glb : IsGLB s a\na_not_mem : ¬a ∈ s\nx : α\nhx : x ∈ s\n⊢ Ici x ⊆ Ioi a\n\ncase refine'_2\nι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\na_glb : IsGLB s a\na_not_mem : ¬a ∈ s\nx : α\nhx : x ∈ Ioi a\n⊢ x ∈ ⋃ (i : α) (_ : i ∈ s), Ici i",
"state_before": "ι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\na_glb : IsGLB s a\na_not_mem : ¬a ∈ s\n⊢ (⋃ (x : α) (_ : x ∈ s), Ici x) = Ioi a",
"tactic": "refine' (iUnion₂_subset fun x hx => _).antisymm fun x hx => _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\na_glb : IsGLB s a\na_not_mem : ¬a ∈ s\nx : α\nhx : x ∈ s\n⊢ Ici x ⊆ Ioi a",
"tactic": "exact Ici_subset_Ioi.mpr (lt_of_le_of_ne (a_glb.1 hx) fun h => (h ▸ a_not_mem) hx)"
},
{
"state_after": "case refine'_2.intro.intro.intro\nι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\na_glb : IsGLB s a\na_not_mem : ¬a ∈ s\nx : α\nhx : x ∈ Ioi a\ny : α\nhys : y ∈ s\nleft✝ : a ≤ y\nhyx : y < x\n⊢ x ∈ ⋃ (i : α) (_ : i ∈ s), Ici i",
"state_before": "case refine'_2\nι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\na_glb : IsGLB s a\na_not_mem : ¬a ∈ s\nx : α\nhx : x ∈ Ioi a\n⊢ x ∈ ⋃ (i : α) (_ : i ∈ s), Ici i",
"tactic": "rcases a_glb.exists_between hx with ⟨y, hys, _, hyx⟩"
},
{
"state_after": "case refine'_2.intro.intro.intro\nι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\na_glb : IsGLB s a\na_not_mem : ¬a ∈ s\nx : α\nhx : x ∈ Ioi a\ny : α\nhys : y ∈ s\nleft✝ : a ≤ y\nhyx : y < x\n⊢ ∃ i j, x ∈ Ici i",
"state_before": "case refine'_2.intro.intro.intro\nι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\na_glb : IsGLB s a\na_not_mem : ¬a ∈ s\nx : α\nhx : x ∈ Ioi a\ny : α\nhys : y ∈ s\nleft✝ : a ≤ y\nhyx : y < x\n⊢ x ∈ ⋃ (i : α) (_ : i ∈ s), Ici i",
"tactic": "apply mem_iUnion₂.mpr"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.intro\nι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\na_glb : IsGLB s a\na_not_mem : ¬a ∈ s\nx : α\nhx : x ∈ Ioi a\ny : α\nhys : y ∈ s\nleft✝ : a ≤ y\nhyx : y < x\n⊢ ∃ i j, x ∈ Ici i",
"tactic": "refine' ⟨y, hys, hyx.le⟩"
}
] |
[
210,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.eta
|
[] |
[
56,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Std/Data/List/Lemmas.lean
|
List.mem_of_find?_eq_some
|
[
{
"state_after": "case pos\nα✝ : Type u_1\np : α✝ → Bool\na b : α✝\nl : List α✝\nh : p b = true\nH : b = a\n⊢ a ∈ b :: l\n\ncase neg\nα✝ : Type u_1\np : α✝ → Bool\na b : α✝\nl : List α✝\nh : ¬p b = true\nH : find? p l = some a\n⊢ a ∈ b :: l",
"state_before": "α✝ : Type u_1\np : α✝ → Bool\na b : α✝\nl : List α✝\nH : find? p (b :: l) = some a\n⊢ a ∈ b :: l",
"tactic": "by_cases h : p b <;> simp [find?, h] at H"
},
{
"state_after": "no goals",
"state_before": "case pos\nα✝ : Type u_1\np : α✝ → Bool\na b : α✝\nl : List α✝\nh : p b = true\nH : b = a\n⊢ a ∈ b :: l",
"tactic": "exact H ▸ .head _"
},
{
"state_after": "no goals",
"state_before": "case neg\nα✝ : Type u_1\np : α✝ → Bool\na b : α✝\nl : List α✝\nh : ¬p b = true\nH : find? p l = some a\n⊢ a ∈ b :: l",
"tactic": "exact .tail _ (mem_of_find?_eq_some H)"
}
] |
[
1272,
45
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1268,
9
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
gcd_greatest_associated
|
[] |
[
676,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
672,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.familyOfBFamily'_enum
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.271765\nβ : Type ?u.271768\nγ : Type ?u.271771\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nr : ι → ι → Prop\ninst✝ : IsWellOrder ι r\no : Ordinal\nho : type r = o\nf : (a : Ordinal) → a < o → α\ni : Ordinal\nhi : i < o\n⊢ i < type r",
"tactic": "rwa [ho]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.271768\nγ : Type ?u.271771\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nr : ι → ι → Prop\ninst✝ : IsWellOrder ι r\no : Ordinal\nho : type r = o\nf : (a : Ordinal) → a < o → α\ni : Ordinal\nhi : i < o\n⊢ familyOfBFamily' r ho f (enum r i (_ : i < type r)) = f i hi",
"tactic": "simp only [familyOfBFamily', typein_enum]"
}
] |
[
1157,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1154,
1
] |
Mathlib/Data/List/Chain.lean
|
List.chain_iff_forall₂
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nR r : α → α → Prop\nl l₁ l₂ : List α\na✝ b a : α\n⊢ Chain R a [] ↔ [] = [] ∨ Forall₂ R (a :: dropLast []) []",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nR r : α → α → Prop\nl✝ l₁ l₂ : List α\na✝ b✝ a b : α\nl : List α\n⊢ Chain R a (b :: l) ↔ b :: l = [] ∨ Forall₂ R (a :: dropLast (b :: l)) (b :: l)",
"tactic": "by_cases h : l = [] <;>\nsimp [@chain_iff_forall₂ b l, *]"
}
] |
[
78,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Tactic/PushNeg.lean
|
Mathlib.Tactic.PushNeg.not_ge_eq
|
[] |
[
31,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
31,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.mem_support_onFinset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.175978\nγ : Type ?u.175981\nι : Type ?u.175984\nM : Type u_2\nM' : Type ?u.175990\nN : Type ?u.175993\nP : Type ?u.175996\nG : Type ?u.175999\nH : Type ?u.176002\nR : Type ?u.176005\nS : Type ?u.176008\ninst✝ : Zero M\ns : Finset α\nf : α → M\nhf : ∀ (a : α), f a ≠ 0 → a ∈ s\na : α\n⊢ a ∈ (onFinset s f hf).support ↔ f a ≠ 0",
"tactic": "rw [Finsupp.mem_support_iff, Finsupp.onFinset_apply]"
}
] |
[
713,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
711,
1
] |
Mathlib/Algebra/AddTorsor.lean
|
Prod.mk_vsub_mk
|
[] |
[
332,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/Algebra/GeomSum.lean
|
geom_sum₂_self
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nα : Type u_1\ninst✝ : CommRing α\nx : α\nn : ℕ\n⊢ ∑ i in range n, x ^ i * x ^ (n - 1 - i) = ∑ i in range n, x ^ (i + (n - 1 - i))",
"tactic": "simp_rw [← pow_add]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nα : Type u_1\ninst✝ : CommRing α\nx : α\nn : ℕ\n⊢ card (range n) • x ^ (n - 1) = ↑n * x ^ (n - 1)",
"tactic": "rw [Finset.card_range, nsmul_eq_mul]"
}
] |
[
156,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.sdiff_eq_self_iff_disjoint
|
[] |
[
2360,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2359,
1
] |
Mathlib/Order/Atoms.lean
|
isCoatomic_dual_iff_isAtomic
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.11708\ninst✝¹ : PartialOrder α\ninst✝ : OrderBot α\nh : IsCoatomic αᵒᵈ\nb : α\n⊢ b = ⊥ ∨ ∃ a, IsAtom a ∧ a ≤ b",
"tactic": "apply h.eq_top_or_exists_le_coatom"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.11708\ninst✝¹ : PartialOrder α\ninst✝ : OrderBot α\nh : IsAtomic α\nb : αᵒᵈ\n⊢ b = ⊤ ∨ ∃ a, IsCoatom a ∧ b ≤ a",
"tactic": "apply h.eq_bot_or_exists_atom_le"
}
] |
[
257,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
255,
1
] |
Mathlib/Algebra/Ring/Prod.lean
|
false_of_nontrivial_of_product_domain
|
[
{
"state_after": "α : Type ?u.54070\nβ : Type ?u.54073\nR✝ : Type ?u.54076\nR' : Type ?u.54079\nS✝ : Type ?u.54082\nS' : Type ?u.54085\nT : Type ?u.54088\nT' : Type ?u.54091\nR : Type u_1\nS : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : Ring S\ninst✝² : IsDomain (R × S)\ninst✝¹ : Nontrivial R\ninst✝ : Nontrivial S\nthis : (0, 1) = 0 ∨ (1, 0) = 0\n⊢ False",
"state_before": "α : Type ?u.54070\nβ : Type ?u.54073\nR✝ : Type ?u.54076\nR' : Type ?u.54079\nS✝ : Type ?u.54082\nS' : Type ?u.54085\nT : Type ?u.54088\nT' : Type ?u.54091\nR : Type u_1\nS : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : Ring S\ninst✝² : IsDomain (R × S)\ninst✝¹ : Nontrivial R\ninst✝ : Nontrivial S\n⊢ False",
"tactic": "have :=\n NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero (show ((0 : R), (1 : S)) * (1, 0) = 0 by simp)"
},
{
"state_after": "α : Type ?u.54070\nβ : Type ?u.54073\nR✝ : Type ?u.54076\nR' : Type ?u.54079\nS✝ : Type ?u.54082\nS' : Type ?u.54085\nT : Type ?u.54088\nT' : Type ?u.54091\nR : Type u_1\nS : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : Ring S\ninst✝² : IsDomain (R × S)\ninst✝¹ : Nontrivial R\ninst✝ : Nontrivial S\nthis : 0 = 0 ∧ 1 = 0 ∨ 1 = 0 ∧ 0 = 0\n⊢ False",
"state_before": "α : Type ?u.54070\nβ : Type ?u.54073\nR✝ : Type ?u.54076\nR' : Type ?u.54079\nS✝ : Type ?u.54082\nS' : Type ?u.54085\nT : Type ?u.54088\nT' : Type ?u.54091\nR : Type u_1\nS : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : Ring S\ninst✝² : IsDomain (R × S)\ninst✝¹ : Nontrivial R\ninst✝ : Nontrivial S\nthis : (0, 1) = 0 ∨ (1, 0) = 0\n⊢ False",
"tactic": "rw [Prod.mk_eq_zero, Prod.mk_eq_zero] at this"
},
{
"state_after": "case inl.intro\nα : Type ?u.54070\nβ : Type ?u.54073\nR✝ : Type ?u.54076\nR' : Type ?u.54079\nS✝ : Type ?u.54082\nS' : Type ?u.54085\nT : Type ?u.54088\nT' : Type ?u.54091\nR : Type u_1\nS : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : Ring S\ninst✝² : IsDomain (R × S)\ninst✝¹ : Nontrivial R\ninst✝ : Nontrivial S\nleft✝ : 0 = 0\nh : 1 = 0\n⊢ False\n\ncase inr.intro\nα : Type ?u.54070\nβ : Type ?u.54073\nR✝ : Type ?u.54076\nR' : Type ?u.54079\nS✝ : Type ?u.54082\nS' : Type ?u.54085\nT : Type ?u.54088\nT' : Type ?u.54091\nR : Type u_1\nS : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : Ring S\ninst✝² : IsDomain (R × S)\ninst✝¹ : Nontrivial R\ninst✝ : Nontrivial S\nh : 1 = 0\nright✝ : 0 = 0\n⊢ False",
"state_before": "α : Type ?u.54070\nβ : Type ?u.54073\nR✝ : Type ?u.54076\nR' : Type ?u.54079\nS✝ : Type ?u.54082\nS' : Type ?u.54085\nT : Type ?u.54088\nT' : Type ?u.54091\nR : Type u_1\nS : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : Ring S\ninst✝² : IsDomain (R × S)\ninst✝¹ : Nontrivial R\ninst✝ : Nontrivial S\nthis : 0 = 0 ∧ 1 = 0 ∨ 1 = 0 ∧ 0 = 0\n⊢ False",
"tactic": "rcases this with (⟨_, h⟩ | ⟨h, _⟩)"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.54070\nβ : Type ?u.54073\nR✝ : Type ?u.54076\nR' : Type ?u.54079\nS✝ : Type ?u.54082\nS' : Type ?u.54085\nT : Type ?u.54088\nT' : Type ?u.54091\nR : Type u_1\nS : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : Ring S\ninst✝² : IsDomain (R × S)\ninst✝¹ : Nontrivial R\ninst✝ : Nontrivial S\n⊢ (0, 1) * (1, 0) = 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nα : Type ?u.54070\nβ : Type ?u.54073\nR✝ : Type ?u.54076\nR' : Type ?u.54079\nS✝ : Type ?u.54082\nS' : Type ?u.54085\nT : Type ?u.54088\nT' : Type ?u.54091\nR : Type u_1\nS : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : Ring S\ninst✝² : IsDomain (R × S)\ninst✝¹ : Nontrivial R\ninst✝ : Nontrivial S\nleft✝ : 0 = 0\nh : 1 = 0\n⊢ False",
"tactic": "exact zero_ne_one h.symm"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nα : Type ?u.54070\nβ : Type ?u.54073\nR✝ : Type ?u.54076\nR' : Type ?u.54079\nS✝ : Type ?u.54082\nS' : Type ?u.54085\nT : Type ?u.54088\nT' : Type ?u.54091\nR : Type u_1\nS : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : Ring S\ninst✝² : IsDomain (R × S)\ninst✝¹ : Nontrivial R\ninst✝ : Nontrivial S\nh : 1 = 0\nright✝ : 0 = 0\n⊢ False",
"tactic": "exact zero_ne_one h.symm"
}
] |
[
344,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/Data/Set/Intervals/Group.lean
|
Set.sub_mem_Ico_iff_right
|
[] |
[
127,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
mem_lowerBounds
|
[] |
[
91,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Topology/Algebra/ConstMulAction.lean
|
Continuous.const_smul
|
[] |
[
112,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Data/Quot.lean
|
Quotient.out_inj
|
[] |
[
406,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.semiconj₂_conj
|
[
{
"state_after": "no goals",
"state_before": "α₁ : Type u_1\nβ₁ : Type u_2\ne : α₁ ≃ β₁\nf : α₁ → α₁ → α₁\nx y : α₁\n⊢ ↑e (f x y) = ↑(arrowCongr e (conj e)) f (↑e x) (↑e y)",
"tactic": "simp [arrowCongr]"
}
] |
[
1849,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1849,
1
] |
Mathlib/Deprecated/Subgroup.lean
|
Group.mclosure_inv_subset
|
[] |
[
623,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
1
] |
Mathlib/LinearAlgebra/Quotient.lean
|
Submodule.Quotient.eq'
|
[] |
[
82,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
11
] |
Mathlib/Algebra/Lie/Basic.lean
|
lie_self
|
[] |
[
151,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/FieldTheory/Laurent.lean
|
RatFunc.laurent_X
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nhdomain : IsDomain R\nr s : R\np q : R[X]\nf : RatFunc R\n⊢ ↑(laurent r) X = X + ↑C r",
"tactic": "rw [← algebraMap_X, laurent_algebraMap, taylor_X, _root_.map_add, algebraMap_C]"
}
] |
[
101,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Order/SymmDiff.lean
|
symmDiff_symmDiff_right'
|
[
{
"state_after": "case e_a\nι : Type ?u.91463\nα : Type u_1\nβ : Type ?u.91469\nπ : ι → Type ?u.91474\ninst✝ : BooleanAlgebra α\na b c d : α\n⊢ a ⊓ b ⊓ c ⊔ a ⊓ bᶜ ⊓ cᶜ ⊔ b ⊓ cᶜ ⊓ aᶜ = a ⊓ b ⊓ c ⊔ a ⊓ bᶜ ⊓ cᶜ ⊔ aᶜ ⊓ b ⊓ cᶜ\n\ncase e_a\nι : Type ?u.91463\nα : Type u_1\nβ : Type ?u.91469\nπ : ι → Type ?u.91474\ninst✝ : BooleanAlgebra α\na b c d : α\n⊢ c ⊓ bᶜ ⊓ aᶜ = aᶜ ⊓ bᶜ ⊓ c",
"state_before": "ι : Type ?u.91463\nα : Type u_1\nβ : Type ?u.91469\nπ : ι → Type ?u.91474\ninst✝ : BooleanAlgebra α\na b c d : α\n⊢ a ⊓ b ⊓ c ⊔ a ⊓ bᶜ ⊓ cᶜ ⊔ b ⊓ cᶜ ⊓ aᶜ ⊔ c ⊓ bᶜ ⊓ aᶜ = a ⊓ b ⊓ c ⊔ a ⊓ bᶜ ⊓ cᶜ ⊔ aᶜ ⊓ b ⊓ cᶜ ⊔ aᶜ ⊓ bᶜ ⊓ c",
"tactic": "congr 1"
},
{
"state_after": "case e_a.e_a\nι : Type ?u.91463\nα : Type u_1\nβ : Type ?u.91469\nπ : ι → Type ?u.91474\ninst✝ : BooleanAlgebra α\na b c d : α\n⊢ b ⊓ cᶜ ⊓ aᶜ = aᶜ ⊓ b ⊓ cᶜ",
"state_before": "case e_a\nι : Type ?u.91463\nα : Type u_1\nβ : Type ?u.91469\nπ : ι → Type ?u.91474\ninst✝ : BooleanAlgebra α\na b c d : α\n⊢ a ⊓ b ⊓ c ⊔ a ⊓ bᶜ ⊓ cᶜ ⊔ b ⊓ cᶜ ⊓ aᶜ = a ⊓ b ⊓ c ⊔ a ⊓ bᶜ ⊓ cᶜ ⊔ aᶜ ⊓ b ⊓ cᶜ",
"tactic": "congr 1"
},
{
"state_after": "no goals",
"state_before": "case e_a.e_a\nι : Type ?u.91463\nα : Type u_1\nβ : Type ?u.91469\nπ : ι → Type ?u.91474\ninst✝ : BooleanAlgebra α\na b c d : α\n⊢ b ⊓ cᶜ ⊓ aᶜ = aᶜ ⊓ b ⊓ cᶜ",
"tactic": "rw [inf_comm, inf_assoc]"
},
{
"state_after": "no goals",
"state_before": "case e_a\nι : Type ?u.91463\nα : Type u_1\nβ : Type ?u.91469\nπ : ι → Type ?u.91474\ninst✝ : BooleanAlgebra α\na b c d : α\n⊢ c ⊓ bᶜ ⊓ aᶜ = aᶜ ⊓ bᶜ ⊓ c",
"tactic": "apply inf_left_right_swap"
}
] |
[
789,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
778,
1
] |
Mathlib/Topology/ContinuousFunction/Basic.lean
|
ContinuousMap.continuousAt
|
[] |
[
149,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
11
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
tendstoLocallyUniformlyOn_iff_filter
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\n⊢ TendstoLocallyUniformlyOn F f p s ↔\n ∀ (x : α),\n x ∈ s →\n ∀ (u : Set (β × β)),\n u ∈ 𝓤 β →\n ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧\n ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\n⊢ TendstoLocallyUniformlyOn F f p s ↔ ∀ (x : α), x ∈ s → TendstoUniformlyOnFilter F f p (𝓝[s] x)",
"tactic": "simp only [TendstoUniformlyOnFilter, eventually_prod_iff]"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\n⊢ TendstoLocallyUniformlyOn F f p s →\n ∀ (x : α),\n x ∈ s →\n ∀ (u : Set (β × β)),\n u ∈ 𝓤 β →\n ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧\n ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u\n\ncase mpr\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\n⊢ (∀ (x : α),\n x ∈ s →\n ∀ (u : Set (β × β)),\n u ∈ 𝓤 β →\n ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧\n ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u) →\n TendstoLocallyUniformlyOn F f p s",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\n⊢ TendstoLocallyUniformlyOn F f p s ↔\n ∀ (x : α),\n x ∈ s →\n ∀ (u : Set (β × β)),\n u ∈ 𝓤 β →\n ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧\n ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh : TendstoLocallyUniformlyOn F f p s\nx : α\nhx : x ∈ s\nu : Set (β × β)\nhu : u ∈ 𝓤 β\n⊢ ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧ ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u",
"state_before": "case mp\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\n⊢ TendstoLocallyUniformlyOn F f p s →\n ∀ (x : α),\n x ∈ s →\n ∀ (u : Set (β × β)),\n u ∈ 𝓤 β →\n ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧\n ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u",
"tactic": "rintro h x hx u hu"
},
{
"state_after": "case mp.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns✝ s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh : TendstoLocallyUniformlyOn F f p s✝\nx : α\nhx : x ∈ s✝\nu : Set (β × β)\nhu : u ∈ 𝓤 β\ns : Set α\nhs1 : s ∈ 𝓝[s✝] x\nhs2 : ∀ᶠ (n : ι) in p, ∀ (y : α), y ∈ s → (f y, F n y) ∈ u\n⊢ ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧ ∃ pb, (∀ᶠ (y : α) in 𝓝[s✝] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u",
"state_before": "case mp\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh : TendstoLocallyUniformlyOn F f p s\nx : α\nhx : x ∈ s\nu : Set (β × β)\nhu : u ∈ 𝓤 β\n⊢ ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧ ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u",
"tactic": "obtain ⟨s, hs1, hs2⟩ := h u hu x hx"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns✝ s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh : TendstoLocallyUniformlyOn F f p s✝\nx : α\nhx : x ∈ s✝\nu : Set (β × β)\nhu : u ∈ 𝓤 β\ns : Set α\nhs1 : s ∈ 𝓝[s✝] x\nhs2 : ∀ᶠ (n : ι) in p, ∀ (y : α), y ∈ s → (f y, F n y) ∈ u\n⊢ ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧ ∃ pb, (∀ᶠ (y : α) in 𝓝[s✝] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u",
"tactic": "exact ⟨_, hs2, _, eventually_of_mem hs1 fun x => id, fun hi y hy => hi y hy⟩"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh :\n ∀ (x : α),\n x ∈ s →\n ∀ (u : Set (β × β)),\n u ∈ 𝓤 β →\n ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧\n ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nx : α\nhx : x ∈ s\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : α), y ∈ t → (f y, F n y) ∈ u",
"state_before": "case mpr\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\n⊢ (∀ (x : α),\n x ∈ s →\n ∀ (u : Set (β × β)),\n u ∈ 𝓤 β →\n ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧\n ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u) →\n TendstoLocallyUniformlyOn F f p s",
"tactic": "rintro h u hu x hx"
},
{
"state_after": "case mpr.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh✝ :\n ∀ (x : α),\n x ∈ s →\n ∀ (u : Set (β × β)),\n u ∈ 𝓤 β →\n ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧\n ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nx : α\nhx : x ∈ s\npa : ι → Prop\nhpa : ∀ᶠ (x : ι) in p, pa x\npb : α → Prop\nhpb : ∀ᶠ (y : α) in 𝓝[s] x, pb y\nh : ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : α), y ∈ t → (f y, F n y) ∈ u",
"state_before": "case mpr\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh :\n ∀ (x : α),\n x ∈ s →\n ∀ (u : Set (β × β)),\n u ∈ 𝓤 β →\n ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧\n ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nx : α\nhx : x ∈ s\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : α), y ∈ t → (f y, F n y) ∈ u",
"tactic": "obtain ⟨pa, hpa, pb, hpb, h⟩ := h x hx u hu"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx✝ : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nh✝ :\n ∀ (x : α),\n x ∈ s →\n ∀ (u : Set (β × β)),\n u ∈ 𝓤 β →\n ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧\n ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nx : α\nhx : x ∈ s\npa : ι → Prop\nhpa : ∀ᶠ (x : ι) in p, pa x\npb : α → Prop\nhpb : ∀ᶠ (y : α) in 𝓝[s] x, pb y\nh : ∀ {x : ι}, pa x → ∀ {y : α}, pb y → (f y, F x y) ∈ u\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ ∀ᶠ (n : ι) in p, ∀ (y : α), y ∈ t → (f y, F n y) ∈ u",
"tactic": "refine' ⟨pb, hpb, eventually_of_mem hpa fun i hi y hy => h hi hy⟩"
}
] |
[
765,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
756,
1
] |
Mathlib/Algebra/Order/UpperLower.lean
|
UpperSet.coe_one
|
[] |
[
149,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_const_mul_Ici
|
[] |
[
617,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
616,
1
] |
Mathlib/Topology/Instances/EReal.lean
|
EReal.mem_nhds_bot_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.16219\ninst✝ : TopologicalSpace α\ns : Set EReal\n⊢ (∃ i, True ∧ Iio ↑i ⊆ s) ↔ ∃ y, Iio ↑y ⊆ s",
"tactic": "simp only [true_and]"
}
] |
[
168,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Data/Real/Irrational.lean
|
Irrational.nat_sub
|
[] |
[
316,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.inter_assoc
|
[] |
[
930,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
929,
1
] |
Mathlib/Computability/EpsilonNFA.lean
|
NFA.toεNFA_correct
|
[
{
"state_after": "α : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS : Set σ\nx : List α\ns : σ\na : α\nM : NFA α σ\n⊢ {x | ∃ S, S ∈ (toεNFA M).accept ∧ S ∈ evalFrom M (toεNFA M).start x} = accepts M",
"state_before": "α : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS : Set σ\nx : List α\ns : σ\na : α\nM : NFA α σ\n⊢ εNFA.accepts (toεNFA M) = accepts M",
"tactic": "rw [εNFA.accepts, εNFA.eval, toεNFA_evalFrom_match]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS : Set σ\nx : List α\ns : σ\na : α\nM : NFA α σ\n⊢ {x | ∃ S, S ∈ (toεNFA M).accept ∧ S ∈ evalFrom M (toεNFA M).start x} = accepts M",
"tactic": "rfl"
}
] |
[
212,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/Data/PFun.lean
|
PFun.preimage_union
|
[] |
[
451,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
450,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.map_map_zero_lt
|
[] |
[
487,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.mex_lt_ord_succ_mk
|
[
{
"state_after": "α : Type ?u.378408\nβ : Type ?u.378411\nγ : Type ?u.378414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\nh : ord (succ (#ι)) ≤ mex f\n⊢ False",
"state_before": "α : Type ?u.378408\nβ : Type ?u.378411\nγ : Type ?u.378414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\n⊢ mex f < ord (succ (#ι))",
"tactic": "by_contra' h"
},
{
"state_after": "α : Type ?u.378408\nβ : Type ?u.378411\nγ : Type ?u.378414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\nh : ord (succ (#ι)) ≤ mex f\n⊢ succ (#ι) ≤ (#ι)",
"state_before": "α : Type ?u.378408\nβ : Type ?u.378411\nγ : Type ?u.378414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\nh : ord (succ (#ι)) ≤ mex f\n⊢ False",
"tactic": "apply (lt_succ (#ι)).not_le"
},
{
"state_after": "α : Type ?u.378408\nβ : Type ?u.378411\nγ : Type ?u.378414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\nh : ord (succ (#ι)) ≤ mex f\nH : ∀ (a : (Quotient.out (ord (succ (#ι)))).α), ∃ i, f i = typein (fun x x_1 => x < x_1) a\n⊢ succ (#ι) ≤ (#ι)",
"state_before": "α : Type ?u.378408\nβ : Type ?u.378411\nγ : Type ?u.378414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\nh : ord (succ (#ι)) ≤ mex f\n⊢ succ (#ι) ≤ (#ι)",
"tactic": "have H := fun a => exists_of_lt_mex ((typein_lt_self a).trans_le h)"
},
{
"state_after": "α : Type ?u.378408\nβ : Type ?u.378411\nγ : Type ?u.378414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\nh : ord (succ (#ι)) ≤ mex f\nH : ∀ (a : (Quotient.out (ord (succ (#ι)))).α), ∃ i, f i = typein (fun x x_1 => x < x_1) a\ng : (Quotient.out (ord (succ (#ι)))).α → ι := fun a => choose (_ : ∃ i, f i = typein (fun x x_1 => x < x_1) a)\n⊢ succ (#ι) ≤ (#ι)",
"state_before": "α : Type ?u.378408\nβ : Type ?u.378411\nγ : Type ?u.378414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\nh : ord (succ (#ι)) ≤ mex f\nH : ∀ (a : (Quotient.out (ord (succ (#ι)))).α), ∃ i, f i = typein (fun x x_1 => x < x_1) a\n⊢ succ (#ι) ≤ (#ι)",
"tactic": "let g : (succ (#ι)).ord.out.α → ι := fun a => Classical.choose (H a)"
},
{
"state_after": "case h.e'_3\nα : Type ?u.378408\nβ : Type ?u.378411\nγ : Type ?u.378414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\nh : ord (succ (#ι)) ≤ mex f\nH : ∀ (a : (Quotient.out (ord (succ (#ι)))).α), ∃ i, f i = typein (fun x x_1 => x < x_1) a\ng : (Quotient.out (ord (succ (#ι)))).α → ι := fun a => choose (_ : ∃ i, f i = typein (fun x x_1 => x < x_1) a)\nhg : Injective g\n⊢ succ (#ι) = (#(Quotient.out (ord (succ (#ι)))).α)",
"state_before": "α : Type ?u.378408\nβ : Type ?u.378411\nγ : Type ?u.378414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\nh : ord (succ (#ι)) ≤ mex f\nH : ∀ (a : (Quotient.out (ord (succ (#ι)))).α), ∃ i, f i = typein (fun x x_1 => x < x_1) a\ng : (Quotient.out (ord (succ (#ι)))).α → ι := fun a => choose (_ : ∃ i, f i = typein (fun x x_1 => x < x_1) a)\nhg : Injective g\n⊢ succ (#ι) ≤ (#ι)",
"tactic": "convert Cardinal.mk_le_of_injective hg"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nα : Type ?u.378408\nβ : Type ?u.378411\nγ : Type ?u.378414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\nh : ord (succ (#ι)) ≤ mex f\nH : ∀ (a : (Quotient.out (ord (succ (#ι)))).α), ∃ i, f i = typein (fun x x_1 => x < x_1) a\ng : (Quotient.out (ord (succ (#ι)))).α → ι := fun a => choose (_ : ∃ i, f i = typein (fun x x_1 => x < x_1) a)\nhg : Injective g\n⊢ succ (#ι) = (#(Quotient.out (ord (succ (#ι)))).α)",
"tactic": "rw [Cardinal.mk_ord_out (succ (#ι))]"
},
{
"state_after": "α : Type ?u.378408\nβ : Type ?u.378411\nγ : Type ?u.378414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\nh : ord (succ (#ι)) ≤ mex f\nH : ∀ (a : (Quotient.out (ord (succ (#ι)))).α), ∃ i, f i = typein (fun x x_1 => x < x_1) a\ng : (Quotient.out (ord (succ (#ι)))).α → ι := fun a => choose (_ : ∃ i, f i = typein (fun x x_1 => x < x_1) a)\na b : (Quotient.out (ord (succ (#ι)))).α\nHf : ∀ (x : (Quotient.out (ord (succ (#ι)))).α), f (g x) = typein (fun x x_1 => x < x_1) x\nh' : f (g a) = f (g b)\n⊢ a = b",
"state_before": "α : Type ?u.378408\nβ : Type ?u.378411\nγ : Type ?u.378414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\nh : ord (succ (#ι)) ≤ mex f\nH : ∀ (a : (Quotient.out (ord (succ (#ι)))).α), ∃ i, f i = typein (fun x x_1 => x < x_1) a\ng : (Quotient.out (ord (succ (#ι)))).α → ι := fun a => choose (_ : ∃ i, f i = typein (fun x x_1 => x < x_1) a)\na b : (Quotient.out (ord (succ (#ι)))).α\nh' : g a = g b\nHf : ∀ (x : (Quotient.out (ord (succ (#ι)))).α), f (g x) = typein (fun x x_1 => x < x_1) x\n⊢ a = b",
"tactic": "apply_fun f at h'"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.378408\nβ : Type ?u.378411\nγ : Type ?u.378414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\nh : ord (succ (#ι)) ≤ mex f\nH : ∀ (a : (Quotient.out (ord (succ (#ι)))).α), ∃ i, f i = typein (fun x x_1 => x < x_1) a\ng : (Quotient.out (ord (succ (#ι)))).α → ι := fun a => choose (_ : ∃ i, f i = typein (fun x x_1 => x < x_1) a)\na b : (Quotient.out (ord (succ (#ι)))).α\nHf : ∀ (x : (Quotient.out (ord (succ (#ι)))).α), f (g x) = typein (fun x x_1 => x < x_1) x\nh' : f (g a) = f (g b)\n⊢ a = b",
"tactic": "rwa [Hf, Hf, typein_inj] at h'"
}
] |
[
2059,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2046,
1
] |
Mathlib/Topology/GDelta.lean
|
isGδ_singleton
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.12419\nγ : Type ?u.12422\nι : Type ?u.12425\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : FirstCountableTopology α\na : α\nU : ℕ → Set α\nhU : ∀ (i : ℕ), a ∈ U i ∧ IsOpen (U i)\nh_basis : HasAntitoneBasis (𝓝 a) fun i => U i\n⊢ IsGδ {a}",
"state_before": "α : Type u_1\nβ : Type ?u.12419\nγ : Type ?u.12422\nι : Type ?u.12425\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : FirstCountableTopology α\na : α\n⊢ IsGδ {a}",
"tactic": "rcases (nhds_basis_opens a).exists_antitone_subbasis with ⟨U, hU, h_basis⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.12419\nγ : Type ?u.12422\nι : Type ?u.12425\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : FirstCountableTopology α\na : α\nU : ℕ → Set α\nhU : ∀ (i : ℕ), a ∈ U i ∧ IsOpen (U i)\nh_basis : HasAntitoneBasis (𝓝 a) fun i => U i\n⊢ IsGδ (⋂ (i : ℕ) (_ : True), U i)",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.12419\nγ : Type ?u.12422\nι : Type ?u.12425\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : FirstCountableTopology α\na : α\nU : ℕ → Set α\nhU : ∀ (i : ℕ), a ∈ U i ∧ IsOpen (U i)\nh_basis : HasAntitoneBasis (𝓝 a) fun i => U i\n⊢ IsGδ {a}",
"tactic": "rw [← biInter_basis_nhds h_basis.toHasBasis]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.12419\nγ : Type ?u.12422\nι : Type ?u.12425\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : FirstCountableTopology α\na : α\nU : ℕ → Set α\nhU : ∀ (i : ℕ), a ∈ U i ∧ IsOpen (U i)\nh_basis : HasAntitoneBasis (𝓝 a) fun i => U i\n⊢ IsGδ (⋂ (i : ℕ) (_ : True), U i)",
"tactic": "exact isGδ_biInter (to_countable _) fun n _ => (hU n).2.isGδ"
}
] |
[
166,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.sup_eq_of_range_eq
|
[] |
[
1333,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1331,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
ContinuousWithinAt.mono
|
[] |
[
698,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
696,
1
] |
Mathlib/Data/Set/Intervals/Monoid.lean
|
Set.image_const_add_Ico
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\ninst✝¹ : OrderedCancelAddCommMonoid M\ninst✝ : ExistsAddOfLE M\na b c d : M\n⊢ (fun x => a + x) '' Ico b c = Ico (a + b) (a + c)",
"tactic": "simp only [add_comm a, image_add_const_Ico]"
}
] |
[
132,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
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